List of space groups
There are 230 space groups in three dimensions, given by a number index, and a full name in Hermann–Mauguin notation, and a short name (international short symbol). The long names are given with spaces for readability. The groups each have a point groups of the unit cell.
Symbols
In Hermann–Mauguin notation, space groups are named by a symbol combining the point group identifier with the uppercase letters describing the lattice type. Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group.
These are the Bravais lattices in three dimensions:
- P primitive
- I body centered (from the German "Innenzentriert")
- F face centered (from the German "Flächenzentriert")
- A centered on A faces only
- B centered on B faces only
- C centered on C faces only
- R rhombohedral
A reflection plane m within the point groups can be replaced by a glide plane, labeled as a, b, or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The d glide is often called the diamond glide plane as it features in the diamond structure.
- , , or glide translation along half the lattice vector of this face
- glide translation along with half a face diagonal
- glide planes with translation along a quarter of a face diagonal.
- two glides with the same glide plane and translation along two (different) half-lattice vectors.
A gyration point can be replaced by a screw axis is noted by a number, n, where the angle of rotation is . The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 21 is a 180° (twofold) rotation followed by a translation of ½ of the lattice vector. 31 is a 120° (threefold) rotation followed by a translation of ⅓ of the lattice vector.
The possible screw axis are: 21, 31, 32, 41, 42, 43, 61, 62, 63, 64, and 65.
In Schoenflies notation, the symbol of a space group is represented by the symbol of corresponding point group with additional superscript. The superscript doesn't give any additional information about symmetry elements of the space group. It is related to the order in which Shoenflies derived space groups.
In Fedorov symbol, the type of space group is denoted as s (symmorphic ), h (hemisymmorphic), or a (asymmorphic). The number is related to the order in which Fedorov derived space groups. There are 73 symmorphic, 54 hemisymmorphic, and 103 asymmorphic space groups. Symmorphic space groups can be obtained as combination of Bravais lattices with corresponding point group. These groups contain the same symmetry elements as the corresponding point groups. Hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. All the other space groups are asymmorphic. Example for point group 4/mmm (): the symmorphic space groups are P4/mmm (, 36s) and I4/mmm (, 37s); hemisymmorphic space groups should contain axial combination 422, these are P4/mcc (, 35h), P4/nbm (, 36h), P4/nnc (, 37h), and I4/mcm (, 38h).
List of Triclinic
Number | Point group | Short name | Full name | Schoenflies | Fedorov | Shubnikov |
---|---|---|---|---|---|---|
1 | 1 | P1 | P 1 | 1s | ||
2 | 1 | P1 | P 1 | 2s |
List of Monoclinic
Simple (P) |
Base (C) |
---|---|
Number | Point group | Short name | Full name(s) | Schoenflies | Fedorov | Shubnikov | |
---|---|---|---|---|---|---|---|
3 | 2 | P2 | P 1 2 1 | P 1 1 2 | 3s | ||
4 | 2 | P21 | P 1 21 1 | P 1 1 21 | 1a | ||
5 | 2 | C2 | C 1 2 1 | B 1 1 2 | 4s | ||
6 | m | Pm | P 1 m 1 | P 1 1 m | 5s | ||
7 | m | Pc | P 1 c 1 | P 1 1 b | 1h | ||
8 | m | Cm | C 1 m 1 | B 1 1 m | 6s | ||
9 | m | Cc | C 1 c 1 | B 1 1 b | 2h | ||
10 | 2/m | P2/m | P 1 2/m 1 | P 1 1 2/m | 7s | ||
11 | 2/m | P21/m | P 1 21/m 1 | P 1 1 21/m | 2a | ||
12 | 2/m | C2/m | C 1 2/m 1 | B 1 1 2/m | 8s | ||
13 | 2/m | P2/c | P 1 2/c 1 | P 1 1 2/b | 3h | ||
14 | 2/m | P21/c | P 1 21/c 1 | P 1 1 21/b | 3a | ||
15 | 2/m | C2/c | C 1 2/c 1 | B 1 1 2/b | 4h |
List of Orthorhombic
Number | Point group | Short name | Full name | Schoenflies | Fedorov | Shubnikov |
---|---|---|---|---|---|---|
16 | 222 | P222 | P 2 2 2 | 9s | ||
17 | 222 | P2221 | P 2 2 21 | 4a | ||
18 | 222 | P21212 | P 21 21 2 | 7a | ||
19 | 222 | P212121 | P 21 21 21 | 8a | ||
20 | 222 | C2221 | C 2 2 21 | 5a | ||
21 | 222 | C222 | C 2 2 2 | 10s | ||
22 | 222 | F222 | F 2 2 2 | 12s | ||
23 | 222 | I222 | I 2 2 2 | 11s | ||
24 | 222 | I212121 | I 21 21 21 | 6a | ||
25 | mm2 | Pmm2 | P m m 2 | 13s | ||
26 | mm2 | Pmc21 | P m c 21 | 9a | ||
27 | mm2 | Pcc2 | P c c 2 | 5h | ||
28 | mm2 | Pma2 | P m a 2 | 6h | ||
29 | mm2 | Pca21 | P c a 21 | 11a | ||
30 | mm2 | Pnc2 | P n c 2 | 7h | ||
31 | mm2 | Pmn21 | P m n 21 | 10a | ||
32 | mm2 | Pba2 | P b a 2 | 9h | ||
33 | mm2 | Pna21 | P n a 21 | 12a | ||
34 | mm2 | Pnn2 | P n n 2 | 8h | ||
35 | mm2 | Cmm2 | C m m 2 | 14s | ||
36 | mm2 | Cmc21 | C m c 21 | 13a | ||
37 | mm2 | Ccc2 | C c c 2 | 10h | ||
38 | mm2 | Amm2 | A m m 2 | 15s | ||
39 | mm2 | Aem2 | A b m 2 | 11h | ||
40 | mm2 | Ama2 | A m a 2 | 12h | ||
41 | mm2 | Aea2 | A b a 2 | 13h | ||
42 | mm2 | Fmm2 | F m m 2 | 17s | ||
43 | mm2 | Fdd2 | F dd2 | 16h | ||
44 | mm2 | Imm2 | I m m 2 | 16s | ||
45 | mm2 | Iba2 | I b a 2 | 15h | ||
46 | mm2 | Ima2 | I m a 2 | 14h | ||
47 | Pmmm | P 2/m 2/m 2/m | 18s | |||
48 | Pnnn | P 2/n 2/n 2/n | 19h | |||
49 | Pccm | P 2/c 2/c 2/m | 17h | |||
50 | Pban | P 2/b 2/a 2/n | 18h | |||
51 | Pmma | P 21/m 2/m 2/a | 14a | |||
52 | Pnna | P 2/n 21/n 2/a | 17a | |||
53 | Pmna | P 2/m 2/n 21/a | 15a | |||
54 | Pcca | P 21/c 2/c 2/a | 16a | |||
55 | Pbam | P 21/b 21/a 2/m | 22a | |||
56 | Pccn | P 21/c 21/c 2/n | 27a | |||
57 | Pbcm | P 2/b 21/c 21/m | 23a | |||
58 | Pnnm | P 21/n 21/n 2/m | 25a | |||
59 | Pmmn | P 21/m 21/m 2/n | 24a | |||
60 | Pbcn | P 21/b 2/c 21/n | 26a | |||
61 | Pbca | P 21/b 21/c 21/a | 29a | |||
62 | Pnma | P 21/n 21/m 21/a | 28a | |||
63 | Cmcm | C 2/m 2/c 21/m | 18a | |||
64 | Cmca | C 2/m 2/c 21/a | 19a | |||
65 | Cmmm | C 2/m 2/m 2/m | 19s | |||
66 | Cccm | C 2/c 2/c 2/m | 20h | |||
67 | Cmme | C 2/m 2/m 2/e | 21h | |||
68 | Ccce | C 2/c 2/c 2/e | 22h | |||
69 | Fmmm | F 2/m 2/m 2/m | 21s | |||
70 | Fddd | F 2/d 2/d 2/d | 24h | |||
71 | Immm | I 2/m 2/m 2/m | 20s | |||
72 | Ibam | I 2/b 2/a 2/m | 23h | |||
73 | Ibca | I 2/b 2/c 2/a | 21a | |||
74 | Imma | I 2/m 2/m 2/a | 20a |
List of Tetragonal
Number | Point group | Short name | Full name | Schoenflies | Fedorov | Shubnikov |
---|---|---|---|---|---|---|
75 | 4 | P4 | P 4 | 22s | ||
76 | 4 | P41 | P 41 | 30a | ||
77 | 4 | P42 | P 42 | 33a | ||
78 | 4 | P43 | P 43 | 31a | ||
79 | 4 | I4 | I 4 | 23s | ||
80 | 4 | I41 | I 41 | 32a | ||
81 | 4 | P4 | P 4 | 26s | ||
82 | 4 | I4 | I 4 | 27s | ||
83 | 4/m | P4/m | P 4/m | 28s | ||
84 | 4/m | P42/m | P 42/m | 41a | ||
85 | 4/m | P4/n | P 4/n | 29h | ||
86 | 4/m | P42/n | P 42/n | 42a | ||
87 | 4/m | I4/m | I 4/m | 29s | ||
88 | 4/m | I41/a | I 41/a | 40a | ||
89 | 422 | P422 | P 4 2 2 | 30s | ||
90 | 422 | P4212 | P4212 | 43a | ||
91 | 422 | P4122 | P 41 2 2 | 44a | ||
92 | 422 | P41212 | P 41 21 2 | 48a | ||
93 | 422 | P4222 | P 42 2 2 | 47a | ||
94 | 422 | P42212 | P 42 21 2 | 50a | ||
95 | 422 | P4322 | P 43 2 2 | 45a | ||
96 | 422 | P43212 | P 43 21 2 | 49a | ||
97 | 422 | I422 | I 4 2 2 | 31s | ||
98 | 422 | I4122 | I 41 2 2 | 46a | ||
99 | 4mm | P4mm | P 4 m m | 24s | ||
100 | 4mm | P4bm | P 4 b m | 26h | ||
101 | 4mm | P42cm | P 42 c m | 37a | ||
102 | 4mm | P42nm | P 42 n m | 38a | ||
103 | 4mm | P4cc | P 4 c c | 25h | ||
104 | 4mm | P4nc | P 4 n c | 27h | ||
105 | 4mm | P42mc | P 42 m c | 36a | ||
106 | 4mm | P42bc | P 42 b c | 39a | ||
107 | 4mm | I4mm | I 4 m m | 25s | ||
108 | 4mm | I4cm | I 4 c m | 28h | ||
109 | 4mm | I41md | I 41 m d | 34a | ||
110 | 4mm | I41cd | I 41 c d | 35a | ||
111 | 42m | P42m | P 4 2 m | 32s | ||
112 | 42m | P42c | P 4 2 c | 30h | ||
113 | 42m | P421m | P 4 21 m | 52a | ||
114 | 42m | P421c | P 4 21 c | 53a | ||
115 | 42m | P4m2 | P 4 m 2 | 33s | ||
116 | 42m | P4c2 | P 4 c 2 | 31h | ||
117 | 42m | P4b2 | P 4 b 2 | 32h | ||
118 | 42m | P4n2 | P 4 n 2 | 33h | ||
119 | 42m | I4m2 | I 4 m 2 | 35s | ||
120 | 42m | I4c2 | I 4 c 2 | 34h | ||
121 | 42m | I42m | I 4 2 m | 34s | ||
122 | 42m | I42d | I 4 2 d | 51a | ||
123 | 4/m 2/m 2/m | P4/mmm | P 4/m 2/m 2/m | 36s | ||
124 | 4/m 2/m 2/m | P4/mcc | P 4/m 2/c 2/c | 35h | ||
125 | 4/m 2/m 2/m | P4/nbm | P 4/n 2/b 2/m | 36h | ||
126 | 4/m 2/m 2/m | P4/nnc | P 4/n 2/n 2/c | 37h | ||
127 | 4/m 2/m 2/m | P4/mbm | P 4/m 21/b 2/m | 54a | ||
128 | 4/m 2/m 2/m | P4/mnc | P 4/m 21/n 2/c | 56a | ||
129 | 4/m 2/m 2/m | P4/nmm | P 4/n 21/m 2/m | 55a | ||
130 | 4/m 2/m 2/m | P4/ncc | P 4/n 21/c 2/c | 57a | ||
131 | 4/m 2/m 2/m | P42/mmc | P 42/m 2/m 2/c | 60a | ||
132 | 4/m 2/m 2/m | P42/mcm | P 42/m 2/c 2/m | 61a | ||
133 | 4/m 2/m 2/m | P42/nbc | P 42/n 2/b 2/c | 63a | ||
134 | 4/m 2/m 2/m | P42/nnm | P 42/n 2/n 2/m | 62a | ||
135 | 4/m 2/m 2/m | P42/mbc | P 42/m 21/b 2/c | 66a | ||
136 | 4/m 2/m 2/m | P42/mnm | P 42/m 21/n 2/m | 65a | ||
137 | 4/m 2/m 2/m | P42/nmc | P 42/n 21/m 2/c | 67a | ||
138 | 4/m 2/m 2/m | P42/ncm | P 42/n 21/c 2/m | 65a | ||
139 | 4/m 2/m 2/m | I4/mmm | I 4/m 2/m 2/m | 37s | ||
140 | 4/m 2/m 2/m | I4/mcm | I 4/m 2/c 2/m | 38h | ||
141 | 4/m 2/m 2/m | I41/amd | I 41/a 2/m 2/d | 59a | ||
142 | 4/m 2/m 2/m | I41/acd | I 41/a 2/c 2/d | 58a |
List of Trigonal
Rhombohedral (R) |
Hexagonal (P) |
---|---|
Number | Point group | Short name | Full name | Schoenflies | Fedorov | Shubnikov |
---|---|---|---|---|---|---|
143 | 3 | P3 | P 3 | 38s | ||
144 | 3 | P31 | P 31 | 68a | ||
145 | 3 | P32 | P 32 | 69a | ||
146 | 3 | R3 | R 3 | 39s | ||
147 | 3 | P3 | P 3 | 51s | ||
148 | 3 | R3 | R 3 | 52s | ||
149 | 32 | P312 | P 3 1 2 | 45s | ||
150 | 32 | P321 | P 3 2 1 | 44s | ||
151 | 32 | P3112 | P 31 1 2 | 72a | ||
152 | 32 | P3121 | P 31 2 1 | 70a | ||
153 | 32 | P3212 | P 32 1 2 | 73a | ||
154 | 32 | P3221 | P 32 2 1 | 71a | ||
155 | 32 | R32 | R 3 2 | 46s | ||
156 | 3m | P3m1 | P 3 m 1 | 40s | ||
157 | 3m | P31m | P 3 1 m | 41s | ||
158 | 3m | P3c1 | P 3 c 1 | 39h | ||
159 | 3m | P31c | P 3 1 c | 40h | ||
160 | 3m | R3m | R 3 m | 42s | ||
161 | 3m | R3c | R 3 c | 41h | ||
162 | 3 2/m | P31m | P 3 1 2/m | 56s | ||
163 | 3 2/m | P31c | P 3 1 2/c | 46h | ||
164 | 3 2/m | P3m1 | P 3 2/m 1 | 55s | ||
165 | 3 2/m | P3c1 | P 3 2/c 1 | 45h | ||
166 | 3 2/m | R3m | R 3 2/m | 57s | ||
167 | 3 2/m | R3c | R 3 2/c | 47h |
List of Hexagonal
Number | Point group | Short name | Full name | Schoenflies | Fedorov | Shubnikov |
---|---|---|---|---|---|---|
168 | 6 | P6 | P 6 | 49s | ||
169 | 6 | P61 | P 61 | 74a | ||
170 | 6 | P65 | P 65 | 75a | ||
171 | 6 | P62 | P 62 | 76a | ||
172 | 6 | P64 | P 64 | 77a | ||
173 | 6 | P63 | P 63 | 78a | ||
174 | 6 | P6 | P 6 | 43s | ||
175 | 6/m | P6/m | P 6/m | 53s | ||
176 | 6/m | P63/m | P 63/m | 81a | ||
177 | 622 | P622 | P 6 2 2 | 54s | ||
178 | 622 | P6122 | P 61 2 2 | 82a | ||
179 | 622 | P6522 | P 65 2 2 | 83a | ||
180 | 622 | P6222 | P 62 2 2 | 84a | ||
181 | 622 | P6422 | P 64 2 2 | 85a | ||
182 | 622 | P6322 | P 63 2 2 | 86a | ||
183 | 6mm | P6mm | P 6 m m | 50s | ||
184 | 6mm | P6cc | P 6 c c | 44h | ||
185 | 6mm | P63cm | P 63 c m | 80a | ||
186 | 6mm | P63mc | P 63 m c | 79a | ||
187 | 6m2 | P6m2 | P 6 m 2 | 48s | ||
188 | 6m2 | P6c2 | P 6 c 2 | 43h | ||
189 | 6m2 | P62m | P 6 2 m | 47s | ||
190 | 6m2 | P62c | P 6 2 c | 42h | ||
191 | 6/m 2/m 2/m | P6/mmm | P 6/m 2/m 2/m | 58s | ||
192 | 6/m 2/m 2/m | P6/mcc | P 6/m 2/c 2/c | 48h | ||
193 | 6/m 2/m 2/m | P63/mcm | P 63/m 2/c 2/m | 87a | ||
194 | 6/m 2/m 2/m | P63/mmc | P 63/m 2/m 2/c | 88a |
List of Cubic
| | | |}
Number | Point group | Short name | Full name | Schoenflies | Fedorov | Shubnikov | Fibrifold |
---|---|---|---|---|---|---|---|
195 | 23 | P23 | P 2 3 | 59s | 2o | ||
196 | 23 | F23 | F 2 3 | 61s | 1o | ||
197 | 23 | I23 | I 2 3 | 60s | 4oo | ||
198 | 23 | P213 | P 21 3 | 89a | 1o/4 | ||
199 | 23 | I213 | I 21 3 | 90a | 2o/4 | ||
200 | 2/m 3 | Pm3 | P 2/m 3 | 62s | 4− | ||
201 | 2/m 3 | Pn3 | P 2/n 3 | 49h | 4+o | ||
202 | 2/m 3 | Fm3 | F 2/m 3 | 64s | 2− | ||
203 | 2/m 3 | Fd3 | F 2/d 3 | 50h | 2+o | ||
204 | 2/m 3 | Im3 | I 2/m 3 | 63s | 8−o | ||
205 | 2/m 3 | Pa3 | P 21/a 3 | 91a | 2−/4 | ||
206 | 2/m 3 | Ia3 | I 21/a 3 | 92a | 4−/4 | ||
207 | 432 | P432 | P 4 3 2 | 68s | 4−o | ||
208 | 432 | P4232 | P 42 3 2 | 98a | 4+ | ||
209 | 432 | F432 | F 4 3 2 | 70s | 2−o | ||
210 | 432 | F4132 | F 41 3 2 | 97a | 2+ | ||
211 | 432 | I432 | I 4 3 2 | 69s | 8+o | ||
212 | 432 | P4332 | P 43 3 2 | 94a | 2+/4 | ||
213 | 432 | P4132 | P 41 3 2 | 95a | 2+/4 | ||
214 | 432 | I4132 | I 41 3 2 | 96a | 4+/4 | ||
215 | 43m | P43m | P 4 3 m | 65s | 2o:2 | ||
216 | 43m | F43m | F 4 3 m | 67s | 1o:2 | ||
217 | 43m | I43m | I 4 3 m | 66s | 4o:2 | ||
218 | 43m | P43n | P 4 3 n | 51h | 4o | ||
219 | 43m | F43c | F 4 3 c | 52h | 2oo | ||
220 | 43m | I43d | I 4 3 d | 93a | 4o/4 | ||
221 | 4/m 3 2/m | Pm3m | P 4/m 3 2/m | 71s | 4−:2 | ||
222 | 4/m 3 2/m | Pn3n | P 4/n 3 2/n | 53h | 8oo | ||
223 | 4/m 3 2/m | Pm3n | P 42/m 3 2/n | 102a | 8o | ||
224 | 4/m 3 2/m | Pn3m | P 42/n 3 2/m | 103a | 4+:2 | ||
225 | 4/m 3 2/m | Fm3m | F 4/m 3 2/m | 73s | 2−:2 | ||
226 | 4/m 3 2/m | Fm3c | F 4/m 3 2/c | 54h | 4−− | ||
227 | 4/m 3 2/m | Fd3m | F 41/d 3 2/m | 100a | 2+:2 | ||
228 | 4/m 3 2/m | Fd3c | F 41/d 3 2/c | 101a | 4++ | ||
229 | 4/m 3 2/m | Im3m | I 4/m 3 2/m | 72s | 8o:2 | ||
230 | 4/m 3 2/m | Ia3d | I 41/a 3 2/d | 99a | 8o/4 |
External links
Wikimedia Commons has media related to Space groups. |
- International Union of Crystallography
- Point Groups and Bravais Lattices
- Full list of 230 crystallographic space groups