Space Group → Bravais Lattice, Point Group

The space group of a crystal uniquely determines the Bravais lattice and the point group. The point group does not uniquely determine the Bravais lattice.

 Space Group 

Bravais lattice

Point group
International / Schoenflies

1 P1

Triclinic

1

C1

2 P1

Triclinic

1

S2 = Ci

3 P2

Simple Monoclinic

2

C2

4 P21

Simple Monoclinic

2

C2

5 C2

Base-Centered Monoclinic

2

C2

6 Pm

Simple Monoclinic

m

C1h = Cs

7 Pc

Simple Monoclinic

m

C1h = Cs

8 Cm

Base-Centered Monoclinic

m

C1h = Cs

9 Cc

Base-Centered Monoclinic

m

C1h = Cs

10 P2/m

Simple Monoclinic

$\frac{2}{m}$

C2h

11 P21/m

Simple Monoclinic

$\frac{2}{m}$

C2h

12 C2/m

Base-Centered Monoclinic

$\frac{2}{m}$

C2h

13 P2/c

Simple Monoclinic

$\frac{2}{m}$

C2h

14 P21/c

Simple Monoclinic

$\frac{2}{m}$

C2h

15 C2/c

Base-Centered Monoclinic

$\frac{2}{m}$

C2h

16 P222

Simple Orthorhombic

222

V = D2

17 P2221

Simple Orthorhombic

222

V = D2

18 P21212

Simple Orthorhombic

222

V = D2

19 P212121

Simple Orthorhombic

222

V = D2

20 C2221

Base-Centered Orthorhombic

222

V = D2

21 C222

Base-Centered Orthorhombic

222

V = D2

22 F222

Face-Centered Orthorhombic

222

V = D2

23 I222

 Body-Centered Orthorhombic 

222

V = D2

24 I212121

Body-Centered Orthorhombic

222

V = D2

25 Pmm2

Simple Orthorhombic

mm2

C2v

26 Pmc21

Simple Orthorhombic

mm2

C2v

27 Pcc2

Simple Orthorhombic

mm2

C2v

28 Pma2

Simple Orthorhombic

mm2

C2v

29 Pca21

Simple Orthorhombic

mm2

C2v

30 Pnc2

Simple Orthorhombic

mm2

C2v

31 Pmn21

Simple Orthorhombic

mm2

C2v

32 Pba2

Simple Orthorhombic

mm2

C2v

33 Pna21

Simple Orthorhombic

mm2

C2v

34 Pnn2

Simple Orthorhombic

mm2

C2v

35 Cmm2

Base-Centered Orthorhombic

mm2

C2v

36 Cmc21

Base-Centered Orthorhombic

mm2

C2v

37 Ccc2

Base-Centered Orthorhombic

mm2

C2v

38 Amm2

Base-Centered Orthorhombic

mm2

C2v

39 Aem2

Base-Centered Orthorhombic

mm2

C2v

40 Ama2

Base-Centered Orthorhombic

mm2

C2v

41 Aea2

Base-Centered Orthorhombic

mm2

C2v

42 Fmm2

Face-Centered Orthorhombic

mm2

C2v

43 Fdd2

Face-Centered Orthorhombic

mm2

C2v

44 Imm2

Body-Centered Orthorhombic

mm2

C2v

45 Iba2

Body-Centered Orthorhombic

mm2

C2v

46 Ima2

Body-Centered Orthorhombic

mm2

C2v

47 Pmmm

Simple Orthorhombic

mmm

Vh = D2h

48 Pnnn

Simple Orthorhombic

mmm

Vh = D2h

49 Pccm

Simple Orthorhombic

mmm

Vh = D2h

50 Pban

Simple Orthorhombic

mmm

Vh = D2h

51 Pmma

Simple Orthorhombic

mmm

Vh = D2h

52 Pnna

Simple Orthorhombic

mmm

Vh = D2h

53 Pmna

Simple Orthorhombic

mmm

Vh = D2h

54 Pcca

Simple Orthorhombic

mmm

Vh = D2h

55 Pbam

Simple Orthorhombic

mmm

Vh = D2h

56 Pccn

Simple Orthorhombic

mmm

Vh = D2h

57 Pbcm

Simple Orthorhombic

mmm

Vh = D2h

58 Pnnm

Simple Orthorhombic

mmm

Vh = D2h

59 Pmmn

Simple Orthorhombic

mmm

Vh = D2h

60 Pbcn

Simple Orthorhombic

mmm

Vh = D2h

61 Pbca

Simple Orthorhombic

mmm

Vh = D2h

62 Pnma

Simple Orthorhombic

mmm

Vh = D2h

63 Cmcm

Base-Centered Orthorhombic

mmm

Vh = D2h

64 Cmce

Base-Centered Orthorhombic

mmm

Vh = D2h

65 Cmmm

Base-Centered Orthorhombic

mmm

Vh = D2h

66 Cccm

Base-Centered Orthorhombic

mmm

Vh = D2h

67 Cmme

Base-Centered Orthorhombic

mmm

Vh = D2h

68 Ccce

Base-Centered Orthorhombic

mmm

Vh = D2h

69 Fmmm

Face-Centered Orthorhombic

mmm

Vh = D2h

70 Fddd

Face-Centered Orthorhombic

mmm

Vh = D2h

71 Immm

Body-Centered Orthorhombic

mmm

Vh = D2h

72 Ibam

Body-Centered Orthorhombic

mmm

Vh = D2h

73 Ibca

Body-Centered Orthorhombic

mmm

Vh = D2h

74 Imma

Body-Centered Orthorhombic

mmm

Vh = D2h

75 P4

Simple Tetragonal

4

C4

76 P41

Simple Tetragonal

4

C4

77 P42

Simple Tetragonal

4

C4

78 P43

Simple Tetragonal

4

C4

79 I4

Body-Centered Tetragonal

4

C4

80 I41

Body-Centered Tetragonal

4

C4

81 P4

Simple Tetragonal

4

S4

82 I4

Body-Centered Tetragonal

4

S4

83 P4/m

Simple Tetragonal

$\frac{4}{m}$

C4h

84 P42/m

Simple Tetragonal

$\frac{4}{m}$

C4h

85 P4/n

Simple Tetragonal

$\frac{4}{m}$

C4h

86 P42/n

Simple Tetragonal

$\frac{4}{m}$

C4h

87 I4/m

Body-Centered Tetragonal

$\frac{4}{m}$

C4h

88 I41/a

Body-Centered Tetragonal

$\frac{4}{m}$

C4h

89 P422

Simple Tetragonal

422

D4

90 P4212

Simple Tetragonal

422

D4

91 P4122

Simple Tetragonal

422

D4

92 P41212

Simple Tetragonal

422

D4

93 P4222

Simple Tetragonal

422

D4

94 P42212

Simple Tetragonal

422

D4

95 P4322

Simple Tetragonal

422

D4

96 P43212

Simple Tetragonal

422

D4

97 I422

Body-Centered Tetragonal

422

D4

98 I4122

Body-Centered Tetragonal

4mm

C4v

99 P4mm

Simple Tetragonal

4mm

C4v

100 P4bm

Simple Tetragonal

4mm

C4v

101 P42cm

Simple Tetragonal

4mm

C4v

102 P42nm

Simple Tetragonal

4mm

C4v

103 P4cc

Simple Tetragonal

4mm

C4v

104 P4nc

Simple Tetragonal

4mm

C4v

105 P42mc

Simple Tetragonal

4mm

C4v

106 P42bc

Simple Tetragonal

4mm

C4v

107 I4mm

Body-Centered Tetragonal

4mm

C4v

108 I4cm

Body-Centered Tetragonal

4mm

C4v

109 I41md

Body-Centered Tetragonal

4mm

C4v

110 I41cd

Body-Centered Tetragonal

4mm

C4v

111 P42m

Simple Tetragonal

42m

Vd = D2d

112 P42c

Simple Tetragonal

42m

Vd = D2d

113 P421m

Simple Tetragonal

42m

Vd = D2d

114 P421c

Simple Tetragonal

42m

Vd = D2d

115 P4m2

Simple Tetragonal

42m

Vd = D2d

116 P4c2

Simple Tetragonal

42m

Vd = D2d

117 P4b2

Simple Tetragonal

42m

Vd = D2d

118 P4n2

Simple Tetragonal

42m

Vd = D2d

119 I4m2

Body-Centered Tetragonal

42m

Vd = D2d

120 I4c2

Body-Centered Tetragonal

42m

Vd = D2d

121 I42m

Body-Centered Tetragonal

42m

Vd = D2d

122 I42d

Body-Centered Tetragonal

42m

Vd = D2d

123 P4/mmm

Simple Tetragonal

$\frac{4}{m}mm$

D4h

124 P4/mcc

Simple Tetragonal

$\frac{4}{m}mm$

D4h

125 P4/nbm

Simple Tetragonal

$\frac{4}{m}mm$

D4h

126 P4/nnc

Simple Tetragonal

$\frac{4}{m}mm$

D4h

127 P4/mbm

Simple Tetragonal

$\frac{4}{m}mm$

D4h

128 P4/mnc

Simple Tetragonal

$\frac{4}{m}mm$

D4h

129 P4/nmm

Simple Tetragonal

$\frac{4}{m}mm$

D4h

130 P4/ncc

Simple Tetragonal

$\frac{4}{m}mm$

D4h

131 P42/mmc

Simple Tetragonal

$\frac{4}{m}mm$

D4h

132 P42/mcm

Simple Tetragonal

$\frac{4}{m}mm$

D4h

133 P42/nbc

Simple Tetragonal

$\frac{4}{m}mm$

D4h

134 P42/nnm

Simple Tetragonal

$\frac{4}{m}mm$

D4h

135 P42/mbc

Simple Tetragonal

$\frac{4}{m}mm$

D4h

136 P42/mnm

Simple Tetragonal

$\frac{4}{m}mm$

D4h

137 P42/nmc

Simple Tetragonal

$\frac{4}{m}mm$

D4h

138 P42/ncm

Simple Tetragonal

$\frac{4}{m}mm$

D4h

139 I4/mmm

Body-Centered Tetragonal

$\frac{4}{m}mm$

D4h

140 I4/mcm

Body-Centered Tetragonal

$\frac{4}{m}mm$

D4h

141 I41/amd

Body-Centered Tetragonal

$\frac{4}{m}mm$

D4h

142 I41/acd

Body-Centered Tetragonal

$\frac{4}{m}mm$

D4h

143 P3

Hexagonal

3

C3

144 P31

Hexagonal

3

C3

145 P32

Hexagonal

3

C3

146 R3

Rhombohedral

3

C3

147 P3

Hexagonal

3

S6 = C6i

148 R3

Rhombohedral

3

S6 = C6i

149 P312

Hexagonal

32

D3

150 P321

Hexagonal

32

D3

151 P3112

Hexagonal

32

D3

152 P3121

Hexagonal

32

D3

153 P3212

Hexagonal

32

D3

154 P3221

Hexagonal

32

D3

155 R32

Rhombohedral

32

D3

156 P3m1

Hexagonal

3m

C3v

157 P31m

Hexagonal

3m

C3v

158 P3c1

Hexagonal

3m

C3v

159 P31c

Hexagonal

3m

C3v

160 R3m

Rhombohedral

3m

C3v

161 R3c

Rhombohedral

3m

C3v

162 P31m

Hexagonal

3m

D3d

163 P31c

Hexagonal

3m

D3d

164 P3m1

Hexagonal

3m

D3d

165 P3c1

Hexagonal

3m

D3d

166 R3m

Rhombohedral

3m

D3d

167 R3c

Rhombohedral

3m

D3d

168 P6

Hexagonal

6

C6

169 P61

Hexagonal

6

C6

170 P65

Hexagonal

6

C6

171 P62

Hexagonal

6

C6

172 P64

Hexagonal

6

C6

173 P63

Hexagonal

6

C6

174 P6

Hexagonal

6

C3h = S3

175 P6/m

Hexagonal

$\frac{6}{m}$

C6h

176 P63/m

Hexagonal

$\frac{6}{m}$

C6h

177 P622

Hexagonal

622

D6

178 P6122

Hexagonal

622

D6

179 P6522

Hexagonal

622

D6

180 P6222

Hexagonal

622

D6

181 P6422

Hexagonal

622

D6

182 P6322

Hexagonal

622

D6

183 P6mm

Hexagonal

6mm

C6v

184 P6cc

Hexagonal

6mm

C6v

185 P63cm

Hexagonal

6mm

C6v

186 P63mc

Hexagonal

6mm

C6v

187 P6m2

Hexagonal

6m2

D3h

188 P6c2

Hexagonal

6m2

D3h

189 P62m

Hexagonal

6m2

D3h

190 P62c

Hexagonal

6m2

D3h

191 P6/mmm

Hexagonal

$\frac{6}{m}mm$

D6h

192 P6/mcc

Hexagonal

$\frac{6}{m}mm$

D6h

193 P63/mcm

Hexagonal

$\frac{6}{m}mm$

D6h

194 P63/mmc

Hexagonal

$\frac{6}{m}mm$

D6h

195 P23

Simple Cubic

23

T

196 F23

Face-Centered Cubic

23

T

197 I23

Body-Centered Cubic

23

T

198 P213

Simple Cubic

23

T

199 I213

Body-Centered Cubic

23

T

200 Pm3

Simple Cubic

m3

Th

201 Pn3

Simple Cubic

m3

Th

202 Fm3

Face-Centered Cubic

m3

Th

203 Fd3

Face-Centered Cubic

m3

Th

204 Im3

Body-Centered Cubic

m3

Th

205 Pa3

Simple Cubic

m3

Th

206 Ia3

Body-Centered Cubic

m3

Th

207 P432

Simple Cubic

432

O

208 P4232

Simple Cubic

432

O

209 F432

Face-Centered Cubic

432

O

210 F4132

Face-Centered Cubic

432

O

211 I432

Body-Centered Cubic

432

O

212 P4332

Simple Cubic

432

O

213 P4132

Simple Cubic

432

O

214 I4132

Body-Centered Cubic

432

O

215 P43m

Simple Cubic

43m

Td

216 F43m

Face-Centered Cubic

43m

Td

217 I43m

Body-Centered Cubic

43m

Td

218 P43n

Simple Cubic

43m

Td

219 F43c

Face-Centered Cubic

43m

Td

220 I43d

Body-Centered Cubic

43m

Td

221 Pm3m

Simple Cubic

m3m

Oh

222 Pn3n

Simple Cubic

m3m

Oh

223 Pm3n

Simple Cubic

m3m

Oh

224 Pn3m

Simple Cubic

m3m

Oh

225 Fm3m

Face-Centered Cubic

m3m

Oh

226 Fm3c

Face-Centered Cubic

m3m

Oh

227 Fd3m

Face-Centered Cubic

m3m

Oh

228 Fd3c

Face-Centered Cubic

m3m

Oh

229 Im3m

Body-Centered Cubic

m3m

Oh

230 Ia3d

Body-Centered Cubic

m3m

Oh


P - Primitive
I - Body centered (Innenzentriert)
F - Face centered
C - Centered on C faces
A - Centered on A faces
R - Rhombohedral