Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
3133 conjugacy classes of subgroups.
The subgroup lattice has 311 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C3×#1007 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C2×Q8×C3⋊Dic3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10595412 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #10595409 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | Q8×#625 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #10595044 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #10595032 | C2 | 1 | 6 | yes | maximal, complemented |
| 8 | #87291 | — | 3 | 2 | no | maximal |
| 9 | C22×C32×Q8 | C22 | 1 | 1 | yes | characteristic |
| 10 | C3×#1007 | — | 2 | 16 | no | — |
| 11 | C2×C3×#239 | C22 | 1 | 8 | yes | — |
| 12 | C2×C3×D3×Q8 | C22 | 1 | 8 | yes | — |
| 13 | C2×C3×#235 | C22 | 1 | 12 | yes | — |
| 14 | Q8×C3⋊Dic3 | C22 | 1 | 4 | yes | — |
| 15 | Q8×C3⋊Dic3 | C22 | 1 | 4 | yes | — |
| 16 | C2×C4⋊(C3⋊Dic3) | C22 | 1 | 3 | yes | — |
| 17 | C2×C4×C3⋊Dic3 | C22 | 1 | 3 | yes | — |
| 18 | #63980 | C22 | 1 | 4 | yes | — |
| 19 | #63980 | C22 | 1 | 12 | yes | — |
| 20 | #63914 | C22 | 1 | 6 | yes | — |
| 21 | #63908 | C22 | 1 | 24 | yes | — |
| 22 | C4×#625 | C22 | 1 | 12 | yes | — |
| 23 | C4×#625 | C22 | 1 | 12 | yes | — |
| 24 | #63873 | C22 | 1 | 6 | yes | — |
| 25 | #63863 | C22 | 1 | 12 | yes | — |
| 26 | #63856 | C22 | 1 | 24 | yes | — |
| 27 | C2×C3×2−1+4 | — | 3 | 2 | no | — |
| 28 | C2×#1007 | D3 | 1 | 2 | yes | complemented |
| 29 | #5834 | — | 3 | 8 | no | — |
| 30 | #5833 | — | 3 | 12 | no | — |
| 31 | #5832 | — | 3 | 12 | no | — |
| 32 | #5831 | — | 3 | 8 | no | — |
| 33 | #5817 | — | 3 | 8 | no | — |
| 34 | Q8×#67 | — | 3 | 8 | no | — |
| 35 | C2×Q8×Dic3 | — | 3 | 2 | no | — |
| 36 | C2×Q8×Dic3 | — | 6 | 1 | no | — |
| 37 | C2×C32×Q8 | D4 | 1 | 8 | yes | complemented |
| 38 | C2×C32×Q8 | C23 | 1 | 4 | yes | — |
| 39 | C22×C32×C4 | C23 | 1 | 3 | yes | — |
| 40 | C22×C3⋊Dic3 | C23 | 1 | 1 | yes | characteristic |
| 41 | C22×C3⋊Dic3 | C23 | 1 | 3 | yes | — |
| 42 | C2×C3×#67 | C23 | 1 | 8 | yes | — |
| 43 | C3×#239 | — | 2 | 32 | no | — |
| 44 | C3×D3×Q8 | — | 2 | 32 | no | — |
| 45 | C3×#235 | — | 2 | 48 | no | — |
| 46 | C2×C3×D12 | C23 | 1 | 12 | yes | — |
| 47 | C2×C3×C4×D3 | C23 | 1 | 24 | yes | — |
| 48 | C2×C3×Dic6 | C23 | 1 | 12 | yes | — |
| 49 | C2×#625 | C23 | 1 | 4 | yes | — |
| 50 | C2×#625 | C23 | 1 | 12 | yes | — |
| 51 | C4⋊(C3⋊Dic3) | C23 | 1 | 12 | yes | — |
| 52 | C4×C3⋊Dic3 | C23 | 1 | 6 | yes | — |
| 53 | C4×C3⋊Dic3 | C23 | 1 | 6 | yes | — |
| 54 | #3724 | C23 | 1 | 12 | yes | — |
| 55 | #3724 | C23 | 1 | 4 | yes | — |
| 56 | (C2×D3)⋊Dic3 | C23 | 1 | 24 | yes | — |
| 57 | (C2×D3)⋊Dic3 | C23 | 1 | 8 | yes | — |
| 58 | #3497 | — | 9 | 1 | no | Sylow |
| 59 | C3×2−1+4 | — | 6 | 16 | no | — |
| 60 | C2×C3×#41 | — | 3 | 8 | no | — |
| 61 | C2×C3×#41 | — | 3 | 12 | no | — |
| 62 | C22×C3×Q8 | — | 2 | 1 | no | — |
| 63 | C22×C3×Q8 | — | 3 | 8 | no | — |
| 64 | C22×C3×Q8 | C2×D3 | 1 | 2 | yes | — |
| 65 | #1007 | — | 2 | 16 | no | — |
| 66 | C2×#239 | C2×D3 | 1 | 8 | yes | — |
| 67 | C2×D3×Q8 | C2×D3 | 1 | 8 | yes | — |
| 68 | C2×#235 | C2×D3 | 1 | 12 | yes | — |
| 69 | #947 | — | 3 | 12 | no | — |
| 70 | #946 | — | 3 | 24 | no | — |
| 71 | Q8×Dic3 | — | 3 | 8 | no | — |
| 72 | Q8×Dic3 | — | 3 | 8 | no | — |
| 73 | Q8×Dic3 | — | 6 | 4 | no | — |
| 74 | Q8×Dic3 | — | 6 | 4 | no | — |
| 75 | #944 | — | 3 | 12 | no | — |
| 76 | #940 | — | 3 | 12 | no | — |
| 77 | #939 | — | 3 | 8 | no | — |
| 78 | #939 | — | 3 | 24 | no | — |
| 79 | #938 | — | 3 | 24 | no | — |
| 80 | #936 | — | 3 | 12 | no | — |
| 81 | #930 | — | 3 | 24 | no | — |
| 82 | C4×#67 | — | 3 | 24 | no | — |
| 83 | C4×#67 | — | 3 | 24 | no | — |
| 84 | C2×C4⋊Dic3 | — | 3 | 6 | no | — |
| 85 | C2×C4⋊Dic3 | — | 6 | 3 | no | — |
| 86 | #924 | — | 3 | 24 | no | — |
| 87 | C2×C4×Dic3 | — | 3 | 6 | no | — |
| 88 | C2×C4×Dic3 | — | 6 | 3 | no | — |
| 89 | C23×C32 | C24 | 1 | 1 | yes | characteristic |
| 90 | C22×C3×D3 | C24 | 1 | 8 | yes | — |
| 91 | C32×Q8 | C2×D4 | 1 | 16 | yes | — |
| 92 | C2×C32×C4 | C2×D4 | 1 | 12 | yes | — |
| 93 | C2×C32×C4 | C24 | 1 | 6 | yes | — |
| 94 | C2×C3⋊Dic3 | — | 2 | 2 | no | — |
| 95 | C2×C3⋊Dic3 | — | 4 | 3 | no | — |
| 96 | C2×C3⋊Dic3 | C24 | 1 | 1 | yes | characteristic |
| 97 | C2×C3⋊Dic3 | C24 | 1 | 6 | yes | — |
| 98 | C2×C3⋊Dic3 | C24 | 1 | 1 | yes | characteristic |
| 99 | C3×#67 | — | 2 | 16 | no | — |
| 100 | C2×C3×Dic3 | C24 | 1 | 8 | yes | — |
| 101 | C3×D12 | — | 2 | 24 | no | — |
| 102 | C3×C4×D3 | — | 2 | 48 | no | — |
| 103 | C3×Dic6 | — | 2 | 24 | no | — |
| 104 | #625 | — | 2 | 8 | no | — |
| 105 | #625 | — | 4 | 12 | no | — |
| 106 | C2×2−1+4 | — | 3 | 2 | no | — |
| 107 | #550 | — | 9 | 4 | no | — |
| 108 | D4×Q8 | — | 9 | 4 | no | — |
| 109 | #548 | — | 9 | 8 | no | — |
| 110 | #537 | — | 9 | 6 | no | — |
| 111 | #532 | — | 9 | 6 | no | — |
| 112 | C2×C4×Q8 | — | 9 | 1 | no | — |
| 113 | C3×#41 | — | 6 | 32 | no | — |
| 114 | C3×#41 | — | 6 | 48 | no | — |
| 115 | C2×C3×Q8 | — | 2 | 8 | no | — |
| 116 | C2×C3×Q8 | — | 2 | 4 | no | — |
| 117 | C2×C3×Q8 | — | 3 | 12 | no | — |
| 118 | C2×C3×Q8 | — | 6 | 32 | no | — |
| 119 | C2×C3×Q8 | #67 | 1 | 16 | yes | complemented |
| 120 | C2×C3×Q8 | C22×D3 | 1 | 8 | yes | — |
| 121 | C2×C3×D4 | — | 3 | 8 | no | — |
| 122 | C2×C3×D4 | — | 3 | 12 | no | — |
| 123 | C22×C3×C4 | — | 2 | 3 | no | — |
| 124 | C22×C3×C4 | — | 3 | 24 | no | — |
| 125 | C22×C3×C4 | C22×D3 | 1 | 6 | yes | — |
| 126 | C2×#67 | — | 3 | 8 | no | — |
| 127 | C2×#67 | — | 3 | 24 | no | — |
| 128 | C2×#67 | C22×D3 | 1 | 8 | yes | — |
| 129 | C22×Dic3 | — | 3 | 2 | no | — |
| 130 | C22×Dic3 | — | 3 | 6 | no | — |
| 131 | C22×Dic3 | — | 6 | 1 | no | — |
| 132 | C22×Dic3 | — | 6 | 3 | no | — |
| 133 | #239 | — | 2 | 32 | no | — |
| 134 | D3×Q8 | — | 2 | 32 | no | — |
| 135 | #235 | — | 2 | 48 | no | — |
| 136 | C2×D12 | C22×D3 | 1 | 12 | yes | — |
| 137 | C2×C4×D3 | C22×D3 | 1 | 24 | yes | — |
| 138 | C2×Dic6 | C22×D3 | 1 | 12 | yes | — |
| 139 | #217 | — | 3 | 24 | no | — |
| 140 | #217 | — | 3 | 8 | no | — |
| 141 | (C2×D3)⋊C4 | — | 3 | 8 | no | — |
| 142 | (C2×D3)⋊C4 | — | 3 | 24 | no | — |
| 143 | C4⋊Dic3 | — | 3 | 24 | no | — |
| 144 | C4⋊Dic3 | — | 6 | 12 | no | — |
| 145 | Dic3⋊C4 | — | 3 | 24 | no | — |
| 146 | Dic3⋊C4 | — | 3 | 8 | no | — |
| 147 | C4×Dic3 | — | 3 | 12 | no | — |
| 148 | C4×Dic3 | — | 3 | 12 | no | — |
| 149 | C4×Dic3 | — | 6 | 6 | no | — |
| 150 | C4×Dic3 | — | 6 | 6 | no | — |
| 151 | C22×C32 | — | 2 | 1 | no | — |
| 152 | C22×C32 | — | 2 | 1 | no | — |
| 153 | C22×C32 | C22×D4 | 1 | 2 | yes | — |
| 154 | C22×C32 | C25 | 1 | 1 | yes | characteristic |
| 155 | C2×C3×D3 | — | 2 | 8 | no | — |
| 156 | C2×C3×D3 | — | 2 | 8 | no | — |
| 157 | C2×C3×D3 | — | 2 | 8 | no | — |
| 158 | C32×C4 | C22×D4 | 1 | 12 | yes | — |
| 159 | C3⋊Dic3 | — | 2 | 1 | no | — |
| 160 | C3⋊Dic3 | — | 2 | 6 | no | — |
| 161 | C3⋊Dic3 | 2−1+4 | 1 | 2 | yes | complemented |
| 162 | C3×Dic3 | — | 2 | 8 | no | — |
| 163 | 2−1+4 | — | 6 | 16 | no | — |
| 164 | C2×#41 | — | 3 | 8 | no | — |
| 165 | C2×#41 | — | 3 | 12 | no | — |
| 166 | C22×Q8 | — | 3 | 8 | no | — |
| 167 | C22×Q8 | D32 | 1 | 1 | yes | characteristic |
| 168 | C4⋊Q8 | — | 9 | 6 | no | — |
| 169 | #127 | — | 9 | 6 | no | — |
| 170 | #124 | — | 9 | 12 | no | — |
| 171 | C22⋊Q8 | — | 9 | 24 | no | — |
| 172 | #121 | — | 9 | 24 | no | — |
| 173 | #121 | — | 9 | 4 | no | — |
| 174 | #121 | — | 9 | 12 | no | — |
| 175 | C4×Q8 | — | 9 | 4 | no | — |
| 176 | C4×Q8 | — | 9 | 4 | no | — |
| 177 | C4×D4 | — | 9 | 12 | no | — |
| 178 | C4×D4 | — | 9 | 12 | no | — |
| 179 | C2×C4⋊C4 | — | 9 | 3 | no | — |
| 180 | C2×C42 | — | 9 | 3 | no | — |
| 181 | C23×C3 | — | 2 | 1 | no | — |
| 182 | C23×C3 | — | 3 | 8 | no | — |
| 183 | C23×C3 | C23×D3 | 1 | 2 | yes | — |
| 184 | C22×D3 | C23×D3 | 1 | 8 | yes | — |
| 185 | C3×Q8 | — | 2 | 16 | no | — |
| 186 | C3×Q8 | — | 6 | 24 | no | — |
| 187 | C3×Q8 | C2×#67 | 1 | 32 | yes | — |
| 188 | C3×D4 | — | 6 | 16 | no | — |
| 189 | C3×D4 | — | 6 | 24 | no | — |
| 190 | C2×C3×C4 | — | 2 | 6 | no | — |
| 191 | C2×C3×C4 | — | 2 | 12 | no | — |
| 192 | C2×C3×C4 | — | 3 | 8 | no | — |
| 193 | C2×C3×C4 | — | 6 | 48 | no | — |
| 194 | C2×C3×C4 | C2×#67 | 1 | 24 | yes | — |
| 195 | C2×C3×C4 | C23×D3 | 1 | 12 | yes | — |
| 196 | #67 | — | 2 | 16 | no | — |
| 197 | #67 | — | 6 | 16 | no | — |
| 198 | #67 | — | 12 | 24 | no | — |
| 199 | C2×Dic3 | — | 3 | 2 | no | — |
| 200 | C2×Dic3 | — | 3 | 2 | no | — |
| 201 | C2×Dic3 | — | 3 | 12 | no | — |
| 202 | C2×Dic3 | — | 6 | 1 | no | — |
| 203 | C2×Dic3 | — | 6 | 1 | no | — |
| 204 | C2×Dic3 | — | 6 | 6 | no | — |
| 205 | C2×Dic3 | — | 6 | 4 | no | — |
| 206 | C2×Dic3 | — | 6 | 4 | no | — |
| 207 | C2×Dic3 | — | 12 | 6 | no | — |
| 208 | C2×Dic3 | — | 12 | 6 | no | — |
| 209 | C2×Dic3 | C23×D3 | 1 | 8 | yes | — |
| 210 | D12 | — | 2 | 24 | no | — |
| 211 | C4×D3 | — | 2 | 48 | no | — |
| 212 | Dic6 | — | 2 | 24 | no | — |
| 213 | C2×C32 | — | 2 | 2 | no | — |
| 214 | C2×C32 | C23×D4 | 1 | 1 | yes | characteristic |
| 215 | C2×C32 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 216 | C2×C32 | #585 | 1 | 1 | yes | characteristic |
| 217 | C3×D3 | — | 4 | 8 | no | — |
| 218 | #41 | — | 6 | 32 | no | — |
| 219 | #41 | — | 6 | 48 | no | — |
| 220 | C2×Q8 | — | 3 | 12 | no | — |
| 221 | C2×Q8 | — | 6 | 32 | no | — |
| 222 | C2×Q8 | #625 | 1 | 8 | yes | complemented |
| 223 | C2×Q8 | C2×D32 | 1 | 4 | yes | — |
| 224 | C2×D4 | — | 3 | 8 | no | — |
| 225 | C2×D4 | — | 3 | 12 | no | — |
| 226 | C2×D4 | — | 9 | 4 | no | — |
| 227 | C2×D4 | — | 9 | 12 | no | — |
| 228 | C22×C4 | — | 3 | 24 | no | — |
| 229 | C22×C4 | — | 9 | 1 | no | — |
| 230 | C22×C4 | — | 9 | 3 | no | — |
| 231 | C22×C4 | C2×D32 | 1 | 3 | yes | — |
| 232 | C4⋊C4 | — | 9 | 4 | no | — |
| 233 | C4⋊C4 | — | 9 | 12 | no | — |
| 234 | C4⋊C4 | — | 9 | 12 | no | — |
| 235 | C22⋊C4 | — | 9 | 8 | no | — |
| 236 | C22⋊C4 | — | 9 | 24 | no | — |
| 237 | C42 | — | 9 | 6 | no | — |
| 238 | C42 | — | 9 | 6 | no | — |
| 239 | C22×C3 | — | 2 | 2 | no | — |
| 240 | C22×C3 | — | 2 | 2 | no | — |
| 241 | C22×C3 | — | 2 | 2 | no | — |
| 242 | C22×C3 | — | 2 | 2 | no | — |
| 243 | C22×C3 | — | 2 | 2 | no | — |
| 244 | C22×C3 | — | 2 | 1 | no | — |
| 245 | C22×C3 | — | 6 | 8 | no | — |
| 246 | C22×C3 | — | 6 | 8 | no | — |
| 247 | C22×C3 | — | 6 | 8 | no | — |
| 248 | C22×C3 | C22×#67 | 1 | 4 | yes | — |
| 249 | C22×C3 | C24×D3 | 1 | 2 | yes | — |
| 250 | C2×D3 | — | 2 | 8 | no | — |
| 251 | C2×D3 | — | 2 | 8 | no | — |
| 252 | C2×D3 | — | 2 | 8 | no | — |
| 253 | C3×C4 | — | 2 | 12 | no | — |
| 254 | C3×C4 | — | 6 | 8 | no | — |
| 255 | C3×C4 | C22×#67 | 1 | 24 | yes | — |
| 256 | Dic3 | — | 2 | 8 | no | — |
| 257 | Dic3 | — | 3 | 4 | no | — |
| 258 | Dic3 | — | 6 | 2 | no | — |
| 259 | Dic3 | — | 6 | 2 | no | — |
| 260 | Dic3 | — | 6 | 2 | no | — |
| 261 | Dic3 | — | 6 | 12 | no | — |
| 262 | Dic3 | — | 12 | 6 | no | — |
| 263 | C32 | #3497 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 264 | C23 | — | 3 | 8 | no | — |
| 265 | C23 | C22×D32 | 1 | 1 | yes | characteristic |
| 266 | Q8 | — | 6 | 24 | no | — |
| 267 | Q8 | C2×#625 | 1 | 16 | yes | — |
| 268 | D4 | — | 6 | 16 | no | — |
| 269 | D4 | — | 6 | 24 | no | — |
| 270 | D4 | — | 18 | 8 | no | — |
| 271 | D4 | — | 36 | 12 | no | — |
| 272 | C2×C4 | — | 3 | 8 | no | — |
| 273 | C2×C4 | — | 6 | 48 | no | — |
| 274 | C2×C4 | — | 9 | 1 | no | — |
| 275 | C2×C4 | — | 9 | 1 | no | — |
| 276 | C2×C4 | — | 9 | 6 | no | — |
| 277 | C2×C4 | — | 18 | 2 | no | — |
| 278 | C2×C4 | — | 36 | 3 | no | — |
| 279 | C2×C4 | C2×#625 | 1 | 12 | yes | — |
| 280 | C2×C4 | C22×D32 | 1 | 6 | yes | — |
| 281 | C2×C3 | — | 2 | 4 | no | — |
| 282 | C2×C3 | — | 2 | 4 | no | — |
| 283 | C2×C3 | — | 2 | 1 | no | — |
| 284 | C2×C3 | — | 2 | 1 | no | — |
| 285 | C2×C3 | — | 2 | 1 | no | — |
| 286 | C2×C3 | — | 12 | 8 | no | — |
| 287 | C2×C3 | D3×2−1+4 | 1 | 2 | yes | — |
| 288 | C2×C3 | #5968 | 1 | 2 | yes | — |
| 289 | C2×C3 | C23×#67 | 1 | 2 | yes | — |
| 290 | D3 | — | 4 | 8 | no | — |
| 291 | C22 | — | 2 | 1 | no | — |
| 292 | C22 | — | 2 | 1 | no | — |
| 293 | C22 | — | 6 | 8 | no | — |
| 294 | C22 | — | 6 | 8 | no | — |
| 295 | C22 | — | 6 | 8 | no | — |
| 296 | C22 | C22×#625 | 1 | 2 | yes | — |
| 297 | C22 | C23×D32 | 1 | 1 | yes | characteristic |
| 298 | C4 | — | 6 | 8 | no | — |
| 299 | C4 | — | 9 | 2 | no | — |
| 300 | C4 | — | 18 | 1 | no | — |
| 301 | C4 | — | 18 | 6 | no | — |
| 302 | C4 | C22×#625 | 1 | 12 | yes | — |
| 303 | C3 | — | 2 | 1 | no | — |
| 304 | C3 | #87291 | 1 | 2 | yes | complemented |
| 305 | C2 | — | 2 | 2 | no | — |
| 306 | C2 | — | 12 | 8 | no | — |
| 307 | C2 | #10596687 | 1 | 1 | yes | characteristic |
| 308 | C2 | #10596689 | 1 | 1 | yes | characteristic |
| 309 | C2 | C23×#625 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D32 |
| commutator, socle, cosocle | C22×C32 | C25 |
| Fitting | C22×C32×Q8 | C22 |
| hypercenter | C22×Q8 | D32 |
| hypocenter | C32 | #3497 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#1007 | C2 | semidirect |
| 2. C2×Q8×C3⋊Dic3 | C2 | semidirect |
| 3. #10595412 | C2 | semidirect |
| 4. #10595409 | C2 | semidirect |
| 5. Q8×#625 | C2 | semidirect |
| 6. #10595044 | C2 | semidirect |
| 7. #10595032 | C2 | semidirect |
| 28. C2×#1007 | D3 | semidirect |
| 37. C2×C32×Q8 | D4 | semidirect |
| 119. C2×C3×Q8 | #67 | semidirect |
| 161. C3⋊Dic3 | 2−1+4 | semidirect |
| 222. C2×Q8 | #625 | semidirect |
| 263. C32 | #3497 | semidirect |
| 304. C3 | #87291 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#1007 | 2 |
| C2×Q8×C3⋊Dic3 | 2 |
| #10595412 | 2 |
| #10595409 | 2 |
| Q8×#625 | 2 |
| #10595044 | 2 |
| #10595032 | 2 |
| #87291 | 3 |