Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
1270 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 1270 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#64316 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#64289 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10595353 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10593375 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #10593374 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #10593360 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #10593349 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C2×#3832 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C3×D3×Q8 | — | 2 | 1 | no | — |
| 10 | #64316 | — | 2 | 4 | no | — |
| 11 | #64316 | C4 | 1 | 8 | yes | complemented |
| 12 | C2×#3799 | — | 2 | 3 | no | — |
| 13 | C2×#3798 | C22 | 1 | 3 | yes | — |
| 14 | C2×#3797 | C22 | 1 | 3 | yes | — |
| 15 | #64289 | — | 2 | 4 | no | — |
| 16 | C2×#3788 | C22 | 1 | 3 | yes | — |
| 17 | C2×#3787 | C22 | 1 | 3 | yes | — |
| 18 | C3×Q8×Dic3 | — | 2 | 1 | no | — |
| 19 | #63900 | C22 | 1 | 3 | yes | — |
| 20 | #63894 | — | 2 | 3 | no | — |
| 21 | #63874 | C22 | 1 | 3 | yes | — |
| 22 | #63740 | C22 | 1 | 3 | yes | — |
| 23 | #63740 | C22 | 1 | 3 | yes | — |
| 24 | #63740 | C22 | 1 | 6 | yes | — |
| 25 | #63739 | C22 | 1 | 1 | yes | characteristic |
| 26 | #63739 | C22 | 1 | 3 | yes | — |
| 27 | C2×#1007 | — | 6 | 1 | no | — |
| 28 | #5574 | — | 6 | 1 | no | — |
| 29 | C2×C32×Q8 | D4 | 1 | 1 | yes | characteristic |
| 30 | #3832 | C2×C4 | 1 | 4 | yes | — |
| 31 | #3832 | D4 | 1 | 4 | yes | — |
| 32 | C2×C4⋊(C3⋊D3) | D4 | 1 | 3 | yes | — |
| 33 | C2×C4×C3⋊D3 | C23 | 1 | 3 | yes | — |
| 34 | C3×D3×Q8 | — | 2 | 8 | no | — |
| 35 | C2×C3×C4×D3 | — | 2 | 3 | no | — |
| 36 | C2×C3×Dic6 | — | 2 | 3 | no | — |
| 37 | C2×#627 | C23 | 1 | 3 | yes | — |
| 38 | C2×#626 | — | 2 | 3 | no | — |
| 39 | C2×#625 | C23 | 1 | 1 | yes | characteristic |
| 40 | C2×#624 | — | 2 | 3 | no | — |
| 41 | C2×#624 | — | 2 | 3 | no | — |
| 42 | #3799 | — | 2 | 24 | no | — |
| 43 | #3798 | — | 2 | 6 | no | — |
| 44 | #3798 | C2×C4 | 1 | 12 | yes | complemented |
| 45 | #3797 | — | 2 | 6 | no | — |
| 46 | #3797 | C2×C4 | 1 | 12 | yes | complemented |
| 47 | C2×#622 | C23 | 1 | 1 | yes | characteristic |
| 48 | C2×#622 | C23 | 1 | 3 | yes | — |
| 49 | #3788 | — | 2 | 6 | no | — |
| 50 | #3787 | — | 2 | 6 | no | — |
| 51 | C3×C4⋊Dic3 | — | 2 | 3 | no | — |
| 52 | C3×C4×Dic3 | — | 2 | 3 | no | — |
| 53 | #3724 | C23 | 1 | 1 | yes | characteristic |
| 54 | #3724 | C23 | 1 | 3 | yes | — |
| 55 | #3722 | — | 2 | 3 | no | — |
| 56 | #2898 | — | 9 | 1 | no | maximal, Sylow |
| 57 | C22×C3×Q8 | — | 6 | 1 | no | — |
| 58 | #1007 | — | 6 | 8 | no | — |
| 59 | #1007 | — | 6 | 8 | no | — |
| 60 | C2×#239 | — | 6 | 1 | no | — |
| 61 | C2×#239 | — | 6 | 1 | no | — |
| 62 | C2×#239 | — | 6 | 3 | no | — |
| 63 | C2×D3×Q8 | — | 2 | 1 | no | — |
| 64 | C2×D3×Q8 | — | 6 | 3 | no | — |
| 65 | C2×#235 | — | 6 | 3 | no | — |
| 66 | C2×#235 | — | 6 | 3 | no | — |
| 67 | C3×C4×Q8 | — | 6 | 1 | no | — |
| 68 | Q8×Dic3 | — | 2 | 1 | no | — |
| 69 | #893 | — | 6 | 3 | no | — |
| 70 | D3×C4⋊C4 | — | 6 | 3 | no | — |
| 71 | C4×D12 | — | 6 | 3 | no | — |
| 72 | #872 | — | 6 | 3 | no | — |
| 73 | C22×C3⋊D3 | C2×D4 | 1 | 3 | yes | — |
| 74 | C22×C3×D3 | — | 2 | 1 | no | — |
| 75 | C32×Q8 | C22⋊C4 | 1 | 4 | yes | — |
| 76 | C2×C32×C4 | C2×D4 | 1 | 3 | yes | — |
| 77 | C2×C3⋊Dic3 | C24 | 1 | 1 | yes | characteristic |
| 78 | C4⋊(C3⋊D3) | C22⋊C4 | 1 | 12 | yes | — |
| 79 | C4×C3⋊D3 | C22×C4 | 1 | 6 | yes | — |
| 80 | C4×C3⋊D3 | C2×D4 | 1 | 6 | yes | — |
| 81 | C2×C3×Dic3 | — | 2 | 3 | no | — |
| 82 | C2×C3×Dic3 | — | 2 | 1 | no | — |
| 83 | C2×C3×Dic3 | — | 2 | 3 | no | — |
| 84 | C3×C4×D3 | — | 2 | 12 | no | — |
| 85 | C3×Dic6 | — | 2 | 12 | no | — |
| 86 | #627 | — | 2 | 3 | no | — |
| 87 | #627 | C22×C4 | 1 | 6 | yes | — |
| 88 | #626 | — | 2 | 12 | no | — |
| 89 | #625 | — | 2 | 1 | no | — |
| 90 | #625 | C22×C4 | 1 | 2 | yes | — |
| 91 | #624 | — | 2 | 12 | no | — |
| 92 | #624 | — | 8 | 3 | no | — |
| 93 | #622 | — | 2 | 1 | no | — |
| 94 | #622 | — | 2 | 3 | no | — |
| 95 | C2×2−1+4 | — | 9 | 1 | no | — |
| 96 | C2×#131 | — | 9 | 1 | no | — |
| 97 | #520 | — | 9 | 1 | no | — |
| 98 | #419 | — | 9 | 3 | no | — |
| 99 | #417 | — | 9 | 3 | no | — |
| 100 | #416 | — | 9 | 3 | no | — |
| 101 | C2×#103 | — | 9 | 3 | no | — |
| 102 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 103 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 104 | C2×C3×Q8 | — | 6 | 3 | no | — |
| 105 | C2×C3×Q8 | — | 6 | 8 | no | — |
| 106 | C22×C3×C4 | — | 6 | 3 | no | — |
| 107 | C2×#67 | — | 6 | 1 | no | — |
| 108 | C2×#67 | — | 6 | 3 | no | — |
| 109 | #239 | — | 6 | 4 | no | — |
| 110 | #239 | — | 6 | 4 | no | — |
| 111 | #239 | — | 6 | 4 | no | — |
| 112 | #239 | — | 6 | 4 | no | — |
| 113 | #239 | — | 6 | 24 | no | — |
| 114 | D3×Q8 | — | 2 | 8 | no | — |
| 115 | D3×Q8 | — | 6 | 12 | no | — |
| 116 | D3×Q8 | — | 6 | 12 | no | — |
| 117 | #235 | — | 6 | 12 | no | — |
| 118 | #235 | — | 6 | 24 | no | — |
| 119 | #235 | — | 6 | 12 | no | — |
| 120 | C2×D12 | — | 6 | 3 | no | — |
| 121 | C2×D12 | — | 6 | 3 | no | — |
| 122 | C2×D12 | — | 6 | 3 | no | — |
| 123 | C2×C4×D3 | — | 2 | 3 | no | — |
| 124 | C2×C4×D3 | — | 6 | 3 | no | — |
| 125 | C2×C4×D3 | — | 6 | 3 | no | — |
| 126 | C2×C4×D3 | — | 6 | 6 | no | — |
| 127 | C2×C4×D3 | — | 6 | 6 | no | — |
| 128 | C2×Dic6 | — | 2 | 3 | no | — |
| 129 | C2×Dic6 | — | 6 | 3 | no | — |
| 130 | C3×C4⋊C4 | — | 6 | 3 | no | — |
| 131 | C3×C42 | — | 6 | 3 | no | — |
| 132 | (C2×D3)⋊C4 | — | 6 | 3 | no | — |
| 133 | (C2×D3)⋊C4 | — | 6 | 3 | no | — |
| 134 | C4⋊Dic3 | — | 2 | 3 | no | — |
| 135 | Dic3⋊C4 | — | 6 | 1 | no | — |
| 136 | Dic3⋊C4 | — | 6 | 3 | no | — |
| 137 | C4×Dic3 | — | 2 | 3 | no | — |
| 138 | C22×C32 | C22×D4 | 1 | 1 | yes | characteristic |
| 139 | C2×C3⋊D3 | — | 2 | 3 | no | — |
| 140 | C2×C3⋊D3 | — | 2 | 3 | no | — |
| 141 | C2×C3⋊D3 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 142 | C2×C3×D3 | — | 2 | 2 | no | — |
| 143 | C2×C3×D3 | — | 4 | 1 | no | — |
| 144 | C2×C3×D3 | — | 4 | 1 | no | — |
| 145 | C32×C4 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 146 | C3⋊Dic3 | C23×C4 | 1 | 1 | yes | characteristic |
| 147 | C3⋊Dic3 | C22×D4 | 1 | 1 | yes | characteristic |
| 148 | C3×Dic3 | — | 2 | 6 | no | — |
| 149 | C3×Dic3 | — | 4 | 3 | no | — |
| 150 | C3×Dic3 | — | 4 | 1 | no | — |
| 151 | 2−1+4 | — | 9 | 8 | no | — |
| 152 | 2−1+4 | — | 18 | 4 | no | — |
| 153 | C2×#41 | — | 9 | 3 | no | — |
| 154 | C2×#41 | — | 9 | 1 | no | — |
| 155 | C2×#41 | — | 18 | 3 | no | — |
| 156 | C22×Q8 | — | 6 | 1 | no | — |
| 157 | C22×Q8 | — | 9 | 3 | no | — |
| 158 | #131 | — | 18 | 4 | no | — |
| 159 | C2×M16 | — | 9 | 3 | no | — |
| 160 | C22×C8 | — | 9 | 3 | no | — |
| 161 | C4×Q8 | — | 6 | 1 | no | — |
| 162 | C4×D4 | — | 18 | 3 | no | — |
| 163 | #117 | — | 9 | 3 | no | — |
| 164 | C2×C4⋊C4 | — | 9 | 3 | no | — |
| 165 | #103 | — | 9 | 3 | no | — |
| 166 | #103 | — | 9 | 6 | no | — |
| 167 | #103 | — | 9 | 3 | no | — |
| 168 | #102 | — | 9 | 1 | no | — |
| 169 | #102 | — | 9 | 3 | no | — |
| 170 | C23×C3 | — | 6 | 1 | no | — |
| 171 | C22×D3 | — | 2 | 1 | no | — |
| 172 | C22×D3 | — | 6 | 3 | no | — |
| 173 | C22×D3 | — | 6 | 3 | no | — |
| 174 | C3×Q8 | — | 2 | 4 | no | — |
| 175 | C3×Q8 | — | 2 | 4 | no | — |
| 176 | C3×Q8 | — | 6 | 12 | no | — |
| 177 | C2×C3×C4 | — | 2 | 3 | no | — |
| 178 | C2×C3×C4 | — | 2 | 3 | no | — |
| 179 | C2×C3×C4 | — | 6 | 12 | no | — |
| 180 | C2×C3×C4 | — | 6 | 3 | no | — |
| 181 | C2×C3×C4 | — | 6 | 1 | no | — |
| 182 | C2×C3×C4 | — | 6 | 3 | no | — |
| 183 | #67 | — | 6 | 2 | no | — |
| 184 | #67 | — | 6 | 12 | no | — |
| 185 | #67 | — | 6 | 2 | no | — |
| 186 | C2×Dic3 | — | 2 | 3 | no | — |
| 187 | C2×Dic3 | — | 2 | 1 | no | — |
| 188 | C2×Dic3 | — | 2 | 3 | no | — |
| 189 | C2×Dic3 | — | 6 | 1 | no | — |
| 190 | C2×Dic3 | — | 6 | 1 | no | — |
| 191 | D12 | — | 6 | 12 | no | — |
| 192 | D12 | — | 6 | 12 | no | — |
| 193 | D12 | — | 6 | 12 | no | — |
| 194 | C4×D3 | — | 2 | 12 | no | — |
| 195 | C4×D3 | — | 6 | 6 | no | — |
| 196 | C4×D3 | — | 6 | 6 | no | — |
| 197 | C4×D3 | — | 6 | 6 | no | — |
| 198 | C4×D3 | — | 6 | 6 | no | — |
| 199 | C4×D3 | — | 6 | 24 | no | — |
| 200 | C4×D3 | — | 24 | 6 | no | — |
| 201 | Dic6 | — | 2 | 12 | no | — |
| 202 | Dic6 | — | 6 | 6 | no | — |
| 203 | Dic6 | — | 6 | 6 | no | — |
| 204 | C2×C32 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 205 | C2×C32 | #577 | 1 | 1 | yes | characteristic |
| 206 | C2×C32 | #578 | 1 | 1 | yes | characteristic |
| 207 | C3⋊D3 | — | 2 | 6 | no | — |
| 208 | C3×D3 | — | 4 | 2 | no | — |
| 209 | #41 | — | 9 | 12 | no | — |
| 210 | #41 | — | 9 | 4 | no | — |
| 211 | #41 | — | 9 | 4 | no | — |
| 212 | #41 | — | 18 | 6 | no | — |
| 213 | #41 | — | 18 | 24 | no | — |
| 214 | C2×Q8 | — | 6 | 8 | no | — |
| 215 | C2×Q8 | — | 6 | 3 | no | — |
| 216 | C2×Q8 | — | 9 | 3 | no | — |
| 217 | C2×Q8 | — | 9 | 12 | no | — |
| 218 | C2×Q8 | — | 18 | 6 | no | — |
| 219 | C2×Q8 | GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 220 | C2×D4 | — | 9 | 1 | no | — |
| 221 | C2×D4 | — | 9 | 3 | no | — |
| 222 | C2×D4 | — | 18 | 3 | no | — |
| 223 | C22×C4 | — | 6 | 3 | no | — |
| 224 | C22×C4 | — | 9 | 3 | no | — |
| 225 | C22×C4 | — | 18 | 3 | no | — |
| 226 | C22×C4 | — | 18 | 3 | no | — |
| 227 | M16 | — | 18 | 6 | no | — |
| 228 | C2×C8 | — | 9 | 1 | no | — |
| 229 | C2×C8 | — | 9 | 3 | no | — |
| 230 | C2×C8 | — | 18 | 6 | no | — |
| 231 | C4⋊C4 | — | 6 | 3 | no | — |
| 232 | C4⋊C4 | — | 9 | 3 | no | — |
| 233 | C4⋊C4 | — | 9 | 1 | no | — |
| 234 | C22⋊C4 | — | 18 | 3 | no | — |
| 235 | C42 | — | 6 | 3 | no | — |
| 236 | C22×C3 | — | 2 | 1 | no | — |
| 237 | C22×C3 | — | 2 | 1 | no | — |
| 238 | C22×C3 | — | 6 | 2 | no | — |
| 239 | C22×C3 | — | 12 | 1 | no | — |
| 240 | C22×C3 | — | 12 | 1 | no | — |
| 241 | C2×D3 | — | 2 | 2 | no | — |
| 242 | C2×D3 | — | 4 | 1 | no | — |
| 243 | C2×D3 | — | 4 | 1 | no | — |
| 244 | C2×D3 | — | 6 | 6 | no | — |
| 245 | C2×D3 | — | 6 | 6 | no | — |
| 246 | C2×D3 | — | 12 | 3 | no | — |
| 247 | C2×D3 | — | 12 | 3 | no | — |
| 248 | C2×D3 | — | 12 | 3 | no | — |
| 249 | C2×D3 | — | 12 | 3 | no | — |
| 250 | C3×C4 | — | 2 | 6 | no | — |
| 251 | C3×C4 | — | 2 | 6 | no | — |
| 252 | C3×C4 | — | 6 | 6 | no | — |
| 253 | C3×C4 | — | 12 | 3 | no | — |
| 254 | C3×C4 | — | 12 | 1 | no | — |
| 255 | Dic3 | — | 2 | 6 | no | — |
| 256 | Dic3 | — | 4 | 3 | no | — |
| 257 | Dic3 | — | 4 | 1 | no | — |
| 258 | Dic3 | — | 6 | 1 | no | — |
| 259 | Dic3 | — | 6 | 1 | no | — |
| 260 | Dic3 | — | 6 | 1 | no | — |
| 261 | Dic3 | — | 6 | 1 | no | — |
| 262 | C32 | #2898 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 263 | C23 | — | 6 | 1 | no | — |
| 264 | C23 | — | 9 | 3 | no | — |
| 265 | Q8 | — | 6 | 12 | no | — |
| 266 | Q8 | — | 9 | 6 | no | — |
| 267 | Q8 | — | 18 | 3 | no | — |
| 268 | Q8 | D32⋊C4 | 1 | 4 | yes | — |
| 269 | D4 | — | 9 | 2 | no | — |
| 270 | D4 | — | 9 | 12 | no | — |
| 271 | D4 | — | 18 | 12 | no | — |
| 272 | D4 | — | 18 | 1 | no | — |
| 273 | C2×C4 | — | 6 | 3 | no | — |
| 274 | C2×C4 | — | 6 | 12 | no | — |
| 275 | C2×C4 | — | 6 | 1 | no | — |
| 276 | C2×C4 | — | 6 | 3 | no | — |
| 277 | C2×C4 | — | 9 | 1 | no | — |
| 278 | C2×C4 | — | 9 | 6 | no | — |
| 279 | C2×C4 | — | 9 | 6 | no | — |
| 280 | C2×C4 | — | 18 | 12 | no | — |
| 281 | C2×C4 | — | 72 | 3 | no | — |
| 282 | C2×C4 | C2×GO+(4,𝔽2) | 1 | 3 | yes | — |
| 283 | C8 | — | 18 | 1 | no | — |
| 284 | C8 | — | 18 | 3 | no | — |
| 285 | C2×C3 | — | 2 | 1 | no | — |
| 286 | C2×C3 | — | 2 | 1 | no | — |
| 287 | C2×C3 | — | 2 | 1 | no | — |
| 288 | C2×C3 | — | 2 | 1 | no | — |
| 289 | C2×C3 | — | 2 | 1 | no | — |
| 290 | C2×C3 | — | 2 | 1 | no | — |
| 291 | C2×C3 | — | 12 | 2 | no | — |
| 292 | D3 | — | 4 | 2 | no | — |
| 293 | D3 | — | 12 | 6 | no | — |
| 294 | D3 | — | 12 | 6 | no | — |
| 295 | C22 | — | 6 | 2 | no | — |
| 296 | C22 | — | 9 | 6 | no | — |
| 297 | C22 | — | 12 | 1 | no | — |
| 298 | C22 | — | 12 | 1 | no | — |
| 299 | C22 | — | 18 | 3 | no | — |
| 300 | C22 | — | 18 | 3 | no | — |
| 301 | C22 | C22×GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 302 | C4 | — | 6 | 6 | no | — |
| 303 | C4 | — | 9 | 1 | no | — |
| 304 | C4 | — | 9 | 1 | no | — |
| 305 | C4 | — | 12 | 1 | no | — |
| 306 | C4 | — | 12 | 3 | no | — |
| 307 | C4 | C2×D32⋊C4 | 1 | 6 | yes | — |
| 308 | C3 | — | 2 | 1 | no | — |
| 309 | C3 | — | 2 | 1 | no | — |
| 310 | C2 | — | 12 | 2 | no | — |
| 311 | C2 | — | 18 | 6 | no | — |
| 312 | C2 | #10596520 | 1 | 1 | yes | characteristic |
| 313 | C2 | #10596521 | 1 | 1 | yes | characteristic |
| 314 | C2 | C22×D32⋊C4 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#64289 | C2 | semidirect |
| 3. #10595353 | C2 | semidirect |
| 4. #10593375 | C2 | semidirect |
| 5. #10593374 | C2 | semidirect |
| 6. #10593360 | C2 | semidirect |
| 7. #10593349 | C2 | semidirect |
| 11. #64316 | C4 | semidirect |
| 44. #3798 | C2×C4 | semidirect |
| 46. #3797 | C2×C4 | semidirect |
| 262. C32 | #2898 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#64316 | 2 |
| C2×#64289 | 2 |
| #10595353 | 2 |
| #10593375 | 2 |
| #10593374 | 2 |
| #10593360 | 2 |
| #10593349 | 2 |
| #2898 | 9 |