← #10691709
Subgroups — #10691709
Order 768.
412 conjugacy classes of subgroups.
The subgroup lattice has 330 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #83330 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #83324 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C3×#1550 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #70524 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×#4482 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #67714 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #67706 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #9270 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C3×#533 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#869 | C22 | 1 | 1 | yes | characteristic |
| 11 | C3×#423 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C3×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 13 | #5079 | — | 2 | 1 | no | — |
| 14 | #5078 | — | 2 | 1 | no | — |
| 15 | #5044 | C4 | 1 | 4 | yes | — |
| 16 | #5024 | — | 2 | 2 | no | — |
| 17 | #5024 | — | 2 | 1 | no | — |
| 18 | Q8×C3⋊C8 | — | 2 | 1 | no | — |
| 19 | #5021 | — | 2 | 2 | no | — |
| 20 | #4999 | — | 2 | 1 | no | — |
| 21 | #4997 | — | 2 | 1 | no | — |
| 22 | #4925 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×C4⋊(C3⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 24 | C4×#208 | C22 | 1 | 1 | yes | characteristic |
| 25 | C3×#333 | — | 2 | 1 | no | — |
| 26 | C3×#332 | — | 2 | 1 | no | — |
| 27 | #4554 | — | 2 | 1 | no | — |
| 28 | #4554 | — | 2 | 1 | no | — |
| 29 | #4482 | — | 2 | 1 | no | — |
| 30 | #4482 | — | 2 | 1 | no | — |
| 31 | #3230 | — | 3 | 1 | no | — |
| 32 | #2962 | — | 3 | 1 | no | — |
| 33 | #1997 | — | 3 | 1 | no | — |
| 34 | #1550 | D3 | 1 | 1 | yes | characteristic |
| 35 | C2×#332 | — | 3 | 1 | no | — |
| 36 | #1301 | — | 3 | 1 | no | — |
| 37 | #1283 | — | 3 | 1 | no | — |
| 38 | C22×Dic6 | — | 2 | 1 | no | — |
| 39 | C2×C3×M16 | — | 2 | 1 | no | — |
| 40 | C22×C3×C8 | — | 2 | 1 | no | — |
| 41 | C3×C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 42 | C3×C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 43 | C3×#125 | — | 2 | 1 | no | — |
| 44 | C3×C22⋊Q8 | — | 2 | 1 | no | — |
| 45 | C3×C22⋊Q8 | C2×C4 | 1 | 2 | yes | — |
| 46 | C3×C4×Q8 | — | 2 | 1 | no | — |
| 47 | C3×C4×Q8 | D4 | 1 | 2 | yes | — |
| 48 | C3×#117 | — | 2 | 1 | no | — |
| 49 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 50 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 51 | C2×C4⋊Dic3 | C2×C4 | 1 | 2 | yes | — |
| 52 | C2×#208 | D4 | 1 | 1 | yes | characteristic |
| 53 | C2×#208 | D4 | 1 | 1 | yes | characteristic |
| 54 | C22×C3⋊C8 | — | 2 | 1 | no | — |
| 55 | #869 | — | 2 | 1 | no | — |
| 56 | #869 | — | 2 | 1 | no | — |
| 57 | #869 | — | 2 | 1 | no | — |
| 58 | #869 | — | 2 | 1 | no | — |
| 59 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 60 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 61 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 62 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 63 | #835 | — | 2 | 4 | no | — |
| 64 | #830 | C2×C4 | 1 | 2 | yes | — |
| 65 | #807 | — | 2 | 4 | no | — |
| 66 | C4⋊(C3⋊C8) | — | 2 | 2 | no | — |
| 67 | C4⋊(C3⋊C8) | — | 2 | 2 | no | — |
| 68 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 69 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 70 | C4⋊(C3⋊C8) | D4 | 1 | 1 | yes | characteristic |
| 71 | C4⋊(C3⋊C8) | D4 | 1 | 1 | yes | characteristic |
| 72 | #803 | — | 2 | 2 | no | — |
| 73 | #803 | — | 2 | 1 | no | — |
| 74 | C4×C3⋊C8 | — | 2 | 2 | no | — |
| 75 | C4×C3⋊C8 | — | 2 | 1 | no | — |
| 76 | C4×C3⋊C8 | — | 2 | 1 | no | — |
| 77 | #533 | C2×D3 | 1 | 1 | yes | characteristic |
| 78 | C2×C4⋊Q8 | — | 3 | 1 | no | — |
| 79 | #500 | — | 6 | 1 | no | — |
| 80 | #487 | — | 6 | 1 | no | — |
| 81 | #484 | — | 3 | 4 | no | — |
| 82 | #477 | — | 6 | 2 | no | — |
| 83 | #446 | — | 6 | 1 | no | — |
| 84 | #446 | — | 6 | 2 | no | — |
| 85 | C8×Q8 | — | 6 | 1 | no | — |
| 86 | #431 | — | 3 | 1 | no | — |
| 87 | #424 | — | 6 | 1 | no | — |
| 88 | #423 | C2×D3 | 1 | 1 | yes | characteristic |
| 89 | C2×C4⋊C8 | — | 3 | 1 | no | — |
| 90 | C2×C4⋊C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 91 | #405 | — | 6 | 1 | no | — |
| 92 | C4×M16 | — | 3 | 1 | no | — |
| 93 | #343 | — | 6 | 1 | no | — |
| 94 | #343 | — | 6 | 1 | no | — |
| 95 | #333 | — | 2 | 1 | no | — |
| 96 | #332 | — | 2 | 1 | no | — |
| 97 | #332 | — | 6 | 1 | no | — |
| 98 | #332 | — | 6 | 1 | no | — |
| 99 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 100 | C2×C3×Q8 | C22⋊C4 | 1 | 2 | yes | — |
| 101 | C22×C3×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 102 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | C22×C3×C4 | #41 | 1 | 1 | yes | characteristic |
| 104 | C22×Dic3 | — | 2 | 2 | no | — |
| 105 | C2×Dic6 | — | 4 | 1 | no | — |
| 106 | C2×Dic6 | — | 4 | 1 | no | — |
| 107 | C2×Dic6 | — | 8 | 1 | no | — |
| 108 | C2×Dic6 | — | 8 | 1 | no | — |
| 109 | C3×M16 | — | 4 | 1 | no | — |
| 110 | C3×M16 | — | 4 | 1 | no | — |
| 111 | C2×C3×C8 | — | 4 | 1 | no | — |
| 112 | C2×C3×C8 | — | 4 | 1 | no | — |
| 113 | C2×C3×C8 | — | 8 | 1 | no | — |
| 114 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 115 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 116 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 117 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 118 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 119 | C3×C4⋊C4 | C22⋊C4 | 1 | 2 | yes | — |
| 120 | C3×C22⋊C4 | — | 2 | 2 | no | — |
| 121 | C3×C42 | — | 2 | 2 | no | — |
| 122 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 123 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 124 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 125 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 126 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 127 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 128 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 129 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 130 | #208 | — | 2 | 1 | no | — |
| 131 | #208 | — | 2 | 1 | no | — |
| 132 | #208 | — | 2 | 1 | no | — |
| 133 | #208 | Q16 | 1 | 2 | yes | — |
| 134 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 135 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 136 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 137 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 138 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 139 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 140 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 141 | C22×Q8 | — | 6 | 1 | no | — |
| 142 | C2×M16 | — | 2 | 1 | no | — |
| 143 | C2×M16 | — | 3 | 1 | no | — |
| 144 | C2×M16 | — | 3 | 1 | no | — |
| 145 | C22×C8 | — | 2 | 1 | no | — |
| 146 | C22×C8 | — | 6 | 1 | no | — |
| 147 | C4⋊Q8 | — | 6 | 1 | no | — |
| 148 | C4⋊Q8 | — | 6 | 1 | no | — |
| 149 | C4⋊Q8 | — | 6 | 1 | no | — |
| 150 | C4⋊Q8 | — | 6 | 1 | no | — |
| 151 | C4⋊Q8 | #67 | 1 | 1 | yes | characteristic |
| 152 | C4⋊Q8 | #67 | 1 | 1 | yes | characteristic |
| 153 | #125 | — | 2 | 1 | no | — |
| 154 | C22⋊Q8 | — | 2 | 1 | no | — |
| 155 | C22⋊Q8 | C4×D3 | 1 | 2 | yes | — |
| 156 | C4×Q8 | — | 2 | 1 | no | — |
| 157 | C4×Q8 | D12 | 1 | 2 | yes | — |
| 158 | #117 | — | 2 | 1 | no | — |
| 159 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 160 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 161 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 162 | #107 | — | 6 | 4 | no | — |
| 163 | C4⋊C8 | — | 2 | 1 | no | — |
| 164 | C4⋊C8 | — | 2 | 1 | no | — |
| 165 | C4⋊C8 | — | 2 | 1 | no | — |
| 166 | C4⋊C8 | — | 2 | 1 | no | — |
| 167 | C4⋊C8 | — | 3 | 1 | no | — |
| 168 | C4⋊C8 | — | 3 | 1 | no | — |
| 169 | C4⋊C8 | — | 6 | 2 | no | — |
| 170 | C4⋊C8 | — | 6 | 2 | no | — |
| 171 | C4⋊C8 | — | 6 | 1 | no | — |
| 172 | C4⋊C8 | — | 6 | 1 | no | — |
| 173 | #103 | — | 6 | 4 | no | — |
| 174 | C22⋊C8 | — | 3 | 2 | no | — |
| 175 | #97 | — | 6 | 2 | no | — |
| 176 | #97 | — | 6 | 1 | no | — |
| 177 | C4×C8 | — | 6 | 2 | no | — |
| 178 | C4×C8 | — | 6 | 1 | no | — |
| 179 | C4×C8 | — | 6 | 1 | no | — |
| 180 | C23×C3 | C4×D4 | 1 | 1 | yes | characteristic |
| 181 | C3×Q8 | — | 4 | 2 | no | — |
| 182 | C3×Q8 | — | 4 | 1 | no | — |
| 183 | C3×Q8 | — | 4 | 1 | no | — |
| 184 | C2×C3×C4 | — | 2 | 1 | no | — |
| 185 | C2×C3×C4 | — | 2 | 1 | no | — |
| 186 | C2×C3×C4 | — | 2 | 2 | no | — |
| 187 | C2×C3×C4 | — | 2 | 2 | no | — |
| 188 | C2×C3×C4 | — | 4 | 1 | no | — |
| 189 | C2×C3×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 190 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 191 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 192 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 193 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 194 | C2×C3×C4 | #123 | 1 | 1 | yes | characteristic |
| 195 | C2×C3×C4 | C2×Q16 | 1 | 1 | yes | characteristic |
| 196 | C2×C3×C4 | #135 | 1 | 1 | yes | characteristic |
| 197 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 198 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 199 | C2×Dic3 | — | 4 | 2 | no | — |
| 200 | C2×Dic3 | — | 8 | 2 | no | — |
| 201 | Dic6 | — | 8 | 2 | no | — |
| 202 | Dic6 | — | 8 | 1 | no | — |
| 203 | Dic6 | — | 8 | 1 | no | — |
| 204 | C3×C8 | — | 8 | 1 | no | — |
| 205 | C3×C8 | — | 8 | 1 | no | — |
| 206 | C3⋊C8 | — | 2 | 2 | no | — |
| 207 | C3⋊C8 | — | 4 | 2 | no | — |
| 208 | C3⋊C8 | — | 4 | 1 | no | — |
| 209 | C3⋊C8 | — | 4 | 1 | no | — |
| 210 | C3⋊C8 | — | 4 | 1 | no | — |
| 211 | C2×Q8 | — | 2 | 1 | no | — |
| 212 | C2×Q8 | — | 12 | 1 | no | — |
| 213 | C2×Q8 | — | 12 | 1 | no | — |
| 214 | C2×Q8 | — | 24 | 1 | no | — |
| 215 | C2×Q8 | — | 24 | 1 | no | — |
| 216 | C2×Q8 | (C2×D3)⋊C4 | 1 | 2 | yes | — |
| 217 | C22×C4 | — | 6 | 2 | no | — |
| 218 | C22×C4 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 219 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 220 | C22×C4 | #237 | 1 | 1 | yes | characteristic |
| 221 | M16 | — | 3 | 2 | no | — |
| 222 | M16 | — | 4 | 1 | no | — |
| 223 | M16 | — | 4 | 1 | no | — |
| 224 | M16 | — | 6 | 1 | no | — |
| 225 | M16 | — | 6 | 1 | no | — |
| 226 | M16 | — | 6 | 1 | no | — |
| 227 | C2×C8 | — | 4 | 1 | no | — |
| 228 | C2×C8 | — | 4 | 1 | no | — |
| 229 | C2×C8 | — | 6 | 1 | no | — |
| 230 | C2×C8 | — | 6 | 1 | no | — |
| 231 | C2×C8 | — | 6 | 1 | no | — |
| 232 | C2×C8 | — | 6 | 1 | no | — |
| 233 | C2×C8 | — | 6 | 2 | no | — |
| 234 | C2×C8 | — | 8 | 1 | no | — |
| 235 | C2×C8 | — | 12 | 1 | no | — |
| 236 | C2×C8 | — | 12 | 1 | no | — |
| 237 | C4⋊C4 | — | 2 | 2 | no | — |
| 238 | C4⋊C4 | — | 2 | 2 | no | — |
| 239 | C4⋊C4 | — | 2 | 1 | no | — |
| 240 | C4⋊C4 | — | 2 | 1 | no | — |
| 241 | C4⋊C4 | — | 2 | 1 | no | — |
| 242 | C4⋊C4 | — | 6 | 2 | no | — |
| 243 | C4⋊C4 | — | 6 | 2 | no | — |
| 244 | C4⋊C4 | — | 12 | 1 | no | — |
| 245 | C4⋊C4 | — | 12 | 1 | no | — |
| 246 | C4⋊C4 | (C2×D3)⋊C4 | 1 | 2 | yes | — |
| 247 | C22⋊C4 | — | 2 | 2 | no | — |
| 248 | C42 | — | 2 | 2 | no | — |
| 249 | C42 | C2×D12 | 1 | 1 | yes | characteristic |
| 250 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 251 | C42 | #237 | 1 | 1 | yes | characteristic |
| 252 | C42 | C2×#67 | 1 | 1 | yes | characteristic |
| 253 | C22×C3 | — | 2 | 1 | no | — |
| 254 | C22×C3 | — | 2 | 1 | no | — |
| 255 | C22×C3 | #386 | 1 | 1 | yes | characteristic |
| 256 | C22×C3 | C4×Q16 | 1 | 1 | yes | characteristic |
| 257 | C22×C3 | #441 | 1 | 1 | yes | characteristic |
| 258 | C3×C4 | — | 2 | 1 | no | — |
| 259 | C3×C4 | — | 2 | 1 | no | — |
| 260 | C3×C4 | — | 2 | 1 | no | — |
| 261 | C3×C4 | — | 4 | 2 | no | — |
| 262 | C3×C4 | — | 4 | 2 | no | — |
| 263 | C3×C4 | #451 | 1 | 1 | yes | characteristic |
| 264 | C3×C4 | #452 | 1 | 1 | yes | characteristic |
| 265 | C3×C4 | #462 | 1 | 1 | yes | characteristic |
| 266 | C3×C4 | #464 | 1 | 1 | yes | characteristic |
| 267 | C3×C4 | #471 | 1 | 1 | yes | characteristic |
| 268 | C3×C4 | #472 | 1 | 1 | yes | characteristic |
| 269 | Dic3 | — | 8 | 2 | no | — |
| 270 | C23 | #67⋊C4 | 1 | 1 | yes | characteristic |
| 271 | Q8 | — | 4 | 2 | no | — |
| 272 | Q8 | — | 4 | 1 | no | — |
| 273 | Q8 | — | 4 | 1 | no | — |
| 274 | Q8 | — | 24 | 2 | no | — |
| 275 | Q8 | — | 24 | 1 | no | — |
| 276 | Q8 | — | 24 | 1 | no | — |
| 277 | C2×C4 | — | 2 | 1 | no | — |
| 278 | C2×C4 | — | 2 | 1 | no | — |
| 279 | C2×C4 | — | 2 | 2 | no | — |
| 280 | C2×C4 | — | 2 | 2 | no | — |
| 281 | C2×C4 | — | 4 | 1 | no | — |
| 282 | C2×C4 | — | 12 | 2 | no | — |
| 283 | C2×C4 | — | 24 | 2 | no | — |
| 284 | C2×C4 | #67⋊C4 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | #882 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | #886 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | #915 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | #916 | 1 | 1 | yes | characteristic |
| 289 | C2×C4 | D3×Q16 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | #918 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | C2×(C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | #939 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | #939 | 1 | 1 | yes | characteristic |
| 294 | C8 | — | 6 | 2 | no | — |
| 295 | C8 | — | 8 | 1 | no | — |
| 296 | C8 | — | 8 | 1 | no | — |
| 297 | C8 | — | 12 | 2 | no | — |
| 298 | C8 | — | 12 | 1 | no | — |
| 299 | C8 | — | 12 | 1 | no | — |
| 300 | C8 | — | 12 | 1 | no | — |
| 301 | C2×C3 | — | 2 | 1 | no | — |
| 302 | C2×C3 | — | 2 | 1 | no | — |
| 303 | C2×C3 | #1878 | 1 | 1 | yes | characteristic |
| 304 | C2×C3 | #1907 | 1 | 1 | yes | characteristic |
| 305 | C2×C3 | #1915 | 1 | 1 | yes | characteristic |
| 306 | C22 | — | 2 | 1 | no | — |
| 307 | C22 | — | 2 | 1 | no | — |
| 308 | C22 | #4789 | 1 | 1 | yes | characteristic |
| 309 | C22 | #4790 | 1 | 1 | yes | characteristic |
| 310 | C22 | #4958 | 1 | 1 | yes | characteristic |
| 311 | C4 | — | 2 | 1 | no | — |
| 312 | C4 | — | 2 | 1 | no | — |
| 313 | C4 | — | 2 | 1 | no | — |
| 314 | C4 | — | 4 | 2 | no | — |
| 315 | C4 | — | 4 | 2 | no | — |
| 316 | C4 | — | 24 | 2 | no | — |
| 317 | C4 | #4808 | 1 | 1 | yes | characteristic |
| 318 | C4 | #4809 | 1 | 1 | yes | characteristic |
| 319 | C4 | #5166 | 1 | 1 | yes | characteristic |
| 320 | C4 | #5182 | 1 | 1 | yes | characteristic |
| 321 | C4 | #5199 | 1 | 1 | yes | characteristic |
| 322 | C4 | #5203 | 1 | 1 | yes | characteristic |
| 323 | C3 | #9270 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 324 | C2 | — | 2 | 1 | no | — |
| 325 | C2 | — | 2 | 1 | no | — |
| 326 | C2 | #70138 | 1 | 1 | yes | characteristic |
| 327 | C2 | #70239 | 1 | 1 | yes | characteristic |
| 328 | C2 | #70303 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta