← #10691698
Subgroups — #10691698
Order 768.
424 conjugacy classes of subgroups.
The subgroup lattice has 328 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #79313 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #79008 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #70524 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #70509 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×#4481 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C3×#1301 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #67744 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #9268 | — | 3 | 1 | no | maximal, Sylow |
| 9 | #5470 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#869 | C22 | 1 | 1 | yes | characteristic |
| 11 | C3×#431 | C22 | 1 | 1 | yes | characteristic |
| 12 | C3×#423 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C3×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 14 | #5102 | — | 2 | 1 | no | — |
| 15 | #5101 | — | 2 | 1 | no | — |
| 16 | C2×C4⋊(C3⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 17 | C4×#208 | C22 | 1 | 1 | yes | characteristic |
| 18 | #4849 | — | 2 | 2 | no | — |
| 19 | #4831 | — | 2 | 1 | no | — |
| 20 | #4830 | — | 2 | 1 | no | — |
| 21 | #4828 | — | 2 | 2 | no | — |
| 22 | #4739 | C4 | 1 | 4 | yes | — |
| 23 | #4699 | — | 2 | 1 | no | — |
| 24 | #4681 | — | 2 | 1 | no | — |
| 25 | #4554 | — | 2 | 1 | no | — |
| 26 | #4554 | — | 2 | 1 | no | — |
| 27 | #4486 | — | 2 | 1 | no | — |
| 28 | #4481 | — | 2 | 1 | no | — |
| 29 | #4481 | — | 2 | 1 | no | — |
| 30 | #4481 | — | 2 | 1 | no | — |
| 31 | #3209 | — | 3 | 1 | no | — |
| 32 | #2962 | — | 3 | 1 | no | — |
| 33 | #1997 | — | 3 | 1 | no | — |
| 34 | #1550 | — | 3 | 1 | no | — |
| 35 | C2×#333 | — | 3 | 1 | no | — |
| 36 | #1301 | D3 | 1 | 1 | yes | characteristic |
| 37 | #1283 | — | 3 | 1 | no | — |
| 38 | C22×Dic6 | C2×C4 | 1 | 2 | yes | — |
| 39 | C2×C3×M16 | — | 2 | 1 | no | — |
| 40 | C22×C3×C8 | — | 2 | 1 | no | — |
| 41 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 42 | #926 | — | 2 | 1 | no | — |
| 43 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 44 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 45 | #924 | — | 2 | 1 | no | — |
| 46 | #924 | C2×C4 | 1 | 2 | yes | — |
| 47 | C2×#208 | D4 | 1 | 1 | yes | characteristic |
| 48 | C2×#208 | D4 | 1 | 1 | yes | characteristic |
| 49 | C22×C3⋊C8 | — | 2 | 1 | no | — |
| 50 | C2×Dic12 | — | 2 | 4 | no | — |
| 51 | #870 | — | 2 | 1 | no | — |
| 52 | #869 | — | 2 | 1 | no | — |
| 53 | #869 | — | 2 | 1 | no | — |
| 54 | #869 | — | 2 | 1 | no | — |
| 55 | #869 | — | 2 | 1 | no | — |
| 56 | #869 | D4 | 1 | 1 | yes | characteristic |
| 57 | #869 | D4 | 1 | 1 | yes | characteristic |
| 58 | C4×Dic6 | — | 2 | 1 | no | — |
| 59 | C4×Dic6 | D4 | 1 | 2 | yes | — |
| 60 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 61 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 62 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 63 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 64 | C3×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 65 | C3×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 66 | C3×C22⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 67 | C3×C4×C8 | — | 2 | 1 | no | — |
| 68 | #816 | — | 2 | 4 | no | — |
| 69 | #815 | — | 2 | 2 | no | — |
| 70 | Dic3⋊C8 | — | 2 | 2 | no | — |
| 71 | Dic3⋊C8 | — | 2 | 2 | no | — |
| 72 | C8×Dic3 | — | 2 | 2 | no | — |
| 73 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 74 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 75 | #803 | — | 2 | 1 | no | — |
| 76 | C4×C3⋊C8 | — | 2 | 1 | no | — |
| 77 | #533 | — | 3 | 1 | no | — |
| 78 | C2×C4⋊Q8 | — | 3 | 1 | no | — |
| 79 | #494 | — | 6 | 1 | no | — |
| 80 | #487 | — | 6 | 1 | no | — |
| 81 | #462 | — | 6 | 2 | no | — |
| 82 | #451 | — | 3 | 4 | no | — |
| 83 | #446 | — | 6 | 2 | no | — |
| 84 | #446 | — | 6 | 1 | no | — |
| 85 | C8×Q8 | — | 6 | 1 | no | — |
| 86 | #431 | C2×D3 | 1 | 1 | yes | characteristic |
| 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 | — | 6 | 1 | no | — |
| 96 | #333 | — | 6 | 1 | no | — |
| 97 | #333 | — | 6 | 1 | no | — |
| 98 | #332 | — | 6 | 1 | no | — |
| 99 | C22×C3×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 100 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 101 | C22×C3×C4 | #41 | 1 | 1 | yes | characteristic |
| 102 | C22×Dic3 | — | 2 | 2 | no | — |
| 103 | C2×Dic6 | — | 2 | 1 | no | — |
| 104 | C2×Dic6 | — | 2 | 2 | no | — |
| 105 | C2×Dic6 | — | 2 | 2 | no | — |
| 106 | C2×Dic6 | — | 8 | 2 | no | — |
| 107 | C2×Dic6 | C22⋊C4 | 1 | 2 | yes | — |
| 108 | C3×M16 | — | 4 | 1 | no | — |
| 109 | C3×M16 | — | 4 | 1 | no | — |
| 110 | C2×C3×C8 | — | 2 | 2 | no | — |
| 111 | C2×C3×C8 | — | 2 | 1 | no | — |
| 112 | C2×C3×C8 | — | 2 | 1 | no | — |
| 113 | C2×C3×C8 | — | 4 | 1 | no | — |
| 114 | C2×C3×C8 | — | 4 | 1 | no | — |
| 115 | C2×C3×C8 | — | 4 | 1 | no | — |
| 116 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 117 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 118 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 119 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 120 | #217 | — | 2 | 2 | no | — |
| 121 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 122 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 123 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 124 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 125 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 126 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 127 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 128 | C4⋊Dic3 | C22⋊C4 | 1 | 2 | yes | — |
| 129 | Dic3⋊C4 | — | 2 | 2 | no | — |
| 130 | Dic3⋊C4 | — | 2 | 2 | no | — |
| 131 | C4×Dic3 | — | 2 | 2 | no | — |
| 132 | #208 | — | 2 | 1 | no | — |
| 133 | #208 | — | 2 | 1 | no | — |
| 134 | #208 | — | 2 | 1 | no | — |
| 135 | #208 | Q16 | 1 | 2 | yes | complemented |
| 136 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 137 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 138 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 139 | C2×C3⋊C8 | — | 8 | 1 | no | — |
| 140 | Dic12 | — | 8 | 4 | no | — |
| 141 | C22×Q8 | — | 3 | 2 | no | — |
| 142 | C2×Q16 | — | 6 | 4 | no | — |
| 143 | C2×M16 | — | 2 | 1 | no | — |
| 144 | C2×M16 | — | 3 | 1 | no | — |
| 145 | C2×M16 | — | 3 | 1 | no | — |
| 146 | C22×C8 | — | 2 | 1 | no | — |
| 147 | C22×C8 | — | 6 | 1 | no | — |
| 148 | C4⋊Q8 | — | 3 | 1 | no | — |
| 149 | C4⋊Q8 | — | 3 | 1 | no | — |
| 150 | C4⋊Q8 | — | 6 | 1 | no | — |
| 151 | C4⋊Q8 | — | 6 | 1 | no | — |
| 152 | C4⋊Q8 | — | 6 | 1 | no | — |
| 153 | C4⋊Q8 | — | 6 | 1 | no | — |
| 154 | #125 | — | 6 | 1 | no | — |
| 155 | C22⋊Q8 | — | 3 | 2 | no | — |
| 156 | C22⋊Q8 | — | 6 | 1 | no | — |
| 157 | C4×Q8 | — | 3 | 2 | no | — |
| 158 | C4×Q8 | — | 6 | 1 | no | — |
| 159 | #117 | — | 6 | 1 | no | — |
| 160 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 161 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 162 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 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 | — | 6 | 2 | no | — |
| 168 | C4⋊C8 | — | 6 | 2 | no | — |
| 169 | C4⋊C8 | — | 6 | 1 | no | — |
| 170 | C4⋊C8 | — | 6 | 1 | no | — |
| 171 | C4⋊C8 | D12 | 1 | 1 | yes | characteristic |
| 172 | C4⋊C8 | D12 | 1 | 1 | yes | characteristic |
| 173 | #103 | — | 6 | 4 | no | — |
| 174 | C22⋊C8 | C4×D3 | 1 | 2 | yes | — |
| 175 | #97 | — | 6 | 2 | no | — |
| 176 | #97 | — | 6 | 1 | no | — |
| 177 | C4×C8 | — | 2 | 1 | no | — |
| 178 | C4×C8 | — | 6 | 2 | no | — |
| 179 | C4×C8 | — | 6 | 1 | no | — |
| 180 | C23×C3 | C4×D4 | 1 | 1 | yes | characteristic |
| 181 | C2×C3×C4 | — | 2 | 1 | no | — |
| 182 | C2×C3×C4 | — | 2 | 1 | no | — |
| 183 | C2×C3×C4 | — | 4 | 1 | no | — |
| 184 | C2×C3×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 185 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 186 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 187 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 188 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 189 | C2×C3×C4 | #123 | 1 | 1 | yes | characteristic |
| 190 | C2×C3×C4 | C2×Q16 | 1 | 1 | yes | characteristic |
| 191 | C2×C3×C4 | #135 | 1 | 1 | yes | characteristic |
| 192 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 193 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 194 | C2×Dic3 | — | 2 | 2 | no | — |
| 195 | C2×Dic3 | — | 2 | 2 | no | — |
| 196 | C2×Dic3 | — | 4 | 2 | no | — |
| 197 | C2×Dic3 | — | 8 | 2 | no | — |
| 198 | Dic6 | — | 4 | 1 | no | — |
| 199 | Dic6 | — | 4 | 1 | no | — |
| 200 | Dic6 | — | 4 | 2 | no | — |
| 201 | Dic6 | — | 4 | 4 | no | — |
| 202 | Dic6 | — | 8 | 2 | no | — |
| 203 | C3×C8 | — | 4 | 2 | no | — |
| 204 | C3×C8 | — | 4 | 1 | no | — |
| 205 | C3×C8 | — | 4 | 1 | no | — |
| 206 | C3×C8 | — | 8 | 1 | no | — |
| 207 | C3⋊C8 | — | 2 | 2 | no | — |
| 208 | C3⋊C8 | — | 4 | 1 | no | — |
| 209 | C3⋊C8 | — | 8 | 1 | no | — |
| 210 | C2×Q8 | — | 3 | 2 | no | — |
| 211 | C2×Q8 | — | 6 | 1 | no | — |
| 212 | C2×Q8 | — | 6 | 2 | no | — |
| 213 | C2×Q8 | — | 6 | 2 | no | — |
| 214 | C2×Q8 | — | 24 | 2 | no | — |
| 215 | C22×C4 | — | 6 | 2 | no | — |
| 216 | C22×C4 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 217 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 218 | C22×C4 | #239 | 1 | 1 | yes | characteristic |
| 219 | Q16 | — | 24 | 4 | no | — |
| 220 | M16 | — | 3 | 2 | no | — |
| 221 | M16 | — | 4 | 1 | no | — |
| 222 | M16 | — | 4 | 1 | no | — |
| 223 | M16 | — | 6 | 1 | no | — |
| 224 | M16 | — | 6 | 1 | no | — |
| 225 | M16 | — | 6 | 1 | no | — |
| 226 | C2×C8 | — | 2 | 2 | no | — |
| 227 | C2×C8 | — | 2 | 1 | no | — |
| 228 | C2×C8 | — | 2 | 1 | no | — |
| 229 | C2×C8 | — | 4 | 1 | no | — |
| 230 | C2×C8 | — | 4 | 1 | no | — |
| 231 | C2×C8 | — | 4 | 1 | no | — |
| 232 | C2×C8 | — | 6 | 1 | no | — |
| 233 | C2×C8 | — | 6 | 1 | no | — |
| 234 | C2×C8 | — | 12 | 1 | no | — |
| 235 | C2×C8 | — | 24 | 1 | no | — |
| 236 | C4⋊C4 | — | 3 | 2 | no | — |
| 237 | C4⋊C4 | — | 6 | 2 | no | — |
| 238 | C4⋊C4 | — | 6 | 1 | no | — |
| 239 | C4⋊C4 | — | 6 | 1 | no | — |
| 240 | C4⋊C4 | — | 6 | 2 | no | — |
| 241 | C4⋊C4 | — | 6 | 1 | no | — |
| 242 | C4⋊C4 | — | 12 | 1 | no | — |
| 243 | C4⋊C4 | — | 12 | 1 | no | — |
| 244 | C4⋊C4 | — | 12 | 1 | no | — |
| 245 | C4⋊C4 | — | 12 | 1 | no | — |
| 246 | C22⋊C4 | — | 6 | 2 | no | — |
| 247 | C42 | — | 6 | 2 | no | — |
| 248 | C42 | C2×D12 | 1 | 1 | yes | characteristic |
| 249 | C42 | #235 | 1 | 1 | yes | characteristic |
| 250 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 251 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 252 | C22×C3 | — | 2 | 1 | no | — |
| 253 | C22×C3 | — | 2 | 1 | no | — |
| 254 | C22×C3 | #386 | 1 | 1 | yes | characteristic |
| 255 | C22×C3 | C4×Q16 | 1 | 1 | yes | characteristic |
| 256 | C22×C3 | #441 | 1 | 1 | yes | characteristic |
| 257 | C3×C4 | — | 2 | 1 | no | — |
| 258 | C3×C4 | — | 2 | 1 | no | — |
| 259 | C3×C4 | — | 2 | 1 | no | — |
| 260 | C3×C4 | #451 | 1 | 1 | yes | characteristic |
| 261 | C3×C4 | #452 | 1 | 1 | yes | characteristic |
| 262 | C3×C4 | #462 | 1 | 1 | yes | characteristic |
| 263 | C3×C4 | #464 | 1 | 1 | yes | characteristic |
| 264 | C3×C4 | #471 | 1 | 1 | yes | characteristic |
| 265 | C3×C4 | #473 | 1 | 1 | yes | characteristic |
| 266 | Dic3 | — | 4 | 2 | no | — |
| 267 | Dic3 | — | 4 | 2 | no | — |
| 268 | Dic3 | — | 8 | 2 | no | — |
| 269 | C23 | #893 | 1 | 1 | yes | characteristic |
| 270 | Q8 | — | 12 | 2 | no | — |
| 271 | Q8 | — | 12 | 4 | no | — |
| 272 | Q8 | — | 12 | 1 | no | — |
| 273 | Q8 | — | 12 | 1 | no | — |
| 274 | Q8 | — | 24 | 2 | no | — |
| 275 | C2×C4 | — | 2 | 1 | no | — |
| 276 | C2×C4 | — | 2 | 1 | no | — |
| 277 | C2×C4 | — | 4 | 1 | no | — |
| 278 | C2×C4 | — | 6 | 2 | no | — |
| 279 | C2×C4 | — | 6 | 2 | no | — |
| 280 | C2×C4 | — | 12 | 2 | no | — |
| 281 | C2×C4 | — | 24 | 2 | no | — |
| 282 | C2×C4 | C4×D12 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | D3×C22⋊C4 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | #882 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | #884 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | #894 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | #895 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | #912 | 1 | 1 | yes | characteristic |
| 289 | C2×C4 | #915 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | #915 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | D3×Q16 | 1 | 1 | yes | characteristic |
| 292 | C8 | — | 4 | 2 | no | — |
| 293 | C8 | — | 4 | 1 | no | — |
| 294 | C8 | — | 4 | 1 | no | — |
| 295 | C8 | — | 6 | 2 | no | — |
| 296 | C8 | — | 8 | 1 | no | — |
| 297 | C8 | — | 12 | 1 | no | — |
| 298 | C8 | — | 24 | 1 | no | — |
| 299 | C2×C3 | — | 2 | 1 | no | — |
| 300 | C2×C3 | — | 2 | 1 | no | — |
| 301 | C2×C3 | #1878 | 1 | 1 | yes | characteristic |
| 302 | C2×C3 | #1908 | 1 | 1 | yes | characteristic |
| 303 | C2×C3 | #1914 | 1 | 1 | yes | characteristic |
| 304 | C22 | — | 2 | 1 | no | — |
| 305 | C22 | — | 2 | 1 | no | — |
| 306 | C22 | #4665 | 1 | 1 | yes | characteristic |
| 307 | C22 | #4853 | 1 | 1 | yes | characteristic |
| 308 | C22 | #4873 | 1 | 1 | yes | characteristic |
| 309 | C4 | — | 2 | 1 | no | — |
| 310 | C4 | — | 2 | 1 | no | — |
| 311 | C4 | — | 2 | 1 | no | — |
| 312 | C4 | — | 12 | 2 | no | — |
| 313 | C4 | — | 12 | 2 | no | — |
| 314 | C4 | — | 24 | 2 | no | — |
| 315 | C4 | #4767 | 1 | 1 | yes | characteristic |
| 316 | C4 | #4783 | 1 | 1 | yes | characteristic |
| 317 | C4 | #4795 | 1 | 1 | yes | characteristic |
| 318 | C4 | #4809 | 1 | 1 | yes | characteristic |
| 319 | C4 | #4896 | 1 | 1 | yes | characteristic |
| 320 | C4 | #4898 | 1 | 1 | yes | characteristic |
| 321 | C3 | #9268 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 322 | C2 | — | 2 | 1 | no | — |
| 323 | C2 | — | 2 | 1 | no | — |
| 324 | C2 | #68908 | 1 | 1 | yes | characteristic |
| 325 | C2 | #68949 | 1 | 1 | yes | characteristic |
| 326 | C2 | #69040 | 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