Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1034 conjugacy classes of subgroups.
The subgroup lattice has 684 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #86177 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×#3171 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #83308 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #83289 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #71747 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #71392 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #71326 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #71248 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #71244 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #69860 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #69859 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #19018 | — | 3 | 1 | no | maximal, Sylow |
| 13 | C3×#532 | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | C3×#529 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#951 | — | 2 | 1 | no | — |
| 16 | C2×#949 | — | 2 | 1 | no | — |
| 17 | #5704 | — | 2 | 2 | no | — |
| 18 | #5700 | C22 | 1 | 2 | yes | — |
| 19 | #5683 | C22 | 1 | 2 | yes | — |
| 20 | #5654 | — | 2 | 2 | no | — |
| 21 | #5644 | — | 2 | 1 | no | — |
| 22 | #5605 | — | 2 | 1 | no | — |
| 23 | #5585 | — | 2 | 1 | no | — |
| 24 | #5552 | — | 2 | 1 | no | — |
| 25 | #5550 | — | 2 | 1 | no | — |
| 26 | D4×D12 | — | 2 | 1 | no | — |
| 27 | #5534 | — | 2 | 1 | no | — |
| 28 | #5525 | — | 2 | 1 | no | — |
| 29 | #5477 | C22 | 1 | 1 | yes | characteristic |
| 30 | C2×#875 | C22 | 1 | 1 | yes | characteristic |
| 31 | C3×#501 | C22 | 1 | 1 | yes | characteristic, complemented |
| 32 | C3×#496 | C22 | 1 | 1 | yes | characteristic, complemented |
| 33 | C3×#490 | C22 | 1 | 1 | yes | characteristic, complemented |
| 34 | C3×#489 | C22 | 1 | 1 | yes | characteristic, complemented |
| 35 | C3×#469 | — | 2 | 1 | no | — |
| 36 | C3×#468 | — | 2 | 1 | no | — |
| 37 | C3×C4×SD16 | — | 2 | 1 | no | — |
| 38 | C3×C4×D8 | — | 2 | 1 | no | — |
| 39 | C3×C4×M16 | C22 | 1 | 1 | yes | characteristic, complemented |
| 40 | #5089 | C22 | 1 | 2 | yes | complemented |
| 41 | #5086 | C22 | 1 | 2 | yes | complemented |
| 42 | #5074 | C22 | 1 | 2 | yes | complemented |
| 43 | #5074 | C22 | 1 | 2 | yes | complemented |
| 44 | #5059 | — | 2 | 2 | no | — |
| 45 | #5037 | — | 2 | 2 | no | — |
| 46 | #5019 | — | 2 | 1 | no | — |
| 47 | #5015 | — | 2 | 1 | no | — |
| 48 | #4926 | C22 | 1 | 1 | yes | characteristic, complemented |
| 49 | C4×#208 | C22 | 1 | 1 | yes | characteristic, complemented |
| 50 | #4543 | C22 | 1 | 2 | yes | complemented |
| 51 | #4540 | C22 | 1 | 2 | yes | complemented |
| 52 | #4464 | C22 | 1 | 2 | yes | complemented |
| 53 | #4464 | C22 | 1 | 2 | yes | complemented |
| 54 | #4464 | C22 | 1 | 2 | yes | complemented |
| 55 | #4464 | C22 | 1 | 2 | yes | complemented |
| 56 | #3558 | — | 3 | 1 | no | — |
| 57 | #3214 | — | 3 | 1 | no | — |
| 58 | #3171 | D3 | 1 | 1 | yes | characteristic, complemented |
| 59 | #3153 | — | 3 | 1 | no | — |
| 60 | #1661 | — | 3 | 2 | no | — |
| 61 | #1659 | — | 3 | 2 | no | — |
| 62 | #1656 | — | 3 | 2 | no | — |
| 63 | #1655 | — | 3 | 2 | no | — |
| 64 | #1529 | — | 3 | 1 | no | — |
| 65 | #1527 | — | 3 | 1 | no | — |
| 66 | #1525 | — | 3 | 1 | no | — |
| 67 | C2×C3×#41 | — | 2 | 1 | no | — |
| 68 | C2×D3×D4 | — | 2 | 2 | no | — |
| 69 | C2×#235 | — | 2 | 1 | no | — |
| 70 | C2×#235 | D4 | 1 | 2 | yes | — |
| 71 | C22×D12 | — | 2 | 1 | no | — |
| 72 | C22×D12 | D4 | 1 | 1 | yes | characteristic |
| 73 | C22×Dic6 | D4 | 1 | 1 | yes | characteristic |
| 74 | C2×C3×SD16 | — | 2 | 1 | no | — |
| 75 | C2×C3×D8 | — | 2 | 1 | no | — |
| 76 | C2×C3×M16 | — | 2 | 1 | no | — |
| 77 | C3×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 78 | C3×#127 | C23 | 1 | 1 | yes | characteristic |
| 79 | C3×#126 | — | 2 | 1 | no | — |
| 80 | C3×#125 | C23 | 1 | 1 | yes | characteristic |
| 81 | C3×#124 | — | 2 | 1 | no | — |
| 82 | C3×#124 | C23 | 1 | 1 | yes | characteristic |
| 83 | C3×#123 | — | 2 | 1 | no | — |
| 84 | C3×C22⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 85 | C3×#121 | — | 2 | 1 | no | — |
| 86 | C3×#121 | — | 2 | 1 | no | — |
| 87 | C3×#121 | D4 | 1 | 1 | yes | characteristic |
| 88 | C3×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 89 | C3×C4×D4 | — | 2 | 1 | no | — |
| 90 | C3×C4×D4 | — | 2 | 1 | no | — |
| 91 | C3×C4×D4 | — | 2 | 1 | no | — |
| 92 | C3×C4×D4 | D4 | 1 | 1 | yes | characteristic |
| 93 | C3×#117 | — | 2 | 1 | no | — |
| 94 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 95 | #951 | — | 4 | 4 | no | — |
| 96 | #949 | — | 4 | 4 | no | — |
| 97 | #947 | — | 2 | 2 | no | — |
| 98 | C2×#216 | — | 2 | 2 | no | — |
| 99 | C2×#215 | — | 2 | 2 | no | — |
| 100 | #939 | — | 2 | 2 | no | — |
| 101 | #937 | — | 2 | 2 | no | — |
| 102 | C2×#214 | — | 2 | 2 | no | — |
| 103 | C2×#214 | — | 2 | 2 | no | — |
| 104 | C2×#213 | — | 2 | 2 | no | — |
| 105 | C2×#213 | — | 2 | 2 | no | — |
| 106 | #930 | — | 2 | 1 | no | — |
| 107 | #930 | — | 2 | 1 | no | — |
| 108 | #930 | — | 2 | 1 | no | — |
| 109 | C2×(C2×D3)⋊C4 | — | 2 | 1 | no | — |
| 110 | C2×(C2×D3)⋊C4 | — | 2 | 1 | no | — |
| 111 | #924 | — | 2 | 1 | no | — |
| 112 | C2×#208 | — | 2 | 1 | no | — |
| 113 | #898 | — | 2 | 2 | no | — |
| 114 | #896 | — | 2 | 2 | no | — |
| 115 | #896 | — | 2 | 2 | no | — |
| 116 | #895 | — | 2 | 2 | no | — |
| 117 | #895 | — | 2 | 2 | no | — |
| 118 | #895 | — | 2 | 2 | no | — |
| 119 | #894 | — | 2 | 2 | no | — |
| 120 | #894 | — | 2 | 2 | no | — |
| 121 | #894 | — | 2 | 2 | no | — |
| 122 | #893 | — | 2 | 2 | no | — |
| 123 | #885 | — | 2 | 2 | no | — |
| 124 | #885 | — | 2 | 2 | no | — |
| 125 | #884 | — | 2 | 2 | no | — |
| 126 | #883 | — | 2 | 2 | no | — |
| 127 | #882 | — | 2 | 2 | no | — |
| 128 | #882 | — | 2 | 2 | no | — |
| 129 | #882 | — | 2 | 2 | no | — |
| 130 | #882 | — | 2 | 2 | no | — |
| 131 | #67⋊C4 | — | 2 | 2 | no | — |
| 132 | D3×C22⋊C4 | — | 2 | 2 | no | — |
| 133 | #875 | — | 2 | 1 | no | — |
| 134 | #875 | — | 2 | 1 | no | — |
| 135 | #875 | — | 2 | 1 | no | — |
| 136 | #875 | D4 | 1 | 2 | yes | — |
| 137 | #875 | C23 | 1 | 2 | yes | — |
| 138 | #875 | C23 | 1 | 2 | yes | — |
| 139 | #874 | — | 2 | 1 | no | — |
| 140 | #874 | D4 | 1 | 1 | yes | characteristic |
| 141 | C4×D12 | — | 2 | 1 | no | — |
| 142 | C4×D12 | — | 2 | 1 | no | — |
| 143 | C4×D12 | — | 2 | 1 | no | — |
| 144 | #869 | D4 | 1 | 1 | yes | characteristic |
| 145 | C4×Dic6 | — | 2 | 1 | no | — |
| 146 | C3×#107 | — | 2 | 1 | no | — |
| 147 | C3×#106 | — | 2 | 1 | no | — |
| 148 | C3×#103 | — | 2 | 1 | no | — |
| 149 | C3×#102 | — | 2 | 1 | no | — |
| 150 | C3×#102 | — | 2 | 1 | no | — |
| 151 | C3×#102 | — | 2 | 1 | no | — |
| 152 | C3×#97 | C23 | 1 | 1 | yes | characteristic |
| 153 | C3×#97 | C23 | 1 | 1 | yes | characteristic |
| 154 | C3×C4×C8 | — | 2 | 1 | no | — |
| 155 | #830 | — | 2 | 1 | no | — |
| 156 | #810 | — | 2 | 2 | no | — |
| 157 | #809 | — | 2 | 2 | no | — |
| 158 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 159 | #803 | C23 | 1 | 2 | yes | — |
| 160 | #803 | C23 | 1 | 2 | yes | — |
| 161 | C4×C3⋊C8 | — | 2 | 1 | no | — |
| 162 | C2×#137 | — | 6 | 1 | no | — |
| 163 | C2×AGL(1,ℤ/8) | — | 6 | 1 | no | — |
| 164 | #562 | — | 3 | 2 | no | — |
| 165 | H(3,𝔽4) | — | 3 | 2 | no | — |
| 166 | #559 | — | 6 | 1 | no | — |
| 167 | #548 | — | 6 | 1 | no | — |
| 168 | #546 | — | 6 | 2 | no | — |
| 169 | #546 | — | 6 | 1 | no | — |
| 170 | D42 | — | 6 | 1 | no | — |
| 171 | #542 | — | 6 | 1 | no | — |
| 172 | #538 | — | 6 | 2 | no | — |
| 173 | #538 | — | 6 | 1 | no | — |
| 174 | #535 | — | 6 | 1 | no | — |
| 175 | #534 | — | 6 | 1 | no | — |
| 176 | #532 | — | 3 | 1 | no | — |
| 177 | #532 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 178 | #529 | C2×D3 | 1 | 1 | yes | characteristic |
| 179 | C2×#124 | — | 3 | 1 | no | — |
| 180 | #501 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 181 | #497 | — | 3 | 2 | no | — |
| 182 | #496 | — | 3 | 2 | no | — |
| 183 | #496 | — | 3 | 2 | no | — |
| 184 | #496 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 185 | #495 | — | 6 | 2 | no | — |
| 186 | #490 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 187 | #489 | — | 3 | 2 | no | — |
| 188 | #489 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 189 | #469 | — | 2 | 1 | no | — |
| 190 | #468 | — | 2 | 1 | no | — |
| 191 | #461 | — | 6 | 1 | no | — |
| 192 | #459 | — | 6 | 1 | no | — |
| 193 | #449 | — | 6 | 2 | no | — |
| 194 | C4×SD16 | — | 2 | 1 | no | — |
| 195 | C4×D8 | — | 2 | 1 | no | — |
| 196 | #432 | — | 3 | 1 | no | — |
| 197 | C4×M16 | — | 3 | 1 | no | — |
| 198 | C4×M16 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 199 | #330 | — | 3 | 2 | no | — |
| 200 | #329 | — | 3 | 2 | no | — |
| 201 | #329 | — | 3 | 2 | no | — |
| 202 | #329 | — | 3 | 2 | no | — |
| 203 | #329 | — | 3 | 2 | no | — |
| 204 | #329 | — | 3 | 2 | no | — |
| 205 | C23×D3 | — | 2 | 1 | no | — |
| 206 | C23×D3 | — | 2 | 2 | no | — |
| 207 | C3×#41 | — | 4 | 2 | no | — |
| 208 | C3×#41 | — | 4 | 2 | no | — |
| 209 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 210 | C2×C3×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 211 | C2×C3×D4 | — | 2 | 1 | no | — |
| 212 | C2×C3×D4 | — | 2 | 1 | no | — |
| 213 | C2×C3×D4 | — | 2 | 1 | no | — |
| 214 | C2×C3×D4 | — | 2 | 1 | no | — |
| 215 | C2×C3×D4 | — | 4 | 1 | no | — |
| 216 | C2×C3×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 217 | C22×C3×C4 | — | 2 | 1 | no | — |
| 218 | C22×C3×C4 | — | 4 | 1 | no | — |
| 219 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 220 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 221 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 222 | C2×#67 | — | 2 | 2 | no | — |
| 223 | C2×#67 | — | 2 | 2 | no | — |
| 224 | C2×#67 | — | 2 | 2 | no | — |
| 225 | C22×Dic3 | — | 2 | 1 | no | — |
| 226 | D3×D4 | — | 8 | 4 | no | — |
| 227 | #235 | — | 8 | 2 | no | — |
| 228 | #235 | — | 8 | 2 | no | — |
| 229 | C2×D12 | — | 2 | 1 | no | — |
| 230 | C2×D12 | — | 2 | 1 | no | — |
| 231 | C2×D12 | — | 2 | 2 | no | — |
| 232 | C2×D12 | — | 2 | 2 | no | — |
| 233 | C2×D12 | — | 2 | 2 | no | — |
| 234 | C2×D12 | — | 2 | 1 | no | — |
| 235 | C2×D12 | — | 4 | 2 | no | — |
| 236 | C2×D12 | — | 8 | 2 | no | — |
| 237 | C2×D12 | C2×D4 | 1 | 2 | yes | — |
| 238 | C2×D12 | C2×D4 | 1 | 2 | yes | — |
| 239 | C2×C4×D3 | — | 2 | 2 | no | — |
| 240 | C2×C4×D3 | — | 2 | 2 | no | — |
| 241 | C2×C4×D3 | — | 2 | 2 | no | — |
| 242 | C2×Dic6 | — | 2 | 1 | no | — |
| 243 | C2×Dic6 | — | 4 | 2 | no | — |
| 244 | C2×Dic6 | C2×D4 | 1 | 2 | yes | — |
| 245 | C2×Dic6 | C2×D4 | 1 | 2 | yes | — |
| 246 | C3×SD16 | — | 4 | 2 | no | — |
| 247 | C3×D8 | — | 4 | 2 | no | — |
| 248 | C3×M16 | — | 4 | 2 | no | — |
| 249 | C2×C3×C8 | — | 2 | 1 | no | — |
| 250 | C2×C3×C8 | — | 2 | 1 | no | — |
| 251 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 252 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 253 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 254 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 255 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 256 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 257 | C3×C4⋊C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 258 | C3×C4⋊C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 259 | C3×C22⋊C4 | — | 2 | 1 | no | — |
| 260 | C3×C22⋊C4 | — | 2 | 1 | no | — |
| 261 | C3×C22⋊C4 | — | 2 | 1 | no | — |
| 262 | C3×C22⋊C4 | — | 2 | 1 | no | — |
| 263 | C3×C22⋊C4 | — | 2 | 1 | no | — |
| 264 | C3×C22⋊C4 | — | 4 | 1 | no | — |
| 265 | C3×C22⋊C4 | — | 4 | 1 | no | — |
| 266 | C3×C42 | — | 2 | 1 | no | — |
| 267 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 268 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 269 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 270 | C3×C42 | C24 | 1 | 1 | yes | characteristic |
| 271 | #217 | — | 2 | 2 | no | — |
| 272 | #216 | — | 4 | 4 | no | — |
| 273 | #215 | — | 4 | 4 | no | — |
| 274 | #214 | — | 4 | 4 | no | — |
| 275 | #214 | — | 8 | 2 | no | — |
| 276 | #213 | — | 4 | 4 | no | — |
| 277 | #213 | — | 8 | 2 | no | — |
| 278 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 279 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 280 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 281 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 282 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 283 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 284 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 285 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 286 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 287 | (C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 288 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 289 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 290 | Dic3⋊C4 | — | 2 | 2 | no | — |
| 291 | Dic3⋊C4 | — | 2 | 2 | no | — |
| 292 | C4×Dic3 | — | 2 | 2 | no | — |
| 293 | #208 | — | 2 | 4 | no | — |
| 294 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 295 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 296 | C2×#41 | — | 2 | 1 | no | — |
| 297 | C2×#41 | — | 3 | 2 | no | — |
| 298 | C2×#41 | — | 6 | 1 | no | — |
| 299 | C22×Q8 | — | 3 | 1 | no | — |
| 300 | C22×D4 | — | 3 | 1 | no | — |
| 301 | C22×D4 | — | 6 | 1 | no | — |
| 302 | C22×D4 | — | 6 | 2 | no | — |
| 303 | #137 | — | 12 | 4 | no | — |
| 304 | AGL(1,ℤ/8) | — | 12 | 4 | no | — |
| 305 | C2×Q16 | — | 6 | 2 | no | — |
| 306 | C2×SD16 | — | 2 | 1 | no | — |
| 307 | C2×SD16 | — | 6 | 2 | no | — |
| 308 | C2×SD16 | — | 6 | 2 | no | — |
| 309 | C2×SD16 | — | 6 | 2 | no | — |
| 310 | C2×D8 | — | 2 | 1 | no | — |
| 311 | C2×D8 | — | 6 | 2 | no | — |
| 312 | C2×D8 | — | 6 | 2 | no | — |
| 313 | C2×M16 | — | 2 | 1 | no | — |
| 314 | C2×M16 | — | 6 | 1 | no | — |
| 315 | C4⋊Q8 | — | 3 | 1 | no | — |
| 316 | C4⋊Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 317 | #127 | — | 3 | 1 | no | — |
| 318 | #127 | — | 6 | 1 | no | — |
| 319 | #127 | C22×D3 | 1 | 1 | yes | characteristic |
| 320 | #126 | — | 2 | 1 | no | — |
| 321 | #126 | — | 6 | 2 | no | — |
| 322 | #125 | C22×D3 | 1 | 1 | yes | characteristic |
| 323 | #124 | — | 2 | 1 | no | — |
| 324 | #124 | — | 3 | 2 | no | — |
| 325 | #124 | — | 3 | 2 | no | — |
| 326 | #124 | — | 3 | 2 | no | — |
| 327 | #124 | — | 6 | 1 | no | — |
| 328 | #124 | — | 6 | 2 | no | — |
| 329 | #124 | — | 6 | 2 | no | — |
| 330 | #124 | — | 6 | 2 | no | — |
| 331 | #124 | — | 6 | 1 | no | — |
| 332 | #124 | — | 6 | 1 | no | — |
| 333 | #124 | C22×D3 | 1 | 1 | yes | characteristic |
| 334 | #123 | — | 2 | 1 | no | — |
| 335 | #123 | — | 6 | 2 | no | — |
| 336 | #123 | — | 6 | 2 | no | — |
| 337 | #123 | — | 6 | 2 | no | — |
| 338 | #123 | — | 6 | 2 | no | — |
| 339 | C22⋊Q8 | — | 6 | 2 | no | — |
| 340 | C22⋊Q8 | — | 6 | 2 | no | — |
| 341 | C22⋊Q8 | — | 6 | 1 | no | — |
| 342 | C22⋊Q8 | #67 | 1 | 1 | yes | characteristic |
| 343 | #121 | — | 2 | 1 | no | — |
| 344 | #121 | — | 2 | 1 | no | — |
| 345 | #121 | — | 6 | 2 | no | — |
| 346 | #121 | — | 6 | 2 | no | — |
| 347 | #121 | — | 6 | 2 | no | — |
| 348 | #121 | — | 6 | 2 | no | — |
| 349 | #121 | — | 6 | 1 | no | — |
| 350 | #121 | — | 6 | 2 | no | — |
| 351 | #121 | — | 6 | 1 | no | — |
| 352 | #121 | — | 6 | 1 | no | — |
| 353 | #121 | #67 | 1 | 1 | yes | characteristic |
| 354 | #120 | — | 6 | 2 | no | — |
| 355 | #120 | — | 6 | 2 | no | — |
| 356 | #120 | — | 6 | 2 | no | — |
| 357 | #120 | — | 6 | 2 | no | — |
| 358 | #120 | — | 6 | 2 | no | — |
| 359 | C4×Q8 | — | 6 | 1 | no | — |
| 360 | C4×Q8 | #67 | 1 | 1 | yes | characteristic |
| 361 | C4×D4 | — | 2 | 1 | no | — |
| 362 | C4×D4 | — | 2 | 1 | no | — |
| 363 | C4×D4 | — | 2 | 1 | no | — |
| 364 | C4×D4 | — | 6 | 1 | no | — |
| 365 | C4×D4 | — | 6 | 2 | no | — |
| 366 | C4×D4 | — | 6 | 2 | no | — |
| 367 | C4×D4 | — | 6 | 1 | no | — |
| 368 | C4×D4 | — | 6 | 1 | no | — |
| 369 | C4×D4 | #67 | 1 | 1 | yes | characteristic |
| 370 | #117 | — | 2 | 1 | no | — |
| 371 | C2×C22⋊C4 | — | 6 | 2 | no | — |
| 372 | C2×C22⋊C4 | — | 6 | 1 | no | — |
| 373 | C2×C22⋊C4 | — | 6 | 1 | no | — |
| 374 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 375 | #107 | — | 2 | 1 | no | — |
| 376 | #106 | — | 2 | 1 | no | — |
| 377 | C4⋊C8 | — | 6 | 1 | no | — |
| 378 | #103 | — | 2 | 1 | no | — |
| 379 | #103 | — | 6 | 2 | no | — |
| 380 | #102 | — | 2 | 1 | no | — |
| 381 | #102 | — | 2 | 1 | no | — |
| 382 | #102 | — | 2 | 1 | no | — |
| 383 | #102 | — | 6 | 2 | no | — |
| 384 | C22⋊C8 | — | 6 | 1 | no | — |
| 385 | #97 | — | 3 | 2 | no | — |
| 386 | #97 | — | 3 | 2 | no | — |
| 387 | #97 | C22×D3 | 1 | 1 | yes | characteristic |
| 388 | #97 | C22×D3 | 1 | 1 | yes | characteristic |
| 389 | C4×C8 | — | 2 | 1 | no | — |
| 390 | C4×C8 | — | 6 | 1 | no | — |
| 391 | C23×C3 | — | 2 | 1 | no | — |
| 392 | C23×C3 | — | 4 | 1 | no | — |
| 393 | C23×C3 | #120 | 1 | 1 | yes | characteristic |
| 394 | C22×D3 | — | 2 | 2 | no | — |
| 395 | C22×D3 | — | 2 | 2 | no | — |
| 396 | C22×D3 | — | 4 | 2 | no | — |
| 397 | C22×D3 | — | 8 | 1 | no | — |
| 398 | C22×D3 | — | 8 | 1 | no | — |
| 399 | C22×D3 | — | 8 | 2 | no | — |
| 400 | C22×D3 | — | 8 | 2 | no | — |
| 401 | C22×D3 | — | 8 | 2 | no | — |
| 402 | C3×Q8 | — | 4 | 1 | no | — |
| 403 | C3×Q8 | — | 4 | 1 | no | — |
| 404 | C3×Q8 | — | 4 | 1 | no | — |
| 405 | C3×D4 | — | 4 | 1 | no | — |
| 406 | C3×D4 | — | 4 | 1 | no | — |
| 407 | C3×D4 | — | 4 | 1 | no | — |
| 408 | C3×D4 | — | 4 | 1 | no | — |
| 409 | C3×D4 | — | 4 | 1 | no | — |
| 410 | C3×D4 | — | 4 | 1 | no | — |
| 411 | C3×D4 | — | 4 | 1 | no | — |
| 412 | C3×D4 | — | 4 | 2 | no | — |
| 413 | C3×D4 | — | 8 | 2 | no | — |
| 414 | C2×C3×C4 | — | 2 | 2 | no | — |
| 415 | C2×C3×C4 | — | 2 | 1 | no | — |
| 416 | C2×C3×C4 | — | 2 | 1 | no | — |
| 417 | C2×C3×C4 | — | 2 | 1 | no | — |
| 418 | C2×C3×C4 | — | 4 | 1 | no | — |
| 419 | C2×C3×C4 | — | 4 | 1 | no | — |
| 420 | C2×C3×C4 | — | 4 | 2 | no | — |
| 421 | C2×C3×C4 | — | 4 | 1 | no | — |
| 422 | C2×C3×C4 | — | 8 | 2 | no | — |
| 423 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 424 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 425 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 426 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 427 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 428 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 429 | #67 | — | 4 | 4 | no | — |
| 430 | #67 | — | 8 | 2 | no | — |
| 431 | #67 | — | 8 | 2 | no | — |
| 432 | C2×Dic3 | — | 2 | 2 | no | — |
| 433 | C2×Dic3 | — | 2 | 2 | no | — |
| 434 | C2×Dic3 | — | 8 | 1 | no | — |
| 435 | D12 | — | 4 | 2 | no | — |
| 436 | D12 | — | 4 | 2 | no | — |
| 437 | D12 | — | 4 | 2 | no | — |
| 438 | D12 | — | 4 | 2 | no | — |
| 439 | D12 | — | 8 | 1 | no | — |
| 440 | D12 | — | 8 | 2 | no | — |
| 441 | D12 | — | 8 | 2 | no | — |
| 442 | D12 | — | 8 | 2 | no | — |
| 443 | C4×D3 | — | 8 | 2 | no | — |
| 444 | C4×D3 | — | 8 | 2 | no | — |
| 445 | C4×D3 | — | 8 | 2 | no | — |
| 446 | Dic6 | — | 4 | 2 | no | — |
| 447 | Dic6 | — | 4 | 2 | no | — |
| 448 | Dic6 | — | 4 | 2 | no | — |
| 449 | C3×C8 | — | 2 | 2 | no | — |
| 450 | C3×C8 | — | 4 | 1 | no | — |
| 451 | C3⋊C8 | — | 2 | 4 | no | — |
| 452 | C3⋊C8 | — | 4 | 2 | no | — |
| 453 | C24 | — | 6 | 2 | no | — |
| 454 | C24 | — | 6 | 1 | no | — |
| 455 | #41 | — | 4 | 2 | no | — |
| 456 | #41 | — | 4 | 2 | no | — |
| 457 | #41 | — | 24 | 2 | no | — |
| 458 | #41 | — | 24 | 2 | no | — |
| 459 | C2×Q8 | — | 2 | 1 | no | — |
| 460 | C2×Q8 | — | 3 | 2 | no | — |
| 461 | C2×Q8 | — | 3 | 2 | no | — |
| 462 | C2×Q8 | — | 6 | 1 | no | — |
| 463 | C2×Q8 | — | 12 | 2 | no | — |
| 464 | C2×Q8 | C2×#67 | 1 | 1 | yes | characteristic |
| 465 | C2×D4 | — | 2 | 1 | no | — |
| 466 | C2×D4 | — | 2 | 1 | no | — |
| 467 | C2×D4 | — | 2 | 1 | no | — |
| 468 | C2×D4 | — | 2 | 1 | no | — |
| 469 | C2×D4 | — | 3 | 2 | no | — |
| 470 | C2×D4 | — | 3 | 2 | no | — |
| 471 | C2×D4 | — | 4 | 1 | no | — |
| 472 | C2×D4 | — | 6 | 2 | no | — |
| 473 | C2×D4 | — | 6 | 2 | no | — |
| 474 | C2×D4 | — | 6 | 2 | no | — |
| 475 | C2×D4 | — | 6 | 2 | no | — |
| 476 | C2×D4 | — | 6 | 2 | no | — |
| 477 | C2×D4 | — | 6 | 2 | no | — |
| 478 | C2×D4 | — | 6 | 1 | no | — |
| 479 | C2×D4 | — | 6 | 1 | no | — |
| 480 | C2×D4 | — | 6 | 1 | no | — |
| 481 | C2×D4 | — | 12 | 2 | no | — |
| 482 | C2×D4 | — | 24 | 2 | no | — |
| 483 | C2×D4 | — | 24 | 4 | no | — |
| 484 | C2×D4 | C2×#67 | 1 | 1 | yes | characteristic |
| 485 | C22×C4 | — | 2 | 1 | no | — |
| 486 | C22×C4 | — | 4 | 1 | no | — |
| 487 | C22×C4 | — | 6 | 2 | no | — |
| 488 | C22×C4 | — | 6 | 2 | no | — |
| 489 | C22×C4 | — | 6 | 1 | no | — |
| 490 | C22×C4 | — | 6 | 2 | no | — |
| 491 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 492 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 493 | C22×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 494 | Q16 | — | 12 | 4 | no | — |
| 495 | SD16 | — | 4 | 2 | no | — |
| 496 | SD16 | — | 12 | 4 | no | — |
| 497 | SD16 | — | 12 | 4 | no | — |
| 498 | SD16 | — | 24 | 2 | no | — |
| 499 | D8 | — | 4 | 2 | no | — |
| 500 | D8 | — | 12 | 4 | no | — |
| 501 | D8 | — | 24 | 2 | no | — |
| 502 | M16 | — | 4 | 2 | no | — |
| 503 | M16 | — | 6 | 4 | no | — |
| 504 | C2×C8 | — | 2 | 1 | no | — |
| 505 | C2×C8 | — | 2 | 1 | no | — |
| 506 | C2×C8 | — | 6 | 2 | no | — |
| 507 | C2×C8 | — | 6 | 2 | no | — |
| 508 | C4⋊C4 | — | 2 | 1 | no | — |
| 509 | C4⋊C4 | — | 2 | 1 | no | — |
| 510 | C4⋊C4 | — | 2 | 1 | no | — |
| 511 | C4⋊C4 | — | 2 | 1 | no | — |
| 512 | C4⋊C4 | — | 2 | 1 | no | — |
| 513 | C4⋊C4 | — | 2 | 1 | no | — |
| 514 | C4⋊C4 | — | 6 | 2 | no | — |
| 515 | C4⋊C4 | — | 6 | 2 | no | — |
| 516 | C4⋊C4 | — | 6 | 1 | no | — |
| 517 | C4⋊C4 | — | 6 | 1 | no | — |
| 518 | C4⋊C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 519 | C4⋊C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 520 | C22⋊C4 | — | 2 | 1 | no | — |
| 521 | C22⋊C4 | — | 2 | 1 | no | — |
| 522 | C22⋊C4 | — | 2 | 1 | no | — |
| 523 | C22⋊C4 | — | 2 | 1 | no | — |
| 524 | C22⋊C4 | — | 2 | 1 | no | — |
| 525 | C22⋊C4 | — | 4 | 1 | no | — |
| 526 | C22⋊C4 | — | 4 | 1 | no | — |
| 527 | C22⋊C4 | — | 6 | 2 | no | — |
| 528 | C22⋊C4 | — | 6 | 2 | no | — |
| 529 | C22⋊C4 | — | 6 | 2 | no | — |
| 530 | C22⋊C4 | — | 6 | 2 | no | — |
| 531 | C22⋊C4 | — | 6 | 2 | no | — |
| 532 | C22⋊C4 | — | 6 | 2 | no | — |
| 533 | C22⋊C4 | — | 6 | 2 | no | — |
| 534 | C22⋊C4 | — | 6 | 2 | no | — |
| 535 | C22⋊C4 | — | 6 | 2 | no | — |
| 536 | C22⋊C4 | — | 6 | 2 | no | — |
| 537 | C22⋊C4 | — | 6 | 2 | no | — |
| 538 | C42 | — | 2 | 1 | no | — |
| 539 | C42 | — | 6 | 2 | no | — |
| 540 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 541 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 542 | C42 | C2×#67 | 1 | 1 | yes | characteristic |
| 543 | C42 | C23×D3 | 1 | 1 | yes | characteristic |
| 544 | C22×C3 | — | 2 | 1 | no | — |
| 545 | C22×C3 | — | 2 | 1 | no | — |
| 546 | C22×C3 | — | 2 | 1 | no | — |
| 547 | C22×C3 | — | 4 | 1 | no | — |
| 548 | C22×C3 | — | 4 | 1 | no | — |
| 549 | C22×C3 | — | 4 | 1 | no | — |
| 550 | C22×C3 | — | 8 | 1 | no | — |
| 551 | C22×C3 | — | 8 | 1 | no | — |
| 552 | C22×C3 | — | 8 | 1 | no | — |
| 553 | C22×C3 | C2×#120 | 1 | 1 | yes | characteristic |
| 554 | C2×D3 | — | 4 | 2 | no | — |
| 555 | C2×D3 | — | 4 | 2 | no | — |
| 556 | C2×D3 | — | 4 | 2 | no | — |
| 557 | C2×D3 | — | 4 | 2 | no | — |
| 558 | C2×D3 | — | 4 | 2 | no | — |
| 559 | C2×D3 | — | 4 | 2 | no | — |
| 560 | C2×D3 | — | 16 | 2 | no | — |
| 561 | C2×D3 | — | 16 | 4 | no | — |
| 562 | C3×C4 | — | 2 | 1 | no | — |
| 563 | C3×C4 | — | 2 | 1 | no | — |
| 564 | C3×C4 | — | 2 | 1 | no | — |
| 565 | C3×C4 | — | 2 | 1 | no | — |
| 566 | C3×C4 | — | 2 | 1 | no | — |
| 567 | C3×C4 | — | 4 | 1 | no | — |
| 568 | C3×C4 | — | 4 | 1 | no | — |
| 569 | C3×C4 | — | 4 | 1 | no | — |
| 570 | C3×C4 | — | 8 | 1 | no | — |
| 571 | C3×C4 | #453 | 1 | 2 | yes | — |
| 572 | Dic3 | — | 4 | 2 | no | — |
| 573 | Dic3 | — | 4 | 2 | no | — |
| 574 | C23 | — | 2 | 1 | no | — |
| 575 | C23 | — | 4 | 1 | no | — |
| 576 | C23 | — | 6 | 2 | no | — |
| 577 | C23 | — | 6 | 2 | no | — |
| 578 | C23 | — | 12 | 2 | no | — |
| 579 | C23 | — | 24 | 1 | no | — |
| 580 | C23 | — | 24 | 1 | no | — |
| 581 | C23 | — | 24 | 2 | no | — |
| 582 | C23 | — | 24 | 2 | no | — |
| 583 | C23 | — | 24 | 2 | no | — |
| 584 | C23 | #937 | 1 | 1 | yes | characteristic |
| 585 | Q8 | — | 4 | 1 | no | — |
| 586 | Q8 | — | 4 | 1 | no | — |
| 587 | Q8 | — | 4 | 1 | no | — |
| 588 | Q8 | — | 12 | 2 | no | — |
| 589 | Q8 | — | 12 | 2 | no | — |
| 590 | Q8 | — | 12 | 2 | no | — |
| 591 | D4 | — | 4 | 1 | no | — |
| 592 | D4 | — | 4 | 1 | no | — |
| 593 | D4 | — | 4 | 1 | no | — |
| 594 | D4 | — | 4 | 1 | no | — |
| 595 | D4 | — | 4 | 1 | no | — |
| 596 | D4 | — | 4 | 1 | no | — |
| 597 | D4 | — | 4 | 1 | no | — |
| 598 | D4 | — | 4 | 2 | no | — |
| 599 | D4 | — | 8 | 2 | no | — |
| 600 | D4 | — | 12 | 2 | no | — |
| 601 | D4 | — | 12 | 2 | no | — |
| 602 | D4 | — | 12 | 4 | no | — |
| 603 | D4 | — | 12 | 2 | no | — |
| 604 | D4 | — | 12 | 2 | no | — |
| 605 | D4 | — | 24 | 1 | no | — |
| 606 | D4 | — | 24 | 2 | no | — |
| 607 | D4 | — | 24 | 2 | no | — |
| 608 | D4 | — | 24 | 2 | no | — |
| 609 | D4 | — | 24 | 2 | no | — |
| 610 | D4 | — | 24 | 2 | no | — |
| 611 | C2×C4 | — | 2 | 2 | no | — |
| 612 | C2×C4 | — | 2 | 1 | no | — |
| 613 | C2×C4 | — | 2 | 1 | no | — |
| 614 | C2×C4 | — | 2 | 1 | no | — |
| 615 | C2×C4 | — | 4 | 1 | no | — |
| 616 | C2×C4 | — | 4 | 1 | no | — |
| 617 | C2×C4 | — | 4 | 2 | no | — |
| 618 | C2×C4 | — | 4 | 1 | no | — |
| 619 | C2×C4 | — | 6 | 2 | no | — |
| 620 | C2×C4 | — | 6 | 2 | no | — |
| 621 | C2×C4 | — | 8 | 2 | no | — |
| 622 | C2×C4 | — | 24 | 2 | no | — |
| 623 | C2×C4 | — | 24 | 2 | no | — |
| 624 | C2×C4 | — | 24 | 2 | no | — |
| 625 | C2×C4 | — | 24 | 1 | no | — |
| 626 | C2×C4 | #937 | 1 | 1 | yes | characteristic |
| 627 | C2×C4 | #937 | 1 | 1 | yes | characteristic |
| 628 | C2×C4 | #937 | 1 | 1 | yes | characteristic |
| 629 | C2×C4 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 630 | C2×C4 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 631 | C2×C4 | C22×#67 | 1 | 1 | yes | characteristic |
| 632 | C8 | — | 2 | 2 | no | — |
| 633 | C8 | — | 4 | 1 | no | — |
| 634 | C8 | — | 6 | 4 | no | — |
| 635 | C8 | — | 12 | 2 | no | — |
| 636 | C2×C3 | — | 4 | 1 | no | — |
| 637 | C2×C3 | — | 8 | 1 | no | — |
| 638 | C2×C3 | — | 16 | 1 | no | — |
| 639 | C2×C3 | C2×#453 | 1 | 1 | yes | characteristic |
| 640 | C2×C3 | #3018 | 1 | 1 | yes | characteristic |
| 641 | C2×C3 | #3019 | 1 | 1 | yes | characteristic |
| 642 | D3 | — | 8 | 2 | no | — |
| 643 | D3 | — | 8 | 2 | no | — |
| 644 | C22 | — | 2 | 1 | no | — |
| 645 | C22 | — | 2 | 1 | no | — |
| 646 | C22 | — | 2 | 1 | no | — |
| 647 | C22 | — | 4 | 1 | no | — |
| 648 | C22 | — | 4 | 1 | no | — |
| 649 | C22 | — | 4 | 1 | no | — |
| 650 | C22 | — | 8 | 1 | no | — |
| 651 | C22 | — | 8 | 1 | no | — |
| 652 | C22 | — | 8 | 1 | no | — |
| 653 | C22 | — | 12 | 2 | no | — |
| 654 | C22 | — | 12 | 2 | no | — |
| 655 | C22 | — | 12 | 2 | no | — |
| 656 | C22 | — | 12 | 2 | no | — |
| 657 | C22 | — | 12 | 2 | no | — |
| 658 | C22 | — | 12 | 2 | no | — |
| 659 | C22 | — | 48 | 2 | no | — |
| 660 | C22 | — | 48 | 4 | no | — |
| 661 | C22 | C2×#937 | 1 | 1 | yes | characteristic |
| 662 | C4 | — | 2 | 1 | no | — |
| 663 | C4 | — | 2 | 1 | no | — |
| 664 | C4 | — | 2 | 1 | no | — |
| 665 | C4 | — | 2 | 1 | no | — |
| 666 | C4 | — | 2 | 1 | no | — |
| 667 | C4 | — | 4 | 1 | no | — |
| 668 | C4 | — | 4 | 1 | no | — |
| 669 | C4 | — | 4 | 1 | no | — |
| 670 | C4 | — | 8 | 1 | no | — |
| 671 | C4 | — | 12 | 2 | no | — |
| 672 | C4 | — | 12 | 2 | no | — |
| 673 | C4 | #5198 | 1 | 2 | yes | — |
| 674 | C3 | #19018 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 675 | C2 | — | 4 | 1 | no | — |
| 676 | C2 | — | 8 | 1 | no | — |
| 677 | C2 | — | 16 | 1 | no | — |
| 678 | C2 | — | 24 | 2 | no | — |
| 679 | C2 | — | 24 | 2 | no | — |
| 680 | C2 | #83322 | 1 | 1 | yes | characteristic |
| 681 | C2 | C2×#5198 | 1 | 1 | yes | characteristic |
| 682 | C2 | #84005 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#937 |
| commutator, cosocle | C3×C42 | C24 |
| Frattini | C42 | C23×D3 |
| Fitting | C3×#3171 | C2 |
| socle | C22×C3 | C2×#120 |
| hypercenter | #3171 | D3 |
| hypocenter | C3 | #19018 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #3171 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #86177 | C2 | semidirect |
| 2. C3×#3171 | C2 | semidirect |
| 3. #83308 | C2 | semidirect |
| 4. #83289 | C2 | semidirect |
| 5. #71747 | C2 | semidirect |
| 6. #71392 | C2 | semidirect |
| 7. #71326 | C2 | semidirect |
| 8. #71248 | C2 | semidirect |
| 9. #71244 | C2 | semidirect |
| 10. #69860 | C2 | semidirect |
| 11. #69859 | C2 | semidirect |
| 13. C3×#532 | C22 | semidirect |
| 31. C3×#501 | C22 | semidirect |
| 32. C3×#496 | C22 | semidirect |
| 33. C3×#490 | C22 | semidirect |
| 34. C3×#489 | C22 | semidirect |
| 39. C3×C4×M16 | C22 | semidirect |
| 40. #5089 | C22 | semidirect |
| 41. #5086 | C22 | semidirect |
| 42. #5074 | C22 | semidirect |
| 43. #5074 | C22 | semidirect |
| 48. #4926 | C22 | semidirect |
| 49. C4×#208 | C22 | semidirect |
| 50. #4543 | C22 | semidirect |
| 51. #4540 | C22 | semidirect |
| 52. #4464 | C22 | semidirect |
| 53. #4464 | C22 | semidirect |
| 54. #4464 | C22 | semidirect |
| 55. #4464 | C22 | semidirect |
| 58. #3171 | D3 | semidirect |
| 177. #532 | C2×D3 | semidirect |
| 180. #501 | C2×D3 | semidirect |
| 184. #496 | C2×D3 | semidirect |
| 186. #490 | C2×D3 | semidirect |
| 188. #489 | C2×D3 | semidirect |
| 198. C4×M16 | C2×D3 | semidirect |
| 674. C3 | #19018 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #86177 | 2 |
| C3×#3171 | 2 |
| #83308 | 2 |
| #83289 | 2 |
| #71747 | 2 |
| #71392 | 2 |
| #71326 | 2 |
| #71248 | 2 |
| #71244 | 2 |
| #69860 | 2 |
| #69859 | 2 |
| #19018 | 3 |