← #10694455
Subgroups — #10694455
Order 768.
454 conjugacy classes of subgroups.
The subgroup lattice has 320 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#5093 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#1987 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#4546 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#4486 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×#4479 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #69490 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C4⋊C4×C3⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #9737 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×C3×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#869 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 12 | C3×C4×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 13 | #5093 | — | 2 | 2 | no | — |
| 14 | #5093 | C4 | 1 | 4 | yes | complemented |
| 15 | C2×#810 | — | 2 | 1 | no | — |
| 16 | C2×#810 | — | 2 | 1 | no | — |
| 17 | C2×C4⋊(C3⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×C4⋊(C3⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×C4×C3⋊C8 | — | 2 | 1 | no | — |
| 20 | C2×C4×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 21 | C3×#340 | — | 2 | 2 | no | — |
| 22 | #4546 | — | 2 | 2 | no | — |
| 23 | #4529 | — | 2 | 2 | no | — |
| 24 | #4524 | — | 2 | 1 | no | — |
| 25 | #4486 | — | 2 | 2 | no | — |
| 26 | #4479 | — | 2 | 2 | no | — |
| 27 | C2×#487 | — | 3 | 1 | no | — |
| 28 | #1988 | — | 3 | 1 | no | — |
| 29 | #1987 | D3 | 1 | 1 | yes | characteristic |
| 30 | C8×C4⋊C4 | — | 3 | 1 | no | — |
| 31 | C2×#335 | — | 3 | 1 | no | — |
| 32 | C2×#333 | — | 3 | 1 | no | — |
| 33 | C2×#332 | — | 3 | 1 | no | — |
| 34 | C22×C3×Q8 | — | 2 | 1 | no | — |
| 35 | C22×Dic6 | — | 2 | 1 | no | — |
| 36 | C22×C3×C8 | — | 2 | 1 | no | — |
| 37 | C3×C4⋊Q8 | — | 2 | 1 | no | — |
| 38 | C3×C4⋊Q8 | — | 2 | 1 | no | — |
| 39 | C3×C4⋊Q8 | — | 2 | 1 | no | — |
| 40 | C3×C4⋊Q8 | C2×C4 | 1 | 2 | yes | — |
| 41 | C2×C3×C4⋊C4 | — | 2 | 1 | no | — |
| 42 | C2×C3×C4⋊C4 | — | 2 | 1 | no | — |
| 43 | C2×C3×C4⋊C4 | D4 | 1 | 2 | yes | — |
| 44 | C2×C3×C42 | — | 2 | 1 | no | — |
| 45 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 47 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 48 | C22×C3⋊C8 | — | 2 | 1 | no | — |
| 49 | C22×C3⋊C8 | — | 2 | 1 | no | — |
| 50 | C22×C3⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 51 | C22×C3⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 52 | #869 | — | 2 | 1 | no | — |
| 53 | #869 | — | 2 | 1 | no | — |
| 54 | #869 | — | 2 | 1 | no | — |
| 55 | #869 | C2×C4 | 1 | 2 | yes | — |
| 56 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 57 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 58 | C3×#95 | — | 2 | 1 | no | — |
| 59 | #810 | — | 2 | 4 | no | — |
| 60 | #810 | — | 2 | 4 | no | — |
| 61 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 62 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 63 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 64 | C4⋊(C3⋊C8) | D4 | 1 | 2 | yes | — |
| 65 | C4×C3⋊C8 | — | 4 | 2 | no | — |
| 66 | C4×C3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 67 | C4×C3⋊C8 | D4 | 1 | 2 | yes | — |
| 68 | C2×C4⋊Q8 | — | 3 | 1 | no | — |
| 69 | C2×C4⋊Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 70 | #487 | — | 3 | 4 | no | — |
| 71 | #487 | — | 6 | 2 | no | — |
| 72 | C2×C4⋊C8 | — | 3 | 1 | no | — |
| 73 | C2×C4⋊C8 | — | 3 | 1 | no | — |
| 74 | C2×C4⋊C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 75 | C2×#103 | — | 6 | 1 | no | — |
| 76 | C2×#103 | — | 6 | 1 | no | — |
| 77 | C2×C4×C8 | — | 3 | 1 | no | — |
| 78 | C2×C4×C8 | — | 6 | 1 | no | — |
| 79 | C4×C4⋊C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 80 | #340 | — | 2 | 2 | no | — |
| 81 | #340 | — | 6 | 2 | no | — |
| 82 | #336 | — | 6 | 1 | no | — |
| 83 | #335 | — | 6 | 2 | no | — |
| 84 | #333 | — | 6 | 2 | no | — |
| 85 | #332 | — | 6 | 2 | no | — |
| 86 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 87 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 88 | C2×C3×Q8 | — | 8 | 2 | no | — |
| 89 | C22×C3×C4 | — | 2 | 2 | no | — |
| 90 | C22×C3×C4 | — | 4 | 1 | no | — |
| 91 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C22×C3×C4 | #41 | 1 | 1 | yes | characteristic |
| 94 | C22×Dic3 | — | 4 | 1 | no | — |
| 95 | C2×Dic6 | — | 2 | 2 | no | — |
| 96 | C2×Dic6 | — | 4 | 1 | no | — |
| 97 | C2×Dic6 | — | 4 | 1 | no | — |
| 98 | C2×Dic6 | — | 4 | 1 | no | — |
| 99 | C2×Dic6 | — | 4 | 1 | no | — |
| 100 | C2×Dic6 | — | 4 | 1 | no | — |
| 101 | C2×C3×C8 | — | 4 | 1 | no | — |
| 102 | C2×C3×C8 | — | 4 | 2 | no | — |
| 103 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 104 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 105 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 106 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 107 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 108 | C3×C4⋊C4 | Q16 | 1 | 4 | yes | — |
| 109 | C3×C42 | — | 4 | 1 | no | — |
| 110 | C3×C42 | — | 4 | 1 | no | — |
| 111 | C3×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 112 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 113 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 114 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 115 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 116 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 117 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 118 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 119 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 120 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 121 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 122 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 123 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 124 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 125 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 126 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 127 | C2×C3⋊C8 | — | 4 | 2 | no | — |
| 128 | C2×C3⋊C8 | C22⋊C4 | 1 | 2 | yes | — |
| 129 | C2×C3⋊C8 | C22⋊C4 | 1 | 2 | yes | — |
| 130 | C22×Q8 | — | 2 | 1 | no | — |
| 131 | C22×Q8 | — | 6 | 1 | no | — |
| 132 | C22×C8 | — | 2 | 1 | no | — |
| 133 | C22×C8 | — | 3 | 1 | no | — |
| 134 | C22×C8 | — | 3 | 1 | no | — |
| 135 | C22×C8 | — | 6 | 1 | no | — |
| 136 | C22×C8 | — | 6 | 1 | no | — |
| 137 | C4⋊Q8 | — | 2 | 1 | no | — |
| 138 | C4⋊Q8 | — | 2 | 1 | no | — |
| 139 | C4⋊Q8 | — | 2 | 1 | no | — |
| 140 | C4⋊Q8 | — | 3 | 2 | no | — |
| 141 | C4⋊Q8 | — | 6 | 1 | no | — |
| 142 | C4⋊Q8 | — | 6 | 1 | no | — |
| 143 | C4⋊Q8 | — | 6 | 1 | no | — |
| 144 | C4⋊Q8 | C4×D3 | 1 | 2 | yes | — |
| 145 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 146 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 147 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 148 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 149 | C2×C4⋊C4 | #67 | 1 | 2 | yes | — |
| 150 | C2×C42 | — | 2 | 1 | no | — |
| 151 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 152 | C4⋊C8 | — | 2 | 1 | no | — |
| 153 | C4⋊C8 | — | 2 | 1 | no | — |
| 154 | C4⋊C8 | — | 3 | 2 | no | — |
| 155 | C4⋊C8 | — | 6 | 1 | no | — |
| 156 | C4⋊C8 | — | 6 | 1 | no | — |
| 157 | C4⋊C8 | — | 6 | 1 | no | — |
| 158 | #103 | — | 6 | 4 | no | — |
| 159 | #103 | — | 6 | 4 | no | — |
| 160 | C4×C8 | — | 3 | 2 | no | — |
| 161 | C4×C8 | — | 3 | 2 | no | — |
| 162 | C4×C8 | — | 12 | 2 | no | — |
| 163 | #95 | — | 2 | 1 | no | — |
| 164 | C23×C3 | #121 | 1 | 1 | yes | characteristic |
| 165 | C3×Q8 | — | 8 | 2 | no | — |
| 166 | C3×Q8 | — | 8 | 2 | no | — |
| 167 | C2×C3×C4 | — | 2 | 2 | no | — |
| 168 | C2×C3×C4 | — | 2 | 4 | no | — |
| 169 | C2×C3×C4 | — | 4 | 2 | no | — |
| 170 | C2×C3×C4 | — | 4 | 2 | no | — |
| 171 | C2×C3×C4 | — | 4 | 2 | no | — |
| 172 | C2×C3×C4 | — | 8 | 2 | no | — |
| 173 | C2×C3×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 174 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 175 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 176 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 177 | C2×C3×C4 | #124 | 1 | 1 | yes | characteristic |
| 178 | C2×C3×C4 | #127 | 1 | 1 | yes | characteristic |
| 179 | C2×C3×C4 | C2×Q16 | 1 | 2 | yes | — |
| 180 | C2×C3×C4 | #135 | 1 | 1 | yes | characteristic |
| 181 | C2×C3×C4 | #135 | 1 | 1 | yes | characteristic |
| 182 | C2×C3×C4 | #135 | 1 | 2 | yes | — |
| 183 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 184 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 185 | C2×Dic3 | — | 4 | 2 | no | — |
| 186 | C2×Dic3 | — | 8 | 1 | no | — |
| 187 | C2×Dic3 | — | 8 | 1 | no | — |
| 188 | Dic6 | — | 4 | 2 | no | — |
| 189 | Dic6 | — | 4 | 2 | no | — |
| 190 | Dic6 | — | 8 | 1 | no | — |
| 191 | Dic6 | — | 8 | 1 | no | — |
| 192 | C3×C8 | — | 8 | 1 | no | — |
| 193 | C3⋊C8 | — | 2 | 2 | no | — |
| 194 | C3⋊C8 | — | 2 | 2 | no | — |
| 195 | C3⋊C8 | — | 4 | 2 | no | — |
| 196 | C3⋊C8 | — | 8 | 1 | no | — |
| 197 | C2×Q8 | — | 2 | 2 | no | — |
| 198 | C2×Q8 | — | 4 | 1 | no | — |
| 199 | C2×Q8 | — | 6 | 2 | no | — |
| 200 | C2×Q8 | — | 8 | 2 | no | — |
| 201 | C2×Q8 | — | 12 | 1 | no | — |
| 202 | C2×Q8 | — | 12 | 1 | no | — |
| 203 | C2×Q8 | — | 12 | 1 | no | — |
| 204 | C2×Q8 | — | 12 | 1 | no | — |
| 205 | C2×Q8 | — | 12 | 1 | no | — |
| 206 | C22×C4 | — | 2 | 2 | no | — |
| 207 | C22×C4 | — | 4 | 1 | no | — |
| 208 | C22×C4 | — | 12 | 1 | no | — |
| 209 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 210 | C22×C4 | #237 | 1 | 1 | yes | characteristic |
| 211 | C22×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 212 | C2×C8 | — | 3 | 2 | no | — |
| 213 | C2×C8 | — | 3 | 2 | no | — |
| 214 | C2×C8 | — | 4 | 1 | no | — |
| 215 | C2×C8 | — | 4 | 2 | no | — |
| 216 | C2×C8 | — | 6 | 1 | no | — |
| 217 | C2×C8 | — | 6 | 1 | no | — |
| 218 | C2×C8 | — | 6 | 1 | no | — |
| 219 | C2×C8 | — | 6 | 1 | no | — |
| 220 | C2×C8 | — | 6 | 2 | no | — |
| 221 | C2×C8 | — | 12 | 1 | no | — |
| 222 | C2×C8 | — | 12 | 1 | no | — |
| 223 | C2×C8 | — | 12 | 1 | no | — |
| 224 | C2×C8 | — | 12 | 2 | no | — |
| 225 | C4⋊C4 | — | 2 | 2 | no | — |
| 226 | C4⋊C4 | — | 2 | 2 | no | — |
| 227 | C4⋊C4 | — | 2 | 2 | no | — |
| 228 | C4⋊C4 | — | 4 | 1 | no | — |
| 229 | C4⋊C4 | — | 4 | 1 | no | — |
| 230 | C4⋊C4 | — | 6 | 2 | no | — |
| 231 | C4⋊C4 | — | 6 | 2 | no | — |
| 232 | C4⋊C4 | — | 12 | 1 | no | — |
| 233 | C4⋊C4 | — | 12 | 1 | no | — |
| 234 | C4⋊C4 | #216 | 1 | 4 | yes | — |
| 235 | C42 | — | 4 | 1 | no | — |
| 236 | C42 | — | 4 | 1 | no | — |
| 237 | C42 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 238 | C42 | #235 | 1 | 1 | yes | characteristic |
| 239 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 240 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 241 | C22×C3 | #390 | 1 | 1 | yes | characteristic |
| 242 | C22×C3 | #463 | 1 | 1 | yes | characteristic |
| 243 | C22×C3 | #464 | 1 | 1 | yes | characteristic |
| 244 | C22×C3 | #467 | 1 | 2 | yes | — |
| 245 | C22×C3 | #471 | 1 | 2 | yes | — |
| 246 | C3×C4 | — | 2 | 1 | no | — |
| 247 | C3×C4 | — | 2 | 1 | no | — |
| 248 | C3×C4 | — | 4 | 4 | no | — |
| 249 | C3×C4 | — | 8 | 2 | no | — |
| 250 | C3×C4 | #416 | 1 | 1 | yes | characteristic |
| 251 | C3×C4 | #417 | 1 | 1 | yes | characteristic |
| 252 | C3×C4 | C4×Q16 | 1 | 2 | yes | — |
| 253 | C3×C4 | #463 | 1 | 1 | yes | characteristic |
| 254 | C3×C4 | #464 | 1 | 1 | yes | characteristic |
| 255 | C3×C4 | #494 | 1 | 1 | yes | characteristic |
| 256 | C3×C4 | #495 | 1 | 1 | yes | characteristic |
| 257 | Dic3 | — | 8 | 2 | no | — |
| 258 | C23 | #939 | 1 | 1 | yes | characteristic |
| 259 | Q8 | — | 8 | 2 | no | — |
| 260 | Q8 | — | 8 | 2 | no | — |
| 261 | Q8 | — | 12 | 2 | no | — |
| 262 | Q8 | — | 12 | 2 | no | — |
| 263 | Q8 | — | 24 | 1 | no | — |
| 264 | Q8 | — | 24 | 1 | no | — |
| 265 | C2×C4 | — | 2 | 2 | no | — |
| 266 | C2×C4 | — | 2 | 4 | no | — |
| 267 | C2×C4 | — | 4 | 2 | no | — |
| 268 | C2×C4 | — | 4 | 2 | no | — |
| 269 | C2×C4 | — | 4 | 2 | no | — |
| 270 | C2×C4 | — | 8 | 2 | no | — |
| 271 | C2×C4 | — | 12 | 2 | no | — |
| 272 | C2×C4 | — | 24 | 1 | no | — |
| 273 | C2×C4 | — | 24 | 1 | no | — |
| 274 | C2×C4 | D3×C22⋊C4 | 1 | 1 | yes | characteristic |
| 275 | C2×C4 | #67⋊C4 | 1 | 1 | yes | characteristic |
| 276 | C2×C4 | #884 | 1 | 1 | yes | characteristic |
| 277 | C2×C4 | #885 | 1 | 1 | yes | characteristic |
| 278 | C2×C4 | #914 | 1 | 1 | yes | characteristic |
| 279 | C2×C4 | #916 | 1 | 1 | yes | characteristic |
| 280 | C2×C4 | #918 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | #919 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | C4×#67 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | #940 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | C2×#216 | 1 | 2 | yes | — |
| 285 | C2×C4 | #950 | 1 | 2 | yes | — |
| 286 | C8 | — | 6 | 2 | no | — |
| 287 | C8 | — | 6 | 2 | no | — |
| 288 | C8 | — | 8 | 1 | no | — |
| 289 | C8 | — | 12 | 2 | no | — |
| 290 | C8 | — | 24 | 1 | no | — |
| 291 | C2×C3 | #1683 | 1 | 2 | yes | — |
| 292 | C2×C3 | #1684 | 1 | 2 | yes | — |
| 293 | C2×C3 | #1969 | 1 | 1 | yes | characteristic |
| 294 | C2×C3 | #1977 | 1 | 1 | yes | characteristic |
| 295 | C2×C3 | #1983 | 1 | 1 | yes | characteristic |
| 296 | C22 | #4957 | 1 | 1 | yes | characteristic |
| 297 | C22 | #5051 | 1 | 2 | yes | — |
| 298 | C22 | #5165 | 1 | 1 | yes | characteristic |
| 299 | C22 | #5185 | 1 | 1 | yes | characteristic |
| 300 | C22 | #5200 | 1 | 2 | yes | — |
| 301 | C4 | — | 2 | 1 | no | — |
| 302 | C4 | — | 2 | 1 | no | — |
| 303 | C4 | — | 4 | 4 | no | — |
| 304 | C4 | — | 8 | 2 | no | — |
| 305 | C4 | — | 24 | 2 | no | — |
| 306 | C4 | #4798 | 1 | 1 | yes | characteristic |
| 307 | C4 | #4803 | 1 | 1 | yes | characteristic |
| 308 | C4 | #4804 | 1 | 1 | yes | characteristic |
| 309 | C4 | #4819 | 1 | 1 | yes | characteristic |
| 310 | C4 | C4×#216 | 1 | 2 | yes | — |
| 311 | C4 | #5059 | 1 | 1 | yes | characteristic |
| 312 | C4 | #5092 | 1 | 1 | yes | characteristic |
| 313 | C3 | #9737 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 314 | C2 | #69677 | 1 | 1 | yes | characteristic |
| 315 | C2 | #70230 | 1 | 1 | yes | characteristic |
| 316 | C2 | #70315 | 1 | 1 | yes | characteristic |
| 317 | C2 | #71135 | 1 | 2 | yes | — |
| 318 | C2 | #71151 | 1 | 2 | yes | — |
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