← #10713685
Subgroups — #10713685
Order 768.
472 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 | #78467 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#1772 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #70690 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #70656 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C2×#4463 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #13259 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×#904 | — | 2 | 1 | no | — |
| 8 | C2×#904 | C22 | 1 | 2 | yes | — |
| 9 | #5477 | — | 2 | 1 | no | — |
| 10 | C2×C3×#108 | C22 | 1 | 1 | yes | characteristic |
| 11 | C3×#427 | C22 | 1 | 1 | yes | characteristic |
| 12 | C3×#405 | — | 2 | 1 | no | — |
| 13 | C2×C3×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#812 | C22 | 1 | 2 | yes | — |
| 15 | #4699 | C4 | 1 | 1 | yes | characteristic, complemented |
| 16 | #4696 | — | 2 | 1 | no | — |
| 17 | #4696 | C4 | 1 | 2 | yes | complemented |
| 18 | #4695 | C4 | 1 | 1 | yes | characteristic, complemented |
| 19 | #4693 | — | 2 | 1 | no | — |
| 20 | C3×#339 | — | 2 | 1 | no | — |
| 21 | #4497 | — | 2 | 4 | no | — |
| 22 | #4488 | — | 2 | 4 | no | — |
| 23 | #4463 | — | 2 | 1 | no | — |
| 24 | #4463 | — | 2 | 1 | no | — |
| 25 | #3147 | — | 3 | 1 | no | — |
| 26 | C2×#364 | — | 3 | 1 | no | — |
| 27 | #2145 | — | 3 | 2 | no | — |
| 28 | #2143 | — | 3 | 2 | no | — |
| 29 | #1772 | D3 | 1 | 1 | yes | characteristic, complemented |
| 30 | C2×#235 | — | 2 | 2 | no | — |
| 31 | C2×C3×M16 | — | 2 | 1 | no | — |
| 32 | C22×C3×C8 | D4 | 1 | 1 | yes | characteristic |
| 33 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 34 | C3×#117 | — | 2 | 1 | no | — |
| 35 | C2×C3×C42 | D4 | 1 | 1 | yes | characteristic |
| 36 | #930 | — | 4 | 2 | no | — |
| 37 | C2×Dic12 | — | 2 | 1 | no | — |
| 38 | C2×Dic12 | C2×C4 | 1 | 2 | yes | — |
| 39 | #904 | — | 2 | 2 | no | — |
| 40 | #904 | — | 2 | 2 | no | — |
| 41 | #904 | — | 2 | 2 | no | — |
| 42 | #904 | — | 2 | 2 | no | — |
| 43 | #904 | — | 4 | 1 | no | — |
| 44 | #904 | — | 4 | 1 | no | — |
| 45 | #904 | — | 4 | 1 | no | — |
| 46 | #904 | — | 4 | 1 | no | — |
| 47 | C2×D24 | — | 2 | 1 | no | — |
| 48 | C2×D24 | C2×C4 | 1 | 2 | yes | — |
| 49 | C2×#204 | — | 2 | 2 | no | — |
| 50 | C2×#204 | — | 2 | 2 | no | — |
| 51 | #875 | — | 2 | 2 | no | — |
| 52 | #874 | — | 2 | 1 | no | — |
| 53 | C4×D12 | — | 4 | 2 | no | — |
| 54 | #869 | — | 2 | 1 | no | — |
| 55 | C3×#108 | — | 2 | 1 | no | — |
| 56 | C3×#108 | D4 | 1 | 2 | yes | complemented |
| 57 | C3×#107 | — | 2 | 1 | no | — |
| 58 | C3×#106 | D4 | 1 | 2 | yes | complemented |
| 59 | C3×#97 | — | 4 | 1 | no | — |
| 60 | C3×C4×C8 | — | 4 | 1 | no | — |
| 61 | C3×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 62 | C3×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 63 | C3×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 64 | C3×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 65 | #812 | — | 2 | 2 | no | — |
| 66 | #812 | — | 2 | 2 | no | — |
| 67 | C2×C3⋊C16 | — | 2 | 2 | no | — |
| 68 | C2×#135 | — | 3 | 2 | no | — |
| 69 | C2×#135 | — | 6 | 1 | no | — |
| 70 | #532 | — | 6 | 1 | no | — |
| 71 | C2×M32 | — | 3 | 2 | no | — |
| 72 | #495 | — | 3 | 2 | no | — |
| 73 | #495 | — | 6 | 1 | no | — |
| 74 | #494 | — | 3 | 1 | no | — |
| 75 | #493 | — | 3 | 1 | no | — |
| 76 | #492 | — | 6 | 1 | no | — |
| 77 | C2×#108 | C2×D3 | 1 | 1 | yes | characteristic |
| 78 | #427 | C2×D3 | 1 | 1 | yes | characteristic |
| 79 | #405 | — | 2 | 1 | no | — |
| 80 | C2×C4×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 81 | #364 | — | 6 | 1 | no | — |
| 82 | #364 | — | 6 | 1 | no | — |
| 83 | #360 | — | 6 | 4 | no | — |
| 84 | #359 | — | 6 | 4 | no | — |
| 85 | #339 | — | 2 | 1 | no | — |
| 86 | C22×C3×C4 | — | 2 | 1 | no | — |
| 87 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 88 | C2×#67 | — | 4 | 2 | no | — |
| 89 | #235 | — | 4 | 2 | no | — |
| 90 | #235 | — | 4 | 2 | no | — |
| 91 | #235 | — | 4 | 2 | no | — |
| 92 | #235 | — | 4 | 2 | no | — |
| 93 | C2×D12 | — | 2 | 2 | no | — |
| 94 | C2×D12 | — | 4 | 2 | no | — |
| 95 | C2×C4×D3 | — | 4 | 2 | no | — |
| 96 | C2×Dic6 | — | 2 | 2 | no | — |
| 97 | C3×M16 | — | 2 | 2 | no | — |
| 98 | C3×M16 | — | 4 | 1 | no | — |
| 99 | C2×C3×C8 | — | 4 | 1 | no | — |
| 100 | C2×C3×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 101 | C2×C3×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 102 | C2×C3×C8 | D8 | 1 | 1 | yes | characteristic |
| 103 | C2×C3×C8 | D8 | 1 | 1 | yes | characteristic |
| 104 | C2×C3×C8 | SD16 | 1 | 1 | yes | characteristic |
| 105 | C2×C3×C8 | SD16 | 1 | 1 | yes | characteristic |
| 106 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 107 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 108 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 109 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 111 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 112 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 113 | C3×C22⋊C4 | — | 4 | 1 | no | — |
| 114 | C3×C42 | — | 2 | 1 | no | — |
| 115 | C3×C42 | — | 4 | 1 | no | — |
| 116 | C3×C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 117 | C3×C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 118 | (C2×D3)⋊C4 | — | 4 | 4 | no | — |
| 119 | C4⋊Dic3 | — | 4 | 2 | no | — |
| 120 | Dic12 | — | 2 | 2 | no | — |
| 121 | Dic12 | — | 2 | 2 | no | — |
| 122 | Dic12 | — | 4 | 1 | no | — |
| 123 | Dic12 | — | 4 | 1 | no | — |
| 124 | D24 | — | 2 | 2 | no | — |
| 125 | D24 | — | 2 | 2 | no | — |
| 126 | D24 | — | 4 | 1 | no | — |
| 127 | D24 | — | 4 | 1 | no | — |
| 128 | #204 | — | 4 | 2 | no | — |
| 129 | #204 | — | 4 | 2 | no | — |
| 130 | #204 | — | 4 | 2 | no | — |
| 131 | #204 | — | 4 | 2 | no | — |
| 132 | C3⋊C16 | — | 4 | 2 | no | — |
| 133 | C2×#41 | — | 6 | 2 | no | — |
| 134 | #135 | — | 6 | 2 | no | — |
| 135 | #135 | — | 6 | 2 | no | — |
| 136 | #135 | — | 6 | 2 | no | — |
| 137 | #135 | — | 6 | 2 | no | — |
| 138 | #135 | — | 12 | 1 | no | — |
| 139 | #135 | — | 12 | 1 | no | — |
| 140 | #135 | — | 12 | 1 | no | — |
| 141 | #135 | — | 12 | 1 | no | — |
| 142 | C2×Q16 | — | 3 | 2 | no | — |
| 143 | C2×Q16 | — | 6 | 1 | no | — |
| 144 | C2×SD16 | — | 6 | 2 | no | — |
| 145 | C2×SD16 | — | 6 | 2 | no | — |
| 146 | C2×D8 | — | 3 | 2 | no | — |
| 147 | C2×D8 | — | 6 | 1 | no | — |
| 148 | C2×M16 | — | 2 | 1 | no | — |
| 149 | C22×C8 | #67 | 1 | 1 | yes | characteristic |
| 150 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 151 | C4⋊Q8 | — | 6 | 1 | no | — |
| 152 | #127 | — | 6 | 1 | no | — |
| 153 | #124 | — | 6 | 2 | no | — |
| 154 | #121 | — | 12 | 2 | no | — |
| 155 | C4×D4 | — | 12 | 2 | no | — |
| 156 | #117 | — | 2 | 1 | no | — |
| 157 | C2×C42 | #67 | 1 | 1 | yes | characteristic |
| 158 | M32 | — | 6 | 2 | no | — |
| 159 | M32 | — | 6 | 2 | no | — |
| 160 | C2×C16 | — | 6 | 2 | no | — |
| 161 | #108 | — | 2 | 1 | no | — |
| 162 | #108 | D12 | 1 | 2 | yes | complemented |
| 163 | #107 | — | 2 | 1 | no | — |
| 164 | #106 | D12 | 1 | 2 | yes | complemented |
| 165 | #97 | — | 4 | 1 | no | — |
| 166 | C4×C8 | — | 4 | 1 | no | — |
| 167 | C4×C8 | C4×D3 | 1 | 1 | yes | characteristic |
| 168 | C4×C8 | C4×D3 | 1 | 1 | yes | characteristic |
| 169 | C4×C8 | D12 | 1 | 1 | yes | characteristic |
| 170 | C4×C8 | D12 | 1 | 1 | yes | characteristic |
| 171 | C23×C3 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 172 | C22×D3 | — | 4 | 2 | no | — |
| 173 | C2×C3×C4 | — | 2 | 1 | no | — |
| 174 | C2×C3×C4 | — | 2 | 1 | no | — |
| 175 | C2×C3×C4 | — | 4 | 2 | no | — |
| 176 | C2×C3×C4 | — | 4 | 2 | no | — |
| 177 | C2×C3×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 178 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 179 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 180 | C2×C3×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 181 | C2×C3×C4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 182 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 183 | #67 | — | 8 | 2 | no | — |
| 184 | #67 | — | 8 | 2 | no | — |
| 185 | C2×Dic3 | — | 4 | 2 | no | — |
| 186 | D12 | — | 4 | 2 | no | — |
| 187 | D12 | — | 4 | 2 | no | — |
| 188 | D12 | — | 8 | 4 | no | — |
| 189 | C4×D3 | — | 8 | 2 | no | — |
| 190 | C4×D3 | — | 8 | 2 | no | — |
| 191 | Dic6 | — | 4 | 2 | no | — |
| 192 | Dic6 | — | 4 | 2 | no | — |
| 193 | C3×C8 | — | 4 | 2 | no | — |
| 194 | C3×C8 | #102 | 1 | 1 | yes | characteristic |
| 195 | C3×C8 | #102 | 1 | 1 | yes | characteristic |
| 196 | C3×C8 | #102 | 1 | 1 | yes | characteristic |
| 197 | C3×C8 | #102 | 1 | 1 | yes | characteristic |
| 198 | C3×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 199 | C3×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 200 | C3×C8 | #124 | 1 | 1 | yes | characteristic |
| 201 | C3×C8 | #127 | 1 | 1 | yes | characteristic |
| 202 | #41 | — | 12 | 2 | no | — |
| 203 | #41 | — | 12 | 2 | no | — |
| 204 | #41 | — | 12 | 2 | no | — |
| 205 | #41 | — | 12 | 2 | no | — |
| 206 | C2×Q8 | — | 6 | 2 | no | — |
| 207 | C2×D4 | — | 6 | 2 | no | — |
| 208 | C2×D4 | — | 12 | 2 | no | — |
| 209 | C2×D4 | — | 12 | 2 | no | — |
| 210 | C22×C4 | — | 2 | 1 | no | — |
| 211 | C22×C4 | — | 12 | 2 | no | — |
| 212 | C22×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 213 | Q16 | — | 6 | 2 | no | — |
| 214 | Q16 | — | 6 | 2 | no | — |
| 215 | Q16 | — | 12 | 1 | no | — |
| 216 | Q16 | — | 12 | 1 | no | — |
| 217 | SD16 | — | 12 | 2 | no | — |
| 218 | SD16 | — | 12 | 2 | no | — |
| 219 | SD16 | — | 12 | 2 | no | — |
| 220 | SD16 | — | 12 | 2 | no | — |
| 221 | D8 | — | 6 | 2 | no | — |
| 222 | D8 | — | 6 | 2 | no | — |
| 223 | D8 | — | 12 | 1 | no | — |
| 224 | D8 | — | 12 | 1 | no | — |
| 225 | M16 | — | 2 | 2 | no | — |
| 226 | M16 | — | 4 | 1 | no | — |
| 227 | C2×C8 | — | 4 | 1 | no | — |
| 228 | C2×C8 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 229 | C2×C8 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 230 | C2×C8 | #213 | 1 | 1 | yes | characteristic |
| 231 | C2×C8 | #213 | 1 | 1 | yes | characteristic |
| 232 | C2×C8 | #215 | 1 | 1 | yes | characteristic |
| 233 | C2×C8 | #215 | 1 | 1 | yes | characteristic |
| 234 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 235 | C2×C8 | C2×D12 | 1 | 1 | yes | characteristic |
| 236 | C2×C8 | C2×D12 | 1 | 1 | yes | characteristic |
| 237 | C2×C8 | C2×D12 | 1 | 1 | yes | characteristic |
| 238 | C2×C8 | #235 | 1 | 1 | yes | characteristic |
| 239 | C2×C8 | #235 | 1 | 1 | yes | characteristic |
| 240 | C4⋊C4 | — | 2 | 2 | no | — |
| 241 | C4⋊C4 | — | 12 | 2 | no | — |
| 242 | C22⋊C4 | — | 4 | 1 | no | — |
| 243 | C22⋊C4 | — | 12 | 4 | no | — |
| 244 | C42 | — | 2 | 1 | no | — |
| 245 | C42 | — | 4 | 1 | no | — |
| 246 | C42 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 247 | C42 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 248 | C16 | — | 12 | 2 | no | — |
| 249 | C22×C3 | — | 4 | 1 | no | — |
| 250 | C22×C3 | #418 | 1 | 1 | yes | characteristic |
| 251 | C22×C3 | #459 | 1 | 1 | yes | characteristic |
| 252 | C22×C3 | #460 | 1 | 1 | yes | characteristic |
| 253 | C2×D3 | — | 8 | 4 | no | — |
| 254 | C2×D3 | — | 8 | 2 | no | — |
| 255 | C3×C4 | — | 2 | 2 | no | — |
| 256 | C3×C4 | — | 4 | 1 | no | — |
| 257 | C3×C4 | — | 8 | 2 | no | — |
| 258 | C3×C4 | #390 | 1 | 1 | yes | characteristic |
| 259 | C3×C4 | C2×#102 | 1 | 1 | yes | characteristic |
| 260 | C3×C4 | #459 | 1 | 1 | yes | characteristic |
| 261 | C3×C4 | #460 | 1 | 1 | yes | characteristic |
| 262 | Dic3 | — | 8 | 2 | no | — |
| 263 | C23 | — | 12 | 2 | no | — |
| 264 | C23 | #949 | 1 | 1 | yes | characteristic |
| 265 | Q8 | — | 12 | 2 | no | — |
| 266 | Q8 | — | 12 | 2 | no | — |
| 267 | D4 | — | 12 | 2 | no | — |
| 268 | D4 | — | 12 | 2 | no | — |
| 269 | D4 | — | 24 | 4 | no | — |
| 270 | D4 | — | 24 | 2 | no | — |
| 271 | D4 | — | 24 | 2 | no | — |
| 272 | C2×C4 | — | 2 | 1 | no | — |
| 273 | C2×C4 | — | 2 | 1 | no | — |
| 274 | C2×C4 | — | 4 | 2 | no | — |
| 275 | C2×C4 | — | 4 | 2 | no | — |
| 276 | C2×C4 | — | 12 | 2 | no | — |
| 277 | C2×C4 | — | 24 | 2 | no | — |
| 278 | C2×C4 | — | 24 | 2 | no | — |
| 279 | C2×C4 | C2×(C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 280 | C2×C4 | #930 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | #930 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | C2×#213 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | C2×#215 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | #949 | 1 | 1 | yes | characteristic |
| 285 | C8 | — | 4 | 2 | no | — |
| 286 | C8 | #809 | 1 | 1 | yes | characteristic |
| 287 | C8 | #809 | 1 | 1 | yes | characteristic |
| 288 | C8 | #809 | 1 | 1 | yes | characteristic |
| 289 | C8 | #809 | 1 | 1 | yes | characteristic |
| 290 | C8 | C4×D12 | 1 | 1 | yes | characteristic |
| 291 | C8 | C4×D12 | 1 | 1 | yes | characteristic |
| 292 | C8 | #874 | 1 | 1 | yes | characteristic |
| 293 | C8 | #875 | 1 | 1 | yes | characteristic |
| 294 | C2×C3 | — | 2 | 1 | no | — |
| 295 | C2×C3 | — | 2 | 1 | no | — |
| 296 | C2×C3 | — | 2 | 1 | no | — |
| 297 | C2×C3 | #1982 | 1 | 1 | yes | characteristic |
| 298 | D3 | — | 16 | 2 | no | — |
| 299 | C22 | — | 4 | 1 | no | — |
| 300 | C22 | — | 24 | 4 | no | — |
| 301 | C22 | — | 24 | 2 | no | — |
| 302 | C22 | #5001 | 1 | 1 | yes | characteristic |
| 303 | C22 | #5015 | 1 | 1 | yes | characteristic |
| 304 | C22 | #5027 | 1 | 1 | yes | characteristic |
| 305 | C4 | — | 2 | 2 | no | — |
| 306 | C4 | — | 4 | 1 | no | — |
| 307 | C4 | — | 8 | 2 | no | — |
| 308 | C4 | — | 24 | 2 | no | — |
| 309 | C4 | #4939 | 1 | 1 | yes | characteristic |
| 310 | C4 | C2×#809 | 1 | 1 | yes | characteristic |
| 311 | C4 | #5015 | 1 | 1 | yes | characteristic |
| 312 | C4 | #5027 | 1 | 1 | yes | characteristic |
| 313 | C3 | #13259 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 314 | C2 | — | 2 | 1 | no | — |
| 315 | C2 | — | 2 | 1 | no | — |
| 316 | C2 | — | 2 | 1 | no | — |
| 317 | C2 | — | 48 | 2 | no | — |
| 318 | C2 | #69487 | 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