← #10713288
Subgroups — #10713288
Order 768.
326 conjugacy classes of subgroups.
The subgroup lattice has 268 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C3×C4⋊C16 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#4505 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#4504 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #69766 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C4×#818 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #67806 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #67806 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #13178 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C22×C3×C16 | C22 | 1 | 1 | yes | characteristic |
| 10 | C22×C3×C16 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#818 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#818 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#818 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#818 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×#817 | — | 2 | 1 | no | — |
| 17 | #4936 | — | 2 | 1 | no | — |
| 18 | C4×C4⋊Dic3 | — | 2 | 1 | no | — |
| 19 | C3×C4⋊C16 | C4 | 1 | 4 | yes | complemented |
| 20 | #4551 | — | 2 | 1 | no | — |
| 21 | #4505 | — | 2 | 2 | no | — |
| 22 | #4504 | — | 2 | 2 | no | — |
| 23 | C2×#367 | — | 3 | 1 | no | — |
| 24 | C2×#366 | — | 3 | 1 | no | — |
| 25 | C2×C4⋊C16 | D3 | 1 | 1 | yes | characteristic |
| 26 | #1846 | — | 3 | 1 | no | — |
| 27 | C4×#107 | — | 3 | 1 | no | — |
| 28 | #1380 | — | 3 | 1 | no | — |
| 29 | #1380 | — | 3 | 1 | no | — |
| 30 | C22×C3×C8 | D4 | 1 | 1 | yes | characteristic |
| 31 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 32 | C2×C3×C42 | D4 | 1 | 1 | yes | characteristic |
| 33 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 34 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 35 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 36 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 37 | C2×C4×Dic3 | — | 4 | 1 | no | — |
| 38 | C2×C3×C16 | — | 2 | 1 | no | — |
| 39 | C2×C3×C16 | — | 2 | 1 | no | — |
| 40 | C2×C3×C16 | — | 2 | 1 | no | — |
| 41 | C2×C3×C16 | — | 2 | 1 | no | — |
| 42 | C2×C3×C16 | C2×C4 | 1 | 2 | yes | — |
| 43 | C2×C3×C16 | C2×C4 | 1 | 2 | yes | — |
| 44 | C3×C4×C8 | C2×C4 | 1 | 2 | yes | — |
| 45 | C3×C4×C8 | Q8 | 1 | 2 | yes | — |
| 46 | C3⋊#95 | — | 4 | 1 | no | — |
| 47 | C3⋊#95 | — | 4 | 1 | no | — |
| 48 | C3⋊#95 | — | 4 | 1 | no | — |
| 49 | #818 | — | 2 | 1 | no | — |
| 50 | #818 | — | 2 | 1 | no | — |
| 51 | #818 | — | 2 | 1 | no | — |
| 52 | #818 | — | 2 | 1 | no | — |
| 53 | #818 | — | 2 | 1 | no | — |
| 54 | #818 | — | 2 | 1 | no | — |
| 55 | #818 | D4 | 1 | 2 | yes | — |
| 56 | #818 | Q8 | 1 | 2 | yes | — |
| 57 | #817 | — | 4 | 2 | no | — |
| 58 | C22×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 59 | C22×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 60 | C2×#107 | — | 3 | 1 | no | — |
| 61 | C2×#107 | — | 3 | 1 | no | — |
| 62 | C2×#107 | — | 3 | 1 | no | — |
| 63 | C2×#107 | — | 3 | 1 | no | — |
| 64 | C2×#106 | — | 6 | 1 | no | — |
| 65 | C2×C4×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 66 | #382 | — | 6 | 1 | no | — |
| 67 | C4×C4⋊C4 | — | 6 | 1 | no | — |
| 68 | #367 | — | 6 | 2 | no | — |
| 69 | #366 | — | 6 | 2 | no | — |
| 70 | C4⋊C16 | Dic3 | 1 | 4 | yes | complemented |
| 71 | #340 | — | 6 | 1 | no | — |
| 72 | C22×C3×C4 | D8 | 1 | 1 | yes | characteristic |
| 73 | C22×C3×C4 | D8 | 1 | 1 | yes | characteristic |
| 74 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C22×Dic3 | — | 4 | 1 | no | — |
| 76 | C22×Dic3 | — | 4 | 1 | no | — |
| 77 | C22×Dic3 | — | 4 | 1 | no | — |
| 78 | C22×Dic3 | — | 4 | 1 | no | — |
| 79 | C2×C3×C8 | — | 2 | 1 | no | — |
| 80 | C2×C3×C8 | — | 2 | 1 | no | — |
| 81 | C2×C3×C8 | C4⋊C4 | 1 | 2 | yes | — |
| 82 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C2×C3×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 85 | C2×C3×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 86 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 87 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 88 | C3×C42 | C4⋊C4 | 1 | 2 | yes | — |
| 89 | C3×C42 | Q16 | 1 | 2 | yes | — |
| 90 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 91 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 92 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 93 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 94 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 95 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 96 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 97 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 98 | C4×Dic3 | — | 8 | 2 | no | — |
| 99 | C3×C16 | — | 2 | 2 | no | — |
| 100 | C3×C16 | — | 2 | 2 | no | — |
| 101 | C22×C8 | D12 | 1 | 1 | yes | characteristic |
| 102 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 103 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 104 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 105 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 106 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 107 | C2×C42 | — | 12 | 1 | no | — |
| 108 | C2×C42 | D12 | 1 | 1 | yes | characteristic |
| 109 | C2×C16 | — | 2 | 1 | no | — |
| 110 | C2×C16 | — | 2 | 1 | no | — |
| 111 | C2×C16 | — | 2 | 1 | no | — |
| 112 | C2×C16 | — | 2 | 1 | no | — |
| 113 | C2×C16 | C2×Dic3 | 1 | 2 | yes | — |
| 114 | C2×C16 | C2×Dic3 | 1 | 2 | yes | — |
| 115 | #107 | — | 3 | 2 | no | — |
| 116 | #107 | — | 3 | 2 | no | — |
| 117 | #107 | — | 6 | 1 | no | — |
| 118 | #107 | — | 6 | 1 | no | — |
| 119 | #107 | — | 6 | 1 | no | — |
| 120 | #107 | — | 6 | 1 | no | — |
| 121 | #107 | — | 6 | 1 | no | — |
| 122 | #107 | — | 6 | 1 | no | — |
| 123 | #106 | — | 12 | 2 | no | — |
| 124 | C4×C8 | Dic6 | 1 | 2 | yes | — |
| 125 | C4×C8 | C2×Dic3 | 1 | 2 | yes | — |
| 126 | #95 | — | 12 | 1 | no | — |
| 127 | #95 | — | 12 | 1 | no | — |
| 128 | #95 | — | 12 | 1 | no | — |
| 129 | C23×C3 | C2×D8 | 1 | 1 | yes | characteristic |
| 130 | C2×C3×C4 | — | 2 | 1 | no | — |
| 131 | C2×C3×C4 | — | 2 | 1 | no | — |
| 132 | C2×C3×C4 | — | 2 | 1 | no | — |
| 133 | C2×C3×C4 | — | 2 | 1 | no | — |
| 134 | C2×C3×C4 | #107 | 1 | 2 | yes | — |
| 135 | C2×C3×C4 | #107 | 1 | 2 | yes | — |
| 136 | C2×C3×C4 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 137 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 138 | C2×C3×C4 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 139 | C2×C3×C4 | C2×Q16 | 1 | 1 | yes | characteristic |
| 140 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 141 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 142 | C2×Dic3 | — | 4 | 2 | no | — |
| 143 | C2×Dic3 | — | 4 | 2 | no | — |
| 144 | C2×Dic3 | — | 8 | 1 | no | — |
| 145 | C2×Dic3 | — | 8 | 1 | no | — |
| 146 | C2×Dic3 | — | 8 | 1 | no | — |
| 147 | C2×Dic3 | — | 8 | 1 | no | — |
| 148 | C2×Dic3 | — | 8 | 1 | no | — |
| 149 | C2×Dic3 | — | 8 | 1 | no | — |
| 150 | C2×Dic3 | — | 8 | 1 | no | — |
| 151 | C2×Dic3 | — | 8 | 1 | no | — |
| 152 | C2×Dic3 | — | 8 | 1 | no | — |
| 153 | C2×Dic3 | — | 8 | 1 | no | — |
| 154 | C3×C8 | — | 2 | 2 | no | — |
| 155 | C3×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 156 | C3×C8 | C4×Q8 | 1 | 1 | yes | characteristic |
| 157 | C3×C8 | #125 | 1 | 1 | yes | characteristic |
| 158 | C3×C8 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 159 | C22×C4 | — | 12 | 1 | no | — |
| 160 | C22×C4 | — | 12 | 1 | no | — |
| 161 | C22×C4 | — | 12 | 1 | no | — |
| 162 | C22×C4 | — | 12 | 1 | no | — |
| 163 | C22×C4 | D24 | 1 | 1 | yes | characteristic |
| 164 | C22×C4 | D24 | 1 | 1 | yes | characteristic |
| 165 | C22×C4 | C2×D12 | 1 | 1 | yes | characteristic |
| 166 | C2×C8 | — | 2 | 1 | no | — |
| 167 | C2×C8 | — | 2 | 1 | no | — |
| 168 | C2×C8 | C4⋊Dic3 | 1 | 2 | yes | — |
| 169 | C2×C8 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 170 | C2×C8 | D3×D4 | 1 | 1 | yes | characteristic |
| 171 | C2×C8 | #237 | 1 | 1 | yes | characteristic |
| 172 | C2×C8 | D3×Q8 | 1 | 1 | yes | characteristic |
| 173 | C2×C8 | #239 | 1 | 1 | yes | characteristic |
| 174 | C2×C8 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 175 | C4⋊C4 | — | 6 | 2 | no | — |
| 176 | C4⋊C4 | — | 6 | 2 | no | — |
| 177 | C4⋊C4 | — | 12 | 1 | no | — |
| 178 | C4⋊C4 | — | 12 | 1 | no | — |
| 179 | C4⋊C4 | — | 12 | 1 | no | — |
| 180 | C4⋊C4 | — | 12 | 1 | no | — |
| 181 | C4⋊C4 | — | 12 | 1 | no | — |
| 182 | C4⋊C4 | — | 12 | 1 | no | — |
| 183 | C42 | — | 24 | 2 | no | — |
| 184 | C42 | Dic12 | 1 | 2 | yes | — |
| 185 | C42 | C4⋊Dic3 | 1 | 2 | yes | — |
| 186 | C16 | — | 2 | 2 | no | — |
| 187 | C16 | — | 2 | 2 | no | — |
| 188 | C22×C3 | C2×#107 | 1 | 1 | yes | characteristic |
| 189 | C22×C3 | #459 | 1 | 1 | yes | characteristic |
| 190 | C22×C3 | #474 | 1 | 1 | yes | characteristic |
| 191 | C22×C3 | #508 | 1 | 1 | yes | characteristic |
| 192 | C22×C3 | #508 | 1 | 1 | yes | characteristic |
| 193 | C22×C3 | #509 | 1 | 1 | yes | characteristic |
| 194 | C22×C3 | #510 | 1 | 1 | yes | characteristic |
| 195 | C3×C4 | — | 2 | 2 | no | — |
| 196 | C3×C4 | — | 2 | 2 | no | — |
| 197 | C3×C4 | #389 | 1 | 1 | yes | characteristic |
| 198 | C3×C4 | #428 | 1 | 1 | yes | characteristic |
| 199 | C3×C4 | #462 | 1 | 1 | yes | characteristic |
| 200 | C3×C4 | #477 | 1 | 1 | yes | characteristic |
| 201 | Dic3 | — | 8 | 2 | no | — |
| 202 | Dic3 | — | 8 | 2 | no | — |
| 203 | Dic3 | — | 16 | 1 | no | — |
| 204 | Dic3 | — | 16 | 1 | no | — |
| 205 | C23 | C2×D24 | 1 | 1 | yes | characteristic |
| 206 | C2×C4 | — | 2 | 1 | no | — |
| 207 | C2×C4 | — | 2 | 1 | no | — |
| 208 | C2×C4 | — | 2 | 1 | no | — |
| 209 | C2×C4 | — | 2 | 1 | no | — |
| 210 | C2×C4 | — | 12 | 2 | no | — |
| 211 | C2×C4 | — | 12 | 2 | no | — |
| 212 | C2×C4 | — | 24 | 1 | no | — |
| 213 | C2×C4 | — | 24 | 1 | no | — |
| 214 | C2×C4 | — | 24 | 1 | no | — |
| 215 | C2×C4 | — | 24 | 1 | no | — |
| 216 | C2×C4 | — | 24 | 1 | no | — |
| 217 | C2×C4 | — | 24 | 1 | no | — |
| 218 | C2×C4 | — | 24 | 1 | no | — |
| 219 | C2×C4 | — | 24 | 1 | no | — |
| 220 | C2×C4 | — | 24 | 1 | no | — |
| 221 | C2×C4 | — | 24 | 1 | no | — |
| 222 | C2×C4 | #818 | 1 | 2 | yes | — |
| 223 | C2×C4 | #818 | 1 | 2 | yes | — |
| 224 | C2×C4 | #895 | 1 | 1 | yes | characteristic |
| 225 | C2×C4 | #897 | 1 | 1 | yes | characteristic |
| 226 | C2×C4 | C2×Dic12 | 1 | 1 | yes | characteristic |
| 227 | C2×C4 | #908 | 1 | 1 | yes | characteristic |
| 228 | C2×C4 | #909 | 1 | 1 | yes | characteristic |
| 229 | C2×C4 | C2×C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 230 | C8 | — | 2 | 2 | no | — |
| 231 | C8 | #888 | 1 | 1 | yes | characteristic |
| 232 | C8 | #890 | 1 | 1 | yes | characteristic |
| 233 | C8 | D4×Dic3 | 1 | 1 | yes | characteristic |
| 234 | C8 | Q8×Dic3 | 1 | 1 | yes | characteristic |
| 235 | C2×C3 | #1947 | 1 | 1 | yes | characteristic |
| 236 | C2×C3 | #2158 | 1 | 1 | yes | characteristic |
| 237 | C2×C3 | #2159 | 1 | 1 | yes | characteristic |
| 238 | C2×C3 | #2210 | 1 | 1 | yes | characteristic |
| 239 | C2×C3 | #2211 | 1 | 1 | yes | characteristic |
| 240 | C2×C3 | #2228 | 1 | 1 | yes | characteristic |
| 241 | C2×C3 | #2229 | 1 | 1 | yes | characteristic |
| 242 | C22 | #4843 | 1 | 1 | yes | characteristic |
| 243 | C22 | #4846 | 1 | 1 | yes | characteristic |
| 244 | C22 | #4904 | 1 | 1 | yes | characteristic |
| 245 | C22 | #4904 | 1 | 1 | yes | characteristic |
| 246 | C22 | #4908 | 1 | 1 | yes | characteristic |
| 247 | C22 | #4909 | 1 | 1 | yes | characteristic |
| 248 | C22 | C2×#818 | 1 | 1 | yes | characteristic |
| 249 | C4 | — | 2 | 2 | no | — |
| 250 | C4 | — | 2 | 2 | no | — |
| 251 | C4 | — | 24 | 2 | no | — |
| 252 | C4 | — | 24 | 2 | no | — |
| 253 | C4 | — | 48 | 1 | no | — |
| 254 | C4 | — | 48 | 1 | no | — |
| 255 | C4 | #4849 | 1 | 1 | yes | characteristic |
| 256 | C4 | #4850 | 1 | 1 | yes | characteristic |
| 257 | C4 | #4984 | 1 | 1 | yes | characteristic |
| 258 | C4 | #5121 | 1 | 1 | yes | characteristic |
| 259 | C3 | #13178 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 260 | C2 | #69288 | 1 | 1 | yes | characteristic |
| 261 | C2 | #69291 | 1 | 1 | yes | characteristic |
| 262 | C2 | #69294 | 1 | 1 | yes | characteristic |
| 263 | C2 | #69297 | 1 | 1 | yes | characteristic |
| 264 | C2 | #70453 | 1 | 1 | yes | characteristic |
| 265 | C2 | #71404 | 1 | 1 | yes | characteristic |
| 266 | C2 | #71428 | 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