← #10950793
Subgroups — #10950793
Order 768.
532 conjugacy classes of subgroups.
The subgroup lattice has 242 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #78569 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C3×#2108 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #71443 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #71413 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #69080 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #69078 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #69060 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C4×D48 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #69045 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #33740 | — | 3 | 1 | no | maximal, Sylow |
| 11 | #5535 | — | 2 | 2 | no | — |
| 12 | C2×C3×M32 | C22 | 1 | 1 | yes | characteristic |
| 13 | C22×C3×C16 | C22 | 1 | 1 | yes | characteristic |
| 14 | C3×#405 | C22 | 1 | 1 | yes | characteristic |
| 15 | #5135 | — | 2 | 2 | no | — |
| 16 | #5115 | C22 | 1 | 4 | yes | — |
| 17 | C2×D48 | C22 | 1 | 2 | yes | — |
| 18 | C2×D48 | C22 | 1 | 2 | yes | — |
| 19 | #4713 | — | 2 | 2 | no | — |
| 20 | #4711 | — | 2 | 2 | no | — |
| 21 | #4695 | C22 | 1 | 2 | yes | — |
| 22 | C4×D24 | C22 | 1 | 4 | yes | — |
| 23 | #4683 | C22 | 1 | 2 | yes | — |
| 24 | C3×#346 | C22 | 1 | 1 | yes | characteristic |
| 25 | C3×#346 | C22 | 1 | 1 | yes | characteristic |
| 26 | C3×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 27 | C3×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 28 | #4509 | C22 | 1 | 4 | yes | — |
| 29 | #4509 | C22 | 1 | 4 | yes | — |
| 30 | #4505 | C22 | 1 | 2 | yes | — |
| 31 | #4504 | C22 | 1 | 2 | yes | — |
| 32 | #3176 | — | 3 | 2 | no | — |
| 33 | #2253 | — | 3 | 1 | no | — |
| 34 | #2246 | — | 3 | 1 | no | — |
| 35 | #2244 | — | 3 | 2 | no | — |
| 36 | #2218 | — | 3 | 2 | no | — |
| 37 | #2215 | — | 3 | 2 | no | — |
| 38 | #2177 | — | 3 | 2 | no | — |
| 39 | C4×D16 | — | 3 | 2 | no | — |
| 40 | #2108 | D3 | 1 | 1 | yes | characteristic, complemented |
| 41 | C2×C3×M16 | D4 | 1 | 1 | yes | characteristic |
| 42 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 43 | C3×#117 | D4 | 1 | 1 | yes | characteristic |
| 44 | #930 | — | 2 | 4 | no | — |
| 45 | C2×D24 | — | 2 | 2 | no | — |
| 46 | C2×D24 | C23 | 1 | 4 | yes | — |
| 47 | #894 | — | 4 | 4 | no | — |
| 48 | #884 | — | 4 | 4 | no | — |
| 49 | #874 | — | 2 | 2 | no | — |
| 50 | C4×D12 | — | 2 | 4 | no | — |
| 51 | #870 | — | 2 | 2 | no | — |
| 52 | C3×M32 | — | 2 | 2 | no | — |
| 53 | C2×C3×C16 | — | 2 | 2 | no | — |
| 54 | C2×C3×C16 | C23 | 1 | 1 | yes | characteristic |
| 55 | C2×C3×C16 | C23 | 1 | 1 | yes | characteristic |
| 56 | C2×C3×C16 | C23 | 1 | 1 | yes | characteristic |
| 57 | C2×C3×C16 | C23 | 1 | 1 | yes | characteristic |
| 58 | C3×#97 | D4 | 1 | 1 | yes | characteristic |
| 59 | C3×#97 | D4 | 1 | 1 | yes | characteristic |
| 60 | C3×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 61 | C3×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 62 | #821 | — | 2 | 4 | no | — |
| 63 | #821 | — | 2 | 4 | no | — |
| 64 | #818 | C23 | 1 | 4 | yes | — |
| 65 | #817 | — | 2 | 2 | no | — |
| 66 | D48 | — | 2 | 4 | no | — |
| 67 | D48 | — | 2 | 4 | no | — |
| 68 | #540 | — | 6 | 2 | no | — |
| 69 | C2×D16 | — | 3 | 2 | no | — |
| 70 | C2×D16 | — | 3 | 2 | no | — |
| 71 | C2×M32 | C2×D3 | 1 | 1 | yes | characteristic |
| 72 | C22×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 73 | #499 | — | 3 | 2 | no | — |
| 74 | #493 | — | 3 | 2 | no | — |
| 75 | #490 | — | 6 | 2 | no | — |
| 76 | #469 | — | 6 | 2 | no | — |
| 77 | #466 | — | 3 | 4 | no | — |
| 78 | #442 | — | 6 | 2 | no | — |
| 79 | C4×D8 | — | 3 | 4 | no | — |
| 80 | #405 | C2×D3 | 1 | 1 | yes | characteristic |
| 81 | #367 | — | 3 | 2 | no | — |
| 82 | #366 | — | 3 | 2 | no | — |
| 83 | #357 | — | 3 | 4 | no | — |
| 84 | #357 | — | 3 | 4 | no | — |
| 85 | #346 | C2×D3 | 1 | 1 | yes | characteristic |
| 86 | #346 | C2×D3 | 1 | 1 | yes | characteristic |
| 87 | C4×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 88 | C4×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 89 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C2×#67 | — | 4 | 4 | no | — |
| 91 | C2×D12 | — | 2 | 4 | no | — |
| 92 | C2×D12 | — | 4 | 4 | no | — |
| 93 | C2×C4×D3 | — | 4 | 4 | no | — |
| 94 | C3×M16 | — | 2 | 2 | no | — |
| 95 | C2×C3×C8 | — | 2 | 1 | no | — |
| 96 | C2×C3×C8 | — | 2 | 1 | no | — |
| 97 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C2×C3×C8 | C24 | 1 | 1 | yes | characteristic |
| 101 | C3×C4⋊C4 | D8 | 1 | 2 | yes | — |
| 102 | C3×C22⋊C4 | D8 | 1 | 2 | yes | — |
| 103 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 105 | (C2×D3)⋊C4 | — | 4 | 4 | no | — |
| 106 | (C2×D3)⋊C4 | — | 4 | 4 | no | — |
| 107 | C4⋊Dic3 | — | 2 | 4 | no | — |
| 108 | Dic3⋊C4 | — | 4 | 4 | no | — |
| 109 | D24 | — | 2 | 4 | no | — |
| 110 | D24 | — | 2 | 4 | no | — |
| 111 | D24 | — | 4 | 4 | no | — |
| 112 | C3×C16 | — | 2 | 1 | no | — |
| 113 | C3×C16 | — | 2 | 1 | no | — |
| 114 | C3×C16 | #41 | 1 | 2 | yes | — |
| 115 | C3×C16 | #41 | 1 | 2 | yes | — |
| 116 | C2×D8 | — | 3 | 4 | no | — |
| 117 | C2×D8 | — | 6 | 2 | no | — |
| 118 | C2×M16 | D12 | 1 | 1 | yes | characteristic |
| 119 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 120 | #127 | — | 6 | 2 | no | — |
| 121 | #125 | — | 6 | 2 | no | — |
| 122 | #123 | — | 12 | 4 | no | — |
| 123 | #121 | — | 6 | 4 | no | — |
| 124 | #121 | — | 12 | 4 | no | — |
| 125 | C4×D4 | — | 6 | 4 | no | — |
| 126 | #117 | D12 | 1 | 1 | yes | characteristic |
| 127 | D16 | — | 6 | 4 | no | — |
| 128 | D16 | — | 6 | 4 | no | — |
| 129 | M32 | — | 2 | 2 | no | — |
| 130 | C2×C16 | — | 2 | 2 | no | — |
| 131 | C2×C16 | C22×D3 | 1 | 1 | yes | characteristic |
| 132 | C2×C16 | C22×D3 | 1 | 1 | yes | characteristic |
| 133 | C2×C16 | C22×D3 | 1 | 1 | yes | characteristic |
| 134 | C2×C16 | C22×D3 | 1 | 1 | yes | characteristic |
| 135 | #107 | — | 3 | 4 | no | — |
| 136 | #106 | — | 6 | 2 | no | — |
| 137 | #102 | — | 6 | 4 | no | — |
| 138 | #102 | — | 6 | 4 | no | — |
| 139 | #97 | D12 | 1 | 1 | yes | characteristic |
| 140 | #97 | D12 | 1 | 1 | yes | characteristic |
| 141 | C4×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 142 | C4×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 143 | C23×C3 | C2×D8 | 1 | 1 | yes | characteristic |
| 144 | C22×D3 | — | 4 | 4 | no | — |
| 145 | C2×C3×C4 | — | 2 | 1 | no | — |
| 146 | C2×C3×C4 | — | 2 | 1 | no | — |
| 147 | C2×C3×C4 | C2×D8 | 1 | 2 | yes | — |
| 148 | C2×C3×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 149 | C2×C3×C4 | C2×D8 | 1 | 2 | yes | — |
| 150 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 151 | #67 | — | 16 | 4 | no | — |
| 152 | C2×Dic3 | — | 4 | 4 | no | — |
| 153 | D12 | — | 4 | 4 | no | — |
| 154 | D12 | — | 4 | 4 | no | — |
| 155 | D12 | — | 16 | 4 | no | — |
| 156 | C4×D3 | — | 16 | 4 | no | — |
| 157 | C3×C8 | — | 2 | 1 | no | — |
| 158 | C3×C8 | — | 2 | 1 | no | — |
| 159 | C3×C8 | #135 | 1 | 2 | yes | — |
| 160 | C3×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 161 | C3×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 162 | C2×D4 | — | 6 | 4 | no | — |
| 163 | C2×D4 | — | 12 | 4 | no | — |
| 164 | C2×D4 | — | 12 | 4 | no | — |
| 165 | C22×C4 | — | 12 | 4 | no | — |
| 166 | C22×C4 | C2×D12 | 1 | 1 | yes | characteristic |
| 167 | D8 | — | 6 | 4 | no | — |
| 168 | D8 | — | 6 | 4 | no | — |
| 169 | D8 | — | 12 | 4 | no | — |
| 170 | M16 | — | 2 | 2 | no | — |
| 171 | C2×C8 | — | 2 | 1 | no | — |
| 172 | C2×C8 | — | 2 | 1 | no | — |
| 173 | C2×C8 | C2×D12 | 1 | 1 | yes | characteristic |
| 174 | C2×C8 | C2×D12 | 1 | 1 | yes | characteristic |
| 175 | C2×C8 | C2×D12 | 1 | 1 | yes | characteristic |
| 176 | C2×C8 | C23×D3 | 1 | 1 | yes | characteristic |
| 177 | C4⋊C4 | — | 6 | 4 | no | — |
| 178 | C4⋊C4 | — | 12 | 4 | no | — |
| 179 | C4⋊C4 | D24 | 1 | 2 | yes | — |
| 180 | C22⋊C4 | — | 12 | 4 | no | — |
| 181 | C22⋊C4 | — | 12 | 4 | no | — |
| 182 | C22⋊C4 | D24 | 1 | 2 | yes | — |
| 183 | C42 | C2×D12 | 1 | 1 | yes | characteristic |
| 184 | C42 | C2×D12 | 1 | 1 | yes | characteristic |
| 185 | C16 | — | 2 | 1 | no | — |
| 186 | C16 | — | 2 | 1 | no | — |
| 187 | C16 | #235 | 1 | 2 | yes | — |
| 188 | C16 | #235 | 1 | 2 | yes | — |
| 189 | C22×C3 | — | 2 | 2 | no | — |
| 190 | C22×C3 | — | 2 | 1 | no | — |
| 191 | C22×C3 | C22×D8 | 1 | 1 | yes | characteristic |
| 192 | C2×D3 | — | 8 | 4 | no | — |
| 193 | C2×D3 | — | 8 | 4 | no | — |
| 194 | C2×D3 | — | 8 | 4 | no | — |
| 195 | C3×C4 | — | 2 | 2 | no | — |
| 196 | C3×C4 | — | 2 | 2 | no | — |
| 197 | C3×C4 | — | 2 | 1 | no | — |
| 198 | C3×C4 | #532 | 1 | 1 | yes | characteristic |
| 199 | C3×C4 | C2×#135 | 1 | 1 | yes | characteristic |
| 200 | Dic3 | — | 8 | 4 | no | — |
| 201 | C23 | — | 12 | 4 | no | — |
| 202 | C23 | C2×D24 | 1 | 1 | yes | characteristic |
| 203 | D4 | — | 12 | 4 | no | — |
| 204 | D4 | — | 12 | 4 | no | — |
| 205 | D4 | — | 48 | 4 | no | — |
| 206 | D4 | — | 48 | 4 | no | — |
| 207 | C2×C4 | — | 2 | 1 | no | — |
| 208 | C2×C4 | — | 2 | 1 | no | — |
| 209 | C2×C4 | — | 12 | 4 | no | — |
| 210 | C2×C4 | — | 48 | 4 | no | — |
| 211 | C2×C4 | C2×D24 | 1 | 2 | yes | — |
| 212 | C2×C4 | C2×D24 | 1 | 2 | yes | — |
| 213 | C2×C4 | C2×D24 | 1 | 1 | yes | characteristic |
| 214 | C2×C4 | C22×D12 | 1 | 1 | yes | characteristic |
| 215 | C8 | — | 2 | 1 | no | — |
| 216 | C8 | — | 2 | 1 | no | — |
| 217 | C8 | #904 | 1 | 2 | yes | — |
| 218 | C8 | C2×#235 | 1 | 1 | yes | characteristic |
| 219 | C8 | C2×#235 | 1 | 1 | yes | characteristic |
| 220 | C2×C3 | — | 4 | 1 | no | — |
| 221 | C2×C3 | #3169 | 1 | 1 | yes | characteristic |
| 222 | C2×C3 | #3415 | 1 | 2 | yes | — |
| 223 | D3 | — | 16 | 4 | no | — |
| 224 | C22 | — | 2 | 2 | no | — |
| 225 | C22 | — | 2 | 1 | no | — |
| 226 | C22 | — | 24 | 4 | no | — |
| 227 | C22 | — | 24 | 4 | no | — |
| 228 | C22 | — | 24 | 4 | no | — |
| 229 | C22 | C22×D24 | 1 | 1 | yes | characteristic |
| 230 | C4 | — | 2 | 2 | no | — |
| 231 | C4 | — | 2 | 2 | no | — |
| 232 | C4 | — | 2 | 1 | no | — |
| 233 | C4 | — | 24 | 4 | no | — |
| 234 | C4 | #5477 | 1 | 1 | yes | characteristic |
| 235 | C4 | C2×#904 | 1 | 1 | yes | characteristic |
| 236 | C3 | #33740 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 237 | C2 | — | 4 | 1 | no | — |
| 238 | C2 | — | 48 | 4 | no | — |
| 239 | C2 | #78470 | 1 | 1 | yes | characteristic |
| 240 | C2 | #82013 | 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