← #10713601
Subgroups — #10713601
Order 768.
410 conjugacy classes of subgroups.
The subgroup lattice has 272 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#4904 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#2108 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #71450 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #71404 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #69897 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13243 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×#904 | C22 | 1 | 2 | yes | — |
| 8 | C2×C3×M32 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×C3×C16 | C22 | 1 | 1 | yes | characteristic |
| 10 | C3×#405 | C22 | 1 | 1 | yes | characteristic |
| 11 | #5139 | — | 2 | 2 | no | — |
| 12 | #5106 | C22 | 1 | 2 | yes | — |
| 13 | #5006 | — | 2 | 2 | no | — |
| 14 | C2×Dic24 | C4 | 1 | 1 | yes | characteristic, complemented |
| 15 | #4904 | — | 2 | 1 | no | — |
| 16 | #4904 | — | 2 | 1 | no | — |
| 17 | #4904 | — | 2 | 1 | no | — |
| 18 | #4904 | — | 2 | 1 | no | — |
| 19 | C2×#800 | C4 | 1 | 2 | yes | complemented |
| 20 | C2×D48 | C4 | 1 | 1 | yes | characteristic, complemented |
| 21 | C3×#346 | — | 2 | 1 | no | — |
| 22 | C3×C4×C16 | — | 2 | 1 | no | — |
| 23 | #4518 | — | 2 | 4 | no | — |
| 24 | #4505 | — | 2 | 1 | no | — |
| 25 | #4504 | — | 2 | 1 | no | — |
| 26 | C2×#508 | — | 3 | 1 | no | — |
| 27 | #2158 | — | 3 | 1 | no | — |
| 28 | #2145 | — | 3 | 2 | no | — |
| 29 | #2108 | D3 | 1 | 1 | yes | characteristic, complemented |
| 30 | #1976 | — | 3 | 2 | no | — |
| 31 | C2×#235 | — | 2 | 2 | no | — |
| 32 | C2×C3×M16 | D4 | 1 | 1 | yes | characteristic |
| 33 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 34 | C3×#117 | D4 | 1 | 1 | yes | characteristic |
| 35 | #926 | — | 2 | 2 | no | — |
| 36 | C2×Dic12 | C2×C4 | 1 | 2 | yes | — |
| 37 | #904 | — | 2 | 2 | no | — |
| 38 | #904 | — | 2 | 2 | no | — |
| 39 | #904 | — | 2 | 2 | no | — |
| 40 | #904 | — | 2 | 2 | no | — |
| 41 | C2×D24 | C2×C4 | 1 | 2 | yes | — |
| 42 | C2×#204 | — | 2 | 2 | no | — |
| 43 | C3×M32 | — | 2 | 1 | no | — |
| 44 | C3×M32 | D4 | 1 | 2 | yes | — |
| 45 | C2×C3×C16 | — | 2 | 1 | no | — |
| 46 | C2×C3×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C2×C3×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | C2×C3×C16 | D4 | 1 | 1 | yes | characteristic |
| 49 | C2×C3×C16 | D4 | 1 | 1 | yes | characteristic |
| 50 | C2×C3×C16 | D4 | 1 | 1 | yes | characteristic |
| 51 | C2×C3×C16 | D4 | 1 | 1 | yes | characteristic |
| 52 | C3×#97 | — | 2 | 1 | no | — |
| 53 | C3×C4×C8 | — | 2 | 1 | no | — |
| 54 | #825 | — | 4 | 4 | no | — |
| 55 | #818 | — | 2 | 2 | no | — |
| 56 | #817 | — | 2 | 2 | no | — |
| 57 | #806 | — | 4 | 4 | no | — |
| 58 | Dic24 | — | 2 | 1 | no | — |
| 59 | Dic24 | — | 2 | 1 | no | — |
| 60 | #800 | — | 2 | 2 | no | — |
| 61 | #800 | — | 2 | 2 | no | — |
| 62 | D48 | — | 2 | 1 | no | — |
| 63 | D48 | — | 2 | 1 | no | — |
| 64 | C2×#135 | — | 3 | 2 | no | — |
| 65 | #508 | — | 6 | 1 | no | — |
| 66 | #508 | — | 6 | 1 | no | — |
| 67 | #508 | — | 6 | 1 | no | — |
| 68 | #508 | — | 6 | 1 | no | — |
| 69 | C2×Q32 | — | 3 | 1 | no | — |
| 70 | C2×SD32 | — | 3 | 2 | no | — |
| 71 | C2×D16 | — | 3 | 1 | no | — |
| 72 | C2×M32 | C2×D3 | 1 | 1 | yes | characteristic |
| 73 | C22×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 74 | #427 | — | 3 | 2 | no | — |
| 75 | #421 | — | 6 | 2 | no | — |
| 76 | #410 | — | 6 | 2 | no | — |
| 77 | #405 | C2×D3 | 1 | 1 | yes | characteristic |
| 78 | #367 | — | 6 | 1 | no | — |
| 79 | #366 | — | 6 | 1 | no | — |
| 80 | #360 | — | 6 | 4 | no | — |
| 81 | #346 | — | 2 | 1 | no | — |
| 82 | C4×C16 | — | 2 | 1 | no | — |
| 83 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C2×#67 | — | 4 | 2 | no | — |
| 85 | #235 | — | 4 | 2 | no | — |
| 86 | #235 | — | 4 | 2 | no | — |
| 87 | #235 | — | 4 | 2 | no | — |
| 88 | #235 | — | 4 | 2 | no | — |
| 89 | C2×D12 | — | 2 | 2 | no | — |
| 90 | C2×C4×D3 | — | 4 | 2 | no | — |
| 91 | C2×Dic6 | — | 2 | 2 | no | — |
| 92 | C3×M16 | — | 2 | 1 | no | — |
| 93 | C3×M16 | D8 | 1 | 2 | yes | — |
| 94 | C2×C3×C8 | — | 2 | 1 | no | — |
| 95 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 96 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 100 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 101 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 102 | C3×C22⋊C4 | D8 | 1 | 2 | yes | — |
| 103 | C3×C42 | — | 2 | 1 | no | — |
| 104 | #217 | — | 4 | 2 | no | — |
| 105 | C4⋊Dic3 | — | 4 | 2 | no | — |
| 106 | C4×Dic3 | — | 4 | 2 | no | — |
| 107 | Dic12 | — | 2 | 2 | no | — |
| 108 | Dic12 | — | 2 | 2 | no | — |
| 109 | D24 | — | 2 | 2 | no | — |
| 110 | D24 | — | 2 | 2 | no | — |
| 111 | #204 | — | 4 | 2 | no | — |
| 112 | #204 | — | 4 | 2 | no | — |
| 113 | C3×C16 | — | 2 | 2 | no | — |
| 114 | C3×C16 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 115 | C3×C16 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 116 | C3×C16 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 117 | C3×C16 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 118 | C2×#41 | — | 6 | 2 | no | — |
| 119 | #135 | — | 6 | 2 | no | — |
| 120 | #135 | — | 6 | 2 | no | — |
| 121 | #135 | — | 6 | 2 | no | — |
| 122 | #135 | — | 6 | 2 | no | — |
| 123 | C2×Q16 | — | 3 | 2 | no | — |
| 124 | C2×SD16 | — | 6 | 2 | no | — |
| 125 | C2×D8 | — | 3 | 2 | no | — |
| 126 | C2×M16 | D12 | 1 | 1 | yes | characteristic |
| 127 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 128 | #117 | — | 6 | 2 | no | — |
| 129 | #117 | D12 | 1 | 1 | yes | characteristic |
| 130 | Q32 | — | 6 | 1 | no | — |
| 131 | Q32 | — | 6 | 1 | no | — |
| 132 | SD32 | — | 6 | 2 | no | — |
| 133 | SD32 | — | 6 | 2 | no | — |
| 134 | D16 | — | 6 | 1 | no | — |
| 135 | D16 | — | 6 | 1 | no | — |
| 136 | M32 | — | 2 | 1 | no | — |
| 137 | M32 | D12 | 1 | 2 | yes | — |
| 138 | C2×C16 | — | 2 | 1 | no | — |
| 139 | C2×C16 | C4×D3 | 1 | 1 | yes | characteristic |
| 140 | C2×C16 | C4×D3 | 1 | 1 | yes | characteristic |
| 141 | C2×C16 | D12 | 1 | 1 | yes | characteristic |
| 142 | C2×C16 | D12 | 1 | 1 | yes | characteristic |
| 143 | C2×C16 | #67 | 1 | 1 | yes | characteristic |
| 144 | C2×C16 | #67 | 1 | 1 | yes | characteristic |
| 145 | #107 | — | 6 | 2 | no | — |
| 146 | #106 | — | 6 | 2 | no | — |
| 147 | #104 | — | 12 | 4 | no | — |
| 148 | #99 | — | 12 | 4 | no | — |
| 149 | #97 | — | 2 | 1 | no | — |
| 150 | C4×C8 | — | 2 | 1 | no | — |
| 151 | C23×C3 | C2×D8 | 1 | 1 | yes | characteristic |
| 152 | C22×D3 | — | 4 | 2 | no | — |
| 153 | C2×C3×C4 | — | 2 | 2 | no | — |
| 154 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 155 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 156 | C2×C3×C4 | #127 | 1 | 1 | yes | characteristic |
| 157 | C2×C3×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 158 | C2×C3×C4 | #135 | 1 | 1 | yes | characteristic |
| 159 | C2×C3×C4 | #135 | 1 | 1 | yes | characteristic |
| 160 | #67 | — | 8 | 2 | no | — |
| 161 | #67 | — | 8 | 2 | no | — |
| 162 | C2×Dic3 | — | 4 | 2 | no | — |
| 163 | C2×Dic3 | — | 8 | 2 | no | — |
| 164 | D12 | — | 4 | 2 | no | — |
| 165 | D12 | — | 4 | 2 | no | — |
| 166 | C4×D3 | — | 8 | 2 | no | — |
| 167 | C4×D3 | — | 8 | 2 | no | — |
| 168 | Dic6 | — | 4 | 2 | no | — |
| 169 | Dic6 | — | 4 | 2 | no | — |
| 170 | C3×C8 | — | 2 | 2 | no | — |
| 171 | C3×C8 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 172 | C3×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 173 | C3×C8 | #121 | 1 | 1 | yes | characteristic |
| 174 | C3×C8 | #124 | 1 | 1 | yes | characteristic |
| 175 | #41 | — | 12 | 2 | no | — |
| 176 | #41 | — | 12 | 2 | no | — |
| 177 | #41 | — | 12 | 2 | no | — |
| 178 | #41 | — | 12 | 2 | no | — |
| 179 | C2×Q8 | — | 6 | 2 | no | — |
| 180 | C2×D4 | — | 6 | 2 | no | — |
| 181 | C2×D4 | — | 12 | 2 | no | — |
| 182 | C22×C4 | — | 12 | 2 | no | — |
| 183 | C22×C4 | C2×D12 | 1 | 1 | yes | characteristic |
| 184 | Q16 | — | 6 | 2 | no | — |
| 185 | Q16 | — | 6 | 2 | no | — |
| 186 | SD16 | — | 12 | 2 | no | — |
| 187 | SD16 | — | 12 | 2 | no | — |
| 188 | D8 | — | 6 | 2 | no | — |
| 189 | D8 | — | 6 | 2 | no | — |
| 190 | M16 | — | 2 | 1 | no | — |
| 191 | M16 | D24 | 1 | 2 | yes | — |
| 192 | C2×C8 | — | 2 | 1 | no | — |
| 193 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 194 | C2×C8 | C2×D12 | 1 | 1 | yes | characteristic |
| 195 | C2×C8 | C2×D12 | 1 | 1 | yes | characteristic |
| 196 | C2×C8 | #235 | 1 | 1 | yes | characteristic |
| 197 | C2×C8 | #235 | 1 | 1 | yes | characteristic |
| 198 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 199 | C4⋊C4 | — | 2 | 1 | no | — |
| 200 | C4⋊C4 | — | 12 | 2 | no | — |
| 201 | C22⋊C4 | — | 12 | 2 | no | — |
| 202 | C22⋊C4 | D24 | 1 | 2 | yes | — |
| 203 | C42 | — | 2 | 1 | no | — |
| 204 | C42 | — | 12 | 2 | no | — |
| 205 | C16 | — | 2 | 2 | no | — |
| 206 | C16 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 207 | C16 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 208 | C16 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 209 | C16 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 210 | C22×C3 | — | 4 | 1 | no | — |
| 211 | C22×C3 | C4×D8 | 1 | 1 | yes | characteristic |
| 212 | C22×C3 | #466 | 1 | 1 | yes | characteristic |
| 213 | C22×C3 | #493 | 1 | 1 | yes | characteristic |
| 214 | C2×D3 | — | 8 | 2 | no | — |
| 215 | C2×D3 | — | 16 | 2 | no | — |
| 216 | C3×C4 | — | 4 | 2 | no | — |
| 217 | C3×C4 | #390 | 1 | 1 | yes | characteristic |
| 218 | C3×C4 | C4×D8 | 1 | 1 | yes | characteristic |
| 219 | C3×C4 | #466 | 1 | 1 | yes | characteristic |
| 220 | C3×C4 | #495 | 1 | 1 | yes | characteristic |
| 221 | Dic3 | — | 8 | 2 | no | — |
| 222 | Dic3 | — | 16 | 2 | no | — |
| 223 | C23 | — | 12 | 2 | no | — |
| 224 | C23 | C2×D24 | 1 | 1 | yes | characteristic |
| 225 | Q8 | — | 12 | 2 | no | — |
| 226 | Q8 | — | 12 | 2 | no | — |
| 227 | D4 | — | 12 | 2 | no | — |
| 228 | D4 | — | 12 | 2 | no | — |
| 229 | D4 | — | 24 | 2 | no | — |
| 230 | D4 | — | 24 | 2 | no | — |
| 231 | C2×C4 | — | 2 | 2 | no | — |
| 232 | C2×C4 | — | 12 | 2 | no | — |
| 233 | C2×C4 | — | 24 | 2 | no | — |
| 234 | C2×C4 | — | 24 | 2 | no | — |
| 235 | C2×C4 | — | 24 | 2 | no | — |
| 236 | C2×C4 | C4×D12 | 1 | 1 | yes | characteristic |
| 237 | C2×C4 | #874 | 1 | 1 | yes | characteristic |
| 238 | C2×C4 | C2×D24 | 1 | 1 | yes | characteristic |
| 239 | C2×C4 | #904 | 1 | 1 | yes | characteristic |
| 240 | C2×C4 | #904 | 1 | 1 | yes | characteristic |
| 241 | C2×C4 | #930 | 1 | 1 | yes | characteristic |
| 242 | C8 | — | 2 | 2 | no | — |
| 243 | C8 | C4×D12 | 1 | 1 | yes | characteristic |
| 244 | C8 | #875 | 1 | 1 | yes | characteristic |
| 245 | C8 | C2×(C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 246 | C8 | #930 | 1 | 1 | yes | characteristic |
| 247 | C2×C3 | — | 2 | 1 | no | — |
| 248 | C2×C3 | — | 2 | 1 | no | — |
| 249 | C2×C3 | — | 2 | 1 | no | — |
| 250 | C2×C3 | #1970 | 1 | 1 | yes | characteristic |
| 251 | D3 | — | 16 | 2 | no | — |
| 252 | C22 | — | 4 | 1 | no | — |
| 253 | C22 | — | 24 | 2 | no | — |
| 254 | C22 | — | 48 | 2 | no | — |
| 255 | C22 | C4×D24 | 1 | 1 | yes | characteristic |
| 256 | C22 | #4695 | 1 | 1 | yes | characteristic |
| 257 | C22 | #5115 | 1 | 1 | yes | characteristic |
| 258 | C4 | — | 4 | 2 | no | — |
| 259 | C4 | — | 24 | 2 | no | — |
| 260 | C4 | — | 48 | 2 | no | — |
| 261 | C4 | C4×D24 | 1 | 1 | yes | characteristic |
| 262 | C4 | #4696 | 1 | 1 | yes | characteristic |
| 263 | C4 | #4939 | 1 | 1 | yes | characteristic |
| 264 | C4 | #5115 | 1 | 1 | yes | characteristic |
| 265 | C3 | #13243 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 266 | C2 | — | 2 | 1 | no | — |
| 267 | C2 | — | 2 | 1 | no | — |
| 268 | C2 | — | 2 | 1 | no | — |
| 269 | C2 | — | 48 | 2 | no | — |
| 270 | C2 | #69776 | 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