← #10713596
Subgroups — #10713596
Order 768.
418 conjugacy classes of subgroups.
The subgroup lattice has 250 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 | C2×C3×#346 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#4510 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C2×#4506 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #69795 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13242 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×#904 | C22 | 1 | 2 | yes | — |
| 8 | C22×C3×C16 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×C3×C16 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C3×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 11 | #5110 | — | 2 | 2 | no | — |
| 12 | C2×#819 | C22 | 1 | 2 | yes | — |
| 13 | C2×#805 | — | 2 | 2 | no | — |
| 14 | C2×Dic24 | C4 | 1 | 1 | yes | characteristic, complemented |
| 15 | #4904 | — | 2 | 2 | no | — |
| 16 | #4904 | — | 2 | 2 | no | — |
| 17 | C2×#800 | C4 | 1 | 2 | yes | complemented |
| 18 | C2×D48 | C4 | 1 | 1 | yes | characteristic, complemented |
| 19 | C3×#346 | — | 2 | 2 | no | — |
| 20 | #4510 | — | 2 | 4 | no | — |
| 21 | #4506 | — | 2 | 2 | no | — |
| 22 | C2×#508 | — | 3 | 1 | no | — |
| 23 | C2×#368 | — | 3 | 1 | no | — |
| 24 | C2×#359 | — | 3 | 2 | no | — |
| 25 | C2×#346 | D3 | 1 | 1 | yes | characteristic, complemented |
| 26 | #1974 | — | 3 | 2 | no | — |
| 27 | C2×#235 | — | 2 | 2 | no | — |
| 28 | C22×C3×C8 | D4 | 1 | 1 | yes | characteristic |
| 29 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 30 | C2×C3×C42 | D4 | 1 | 1 | yes | characteristic |
| 31 | C2×#208 | — | 2 | 2 | no | — |
| 32 | C2×Dic12 | C2×C4 | 1 | 2 | yes | — |
| 33 | #904 | — | 2 | 2 | no | — |
| 34 | #904 | — | 2 | 2 | no | — |
| 35 | #904 | — | 2 | 2 | no | — |
| 36 | #904 | — | 2 | 2 | no | — |
| 37 | C2×D24 | C2×C4 | 1 | 2 | yes | — |
| 38 | C2×#204 | — | 2 | 2 | no | — |
| 39 | C2×C3×C16 | — | 2 | 1 | no | — |
| 40 | C2×C3×C16 | — | 2 | 1 | no | — |
| 41 | C2×C3×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C2×C3×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | C2×C3×C16 | D4 | 1 | 2 | yes | — |
| 44 | C2×C3×C16 | D4 | 1 | 2 | yes | — |
| 45 | C2×C3×C16 | D4 | 1 | 2 | yes | — |
| 46 | C3×C4×C8 | — | 2 | 1 | no | — |
| 47 | C3×C4×C8 | — | 2 | 1 | no | — |
| 48 | (C2×D3)⋊C8 | — | 4 | 4 | no | — |
| 49 | #819 | — | 2 | 2 | no | — |
| 50 | #819 | — | 2 | 2 | no | — |
| 51 | #805 | — | 4 | 4 | no | — |
| 52 | Dic24 | — | 2 | 2 | no | — |
| 53 | #800 | — | 2 | 4 | no | — |
| 54 | D48 | — | 2 | 2 | no | — |
| 55 | C2×#135 | — | 3 | 2 | no | — |
| 56 | #508 | — | 6 | 2 | no | — |
| 57 | #508 | — | 6 | 2 | no | — |
| 58 | C2×Q32 | — | 3 | 1 | no | — |
| 59 | C2×SD32 | — | 3 | 2 | no | — |
| 60 | C2×D16 | — | 3 | 1 | no | — |
| 61 | C22×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 62 | C22×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 63 | C2×#108 | — | 3 | 2 | no | — |
| 64 | C2×#104 | — | 6 | 2 | no | — |
| 65 | #408 | — | 6 | 2 | no | — |
| 66 | C2×C4×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 67 | #368 | — | 6 | 2 | no | — |
| 68 | #359 | — | 6 | 4 | no | — |
| 69 | #346 | — | 2 | 2 | no | — |
| 70 | C22×C3×C4 | — | 2 | 1 | no | — |
| 71 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 72 | C2×#67 | — | 4 | 2 | no | — |
| 73 | #235 | — | 4 | 2 | no | — |
| 74 | #235 | — | 4 | 2 | no | — |
| 75 | #235 | — | 4 | 2 | no | — |
| 76 | #235 | — | 4 | 2 | no | — |
| 77 | C2×D12 | — | 2 | 2 | no | — |
| 78 | C2×C4×D3 | — | 4 | 2 | no | — |
| 79 | C2×Dic6 | — | 2 | 2 | no | — |
| 80 | C2×C3×C8 | — | 2 | 1 | no | — |
| 81 | C2×C3×C8 | — | 2 | 1 | no | — |
| 82 | C2×C3×C8 | D8 | 1 | 2 | yes | — |
| 83 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 84 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 88 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 89 | C3×C42 | — | 2 | 1 | no | — |
| 90 | C3×C42 | D8 | 1 | 2 | yes | — |
| 91 | #208 | — | 4 | 2 | no | — |
| 92 | #208 | — | 4 | 2 | no | — |
| 93 | C2×C3⋊C8 | — | 4 | 2 | no | — |
| 94 | Dic12 | — | 2 | 2 | no | — |
| 95 | Dic12 | — | 2 | 2 | no | — |
| 96 | D24 | — | 2 | 2 | no | — |
| 97 | D24 | — | 2 | 2 | no | — |
| 98 | #204 | — | 4 | 2 | no | — |
| 99 | #204 | — | 4 | 2 | no | — |
| 100 | C3×C16 | — | 2 | 2 | no | — |
| 101 | C3×C16 | C22⋊C4 | 1 | 2 | yes | — |
| 102 | C3×C16 | C22⋊C4 | 1 | 2 | yes | — |
| 103 | C2×#41 | — | 6 | 2 | no | — |
| 104 | #135 | — | 6 | 2 | no | — |
| 105 | #135 | — | 6 | 2 | no | — |
| 106 | #135 | — | 6 | 2 | no | — |
| 107 | #135 | — | 6 | 2 | no | — |
| 108 | C2×Q16 | — | 3 | 2 | no | — |
| 109 | C2×SD16 | — | 6 | 2 | no | — |
| 110 | C2×D8 | — | 3 | 2 | no | — |
| 111 | C2×M16 | — | 6 | 2 | no | — |
| 112 | C22×C8 | D12 | 1 | 1 | yes | characteristic |
| 113 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 114 | C2×C42 | D12 | 1 | 1 | yes | characteristic |
| 115 | Q32 | — | 6 | 2 | no | — |
| 116 | SD32 | — | 6 | 4 | no | — |
| 117 | D16 | — | 6 | 2 | no | — |
| 118 | C2×C16 | — | 2 | 1 | no | — |
| 119 | C2×C16 | — | 2 | 1 | no | — |
| 120 | C2×C16 | C4×D3 | 1 | 1 | yes | characteristic |
| 121 | C2×C16 | C4×D3 | 1 | 1 | yes | characteristic |
| 122 | C2×C16 | D12 | 1 | 2 | yes | — |
| 123 | C2×C16 | D12 | 1 | 2 | yes | — |
| 124 | C2×C16 | #67 | 1 | 2 | yes | — |
| 125 | #108 | — | 6 | 2 | no | — |
| 126 | #108 | — | 6 | 2 | no | — |
| 127 | #104 | — | 12 | 4 | no | — |
| 128 | C22⋊C8 | — | 12 | 4 | no | — |
| 129 | C4×C8 | — | 2 | 1 | no | — |
| 130 | C4×C8 | — | 2 | 1 | no | — |
| 131 | C23×C3 | #135 | 1 | 1 | yes | characteristic |
| 132 | C22×D3 | — | 4 | 2 | no | — |
| 133 | C2×C3×C4 | — | 2 | 2 | no | — |
| 134 | C2×C3×C4 | — | 4 | 2 | no | — |
| 135 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 136 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 137 | C2×C3×C4 | #127 | 1 | 1 | yes | characteristic |
| 138 | C2×C3×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 139 | C2×C3×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 140 | C2×C3×C4 | #135 | 1 | 1 | yes | characteristic |
| 141 | #67 | — | 8 | 2 | no | — |
| 142 | #67 | — | 8 | 2 | no | — |
| 143 | C2×Dic3 | — | 4 | 2 | no | — |
| 144 | D12 | — | 4 | 2 | no | — |
| 145 | D12 | — | 4 | 2 | no | — |
| 146 | C4×D3 | — | 8 | 2 | no | — |
| 147 | C4×D3 | — | 8 | 2 | no | — |
| 148 | Dic6 | — | 4 | 2 | no | — |
| 149 | Dic6 | — | 4 | 2 | no | — |
| 150 | C3×C8 | — | 2 | 2 | no | — |
| 151 | C3×C8 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 152 | C3×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 153 | C3×C8 | #121 | 1 | 1 | yes | characteristic |
| 154 | C3×C8 | #124 | 1 | 1 | yes | characteristic |
| 155 | C3⋊C8 | — | 8 | 2 | no | — |
| 156 | #41 | — | 12 | 2 | no | — |
| 157 | #41 | — | 12 | 2 | no | — |
| 158 | #41 | — | 12 | 2 | no | — |
| 159 | #41 | — | 12 | 2 | no | — |
| 160 | C2×Q8 | — | 6 | 2 | no | — |
| 161 | C2×D4 | — | 6 | 2 | no | — |
| 162 | C2×D4 | — | 12 | 2 | no | — |
| 163 | C22×C4 | — | 2 | 1 | no | — |
| 164 | C22×C4 | — | 12 | 2 | no | — |
| 165 | C22×C4 | C2×D12 | 1 | 1 | yes | characteristic |
| 166 | Q16 | — | 6 | 2 | no | — |
| 167 | Q16 | — | 6 | 2 | no | — |
| 168 | SD16 | — | 12 | 2 | no | — |
| 169 | SD16 | — | 12 | 2 | no | — |
| 170 | D8 | — | 6 | 2 | no | — |
| 171 | D8 | — | 6 | 2 | no | — |
| 172 | M16 | — | 12 | 2 | no | — |
| 173 | M16 | — | 12 | 2 | no | — |
| 174 | C2×C8 | — | 2 | 1 | no | — |
| 175 | C2×C8 | — | 2 | 1 | no | — |
| 176 | C2×C8 | — | 12 | 2 | no | — |
| 177 | C2×C8 | D24 | 1 | 2 | yes | — |
| 178 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 179 | C2×C8 | C2×D12 | 1 | 1 | yes | characteristic |
| 180 | C2×C8 | C2×D12 | 1 | 1 | yes | characteristic |
| 181 | C2×C8 | #235 | 1 | 1 | yes | characteristic |
| 182 | C2×C8 | #235 | 1 | 1 | yes | characteristic |
| 183 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 184 | C42 | — | 2 | 1 | no | — |
| 185 | C42 | D24 | 1 | 2 | yes | — |
| 186 | C16 | — | 2 | 2 | no | — |
| 187 | C16 | (C2×D3)⋊C4 | 1 | 2 | yes | — |
| 188 | C16 | (C2×D3)⋊C4 | 1 | 2 | yes | — |
| 189 | C22×C3 | — | 2 | 2 | no | — |
| 190 | C22×C3 | C4×D8 | 1 | 1 | yes | characteristic |
| 191 | C22×C3 | #466 | 1 | 1 | yes | characteristic |
| 192 | C22×C3 | #495 | 1 | 1 | yes | characteristic |
| 193 | C2×D3 | — | 8 | 4 | no | — |
| 194 | C2×D3 | — | 8 | 2 | no | — |
| 195 | C3×C4 | — | 4 | 2 | no | — |
| 196 | C3×C4 | #390 | 1 | 1 | yes | characteristic |
| 197 | C3×C4 | C4×D8 | 1 | 1 | yes | characteristic |
| 198 | C3×C4 | #466 | 1 | 1 | yes | characteristic |
| 199 | C3×C4 | #493 | 1 | 1 | yes | characteristic |
| 200 | Dic3 | — | 8 | 2 | no | — |
| 201 | C23 | — | 12 | 2 | no | — |
| 202 | C23 | #904 | 1 | 1 | yes | characteristic |
| 203 | Q8 | — | 12 | 2 | no | — |
| 204 | Q8 | — | 12 | 2 | no | — |
| 205 | D4 | — | 12 | 2 | no | — |
| 206 | D4 | — | 12 | 2 | no | — |
| 207 | D4 | — | 24 | 2 | no | — |
| 208 | D4 | — | 24 | 2 | no | — |
| 209 | C2×C4 | — | 2 | 2 | no | — |
| 210 | C2×C4 | — | 4 | 2 | no | — |
| 211 | C2×C4 | — | 12 | 2 | no | — |
| 212 | C2×C4 | — | 24 | 2 | no | — |
| 213 | C2×C4 | — | 24 | 2 | no | — |
| 214 | C2×C4 | C4×D12 | 1 | 1 | yes | characteristic |
| 215 | C2×C4 | #874 | 1 | 1 | yes | characteristic |
| 216 | C2×C4 | C2×D24 | 1 | 1 | yes | characteristic |
| 217 | C2×C4 | C2×D24 | 1 | 1 | yes | characteristic |
| 218 | C2×C4 | #904 | 1 | 1 | yes | characteristic |
| 219 | C2×C4 | #930 | 1 | 1 | yes | characteristic |
| 220 | C8 | — | 2 | 2 | no | — |
| 221 | C8 | — | 24 | 2 | no | — |
| 222 | C8 | C4×D12 | 1 | 1 | yes | characteristic |
| 223 | C8 | #875 | 1 | 1 | yes | characteristic |
| 224 | C8 | C2×(C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 225 | C8 | #930 | 1 | 1 | yes | characteristic |
| 226 | C2×C3 | — | 2 | 1 | no | — |
| 227 | C2×C3 | — | 2 | 1 | no | — |
| 228 | C2×C3 | #1970 | 1 | 1 | yes | characteristic |
| 229 | C2×C3 | #2179 | 1 | 2 | yes | — |
| 230 | D3 | — | 16 | 2 | no | — |
| 231 | C22 | — | 2 | 2 | no | — |
| 232 | C22 | — | 24 | 2 | no | — |
| 233 | C22 | — | 24 | 4 | no | — |
| 234 | C22 | C4×D24 | 1 | 1 | yes | characteristic |
| 235 | C22 | #4696 | 1 | 1 | yes | characteristic |
| 236 | C22 | #5115 | 1 | 1 | yes | characteristic |
| 237 | C4 | — | 4 | 2 | no | — |
| 238 | C4 | — | 24 | 2 | no | — |
| 239 | C4 | C4×D24 | 1 | 1 | yes | characteristic |
| 240 | C4 | #4695 | 1 | 1 | yes | characteristic |
| 241 | C4 | #4939 | 1 | 1 | yes | characteristic |
| 242 | C4 | #5115 | 1 | 1 | yes | characteristic |
| 243 | C3 | #13242 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 244 | C2 | — | 2 | 1 | no | — |
| 245 | C2 | — | 2 | 1 | no | — |
| 246 | C2 | — | 48 | 2 | no | — |
| 247 | C2 | #69085 | 1 | 2 | yes | — |
| 248 | 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