← #10710607
Subgroups — #10710607
Order 768.
318 conjugacy classes of subgroups.
The subgroup lattice has 248 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C3×#347 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #71428 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#4513 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #69829 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #69814 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #67819 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #67817 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #12677 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×C3×M32 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C3×M32 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×#97 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#819 | C22 | 1 | 1 | yes | characteristic |
| 13 | #5106 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#818 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#817 | C22 | 1 | 1 | yes | characteristic |
| 16 | #5101 | — | 2 | 1 | no | — |
| 17 | C4×C4⋊Dic3 | — | 2 | 1 | no | — |
| 18 | C3×#347 | C4 | 1 | 4 | yes | complemented |
| 19 | #4551 | — | 2 | 1 | no | — |
| 20 | #4523 | — | 2 | 1 | no | — |
| 21 | #4513 | C4 | 1 | 4 | yes | complemented |
| 22 | #4505 | C4 | 1 | 2 | yes | complemented |
| 23 | #4504 | C4 | 1 | 2 | yes | complemented |
| 24 | #2159 | — | 3 | 1 | no | — |
| 25 | C2×#365 | — | 3 | 1 | no | — |
| 26 | C2×#347 | D3 | 1 | 1 | yes | characteristic |
| 27 | #1776 | — | 3 | 1 | no | — |
| 28 | #1773 | — | 3 | 1 | no | — |
| 29 | #1387 | — | 3 | 1 | no | — |
| 30 | #1386 | — | 3 | 1 | no | — |
| 31 | C22×C3×C8 | D4 | 1 | 1 | yes | characteristic |
| 32 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 33 | C2×C3×C42 | D4 | 1 | 1 | yes | characteristic |
| 34 | #926 | — | 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×#208 | — | 2 | 1 | no | — |
| 39 | C3×M32 | — | 2 | 1 | no | — |
| 40 | C3×M32 | C2×C4 | 1 | 2 | yes | — |
| 41 | C3×M32 | C2×C4 | 1 | 2 | yes | — |
| 42 | C3×M32 | C2×C4 | 1 | 2 | yes | — |
| 43 | C2×C3×C16 | — | 2 | 1 | no | — |
| 44 | C2×C3×C16 | C2×C4 | 1 | 2 | yes | — |
| 45 | C3×#97 | C2×C4 | 1 | 2 | yes | — |
| 46 | C3×#97 | Q8 | 1 | 2 | yes | — |
| 47 | C3⋊#95 | — | 4 | 1 | no | — |
| 48 | #819 | — | 2 | 1 | no | — |
| 49 | #819 | C2×C4 | 1 | 2 | yes | — |
| 50 | #818 | — | 2 | 1 | no | — |
| 51 | #818 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | #818 | C2×C4 | 1 | 2 | yes | — |
| 53 | #818 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | #817 | — | 2 | 1 | no | — |
| 55 | #817 | — | 2 | 1 | no | — |
| 56 | #817 | C2×C4 | 1 | 2 | yes | — |
| 57 | #815 | — | 4 | 1 | no | — |
| 58 | C8×Dic3 | — | 4 | 1 | no | — |
| 59 | C2×M32 | C2×D3 | 1 | 1 | yes | characteristic |
| 60 | C2×M32 | C2×D3 | 1 | 1 | yes | characteristic |
| 61 | C2×#108 | — | 3 | 1 | no | — |
| 62 | #427 | — | 3 | 1 | no | — |
| 63 | C2×#107 | — | 3 | 1 | no | — |
| 64 | C2×#106 | — | 3 | 1 | no | — |
| 65 | #405 | — | 6 | 1 | no | — |
| 66 | C2×#97 | C2×D3 | 1 | 1 | yes | characteristic |
| 67 | C4×C4⋊C4 | — | 6 | 1 | no | — |
| 68 | #367 | — | 3 | 2 | no | — |
| 69 | #366 | — | 3 | 2 | no | — |
| 70 | #365 | — | 3 | 4 | no | — |
| 71 | #347 | Dic3 | 1 | 4 | yes | complemented |
| 72 | #340 | — | 6 | 1 | no | — |
| 73 | #339 | — | 6 | 1 | no | — |
| 74 | C22×C3×C4 | — | 2 | 1 | no | — |
| 75 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 76 | C22×Dic3 | — | 4 | 1 | no | — |
| 77 | C22×Dic3 | — | 4 | 1 | no | — |
| 78 | C2×C3×C8 | — | 2 | 1 | no | — |
| 79 | C2×C3×C8 | — | 2 | 1 | no | — |
| 80 | C2×C3×C8 | C4⋊C4 | 1 | 2 | yes | — |
| 81 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 82 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 84 | C2×C3×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 85 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 86 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 87 | C3×C42 | — | 2 | 1 | no | — |
| 88 | C3×C42 | C4⋊C4 | 1 | 2 | yes | — |
| 89 | #217 | — | 4 | 1 | no | — |
| 90 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 91 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 92 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 93 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 94 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 95 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 96 | C4×Dic3 | — | 4 | 1 | no | — |
| 97 | C4×Dic3 | — | 8 | 2 | no | — |
| 98 | #208 | — | 2 | 2 | no | — |
| 99 | #208 | — | 4 | 1 | no | — |
| 100 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 101 | C3×C16 | — | 2 | 2 | no | — |
| 102 | C3×C16 | C42 | 1 | 4 | yes | complemented |
| 103 | C2×M16 | — | 6 | 1 | no | — |
| 104 | C22×C8 | D12 | 1 | 1 | yes | characteristic |
| 105 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 106 | #117 | — | 6 | 1 | no | — |
| 107 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 108 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 109 | C2×C42 | — | 12 | 1 | no | — |
| 110 | C2×C42 | D12 | 1 | 1 | yes | characteristic |
| 111 | M32 | — | 2 | 1 | no | — |
| 112 | M32 | C4×D3 | 1 | 2 | yes | — |
| 113 | M32 | C2×Dic3 | 1 | 2 | yes | — |
| 114 | M32 | C2×Dic3 | 1 | 2 | yes | — |
| 115 | C2×C16 | — | 2 | 1 | no | — |
| 116 | C2×C16 | C4×D3 | 1 | 2 | yes | — |
| 117 | #108 | — | 3 | 2 | no | — |
| 118 | #108 | — | 6 | 1 | no | — |
| 119 | #107 | — | 3 | 1 | no | — |
| 120 | #107 | — | 3 | 2 | no | — |
| 121 | #107 | — | 3 | 1 | no | — |
| 122 | #107 | — | 6 | 1 | no | — |
| 123 | #106 | — | 3 | 2 | no | — |
| 124 | #106 | — | 6 | 1 | no | — |
| 125 | #106 | — | 6 | 1 | no | — |
| 126 | #97 | — | 12 | 1 | no | — |
| 127 | #97 | Dic6 | 1 | 2 | yes | — |
| 128 | #97 | C2×Dic3 | 1 | 2 | yes | — |
| 129 | C4×C8 | — | 12 | 1 | no | — |
| 130 | #95 | — | 12 | 1 | no | — |
| 131 | C23×C3 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 132 | C2×C3×C4 | — | 2 | 2 | no | — |
| 133 | C2×C3×C4 | — | 4 | 1 | no | — |
| 134 | C2×C3×C4 | — | 4 | 1 | no | — |
| 135 | C2×C3×C4 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 136 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 137 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 138 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 139 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 140 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 141 | C2×Dic3 | — | 4 | 2 | no | — |
| 142 | C2×Dic3 | — | 4 | 1 | no | — |
| 143 | C2×Dic3 | — | 4 | 1 | no | — |
| 144 | C2×Dic3 | — | 4 | 2 | 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 | C3×C8 | — | 2 | 2 | no | — |
| 150 | C3×C8 | C2×C42 | 1 | 1 | yes | characteristic |
| 151 | C3×C8 | #117 | 1 | 1 | yes | characteristic |
| 152 | C3×C8 | C4×Q8 | 1 | 1 | yes | characteristic |
| 153 | C3×C8 | C4×Q8 | 1 | 1 | yes | characteristic |
| 154 | C3⋊C8 | — | 4 | 2 | no | — |
| 155 | C22×C4 | — | 2 | 1 | no | — |
| 156 | C22×C4 | — | 12 | 1 | no | — |
| 157 | C22×C4 | — | 12 | 1 | no | — |
| 158 | C22×C4 | C2×D12 | 1 | 1 | yes | characteristic |
| 159 | M16 | — | 6 | 2 | no | — |
| 160 | M16 | — | 12 | 1 | no | — |
| 161 | C2×C8 | — | 2 | 1 | no | — |
| 162 | C2×C8 | — | 2 | 1 | no | — |
| 163 | C2×C8 | — | 12 | 1 | no | — |
| 164 | C2×C8 | C4⋊Dic3 | 1 | 2 | yes | — |
| 165 | C2×C8 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 166 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 167 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 168 | C2×C8 | #235 | 1 | 1 | yes | characteristic |
| 169 | C2×C8 | #235 | 1 | 1 | yes | characteristic |
| 170 | C2×C8 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 171 | C4⋊C4 | — | 6 | 2 | no | — |
| 172 | C4⋊C4 | — | 6 | 1 | no | — |
| 173 | C4⋊C4 | — | 6 | 1 | no | — |
| 174 | C4⋊C4 | — | 6 | 2 | no | — |
| 175 | C4⋊C4 | — | 12 | 1 | no | — |
| 176 | C4⋊C4 | — | 12 | 1 | no | — |
| 177 | C22⋊C4 | — | 12 | 1 | no | — |
| 178 | C42 | — | 2 | 1 | no | — |
| 179 | C42 | — | 12 | 1 | no | — |
| 180 | C42 | — | 24 | 2 | no | — |
| 181 | C42 | C4⋊Dic3 | 1 | 2 | yes | — |
| 182 | C16 | — | 2 | 2 | no | — |
| 183 | C16 | C4×Dic3 | 1 | 4 | yes | complemented |
| 184 | C22×C3 | — | 2 | 1 | no | — |
| 185 | C22×C3 | — | 2 | 1 | no | — |
| 186 | C22×C3 | #428 | 1 | 1 | yes | characteristic |
| 187 | C22×C3 | #440 | 1 | 1 | yes | characteristic |
| 188 | C22×C3 | #442 | 1 | 1 | yes | characteristic |
| 189 | C3×C4 | — | 4 | 2 | no | — |
| 190 | C3×C4 | C4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 191 | C3×C4 | #428 | 1 | 1 | yes | characteristic |
| 192 | C3×C4 | #440 | 1 | 1 | yes | characteristic |
| 193 | C3×C4 | #441 | 1 | 1 | yes | characteristic |
| 194 | Dic3 | — | 8 | 2 | no | — |
| 195 | Dic3 | — | 8 | 1 | no | — |
| 196 | Dic3 | — | 8 | 1 | no | — |
| 197 | Dic3 | — | 8 | 2 | no | — |
| 198 | C23 | #908 | 1 | 1 | yes | characteristic |
| 199 | C2×C4 | — | 2 | 2 | no | — |
| 200 | C2×C4 | — | 4 | 1 | no | — |
| 201 | C2×C4 | — | 4 | 1 | no | — |
| 202 | C2×C4 | — | 12 | 2 | no | — |
| 203 | C2×C4 | — | 12 | 1 | no | — |
| 204 | C2×C4 | — | 12 | 1 | no | — |
| 205 | C2×C4 | — | 12 | 2 | no | — |
| 206 | C2×C4 | — | 24 | 1 | no | — |
| 207 | C2×C4 | — | 24 | 1 | no | — |
| 208 | C2×C4 | — | 24 | 1 | no | — |
| 209 | C2×C4 | — | 24 | 1 | no | — |
| 210 | C2×C4 | C4×D12 | 1 | 1 | yes | characteristic |
| 211 | C2×C4 | C4×D12 | 1 | 1 | yes | characteristic |
| 212 | C2×C4 | #908 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | #909 | 1 | 1 | yes | characteristic |
| 214 | C2×C4 | #909 | 1 | 1 | yes | characteristic |
| 215 | C2×C4 | C2×C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 216 | C8 | — | 2 | 2 | no | — |
| 217 | C8 | — | 12 | 2 | no | — |
| 218 | C8 | C4×Dic6 | 1 | 1 | yes | characteristic |
| 219 | C8 | C4×Dic6 | 1 | 1 | yes | characteristic |
| 220 | C8 | C2×C4×Dic3 | 1 | 1 | yes | characteristic |
| 221 | C8 | #926 | 1 | 1 | yes | characteristic |
| 222 | C2×C3 | — | 2 | 1 | no | — |
| 223 | C2×C3 | — | 2 | 1 | no | — |
| 224 | C2×C3 | #1776 | 1 | 1 | yes | characteristic |
| 225 | C2×C3 | #2180 | 1 | 1 | yes | characteristic |
| 226 | C2×C3 | AGL(1,ℤ/16) | 1 | 1 | yes | characteristic |
| 227 | C22 | — | 2 | 1 | no | — |
| 228 | C22 | — | 2 | 1 | no | — |
| 229 | C22 | #4710 | 1 | 1 | yes | characteristic |
| 230 | C22 | #4711 | 1 | 1 | yes | characteristic |
| 231 | C22 | #5121 | 1 | 1 | yes | characteristic |
| 232 | C4 | — | 4 | 2 | no | — |
| 233 | C4 | — | 24 | 2 | no | — |
| 234 | C4 | — | 24 | 1 | no | — |
| 235 | C4 | — | 24 | 2 | no | — |
| 236 | C4 | — | 24 | 1 | no | — |
| 237 | C4 | #4710 | 1 | 1 | yes | characteristic |
| 238 | C4 | #4716 | 1 | 1 | yes | characteristic |
| 239 | C4 | C4×C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 240 | C4 | #5121 | 1 | 1 | yes | characteristic |
| 241 | C3 | #12677 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 242 | C2 | — | 2 | 1 | no | — |
| 243 | C2 | — | 2 | 1 | no | — |
| 244 | C2 | #69090 | 1 | 1 | yes | characteristic |
| 245 | C2 | #69091 | 1 | 1 | yes | characteristic |
| 246 | C2 | #69814 | 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