← #10763309
Subgroups — #10763309
Order 768.
954 conjugacy classes of subgroups.
The subgroup lattice has 338 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#5688 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | D3×#432 | C2 | 1 | 2 | yes | maximal |
| 3 | C3×#1534 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #71736 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #70814 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #68365 | C2 | 1 | 8 | yes | maximal, complemented |
| 7 | #17676 | — | 3 | 1 | no | maximal, Sylow |
| 8 | C2×C3×#125 | C22 | 1 | 1 | yes | characteristic |
| 9 | #5688 | — | 2 | 2 | no | — |
| 10 | #5688 | — | 2 | 2 | no | — |
| 11 | #5688 | C4 | 1 | 4 | yes | — |
| 12 | #5688 | C22 | 1 | 4 | yes | — |
| 13 | C2×#896 | — | 2 | 2 | no | — |
| 14 | C2×#892 | C4 | 1 | 2 | yes | — |
| 15 | C2×#888 | C4 | 1 | 2 | yes | — |
| 16 | C2×#887 | — | 2 | 2 | no | — |
| 17 | C2×C42×D3 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#870 | C22 | 1 | 1 | yes | characteristic |
| 19 | C3×#432 | C22 | 1 | 2 | yes | — |
| 20 | #4926 | C22 | 1 | 2 | yes | — |
| 21 | #4835 | — | 2 | 2 | no | — |
| 22 | D3×C4⋊C8 | — | 2 | 2 | no | — |
| 23 | D3×C22⋊C8 | — | 2 | 2 | no | — |
| 24 | #4719 | — | 2 | 2 | no | — |
| 25 | #4705 | C22 | 1 | 4 | yes | — |
| 26 | D3×#97 | C22 | 1 | 4 | yes | — |
| 27 | C3×#330 | C22 | 1 | 4 | yes | complemented |
| 28 | #4543 | C22 | 1 | 4 | yes | complemented |
| 29 | #4465 | C22 | 1 | 8 | yes | complemented |
| 30 | C2×#533 | — | 3 | 1 | no | — |
| 31 | C2×#432 | — | 3 | 2 | no | — |
| 32 | #1534 | — | 3 | 2 | no | — |
| 33 | #1534 | — | 3 | 1 | no | — |
| 34 | #1534 | D3 | 1 | 1 | yes | characteristic, complemented |
| 35 | #1531 | — | 3 | 8 | no | — |
| 36 | C22×C4×D3 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C22×C4×D3 | D4 | 1 | 2 | yes | — |
| 38 | C22×Dic6 | — | 2 | 2 | no | — |
| 39 | C3×#125 | — | 2 | 2 | no | — |
| 40 | C3×#125 | C2×C4 | 1 | 2 | yes | — |
| 41 | C3×#125 | C23 | 1 | 2 | yes | — |
| 42 | C2×C3×C4⋊C4 | — | 2 | 2 | no | — |
| 43 | C2×C3×C4⋊C4 | C2×C4 | 1 | 2 | yes | — |
| 44 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 45 | C2×(C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 46 | C2×C4⋊Dic3 | C2×C4 | 1 | 2 | yes | — |
| 47 | C2×Dic3⋊C4 | — | 2 | 2 | no | — |
| 48 | C2×Dic3⋊C4 | — | 2 | 2 | no | — |
| 49 | C2×C4×Dic3 | — | 2 | 2 | no | — |
| 50 | C2×C4×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | C2×C4×Dic3 | D4 | 1 | 2 | yes | — |
| 52 | C2×D3×C8 | — | 2 | 4 | no | — |
| 53 | #896 | — | 2 | 4 | no | — |
| 54 | #896 | — | 2 | 4 | no | — |
| 55 | #896 | — | 2 | 4 | no | — |
| 56 | #896 | — | 2 | 4 | no | — |
| 57 | #892 | — | 2 | 4 | no | — |
| 58 | #892 | C2×C4 | 1 | 8 | yes | — |
| 59 | #888 | — | 2 | 2 | no | — |
| 60 | #888 | — | 2 | 2 | no | — |
| 61 | #888 | C2×C4 | 1 | 4 | yes | — |
| 62 | #888 | C2×C4 | 1 | 4 | yes | — |
| 63 | #887 | — | 2 | 4 | no | — |
| 64 | #887 | — | 2 | 4 | no | — |
| 65 | #887 | — | 2 | 4 | no | — |
| 66 | #887 | — | 2 | 4 | no | — |
| 67 | C42×D3 | C2×C4 | 1 | 2 | yes | — |
| 68 | C42×D3 | D4 | 1 | 2 | yes | — |
| 69 | C42×D3 | D4 | 1 | 2 | yes | — |
| 70 | C42×D3 | C23 | 1 | 2 | yes | — |
| 71 | #870 | — | 2 | 2 | no | — |
| 72 | #870 | C2×C4 | 1 | 2 | yes | — |
| 73 | #870 | C23 | 1 | 2 | yes | — |
| 74 | C3×C4⋊C8 | — | 2 | 2 | no | — |
| 75 | C3×C22⋊C8 | — | 2 | 2 | no | — |
| 76 | C3×#97 | C23 | 1 | 4 | yes | — |
| 77 | #830 | — | 2 | 2 | no | — |
| 78 | (C2×D3)⋊C8 | — | 2 | 4 | no | — |
| 79 | #815 | — | 2 | 4 | no | — |
| 80 | Dic3⋊C8 | — | 2 | 4 | no | — |
| 81 | C4⋊(C3⋊C8) | — | 2 | 2 | no | — |
| 82 | #803 | C23 | 1 | 4 | yes | — |
| 83 | #533 | — | 3 | 4 | no | — |
| 84 | #533 | — | 3 | 4 | no | — |
| 85 | #533 | — | 6 | 2 | no | — |
| 86 | #533 | — | 6 | 2 | no | — |
| 87 | C2×C4⋊Q8 | — | 3 | 2 | no | — |
| 88 | C2×#125 | — | 3 | 1 | no | — |
| 89 | C2×#125 | C2×D3 | 1 | 1 | yes | characteristic |
| 90 | C2×C22⋊Q8 | — | 6 | 2 | no | — |
| 91 | C2×C4×Q8 | — | 6 | 2 | no | — |
| 92 | C2×#117 | — | 3 | 2 | no | — |
| 93 | C22×C42 | — | 3 | 1 | no | — |
| 94 | #432 | — | 3 | 2 | no | — |
| 95 | #432 | — | 3 | 4 | no | — |
| 96 | #432 | — | 6 | 2 | no | — |
| 97 | #432 | — | 6 | 2 | no | — |
| 98 | #432 | C2×D3 | 1 | 2 | yes | — |
| 99 | C2×C4⋊C8 | — | 6 | 2 | no | — |
| 100 | C2×C22⋊C8 | — | 6 | 2 | no | — |
| 101 | C2×#97 | — | 3 | 4 | no | — |
| 102 | #330 | — | 3 | 4 | no | — |
| 103 | #330 | — | 3 | 8 | no | — |
| 104 | #330 | C2×D3 | 1 | 4 | yes | complemented |
| 105 | C23×D3 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 106 | C22×C3×C4 | — | 2 | 2 | no | — |
| 107 | C22×C3×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 108 | C22×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 109 | C22×Dic3 | — | 2 | 2 | no | — |
| 110 | C22×Dic3 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 111 | C22×Dic3 | C22⋊C4 | 1 | 2 | yes | — |
| 112 | C2×C4×D3 | — | 2 | 4 | no | — |
| 113 | C2×C4×D3 | — | 4 | 4 | no | — |
| 114 | C2×C4×D3 | C22⋊C4 | 1 | 2 | yes | — |
| 115 | C2×C4×D3 | C22⋊C4 | 1 | 2 | yes | — |
| 116 | C2×C4×D3 | C22⋊C4 | 1 | 2 | yes | — |
| 117 | C2×C4×D3 | C22×C4 | 1 | 2 | yes | — |
| 118 | C2×C4×D3 | C2×D4 | 1 | 2 | yes | — |
| 119 | C2×C4×D3 | C2×D4 | 1 | 2 | yes | — |
| 120 | C2×Dic6 | — | 2 | 4 | no | — |
| 121 | C2×Dic6 | — | 2 | 4 | no | — |
| 122 | C2×Dic6 | — | 4 | 8 | no | — |
| 123 | C2×C3×C8 | — | 2 | 4 | no | — |
| 124 | C3×C4⋊C4 | — | 2 | 4 | no | — |
| 125 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 126 | C3×C4⋊C4 | — | 2 | 4 | no | — |
| 127 | C3×C4⋊C4 | C22×C4 | 1 | 4 | yes | — |
| 128 | C3×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 129 | C3×C42 | C2×D4 | 1 | 2 | yes | — |
| 130 | C3×C42 | C24 | 1 | 1 | yes | characteristic |
| 131 | (C2×D3)⋊C4 | — | 2 | 8 | no | — |
| 132 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 133 | C4⋊Dic3 | C22×C4 | 1 | 4 | yes | — |
| 134 | Dic3⋊C4 | — | 2 | 4 | no | — |
| 135 | Dic3⋊C4 | — | 2 | 4 | no | — |
| 136 | Dic3⋊C4 | — | 2 | 4 | no | — |
| 137 | Dic3⋊C4 | — | 2 | 4 | no | — |
| 138 | C4×Dic3 | — | 2 | 8 | no | — |
| 139 | C4×Dic3 | C22⋊C4 | 1 | 2 | yes | — |
| 140 | C4×Dic3 | C22⋊C4 | 1 | 2 | yes | — |
| 141 | C4×Dic3 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 142 | C4×Dic3 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 143 | C4×Dic3 | C22×C4 | 1 | 2 | yes | — |
| 144 | C4×Dic3 | C2×D4 | 1 | 2 | yes | — |
| 145 | C4×Dic3 | C2×D4 | 1 | 2 | yes | — |
| 146 | C2×C3⋊C8 | — | 2 | 4 | no | — |
| 147 | D3×C8 | — | 4 | 8 | no | — |
| 148 | C22×Q8 | — | 6 | 2 | no | — |
| 149 | C23×C4 | — | 3 | 1 | no | — |
| 150 | C23×C4 | — | 3 | 2 | no | — |
| 151 | C22×C8 | — | 6 | 4 | no | — |
| 152 | C4⋊Q8 | — | 3 | 4 | no | — |
| 153 | C4⋊Q8 | — | 3 | 4 | no | — |
| 154 | C4⋊Q8 | — | 6 | 2 | no | — |
| 155 | C4⋊Q8 | — | 6 | 2 | no | — |
| 156 | #125 | — | 2 | 2 | no | — |
| 157 | #125 | — | 3 | 2 | no | — |
| 158 | #125 | — | 3 | 2 | no | — |
| 159 | #125 | — | 6 | 2 | no | — |
| 160 | #125 | C4×D3 | 1 | 2 | yes | — |
| 161 | #125 | C22×D3 | 1 | 2 | yes | — |
| 162 | C22⋊Q8 | — | 6 | 4 | no | — |
| 163 | C22⋊Q8 | — | 6 | 4 | no | — |
| 164 | C22⋊Q8 | — | 6 | 4 | no | — |
| 165 | C22⋊Q8 | — | 6 | 4 | no | — |
| 166 | C4×Q8 | — | 6 | 4 | no | — |
| 167 | C4×Q8 | — | 6 | 4 | no | — |
| 168 | C4×Q8 | — | 6 | 4 | no | — |
| 169 | C4×Q8 | — | 6 | 4 | no | — |
| 170 | #117 | — | 3 | 8 | no | — |
| 171 | #117 | — | 6 | 4 | no | — |
| 172 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 173 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 174 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 175 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 176 | C2×C4⋊C4 | C4×D3 | 1 | 2 | yes | — |
| 177 | C2×C22⋊C4 | — | 6 | 2 | no | — |
| 178 | C2×C42 | — | 3 | 1 | no | — |
| 179 | C2×C42 | — | 3 | 2 | no | — |
| 180 | C2×C42 | — | 3 | 2 | no | — |
| 181 | C2×C42 | — | 3 | 2 | no | — |
| 182 | C2×C42 | — | 3 | 2 | no | — |
| 183 | C2×C42 | — | 3 | 2 | no | — |
| 184 | C2×C42 | — | 6 | 2 | no | — |
| 185 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 186 | C4⋊C8 | — | 2 | 2 | no | — |
| 187 | C4⋊C8 | — | 6 | 4 | no | — |
| 188 | C4⋊C8 | — | 6 | 2 | no | — |
| 189 | C22⋊C8 | — | 2 | 2 | no | — |
| 190 | C22⋊C8 | — | 6 | 4 | no | — |
| 191 | C22⋊C8 | — | 6 | 2 | no | — |
| 192 | #97 | — | 3 | 4 | no | — |
| 193 | #97 | — | 6 | 4 | no | — |
| 194 | #97 | C22×D3 | 1 | 4 | yes | — |
| 195 | C23×C3 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 196 | C22×D3 | — | 4 | 2 | no | — |
| 197 | C22×D3 | #101 | 1 | 4 | yes | — |
| 198 | C22×D3 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 199 | C2×C3×C4 | — | 2 | 2 | no | — |
| 200 | C2×C3×C4 | — | 2 | 4 | no | — |
| 201 | C2×C3×C4 | — | 4 | 2 | no | — |
| 202 | C2×C3×C4 | — | 4 | 4 | no | — |
| 203 | C2×C3×C4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 204 | C2×C3×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 205 | C2×C3×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 206 | C2×C3×C4 | C22×D4 | 1 | 2 | yes | — |
| 207 | C2×Dic3 | — | 2 | 2 | no | — |
| 208 | C2×Dic3 | — | 2 | 4 | no | — |
| 209 | C2×Dic3 | — | 2 | 4 | no | — |
| 210 | C2×Dic3 | — | 4 | 4 | no | — |
| 211 | C2×Dic3 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 212 | C2×Dic3 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 213 | C2×Dic3 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 214 | C4×D3 | — | 2 | 4 | no | — |
| 215 | C4×D3 | — | 2 | 4 | no | — |
| 216 | C4×D3 | — | 2 | 4 | no | — |
| 217 | C4×D3 | — | 2 | 4 | no | — |
| 218 | C4×D3 | — | 4 | 2 | no | — |
| 219 | C4×D3 | — | 4 | 2 | no | — |
| 220 | Dic6 | — | 4 | 8 | no | — |
| 221 | Dic6 | — | 4 | 8 | no | — |
| 222 | C3×C8 | — | 4 | 4 | no | — |
| 223 | C3⋊C8 | — | 4 | 4 | no | — |
| 224 | C24 | — | 3 | 1 | no | — |
| 225 | C2×Q8 | — | 6 | 4 | no | — |
| 226 | C2×Q8 | — | 6 | 4 | no | — |
| 227 | C2×Q8 | — | 12 | 8 | no | — |
| 228 | C22×C4 | — | 2 | 2 | no | — |
| 229 | C22×C4 | — | 3 | 1 | no | — |
| 230 | C22×C4 | — | 3 | 2 | no | — |
| 231 | C22×C4 | — | 3 | 2 | no | — |
| 232 | C22×C4 | — | 3 | 2 | no | — |
| 233 | C22×C4 | — | 3 | 2 | no | — |
| 234 | C22×C4 | — | 3 | 2 | no | — |
| 235 | C22×C4 | — | 3 | 2 | no | — |
| 236 | C22×C4 | — | 3 | 2 | no | — |
| 237 | C22×C4 | — | 6 | 4 | no | — |
| 238 | C22×C4 | — | 6 | 2 | no | — |
| 239 | C22×C4 | — | 12 | 4 | no | — |
| 240 | C22×C4 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 241 | C22×C4 | D3×D4 | 1 | 2 | yes | — |
| 242 | C2×C8 | — | 2 | 4 | no | — |
| 243 | C2×C8 | — | 6 | 4 | no | — |
| 244 | C2×C8 | — | 12 | 8 | no | — |
| 245 | C4⋊C4 | — | 2 | 4 | no | — |
| 246 | C4⋊C4 | — | 2 | 2 | no | — |
| 247 | C4⋊C4 | — | 2 | 4 | no | — |
| 248 | C4⋊C4 | — | 3 | 4 | no | — |
| 249 | C4⋊C4 | — | 6 | 4 | no | — |
| 250 | C4⋊C4 | — | 6 | 4 | no | — |
| 251 | C4⋊C4 | — | 6 | 2 | no | — |
| 252 | C4⋊C4 | — | 6 | 4 | no | — |
| 253 | C4⋊C4 | — | 6 | 4 | no | — |
| 254 | C4⋊C4 | C2×C4×D3 | 1 | 4 | yes | — |
| 255 | C22⋊C4 | — | 6 | 8 | no | — |
| 256 | C42 | — | 3 | 2 | no | — |
| 257 | C42 | — | 3 | 2 | no | — |
| 258 | C42 | — | 3 | 1 | no | — |
| 259 | C42 | — | 3 | 1 | no | — |
| 260 | C42 | — | 3 | 2 | no | — |
| 261 | C42 | — | 3 | 2 | no | — |
| 262 | C42 | — | 3 | 2 | no | — |
| 263 | C42 | — | 6 | 8 | no | — |
| 264 | C42 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 265 | C42 | D3×D4 | 1 | 2 | yes | — |
| 266 | C42 | C23×D3 | 1 | 1 | yes | characteristic |
| 267 | C22×C3 | — | 2 | 2 | no | — |
| 268 | C22×C3 | C2×#101 | 1 | 2 | yes | — |
| 269 | C22×C3 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 270 | C2×D3 | — | 2 | 2 | no | — |
| 271 | C2×D3 | — | 2 | 2 | no | — |
| 272 | C2×D3 | — | 4 | 4 | no | — |
| 273 | C2×D3 | C2×#101 | 1 | 4 | yes | — |
| 274 | C3×C4 | — | 2 | 1 | no | — |
| 275 | C3×C4 | — | 2 | 1 | no | — |
| 276 | C3×C4 | — | 2 | 2 | no | — |
| 277 | C3×C4 | — | 2 | 2 | no | — |
| 278 | C3×C4 | — | 4 | 4 | no | — |
| 279 | Dic3 | — | 2 | 2 | no | — |
| 280 | Dic3 | — | 2 | 2 | no | — |
| 281 | Dic3 | — | 4 | 4 | no | — |
| 282 | Dic3 | #421 | 1 | 4 | yes | — |
| 283 | C23 | — | 3 | 2 | no | — |
| 284 | C23 | — | 3 | 4 | no | — |
| 285 | C23 | — | 12 | 2 | no | — |
| 286 | C23 | D3×C22⋊C4 | 1 | 1 | yes | characteristic |
| 287 | Q8 | — | 12 | 8 | no | — |
| 288 | Q8 | — | 12 | 8 | no | — |
| 289 | C2×C4 | — | 2 | 2 | no | — |
| 290 | C2×C4 | — | 2 | 4 | no | — |
| 291 | C2×C4 | — | 3 | 2 | no | — |
| 292 | C2×C4 | — | 3 | 2 | no | — |
| 293 | C2×C4 | — | 3 | 2 | no | — |
| 294 | C2×C4 | — | 4 | 2 | no | — |
| 295 | C2×C4 | — | 4 | 4 | no | — |
| 296 | C2×C4 | — | 6 | 4 | no | — |
| 297 | C2×C4 | — | 6 | 4 | no | — |
| 298 | C2×C4 | — | 6 | 4 | no | — |
| 299 | C2×C4 | — | 6 | 4 | no | — |
| 300 | C2×C4 | — | 6 | 4 | no | — |
| 301 | C2×C4 | — | 6 | 2 | no | — |
| 302 | C2×C4 | — | 6 | 4 | no | — |
| 303 | C2×C4 | — | 12 | 2 | no | — |
| 304 | C2×C4 | — | 12 | 2 | no | — |
| 305 | C2×C4 | — | 12 | 4 | no | — |
| 306 | C2×C4 | D3×C22⋊C4 | 1 | 2 | yes | — |
| 307 | C2×C4 | D3×C22⋊C4 | 1 | 1 | yes | characteristic |
| 308 | C2×C4 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 309 | C2×C4 | C2×D3×D4 | 1 | 2 | yes | — |
| 310 | C8 | — | 4 | 4 | no | — |
| 311 | C8 | — | 12 | 4 | no | — |
| 312 | C2×C3 | — | 2 | 2 | no | — |
| 313 | C2×C3 | C22×#101 | 1 | 1 | yes | characteristic |
| 314 | C2×C3 | C2×#421 | 1 | 2 | yes | — |
| 315 | D3 | — | 2 | 4 | no | — |
| 316 | C22 | — | 2 | 2 | no | — |
| 317 | C22 | — | 3 | 4 | no | — |
| 318 | C22 | — | 6 | 2 | no | — |
| 319 | C22 | — | 6 | 2 | no | — |
| 320 | C22 | — | 12 | 4 | no | — |
| 321 | C22 | D3×#101 | 1 | 2 | yes | — |
| 322 | C22 | C2×D3×C22⋊C4 | 1 | 1 | yes | characteristic |
| 323 | C4 | — | 2 | 1 | no | — |
| 324 | C4 | — | 2 | 1 | no | — |
| 325 | C4 | — | 2 | 2 | no | — |
| 326 | C4 | — | 2 | 2 | no | — |
| 327 | C4 | — | 3 | 4 | no | — |
| 328 | C4 | — | 4 | 4 | no | — |
| 329 | C4 | — | 6 | 2 | no | — |
| 330 | C4 | — | 6 | 2 | no | — |
| 331 | C4 | — | 12 | 4 | no | — |
| 332 | C3 | #17676 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 333 | C2 | — | 2 | 2 | no | — |
| 334 | C2 | — | 6 | 4 | no | — |
| 335 | C2 | C2×D3×#101 | 1 | 1 | yes | characteristic |
| 336 | C2 | D3×#421 | 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