← #10950796
Subgroups — #10950796
Order 768.
484 conjugacy classes of subgroups.
The subgroup lattice has 312 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C3×#3177 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #78577 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #71354 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #71353 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #71312 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #71311 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #71083 | C2 | 1 | 2 | yes | maximal |
| 8 | #71081 | C2 | 1 | 2 | yes | maximal |
| 9 | #70909 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | C4×#829 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #69881 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #33741 | — | 3 | 1 | no | maximal, Sylow |
| 13 | C3×#541 | — | 2 | 1 | no | — |
| 14 | #5518 | — | 2 | 1 | no | — |
| 15 | C3×#499 | C22 | 1 | 1 | yes | characteristic |
| 16 | C3×#494 | C22 | 1 | 1 | yes | characteristic |
| 17 | C3×#491 | — | 2 | 1 | no | — |
| 18 | C3×#470 | — | 2 | 1 | no | — |
| 19 | C3×#467 | C22 | 1 | 2 | yes | — |
| 20 | C3×#441 | — | 2 | 1 | no | — |
| 21 | C3×C4×Q16 | C22 | 1 | 2 | yes | — |
| 22 | C3×#405 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×#829 | C22 | 1 | 2 | yes | — |
| 24 | C2×#829 | C22 | 1 | 2 | yes | — |
| 25 | #5137 | — | 2 | 1 | no | — |
| 26 | #5116 | C22 | 1 | 2 | yes | — |
| 27 | C2×#812 | C22 | 1 | 1 | yes | characteristic |
| 28 | C22×C3⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 29 | #4716 | — | 2 | 1 | no | — |
| 30 | #4703 | — | 2 | 1 | no | — |
| 31 | #4699 | C22 | 1 | 1 | yes | characteristic |
| 32 | C4×Dic12 | C22 | 1 | 2 | yes | — |
| 33 | #4683 | C22 | 1 | 1 | yes | characteristic |
| 34 | #4564 | C22 | 1 | 2 | yes | — |
| 35 | #4564 | C22 | 1 | 2 | yes | — |
| 36 | #4492 | C22 | 1 | 2 | yes | — |
| 37 | #4492 | C22 | 1 | 2 | yes | — |
| 38 | #4490 | C22 | 1 | 2 | yes | — |
| 39 | #4489 | C22 | 1 | 2 | yes | — |
| 40 | #4461 | C22 | 1 | 1 | yes | characteristic |
| 41 | #4461 | C22 | 1 | 1 | yes | characteristic |
| 42 | C4×C3⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 43 | C4×C3⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 44 | #3177 | — | 3 | 1 | no | — |
| 45 | #3177 | D3 | 1 | 1 | yes | characteristic |
| 46 | #2253 | — | 3 | 1 | no | — |
| 47 | #2247 | — | 3 | 1 | no | — |
| 48 | #2245 | — | 3 | 1 | no | — |
| 49 | #2245 | — | 3 | 1 | no | — |
| 50 | #2219 | — | 3 | 2 | no | — |
| 51 | #2216 | — | 3 | 2 | no | — |
| 52 | #2176 | — | 3 | 2 | no | — |
| 53 | C4×Q32 | — | 3 | 2 | no | — |
| 54 | #2108 | — | 3 | 1 | no | — |
| 55 | C2×C3×Q16 | — | 2 | 1 | no | — |
| 56 | C2×C3×Q16 | C23 | 1 | 2 | yes | — |
| 57 | C2×C3×M16 | D4 | 1 | 1 | yes | characteristic |
| 58 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 59 | C3×C4⋊Q8 | — | 2 | 1 | no | — |
| 60 | C3×#126 | — | 4 | 2 | no | — |
| 61 | C3×#125 | — | 2 | 1 | no | — |
| 62 | C3×#125 | — | 4 | 2 | no | — |
| 63 | C3×C22⋊Q8 | — | 2 | 2 | no | — |
| 64 | C3×C4×Q8 | — | 2 | 2 | no | — |
| 65 | C3×#117 | D4 | 1 | 1 | yes | characteristic |
| 66 | #924 | — | 2 | 2 | no | — |
| 67 | C2×Dic12 | — | 2 | 1 | no | — |
| 68 | C2×Dic12 | C23 | 1 | 2 | yes | — |
| 69 | #889 | — | 4 | 2 | no | — |
| 70 | #879 | — | 4 | 2 | no | — |
| 71 | #870 | — | 2 | 1 | no | — |
| 72 | #869 | — | 2 | 1 | no | — |
| 73 | C4×Dic6 | — | 2 | 2 | no | — |
| 74 | C3×#107 | C23 | 1 | 2 | yes | — |
| 75 | C3×#106 | — | 2 | 1 | no | — |
| 76 | C3×#103 | — | 2 | 2 | no | — |
| 77 | C3×#103 | — | 2 | 2 | no | — |
| 78 | C3×#97 | D4 | 1 | 1 | yes | characteristic |
| 79 | C3×#97 | D4 | 1 | 1 | yes | characteristic |
| 80 | C3×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 81 | C3×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 82 | #829 | — | 2 | 4 | no | — |
| 83 | #829 | — | 2 | 4 | no | — |
| 84 | #818 | C23 | 1 | 2 | yes | — |
| 85 | #817 | — | 2 | 1 | no | — |
| 86 | #816 | — | 2 | 2 | no | — |
| 87 | #816 | — | 2 | 2 | no | — |
| 88 | #812 | — | 2 | 2 | no | — |
| 89 | C2×C3⋊C16 | — | 2 | 2 | no | — |
| 90 | C2×C3⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 91 | C2×C3⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 92 | C2×C3⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 93 | C2×C3⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 94 | #541 | — | 2 | 1 | no | — |
| 95 | #541 | — | 6 | 1 | no | — |
| 96 | C2×Q32 | — | 3 | 2 | no | — |
| 97 | C2×Q32 | — | 3 | 2 | no | — |
| 98 | C2×M32 | — | 3 | 1 | no | — |
| 99 | C22×C16 | — | 3 | 1 | no | — |
| 100 | #499 | — | 3 | 1 | no | — |
| 101 | #499 | C2×D3 | 1 | 1 | yes | characteristic |
| 102 | #494 | — | 3 | 1 | no | — |
| 103 | #494 | C2×D3 | 1 | 1 | yes | characteristic |
| 104 | #491 | — | 2 | 1 | no | — |
| 105 | #491 | — | 6 | 1 | no | — |
| 106 | #470 | — | 2 | 1 | no | — |
| 107 | #470 | — | 6 | 1 | no | — |
| 108 | #467 | — | 3 | 2 | no | — |
| 109 | #467 | C2×D3 | 1 | 2 | yes | — |
| 110 | #441 | — | 2 | 1 | no | — |
| 111 | #441 | — | 6 | 1 | no | — |
| 112 | C4×Q16 | — | 3 | 2 | no | — |
| 113 | C4×Q16 | C2×D3 | 1 | 2 | yes | — |
| 114 | #405 | C2×D3 | 1 | 1 | yes | characteristic |
| 115 | #367 | — | 3 | 2 | no | — |
| 116 | #366 | — | 3 | 2 | no | — |
| 117 | #358 | — | 3 | 2 | no | — |
| 118 | #358 | — | 3 | 2 | no | — |
| 119 | #358 | — | 3 | 2 | no | — |
| 120 | #358 | — | 3 | 2 | no | — |
| 121 | #346 | — | 3 | 1 | no | — |
| 122 | #346 | — | 3 | 1 | no | — |
| 123 | C4×C16 | — | 3 | 1 | no | — |
| 124 | C4×C16 | — | 3 | 1 | no | — |
| 125 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 126 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 127 | C2×Dic6 | — | 2 | 2 | no | — |
| 128 | C3×Q16 | — | 2 | 2 | no | — |
| 129 | C3×Q16 | — | 2 | 2 | no | — |
| 130 | C3×Q16 | — | 4 | 2 | no | — |
| 131 | C3×M16 | — | 2 | 2 | no | — |
| 132 | C2×C3×C8 | — | 2 | 1 | no | — |
| 133 | C2×C3×C8 | — | 2 | 1 | no | — |
| 134 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 135 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 136 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 137 | C2×C3×C8 | C24 | 1 | 1 | yes | characteristic |
| 138 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 139 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 140 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 141 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 142 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 143 | C3×C4⋊C4 | D8 | 1 | 2 | yes | — |
| 144 | C3×C22⋊C4 | — | 4 | 2 | no | — |
| 145 | C3×C22⋊C4 | D8 | 1 | 2 | yes | — |
| 146 | C3×C42 | — | 4 | 2 | no | — |
| 147 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 148 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 149 | #217 | — | 4 | 2 | no | — |
| 150 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 151 | C4⋊Dic3 | — | 4 | 2 | no | — |
| 152 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 153 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 154 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 155 | C4×Dic3 | — | 4 | 2 | no | — |
| 156 | Dic12 | — | 2 | 2 | no | — |
| 157 | Dic12 | — | 2 | 2 | no | — |
| 158 | Dic12 | — | 4 | 2 | no | — |
| 159 | C3⋊C16 | — | 2 | 1 | no | — |
| 160 | C3⋊C16 | — | 2 | 1 | no | — |
| 161 | C3⋊C16 | #41 | 1 | 2 | yes | — |
| 162 | C3⋊C16 | #41 | 1 | 2 | yes | — |
| 163 | C2×Q16 | — | 2 | 1 | no | — |
| 164 | C2×Q16 | — | 3 | 2 | no | — |
| 165 | C2×Q16 | — | 6 | 1 | no | — |
| 166 | C2×Q16 | C22×D3 | 1 | 2 | yes | — |
| 167 | C2×M16 | #67 | 1 | 1 | yes | characteristic |
| 168 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 169 | C4⋊Q8 | — | 2 | 1 | no | — |
| 170 | C4⋊Q8 | — | 6 | 1 | no | — |
| 171 | #126 | — | 4 | 2 | no | — |
| 172 | #126 | — | 12 | 2 | no | — |
| 173 | #125 | — | 2 | 1 | no | — |
| 174 | #125 | — | 4 | 2 | no | — |
| 175 | #125 | — | 6 | 1 | no | — |
| 176 | #125 | — | 12 | 2 | no | — |
| 177 | C22⋊Q8 | — | 2 | 2 | no | — |
| 178 | C22⋊Q8 | — | 6 | 2 | no | — |
| 179 | C4×Q8 | — | 2 | 2 | no | — |
| 180 | C4×Q8 | — | 6 | 2 | no | — |
| 181 | #117 | #67 | 1 | 1 | yes | characteristic |
| 182 | Q32 | — | 6 | 4 | no | — |
| 183 | Q32 | — | 6 | 4 | no | — |
| 184 | M32 | — | 6 | 2 | no | — |
| 185 | C2×C16 | — | 3 | 1 | no | — |
| 186 | C2×C16 | — | 3 | 1 | no | — |
| 187 | C2×C16 | — | 3 | 1 | no | — |
| 188 | C2×C16 | — | 3 | 1 | no | — |
| 189 | C2×C16 | — | 6 | 2 | no | — |
| 190 | #107 | — | 3 | 2 | no | — |
| 191 | #107 | C22×D3 | 1 | 2 | yes | — |
| 192 | #106 | — | 2 | 1 | no | — |
| 193 | #106 | — | 6 | 1 | no | — |
| 194 | #103 | — | 2 | 2 | no | — |
| 195 | #103 | — | 2 | 2 | no | — |
| 196 | #103 | — | 6 | 2 | no | — |
| 197 | #103 | — | 6 | 2 | no | — |
| 198 | #97 | #67 | 1 | 1 | yes | characteristic |
| 199 | #97 | #67 | 1 | 1 | yes | characteristic |
| 200 | C4×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 201 | C4×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 202 | C23×C3 | C2×D8 | 1 | 1 | yes | characteristic |
| 203 | C3×Q8 | — | 4 | 2 | no | — |
| 204 | C3×Q8 | — | 4 | 2 | no | — |
| 205 | C2×C3×C4 | — | 2 | 1 | no | — |
| 206 | C2×C3×C4 | — | 2 | 1 | no | — |
| 207 | C2×C3×C4 | — | 4 | 2 | no | — |
| 208 | C2×C3×C4 | — | 4 | 2 | no | — |
| 209 | C2×C3×C4 | C2×D8 | 1 | 2 | yes | — |
| 210 | C2×C3×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 211 | C2×C3×C4 | C2×D8 | 1 | 2 | yes | — |
| 212 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 213 | C2×Dic3 | — | 4 | 2 | no | — |
| 214 | C2×Dic3 | — | 4 | 2 | no | — |
| 215 | Dic6 | — | 4 | 2 | no | — |
| 216 | Dic6 | — | 4 | 2 | no | — |
| 217 | C3×C8 | — | 2 | 1 | no | — |
| 218 | C3×C8 | — | 2 | 1 | no | — |
| 219 | C3×C8 | #135 | 1 | 2 | yes | — |
| 220 | C3×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 221 | C3×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 222 | C2×Q8 | — | 2 | 2 | no | — |
| 223 | C2×Q8 | — | 6 | 2 | no | — |
| 224 | C22×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 225 | Q16 | — | 2 | 2 | no | — |
| 226 | Q16 | — | 2 | 2 | no | — |
| 227 | Q16 | — | 4 | 2 | no | — |
| 228 | Q16 | — | 6 | 2 | no | — |
| 229 | Q16 | — | 6 | 2 | no | — |
| 230 | Q16 | — | 12 | 2 | no | — |
| 231 | M16 | — | 2 | 2 | no | — |
| 232 | C2×C8 | — | 2 | 1 | no | — |
| 233 | C2×C8 | — | 2 | 1 | no | — |
| 234 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 235 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 236 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 237 | C2×C8 | C23×D3 | 1 | 1 | yes | characteristic |
| 238 | C4⋊C4 | — | 2 | 2 | no | — |
| 239 | C4⋊C4 | — | 4 | 2 | no | — |
| 240 | C4⋊C4 | — | 4 | 2 | no | — |
| 241 | C4⋊C4 | — | 4 | 2 | no | — |
| 242 | C4⋊C4 | — | 4 | 2 | no | — |
| 243 | C4⋊C4 | — | 6 | 2 | no | — |
| 244 | C4⋊C4 | — | 12 | 2 | no | — |
| 245 | C4⋊C4 | — | 12 | 2 | no | — |
| 246 | C4⋊C4 | — | 12 | 2 | no | — |
| 247 | C4⋊C4 | — | 12 | 2 | no | — |
| 248 | C4⋊C4 | #213 | 1 | 2 | yes | — |
| 249 | C22⋊C4 | — | 4 | 2 | no | — |
| 250 | C22⋊C4 | — | 12 | 2 | no | — |
| 251 | C22⋊C4 | #213 | 1 | 2 | yes | — |
| 252 | C42 | — | 4 | 2 | no | — |
| 253 | C42 | — | 12 | 2 | no | — |
| 254 | C42 | C2×#67 | 1 | 1 | yes | characteristic |
| 255 | C42 | C2×#67 | 1 | 1 | yes | characteristic |
| 256 | C16 | — | 3 | 2 | no | — |
| 257 | C16 | — | 3 | 2 | no | — |
| 258 | C16 | — | 6 | 1 | no | — |
| 259 | C16 | — | 6 | 1 | no | — |
| 260 | C22×C3 | — | 2 | 2 | no | — |
| 261 | C22×C3 | — | 2 | 1 | no | — |
| 262 | C22×C3 | C22×D8 | 1 | 1 | yes | characteristic |
| 263 | C3×C4 | — | 2 | 2 | no | — |
| 264 | C3×C4 | — | 2 | 2 | no | — |
| 265 | C3×C4 | — | 2 | 1 | no | — |
| 266 | C3×C4 | — | 8 | 2 | no | — |
| 267 | C3×C4 | — | 8 | 2 | no | — |
| 268 | C3×C4 | #532 | 1 | 1 | yes | characteristic |
| 269 | C3×C4 | C2×#135 | 1 | 1 | yes | characteristic |
| 270 | Dic3 | — | 8 | 2 | no | — |
| 271 | Dic3 | — | 8 | 2 | no | — |
| 272 | C23 | C2×#213 | 1 | 1 | yes | characteristic |
| 273 | Q8 | — | 4 | 2 | no | — |
| 274 | Q8 | — | 4 | 2 | no | — |
| 275 | Q8 | — | 12 | 2 | no | — |
| 276 | Q8 | — | 12 | 2 | no | — |
| 277 | C2×C4 | — | 2 | 1 | no | — |
| 278 | C2×C4 | — | 2 | 1 | no | — |
| 279 | C2×C4 | — | 4 | 2 | no | — |
| 280 | C2×C4 | — | 4 | 2 | no | — |
| 281 | C2×C4 | — | 12 | 2 | no | — |
| 282 | C2×C4 | — | 12 | 2 | no | — |
| 283 | C2×C4 | C2×#213 | 1 | 2 | yes | — |
| 284 | C2×C4 | C2×#213 | 1 | 2 | yes | — |
| 285 | C2×C4 | C2×#213 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | C22×#67 | 1 | 1 | yes | characteristic |
| 287 | C8 | — | 2 | 1 | no | — |
| 288 | C8 | — | 2 | 1 | no | — |
| 289 | C8 | #950 | 1 | 2 | yes | — |
| 290 | C8 | D3×#41 | 1 | 1 | yes | characteristic |
| 291 | C8 | D3×#41 | 1 | 1 | yes | characteristic |
| 292 | C2×C3 | — | 4 | 1 | no | — |
| 293 | C2×C3 | #3169 | 1 | 1 | yes | characteristic |
| 294 | C2×C3 | #3417 | 1 | 2 | yes | — |
| 295 | C22 | — | 2 | 2 | no | — |
| 296 | C22 | — | 2 | 1 | no | — |
| 297 | C22 | C22×#213 | 1 | 1 | yes | characteristic |
| 298 | C4 | — | 2 | 2 | no | — |
| 299 | C4 | — | 2 | 2 | no | — |
| 300 | C4 | — | 2 | 1 | no | — |
| 301 | C4 | — | 8 | 2 | no | — |
| 302 | C4 | — | 8 | 2 | no | — |
| 303 | C4 | — | 24 | 2 | no | — |
| 304 | C4 | — | 24 | 2 | no | — |
| 305 | C4 | C2×#950 | 1 | 1 | yes | characteristic |
| 306 | C4 | #5832 | 1 | 1 | yes | characteristic |
| 307 | C3 | #33741 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 308 | C2 | — | 4 | 1 | no | — |
| 309 | C2 | #83291 | 1 | 1 | yes | characteristic |
| 310 | C2 | #84017 | 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