← #81226
Subgroups — #81226
Order 384.
320 conjugacy classes of subgroups.
The subgroup lattice has 300 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #5685 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #5569 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C3×#479 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #5065 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C4×#216 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #4876 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #4875 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #4857 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #4856 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #4831 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | #4829 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #4799 | C2 | 1 | 1 | yes | maximal, characteristic |
| 13 | #4793 | C2 | 1 | 1 | yes | maximal, characteristic |
| 14 | #4789 | C2 | 1 | 1 | yes | maximal, characteristic |
| 15 | #4789 | C2 | 1 | 1 | yes | maximal, characteristic |
| 16 | #3390 | — | 3 | 1 | no | maximal, Sylow |
| 17 | C3×#125 | C22 | 1 | 1 | yes | characteristic |
| 18 | C3×C4×Q8 | C22 | 1 | 1 | yes | characteristic |
| 19 | Q8×Dic3 | C22 | 1 | 1 | yes | characteristic |
| 20 | Q8×Dic3 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#216 | C22 | 1 | 1 | yes | characteristic |
| 22 | #890 | — | 2 | 1 | no | — |
| 23 | #890 | C22 | 1 | 1 | yes | characteristic |
| 24 | #890 | C22 | 1 | 1 | yes | characteristic |
| 25 | #889 | — | 2 | 1 | no | — |
| 26 | #889 | — | 2 | 1 | no | — |
| 27 | #888 | — | 2 | 1 | no | — |
| 28 | #888 | — | 2 | 1 | no | — |
| 29 | #888 | C22 | 1 | 1 | yes | characteristic |
| 30 | #888 | C22 | 1 | 1 | yes | characteristic |
| 31 | #887 | — | 2 | 1 | no | — |
| 32 | #887 | C22 | 1 | 1 | yes | characteristic |
| 33 | #887 | C22 | 1 | 1 | yes | characteristic |
| 34 | #870 | — | 2 | 1 | no | — |
| 35 | #870 | C22 | 1 | 1 | yes | characteristic |
| 36 | C4×Dic6 | — | 2 | 1 | no | — |
| 37 | C4×Dic6 | C22 | 1 | 1 | yes | characteristic |
| 38 | C4×Dic6 | C22 | 1 | 1 | yes | characteristic |
| 39 | C3×#107 | C22 | 1 | 1 | yes | characteristic |
| 40 | C3×#106 | C22 | 1 | 1 | yes | characteristic |
| 41 | C3×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 42 | C3×#103 | C22 | 1 | 1 | yes | characteristic |
| 43 | C3×#103 | C22 | 1 | 1 | yes | characteristic |
| 44 | #835 | C22 | 1 | 1 | yes | characteristic |
| 45 | #818 | C22 | 1 | 1 | yes | characteristic |
| 46 | #817 | C22 | 1 | 1 | yes | characteristic |
| 47 | #816 | C22 | 1 | 1 | yes | characteristic |
| 48 | #816 | C22 | 1 | 1 | yes | characteristic |
| 49 | #815 | C22 | 1 | 1 | yes | characteristic |
| 50 | #815 | C22 | 1 | 1 | yes | characteristic |
| 51 | Dic3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 52 | Dic3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 53 | #810 | C22 | 1 | 1 | yes | characteristic |
| 54 | #808 | C22 | 1 | 1 | yes | characteristic |
| 55 | #808 | C22 | 1 | 1 | yes | characteristic |
| 56 | #807 | C22 | 1 | 1 | yes | characteristic |
| 57 | #807 | C22 | 1 | 1 | yes | characteristic |
| 58 | #807 | C22 | 1 | 1 | yes | characteristic |
| 59 | C4×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 60 | #557 | — | 3 | 1 | no | — |
| 61 | #557 | — | 3 | 1 | no | — |
| 62 | #501 | — | 3 | 1 | no | — |
| 63 | #501 | — | 3 | 1 | no | — |
| 64 | #499 | — | 3 | 1 | no | — |
| 65 | #479 | — | 3 | 1 | no | — |
| 66 | #479 | — | 3 | 1 | no | — |
| 67 | #479 | — | 3 | 1 | no | — |
| 68 | #479 | D3 | 1 | 1 | yes | characteristic |
| 69 | #477 | — | 3 | 1 | no | — |
| 70 | #477 | — | 3 | 1 | no | — |
| 71 | #446 | — | 3 | 1 | no | — |
| 72 | #441 | — | 3 | 1 | no | — |
| 73 | #441 | — | 3 | 1 | no | — |
| 74 | C4×Q16 | — | 3 | 1 | no | — |
| 75 | C2×C3×Q8 | C23 | 1 | 1 | yes | characteristic |
| 76 | C2×Dic6 | — | 2 | 1 | no | — |
| 77 | C2×Dic6 | — | 2 | 1 | no | — |
| 78 | C2×Dic6 | D4 | 1 | 1 | yes | characteristic |
| 79 | C2×Dic6 | C23 | 1 | 1 | yes | characteristic |
| 80 | C2×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 81 | C2×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 82 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 83 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 84 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 85 | C3×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 86 | C3×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 87 | C3×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 88 | C3×C42 | — | 2 | 1 | no | — |
| 89 | C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 90 | #216 | Q8 | 1 | 4 | yes | complemented |
| 91 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 92 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 93 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 94 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 95 | C4⋊Dic3 | D4 | 1 | 1 | yes | characteristic |
| 96 | C4⋊Dic3 | C23 | 1 | 1 | yes | characteristic |
| 97 | C4⋊Dic3 | C23 | 1 | 1 | yes | characteristic |
| 98 | C4⋊Dic3 | C23 | 1 | 1 | yes | characteristic |
| 99 | Dic3⋊C4 | — | 2 | 1 | no | — |
| 100 | Dic3⋊C4 | — | 2 | 1 | no | — |
| 101 | Dic3⋊C4 | — | 2 | 1 | no | — |
| 102 | Dic3⋊C4 | — | 2 | 1 | no | — |
| 103 | Dic3⋊C4 | — | 2 | 1 | no | — |
| 104 | Dic3⋊C4 | — | 2 | 1 | no | — |
| 105 | Dic3⋊C4 | — | 2 | 1 | no | — |
| 106 | Dic3⋊C4 | — | 2 | 1 | no | — |
| 107 | Dic3⋊C4 | — | 2 | 1 | no | — |
| 108 | Dic3⋊C4 | D4 | 1 | 1 | yes | characteristic |
| 109 | Dic3⋊C4 | D4 | 1 | 1 | yes | characteristic |
| 110 | C4×Dic3 | — | 2 | 1 | no | — |
| 111 | C4×Dic3 | — | 2 | 1 | no | — |
| 112 | C4×Dic3 | — | 2 | 1 | no | — |
| 113 | C4×Dic3 | — | 2 | 1 | no | — |
| 114 | C4×Dic3 | — | 2 | 1 | no | — |
| 115 | C4×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 116 | C4×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 117 | C2×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 118 | C2×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 119 | C2×Q16 | — | 3 | 1 | no | — |
| 120 | C4⋊Q8 | — | 3 | 1 | no | — |
| 121 | C4⋊Q8 | — | 3 | 1 | no | — |
| 122 | C4⋊Q8 | — | 6 | 1 | no | — |
| 123 | C4⋊Q8 | — | 6 | 1 | no | — |
| 124 | #125 | — | 3 | 1 | no | — |
| 125 | #125 | — | 3 | 1 | no | — |
| 126 | #125 | — | 3 | 1 | no | — |
| 127 | #125 | — | 6 | 1 | no | — |
| 128 | #125 | — | 6 | 1 | no | — |
| 129 | #125 | — | 6 | 1 | no | — |
| 130 | #125 | — | 6 | 1 | no | — |
| 131 | #125 | C2×D3 | 1 | 1 | yes | characteristic |
| 132 | C4×Q8 | — | 3 | 1 | no | — |
| 133 | C4×Q8 | — | 3 | 1 | no | — |
| 134 | C4×Q8 | — | 3 | 1 | no | — |
| 135 | C4×Q8 | — | 3 | 1 | no | — |
| 136 | C4×Q8 | — | 3 | 1 | no | — |
| 137 | C4×Q8 | — | 3 | 1 | no | — |
| 138 | C4×Q8 | — | 6 | 1 | no | — |
| 139 | C4×Q8 | — | 6 | 1 | no | — |
| 140 | C4×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 141 | #107 | — | 3 | 1 | no | — |
| 142 | #107 | — | 3 | 1 | no | — |
| 143 | #107 | — | 3 | 1 | no | — |
| 144 | #107 | — | 3 | 1 | no | — |
| 145 | #107 | C2×D3 | 1 | 1 | yes | characteristic |
| 146 | #106 | — | 3 | 1 | no | — |
| 147 | #106 | — | 3 | 1 | no | — |
| 148 | #106 | — | 3 | 1 | no | — |
| 149 | #106 | C2×D3 | 1 | 1 | yes | characteristic |
| 150 | C4⋊C8 | — | 3 | 1 | no | — |
| 151 | C4⋊C8 | — | 3 | 1 | no | — |
| 152 | C4⋊C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 153 | #103 | — | 3 | 1 | no | — |
| 154 | #103 | — | 3 | 1 | no | — |
| 155 | #103 | — | 3 | 1 | no | — |
| 156 | #103 | — | 3 | 1 | no | — |
| 157 | #103 | C2×D3 | 1 | 1 | yes | characteristic |
| 158 | #103 | C2×D3 | 1 | 1 | yes | characteristic |
| 159 | #97 | — | 3 | 1 | no | — |
| 160 | #97 | — | 3 | 1 | no | — |
| 161 | C4×C8 | — | 3 | 1 | no | — |
| 162 | C3×Q8 | — | 2 | 1 | no | — |
| 163 | C3×Q8 | C2×Q8 | 1 | 2 | yes | — |
| 164 | C2×C3×C4 | — | 2 | 1 | no | — |
| 165 | C2×C3×C4 | — | 2 | 1 | no | — |
| 166 | C2×C3×C4 | — | 2 | 1 | no | — |
| 167 | C2×C3×C4 | — | 2 | 1 | no | — |
| 168 | C2×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 169 | C2×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 170 | C2×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 171 | C2×Dic3 | — | 2 | 1 | no | — |
| 172 | C2×Dic3 | — | 2 | 1 | no | — |
| 173 | C2×Dic3 | — | 2 | 1 | no | — |
| 174 | C2×Dic3 | — | 2 | 1 | no | — |
| 175 | C2×Dic3 | C2×D4 | 1 | 1 | yes | characteristic |
| 176 | C2×Dic3 | C2×D4 | 1 | 1 | yes | characteristic |
| 177 | C2×Dic3 | C2×D4 | 1 | 1 | yes | characteristic |
| 178 | C2×Dic3 | C2×D4 | 1 | 1 | yes | characteristic |
| 179 | Dic6 | — | 2 | 1 | no | — |
| 180 | Dic6 | — | 4 | 1 | no | — |
| 181 | Dic6 | — | 4 | 2 | no | — |
| 182 | Dic6 | — | 4 | 2 | no | — |
| 183 | Dic6 | C2×Q8 | 1 | 2 | yes | — |
| 184 | C3×C8 | — | 2 | 1 | no | — |
| 185 | C3×C8 | — | 2 | 1 | no | — |
| 186 | C3⋊C8 | — | 2 | 1 | no | — |
| 187 | C3⋊C8 | C2×Q8 | 1 | 2 | yes | — |
| 188 | C2×Q8 | — | 3 | 1 | no | — |
| 189 | C2×Q8 | — | 3 | 1 | no | — |
| 190 | C2×Q8 | — | 6 | 1 | no | — |
| 191 | C2×Q8 | — | 6 | 1 | no | — |
| 192 | C2×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 193 | Q16 | — | 3 | 4 | no | — |
| 194 | C2×C8 | — | 3 | 1 | no | — |
| 195 | C2×C8 | — | 3 | 1 | no | — |
| 196 | C2×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 197 | C2×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 198 | C4⋊C4 | — | 2 | 1 | no | — |
| 199 | C4⋊C4 | — | 2 | 1 | no | — |
| 200 | C4⋊C4 | — | 2 | 1 | no | — |
| 201 | C4⋊C4 | — | 3 | 1 | no | — |
| 202 | C4⋊C4 | — | 3 | 1 | no | — |
| 203 | C4⋊C4 | — | 3 | 1 | no | — |
| 204 | C4⋊C4 | — | 3 | 1 | no | — |
| 205 | C4⋊C4 | — | 3 | 1 | no | — |
| 206 | C4⋊C4 | — | 3 | 1 | no | — |
| 207 | C4⋊C4 | — | 6 | 1 | no | — |
| 208 | C4⋊C4 | — | 6 | 1 | no | — |
| 209 | C4⋊C4 | — | 6 | 1 | no | — |
| 210 | C4⋊C4 | — | 6 | 1 | no | — |
| 211 | C4⋊C4 | — | 6 | 1 | no | — |
| 212 | C4⋊C4 | — | 6 | 1 | no | — |
| 213 | C4⋊C4 | — | 6 | 1 | no | — |
| 214 | C4⋊C4 | — | 6 | 1 | no | — |
| 215 | C4⋊C4 | — | 6 | 1 | no | — |
| 216 | C4⋊C4 | — | 6 | 1 | no | — |
| 217 | C4⋊C4 | — | 6 | 1 | no | — |
| 218 | C4⋊C4 | — | 6 | 1 | no | — |
| 219 | C4⋊C4 | — | 6 | 1 | no | — |
| 220 | C4⋊C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 221 | C4⋊C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 222 | C4⋊C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 223 | C42 | — | 2 | 1 | no | — |
| 224 | C42 | — | 3 | 1 | no | — |
| 225 | C42 | — | 3 | 1 | no | — |
| 226 | C42 | — | 6 | 1 | no | — |
| 227 | C42 | — | 6 | 1 | no | — |
| 228 | C42 | — | 6 | 1 | no | — |
| 229 | C42 | — | 6 | 1 | no | — |
| 230 | C42 | — | 6 | 1 | no | — |
| 231 | C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 232 | C22×C3 | C22×D4 | 1 | 1 | yes | characteristic |
| 233 | C3×C4 | — | 2 | 2 | no | — |
| 234 | C3×C4 | — | 2 | 1 | no | — |
| 235 | C3×C4 | — | 2 | 1 | no | — |
| 236 | C3×C4 | — | 4 | 1 | no | — |
| 237 | C3×C4 | — | 4 | 1 | no | — |
| 238 | C3×C4 | — | 4 | 1 | no | — |
| 239 | C3×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 240 | C3×C4 | 2−1+4 | 1 | 1 | yes | characteristic |
| 241 | Dic3 | — | 2 | 2 | no | — |
| 242 | Dic3 | — | 2 | 1 | no | — |
| 243 | Dic3 | — | 2 | 1 | no | — |
| 244 | Dic3 | — | 2 | 1 | no | — |
| 245 | Dic3 | — | 2 | 1 | no | — |
| 246 | Dic3 | — | 4 | 1 | no | — |
| 247 | Dic3 | — | 4 | 1 | no | — |
| 248 | Dic3 | — | 4 | 1 | no | — |
| 249 | Q8 | — | 2 | 1 | no | — |
| 250 | Q8 | — | 3 | 2 | no | — |
| 251 | Q8 | — | 6 | 1 | no | — |
| 252 | Q8 | — | 12 | 1 | no | — |
| 253 | Q8 | — | 12 | 2 | no | — |
| 254 | Q8 | — | 12 | 2 | no | — |
| 255 | Q8 | D3×Q8 | 1 | 2 | yes | — |
| 256 | C2×C4 | — | 2 | 1 | no | — |
| 257 | C2×C4 | — | 2 | 1 | no | — |
| 258 | C2×C4 | — | 2 | 1 | no | — |
| 259 | C2×C4 | — | 2 | 1 | no | — |
| 260 | C2×C4 | — | 3 | 1 | no | — |
| 261 | C2×C4 | — | 3 | 1 | no | — |
| 262 | C2×C4 | — | 3 | 1 | no | — |
| 263 | C2×C4 | — | 3 | 1 | no | — |
| 264 | C2×C4 | — | 6 | 1 | no | — |
| 265 | C2×C4 | — | 6 | 1 | no | — |
| 266 | C2×C4 | — | 6 | 1 | no | — |
| 267 | C2×C4 | — | 6 | 1 | no | — |
| 268 | C2×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 269 | C2×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 270 | C2×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 271 | C8 | — | 2 | 1 | no | — |
| 272 | C8 | — | 2 | 1 | no | — |
| 273 | C8 | — | 3 | 2 | no | — |
| 274 | C8 | — | 6 | 1 | no | — |
| 275 | C2×C3 | D4×Q8 | 1 | 1 | yes | characteristic |
| 276 | C2×C3 | #575 | 1 | 1 | yes | characteristic |
| 277 | C2×C3 | #578 | 1 | 1 | yes | characteristic |
| 278 | C22 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 279 | C4 | — | 2 | 2 | no | — |
| 280 | C4 | — | 2 | 1 | no | — |
| 281 | C4 | — | 2 | 1 | no | — |
| 282 | C4 | — | 4 | 1 | no | — |
| 283 | C4 | — | 4 | 1 | no | — |
| 284 | C4 | — | 4 | 1 | no | — |
| 285 | C4 | — | 6 | 2 | no | — |
| 286 | C4 | — | 6 | 1 | no | — |
| 287 | C4 | — | 6 | 1 | no | — |
| 288 | C4 | — | 6 | 1 | no | — |
| 289 | C4 | — | 6 | 1 | no | — |
| 290 | C4 | — | 12 | 1 | no | — |
| 291 | C4 | — | 12 | 1 | no | — |
| 292 | C4 | — | 12 | 1 | no | — |
| 293 | C4 | C2×D3×Q8 | 1 | 1 | yes | characteristic |
| 294 | C4 | #1010 | 1 | 1 | yes | characteristic |
| 295 | C3 | #3390 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 296 | C2 | #5632 | 1 | 1 | yes | characteristic |
| 297 | C2 | #5768 | 1 | 1 | yes | characteristic |
| 298 | C2 | #5779 | 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