← #28481823
Subgroups — #28481823
Order 15104.
466 conjugacy classes of subgroups.
The subgroup lattice has 274 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #21424234 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #21422844 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #21422521 | C2 | 1 | 2 | yes | maximal |
| 4 | #21421993 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C2×C8×D4×C59 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #16837455 | C22 | 1 | 2 | yes | — |
| 7 | #16837426 | C22 | 1 | 2 | yes | — |
| 8 | #16837167 | — | 2 | 1 | no | — |
| 9 | #16837167 | — | 2 | 1 | no | — |
| 10 | #16837155 | — | 2 | 1 | no | — |
| 11 | #16837153 | — | 2 | 1 | no | — |
| 12 | D4⋊(C59⋊C8) | — | 2 | 4 | no | — |
| 13 | C23×C8×C59 | — | 2 | 1 | no | — |
| 14 | C2×C4×D4×C59 | C22 | 1 | 1 | yes | characteristic |
| 15 | C8×D4×C59 | — | 2 | 1 | no | — |
| 16 | C8×D4×C59 | — | 2 | 1 | no | — |
| 17 | C8×D4×C59 | C4 | 1 | 2 | yes | — |
| 18 | C8×D4×C59 | C4 | 1 | 2 | yes | — |
| 19 | C2×C4⋊C8×C59 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×C22⋊C8×C59 | — | 2 | 1 | no | — |
| 21 | C2×C4×C8×C59 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×#11745664 | — | 2 | 2 | no | — |
| 23 | C4⋊(C59⋊C8) | — | 2 | 2 | no | — |
| 24 | C4⋊(C59⋊C8) | — | 2 | 2 | no | — |
| 25 | C4⋊(C59⋊C8) | — | 2 | 2 | no | — |
| 26 | C22⋊(C59⋊C8) | — | 2 | 2 | no | — |
| 27 | C4×C59⋊C8 | — | 2 | 2 | no | — |
| 28 | C22×D4×C59 | Q8 | 1 | 1 | yes | characteristic |
| 29 | C23×C4×C59 | — | 2 | 1 | no | — |
| 30 | C22×C8×C59 | — | 2 | 1 | no | — |
| 31 | C22×C8×C59 | — | 2 | 2 | no | — |
| 32 | C22×C8×C59 | — | 2 | 2 | no | — |
| 33 | C22×C8×C59 | — | 4 | 4 | no | — |
| 34 | C22×C8×C59 | D4 | 1 | 1 | yes | characteristic |
| 35 | C22×C8×C59 | D4 | 1 | 1 | yes | characteristic |
| 36 | C4×D4×C59 | — | 2 | 1 | no | — |
| 37 | C4×D4×C59 | — | 2 | 1 | no | — |
| 38 | C4×D4×C59 | C2×C4 | 1 | 2 | yes | — |
| 39 | C4×D4×C59 | D4 | 1 | 2 | yes | — |
| 40 | C2×C4⋊C4×C59 | Q8 | 1 | 1 | yes | characteristic |
| 41 | C2×C22⋊C4×C59 | — | 2 | 1 | no | — |
| 42 | C2×C42×C59 | C23 | 1 | 1 | yes | characteristic |
| 43 | C4⋊C8×C59 | — | 2 | 1 | no | — |
| 44 | C4⋊C8×C59 | C2×C4 | 1 | 2 | yes | — |
| 45 | C22⋊C8×C59 | — | 2 | 2 | no | — |
| 46 | C22⋊C8×C59 | — | 2 | 2 | no | — |
| 47 | C4×C8×C59 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | C4×C8×C59 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C4×C8×C59 | D4 | 1 | 1 | yes | characteristic |
| 50 | C4×C8×C59 | D4 | 1 | 1 | yes | characteristic |
| 51 | C24×C59 | — | 2 | 1 | no | — |
| 52 | C2×D4×C59 | — | 2 | 4 | no | — |
| 53 | C2×D4×C59 | — | 2 | 1 | no | — |
| 54 | C2×D4×C59 | C4⋊C4 | 1 | 2 | yes | — |
| 55 | C22×C4×C59 | — | 2 | 1 | no | — |
| 56 | C22×C4×C59 | — | 2 | 2 | no | — |
| 57 | C22×C4×C59 | — | 2 | 2 | no | — |
| 58 | C22×C4×C59 | — | 4 | 2 | no | — |
| 59 | C22×C4×C59 | — | 4 | 2 | no | — |
| 60 | C22×C4×C59 | C2×D4 | 1 | 1 | yes | characteristic |
| 61 | C22×C4×C59 | C2×Q8 | 1 | 1 | yes | characteristic |
| 62 | C22×C4×C59 | #41 | 1 | 1 | yes | characteristic |
| 63 | C2×C8×C59 | — | 2 | 2 | no | — |
| 64 | C2×C8×C59 | — | 2 | 2 | no | — |
| 65 | C2×C8×C59 | — | 2 | 2 | no | — |
| 66 | C2×C8×C59 | — | 4 | 4 | no | — |
| 67 | C2×C8×C59 | — | 4 | 4 | no | — |
| 68 | C2×C8×C59 | — | 4 | 2 | no | — |
| 69 | C2×C8×C59 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 70 | C2×C8×C59 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 71 | C2×C8×C59 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 72 | C2×C8×C59 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 73 | C4⋊C4×C59 | — | 2 | 1 | no | — |
| 74 | C4⋊C4×C59 | C4⋊C4 | 1 | 2 | yes | — |
| 75 | C22⋊C4×C59 | — | 2 | 2 | no | — |
| 76 | C22⋊C4×C59 | — | 2 | 2 | no | — |
| 77 | C42×C59 | C22×C4 | 1 | 1 | yes | characteristic |
| 78 | C42×C59 | C2×D4 | 1 | 1 | yes | characteristic |
| 79 | C42×C59 | C2×D4 | 1 | 1 | yes | characteristic |
| 80 | C42×C59 | #41 | 1 | 1 | yes | characteristic |
| 81 | #11745664 | — | 4 | 2 | no | — |
| 82 | #11745664 | — | 4 | 2 | no | — |
| 83 | C2×C59⋊C8 | — | 4 | 2 | no | — |
| 84 | C2×C59⋊C8 | — | 4 | 2 | no | — |
| 85 | C23×C59 | — | 2 | 4 | no | — |
| 86 | C23×C59 | — | 2 | 2 | no | — |
| 87 | C23×C59 | — | 4 | 2 | no | — |
| 88 | C23×C59 | — | 4 | 2 | no | — |
| 89 | C23×C59 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 90 | D4×C59 | — | 2 | 4 | no | — |
| 91 | D4×C59 | — | 2 | 2 | no | — |
| 92 | D4×C59 | #108 | 1 | 4 | yes | — |
| 93 | C2×C4×C59 | — | 2 | 2 | no | — |
| 94 | C2×C4×C59 | — | 2 | 2 | no | — |
| 95 | C2×C4×C59 | — | 4 | 1 | no | — |
| 96 | C2×C4×C59 | — | 4 | 2 | no | — |
| 97 | C2×C4×C59 | — | 4 | 2 | no | — |
| 98 | C2×C4×C59 | — | 4 | 2 | no | — |
| 99 | C2×C4×C59 | — | 4 | 4 | no | — |
| 100 | C2×C4×C59 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 101 | C2×C4×C59 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 102 | C2×C4×C59 | #117 | 1 | 1 | yes | characteristic |
| 103 | C2×C4×C59 | #121 | 1 | 1 | yes | characteristic |
| 104 | C2×C4×C59 | #121 | 1 | 1 | yes | characteristic |
| 105 | C2×C4×C59 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 106 | C2×C4×C59 | #135 | 1 | 2 | yes | — |
| 107 | C2×C4×C59 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 108 | C8×C59 | — | 2 | 1 | no | — |
| 109 | C8×C59 | — | 2 | 1 | no | — |
| 110 | C8×C59 | — | 2 | 1 | no | — |
| 111 | C8×C59 | — | 2 | 1 | no | — |
| 112 | C8×C59 | — | 4 | 2 | no | — |
| 113 | C59⋊C8 | — | 8 | 2 | no | — |
| 114 | C59⋊C8 | — | 8 | 2 | no | — |
| 115 | #8660 | — | 59 | 1 | no | maximal, Sylow |
| 116 | C22×C59 | — | 2 | 1 | no | — |
| 117 | C22×C59 | — | 2 | 1 | no | — |
| 118 | C22×C59 | — | 2 | 4 | no | — |
| 119 | C22×C59 | — | 4 | 8 | no | — |
| 120 | C22×C59 | — | 4 | 2 | no | — |
| 121 | C22×C59 | — | 4 | 2 | no | — |
| 122 | C22×C59 | #380 | 1 | 1 | yes | characteristic |
| 123 | C22×C59 | #478 | 1 | 2 | yes | — |
| 124 | C4×C59 | — | 2 | 1 | no | — |
| 125 | C4×C59 | — | 2 | 1 | no | — |
| 126 | C4×C59 | — | 2 | 1 | no | — |
| 127 | C4×C59 | — | 4 | 2 | no | — |
| 128 | C4×C59 | #416 | 1 | 1 | yes | characteristic |
| 129 | C4×C59 | #418 | 1 | 1 | yes | characteristic |
| 130 | C4×C59 | C2×#108 | 1 | 2 | yes | — |
| 131 | C4×C59 | #463 | 1 | 2 | yes | — |
| 132 | C2×C8×D4 | D59 | 1 | 1 | yes | characteristic |
| 133 | #1837 | — | 59 | 1 | no | — |
| 134 | #1564 | — | 59 | 1 | no | — |
| 135 | #1486 | — | 59 | 2 | no | — |
| 136 | #1301 | — | 59 | 2 | no | — |
| 137 | C2×C59 | — | 2 | 2 | no | — |
| 138 | C2×C59 | — | 4 | 4 | no | — |
| 139 | C2×C59 | #1804 | 1 | 1 | yes | characteristic |
| 140 | C2×C59 | #1810 | 1 | 1 | yes | characteristic |
| 141 | C2×C59 | #1814 | 1 | 1 | yes | characteristic |
| 142 | C23×C8 | — | 2 | 1 | no | — |
| 143 | C2×C4×D4 | C2×D59 | 1 | 1 | yes | characteristic |
| 144 | C8×D4 | — | 2 | 1 | no | — |
| 145 | C8×D4 | — | 2 | 1 | no | — |
| 146 | C8×D4 | Dic59 | 1 | 2 | yes | — |
| 147 | C8×D4 | Dic59 | 1 | 2 | yes | — |
| 148 | #431 | — | 59 | 2 | no | — |
| 149 | #423 | — | 59 | 2 | no | — |
| 150 | C2×C4⋊C8 | C2×D59 | 1 | 1 | yes | characteristic |
| 151 | C2×C22⋊C8 | — | 2 | 1 | no | — |
| 152 | C2×C4×C8 | C2×D59 | 1 | 1 | yes | characteristic |
| 153 | #341 | — | 118 | 1 | no | — |
| 154 | #341 | — | 118 | 1 | no | — |
| 155 | #335 | — | 118 | 1 | no | — |
| 156 | #334 | — | 118 | 1 | no | — |
| 157 | D4⋊C8 | — | 118 | 4 | no | — |
| 158 | C59 | #8660 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 159 | C22×D4 | Dic118 | 1 | 1 | yes | characteristic |
| 160 | C23×C4 | — | 2 | 1 | no | — |
| 161 | C2×M16 | — | 118 | 2 | no | — |
| 162 | C22×C8 | — | 2 | 1 | no | — |
| 163 | C22×C8 | — | 2 | 2 | no | — |
| 164 | C22×C8 | — | 2 | 2 | no | — |
| 165 | C22×C8 | — | 4 | 4 | no | — |
| 166 | C22×C8 | C22⋊D59 | 1 | 1 | yes | characteristic |
| 167 | C22×C8 | C22⋊D59 | 1 | 1 | yes | characteristic |
| 168 | C4×D4 | — | 2 | 1 | no | — |
| 169 | C4×D4 | — | 2 | 1 | no | — |
| 170 | C4×D4 | D236 | 1 | 2 | yes | — |
| 171 | C4×D4 | C2×Dic59 | 1 | 2 | yes | — |
| 172 | C2×C4⋊C4 | Dic118 | 1 | 1 | yes | characteristic |
| 173 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 174 | C2×C42 | C22×D59 | 1 | 1 | yes | characteristic |
| 175 | C4⋊C8 | — | 2 | 1 | no | — |
| 176 | C4⋊C8 | — | 118 | 2 | no | — |
| 177 | C4⋊C8 | — | 118 | 2 | no | — |
| 178 | C4⋊C8 | — | 118 | 2 | no | — |
| 179 | C4⋊C8 | C2×Dic59 | 1 | 2 | yes | — |
| 180 | C22⋊C8 | — | 2 | 2 | no | — |
| 181 | C22⋊C8 | — | 2 | 2 | no | — |
| 182 | C22⋊C8 | — | 118 | 2 | no | — |
| 183 | C4×C8 | — | 118 | 2 | no | — |
| 184 | C4×C8 | C2×Dic59 | 1 | 1 | yes | characteristic |
| 185 | C4×C8 | C2×Dic59 | 1 | 1 | yes | characteristic |
| 186 | C4×C8 | C22⋊D59 | 1 | 1 | yes | characteristic |
| 187 | C4×C8 | C22⋊D59 | 1 | 1 | yes | characteristic |
| 188 | C24 | — | 2 | 1 | no | — |
| 189 | C2×D4 | — | 2 | 4 | no | — |
| 190 | C2×D4 | — | 2 | 1 | no | — |
| 191 | C2×D4 | C4⋊Dic59 | 1 | 2 | yes | — |
| 192 | C22×C4 | — | 2 | 1 | no | — |
| 193 | C22×C4 | — | 2 | 2 | no | — |
| 194 | C22×C4 | — | 2 | 2 | no | — |
| 195 | C22×C4 | — | 4 | 2 | no | — |
| 196 | C22×C4 | — | 4 | 2 | no | — |
| 197 | C22×C4 | C2×Dic118 | 1 | 1 | yes | characteristic |
| 198 | C22×C4 | #11745685 | 1 | 1 | yes | characteristic |
| 199 | C22×C4 | C2×C22⋊D59 | 1 | 1 | yes | characteristic |
| 200 | M16 | — | 236 | 2 | no | — |
| 201 | M16 | — | 236 | 2 | no | — |
| 202 | C2×C8 | — | 2 | 2 | no | — |
| 203 | C2×C8 | — | 2 | 2 | no | — |
| 204 | C2×C8 | — | 2 | 2 | no | — |
| 205 | C2×C8 | — | 4 | 4 | no | — |
| 206 | C2×C8 | — | 4 | 4 | no | — |
| 207 | C2×C8 | — | 4 | 2 | no | — |
| 208 | C2×C8 | — | 236 | 2 | no | — |
| 209 | C2×C8 | — | 236 | 2 | no | — |
| 210 | C2×C8 | C22⋊Dic59 | 1 | 1 | yes | characteristic |
| 211 | C2×C8 | C22⋊Dic59 | 1 | 1 | yes | characteristic |
| 212 | C2×C8 | C22⋊Dic59 | 1 | 1 | yes | characteristic |
| 213 | C2×C8 | C22⋊Dic59 | 1 | 1 | yes | characteristic |
| 214 | C4⋊C4 | — | 2 | 1 | no | — |
| 215 | C4⋊C4 | C4⋊Dic59 | 1 | 2 | yes | — |
| 216 | C22⋊C4 | — | 2 | 2 | no | — |
| 217 | C22⋊C4 | — | 2 | 2 | no | — |
| 218 | C42 | C2×D236 | 1 | 1 | yes | characteristic |
| 219 | C42 | #11745685 | 1 | 1 | yes | characteristic |
| 220 | C42 | C22×Dic59 | 1 | 1 | yes | characteristic |
| 221 | C42 | C2×C22⋊D59 | 1 | 1 | yes | characteristic |
| 222 | C23 | — | 2 | 4 | no | — |
| 223 | C23 | — | 2 | 2 | no | — |
| 224 | C23 | — | 4 | 2 | no | — |
| 225 | C23 | — | 4 | 2 | no | — |
| 226 | C23 | #14273706 | 1 | 1 | yes | characteristic |
| 227 | D4 | — | 2 | 4 | no | — |
| 228 | D4 | — | 2 | 2 | no | — |
| 229 | D4 | #14273663 | 1 | 4 | yes | — |
| 230 | C2×C4 | — | 2 | 2 | no | — |
| 231 | C2×C4 | — | 2 | 2 | no | — |
| 232 | C2×C4 | — | 4 | 1 | no | — |
| 233 | C2×C4 | — | 4 | 2 | no | — |
| 234 | C2×C4 | — | 4 | 2 | no | — |
| 235 | C2×C4 | — | 4 | 2 | no | — |
| 236 | C2×C4 | — | 4 | 4 | no | — |
| 237 | C2×C4 | C2×C22⋊Dic59 | 1 | 1 | yes | characteristic |
| 238 | C2×C4 | C2×C4⋊Dic59 | 1 | 1 | yes | characteristic |
| 239 | C2×C4 | #14273686 | 1 | 1 | yes | characteristic |
| 240 | C2×C4 | #14273701 | 1 | 1 | yes | characteristic |
| 241 | C2×C4 | #14273701 | 1 | 1 | yes | characteristic |
| 242 | C2×C4 | #14273706 | 1 | 1 | yes | characteristic |
| 243 | C2×C4 | #14273747 | 1 | 2 | yes | — |
| 244 | C2×C4 | #14273753 | 1 | 2 | yes | — |
| 245 | C8 | — | 2 | 1 | no | — |
| 246 | C8 | — | 2 | 1 | no | — |
| 247 | C8 | — | 2 | 1 | no | — |
| 248 | C8 | — | 2 | 1 | no | — |
| 249 | C8 | — | 4 | 2 | no | — |
| 250 | C8 | — | 472 | 2 | no | — |
| 251 | C8 | — | 472 | 2 | no | — |
| 252 | C22 | — | 2 | 1 | no | — |
| 253 | C22 | — | 2 | 1 | no | — |
| 254 | C22 | — | 2 | 4 | no | — |
| 255 | C22 | — | 4 | 8 | no | — |
| 256 | C22 | — | 4 | 2 | no | — |
| 257 | C22 | — | 4 | 2 | no | — |
| 258 | C22 | #16837262 | 1 | 1 | yes | characteristic |
| 259 | C22 | #16837702 | 1 | 2 | yes | — |
| 260 | C4 | — | 2 | 1 | no | — |
| 261 | C4 | — | 2 | 1 | no | — |
| 262 | C4 | — | 2 | 1 | no | — |
| 263 | C4 | — | 4 | 2 | no | — |
| 264 | C4 | #16837398 | 1 | 1 | yes | characteristic |
| 265 | C4 | #16837407 | 1 | 1 | yes | characteristic |
| 266 | C4 | C2×#14273663 | 1 | 2 | yes | — |
| 267 | C4 | #16837617 | 1 | 2 | yes | — |
| 268 | C2 | — | 2 | 2 | no | — |
| 269 | C2 | — | 4 | 4 | no | — |
| 270 | C2 | #21424106 | 1 | 1 | yes | characteristic |
| 271 | C2 | #21424134 | 1 | 1 | yes | characteristic |
| 272 | C2 | #21424150 | 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