← #225796
Subgroups — #225796
Order 512.
517 conjugacy classes of subgroups.
The subgroup lattice has 340 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #18059 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #18055 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #17886 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #17841 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #9634 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #9634 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #8901 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #8900 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #8824 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #8672 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | C8×#108 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #2991 | C22 | 1 | 2 | yes | — |
| 13 | #2990 | C22 | 1 | 2 | yes | — |
| 14 | #2965 | C22 | 1 | 2 | yes | — |
| 15 | C4×#135 | C22 | 1 | 1 | yes | characteristic |
| 16 | #1837 | C22 | 1 | 2 | yes | — |
| 17 | C4×#108 | C22 | 1 | 1 | yes | characteristic |
| 18 | #1591 | — | 2 | 1 | no | — |
| 19 | #1591 | C4 | 1 | 1 | yes | characteristic |
| 20 | #1591 | C4 | 1 | 1 | yes | characteristic |
| 21 | #1590 | C4 | 1 | 2 | yes | — |
| 22 | #1590 | C4 | 1 | 2 | yes | — |
| 23 | #1589 | — | 2 | 1 | no | — |
| 24 | #1589 | C4 | 1 | 1 | yes | characteristic |
| 25 | #1589 | C4 | 1 | 1 | yes | characteristic |
| 26 | #1588 | — | 2 | 2 | no | — |
| 27 | #1585 | — | 2 | 2 | no | — |
| 28 | C8×Q16 | — | 2 | 1 | no | — |
| 29 | C8×D8 | — | 2 | 1 | no | — |
| 30 | #1574 | C22 | 1 | 2 | yes | — |
| 31 | #1574 | C22 | 1 | 2 | yes | — |
| 32 | #1572 | — | 2 | 2 | no | — |
| 33 | #1570 | — | 2 | 2 | no | — |
| 34 | #1569 | — | 2 | 2 | no | — |
| 35 | #1567 | C22 | 1 | 1 | yes | characteristic |
| 36 | #1567 | C22 | 1 | 1 | yes | characteristic |
| 37 | #1566 | C22 | 1 | 2 | yes | — |
| 38 | #1565 | — | 2 | 2 | no | — |
| 39 | #1564 | C22 | 1 | 1 | yes | characteristic |
| 40 | #1564 | C22 | 1 | 1 | yes | characteristic |
| 41 | #1489 | C22 | 1 | 2 | yes | — |
| 42 | #1489 | C22 | 1 | 2 | yes | — |
| 43 | #1476 | C22 | 1 | 2 | yes | — |
| 44 | C8×M16 | C22 | 1 | 1 | yes | characteristic |
| 45 | C8×M16 | C22 | 1 | 1 | yes | characteristic |
| 46 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 47 | #1299 | C22 | 1 | 2 | yes | — |
| 48 | #1299 | C22 | 1 | 2 | yes | — |
| 49 | #1278 | C22 | 1 | 1 | yes | characteristic |
| 50 | #1278 | C22 | 1 | 1 | yes | characteristic |
| 51 | C2×#135 | Q8 | 1 | 1 | yes | characteristic |
| 52 | C4×#41 | C23 | 1 | 2 | yes | — |
| 53 | #446 | — | 2 | 2 | no | — |
| 54 | #446 | C2×C4 | 1 | 2 | yes | — |
| 55 | #446 | C2×C4 | 1 | 2 | yes | — |
| 56 | C8×Q8 | — | 2 | 2 | no | — |
| 57 | C4×Q16 | C2×C4 | 1 | 1 | yes | characteristic |
| 58 | C4×Q16 | D4 | 1 | 1 | yes | characteristic |
| 59 | C4×SD16 | C2×C4 | 1 | 2 | yes | — |
| 60 | C4×SD16 | D4 | 1 | 2 | yes | — |
| 61 | C4×D8 | C2×C4 | 1 | 1 | yes | characteristic |
| 62 | C4×D8 | D4 | 1 | 1 | yes | characteristic |
| 63 | #436 | — | 2 | 2 | no | — |
| 64 | #436 | — | 4 | 2 | no | — |
| 65 | #436 | C2×C4 | 1 | 2 | yes | — |
| 66 | #436 | C2×C4 | 1 | 2 | yes | — |
| 67 | #435 | — | 2 | 2 | no | — |
| 68 | #435 | — | 2 | 2 | no | — |
| 69 | #435 | — | 4 | 2 | no | — |
| 70 | C8×D4 | — | 2 | 2 | no | — |
| 71 | #432 | — | 2 | 2 | no | — |
| 72 | #431 | — | 2 | 2 | no | — |
| 73 | #431 | — | 2 | 2 | no | — |
| 74 | #431 | C23 | 1 | 2 | yes | — |
| 75 | #431 | C23 | 1 | 1 | yes | characteristic |
| 76 | #431 | C23 | 1 | 1 | yes | characteristic |
| 77 | C2×#108 | — | 2 | 1 | no | — |
| 78 | #427 | Q8 | 1 | 1 | yes | characteristic |
| 79 | #423 | — | 2 | 2 | no | — |
| 80 | #423 | C23 | 1 | 2 | yes | — |
| 81 | #423 | C23 | 1 | 2 | yes | — |
| 82 | C2×C4⋊C8 | — | 2 | 2 | no | — |
| 83 | #416 | Q8 | 1 | 2 | yes | — |
| 84 | C4×M16 | — | 2 | 2 | no | — |
| 85 | C4×M16 | C23 | 1 | 1 | yes | characteristic |
| 86 | C4×M16 | C23 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 90 | #341 | — | 2 | 1 | no | — |
| 91 | #341 | — | 4 | 2 | no | — |
| 92 | #335 | — | 2 | 2 | no | — |
| 93 | #335 | — | 2 | 2 | no | — |
| 94 | #335 | — | 2 | 1 | no | — |
| 95 | #335 | — | 2 | 1 | no | — |
| 96 | #335 | — | 2 | 1 | no | — |
| 97 | #335 | — | 2 | 1 | no | — |
| 98 | #335 | C2×C4 | 1 | 1 | yes | characteristic |
| 99 | #335 | C2×C4 | 1 | 1 | yes | characteristic |
| 100 | #335 | D4 | 1 | 1 | yes | characteristic |
| 101 | #335 | D4 | 1 | 1 | yes | characteristic |
| 102 | #334 | — | 2 | 2 | no | — |
| 103 | #334 | — | 2 | 2 | no | — |
| 104 | #334 | — | 2 | 2 | no | — |
| 105 | #334 | — | 2 | 2 | no | — |
| 106 | #334 | C2×C4 | 1 | 2 | yes | — |
| 107 | #334 | D4 | 1 | 1 | yes | characteristic |
| 108 | #334 | D4 | 1 | 1 | yes | characteristic |
| 109 | Q8⋊C8 | — | 2 | 2 | no | — |
| 110 | Q8⋊C8 | — | 2 | 2 | no | — |
| 111 | Q8⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 112 | Q8⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 113 | D4⋊C8 | — | 2 | 2 | no | — |
| 114 | D4⋊C8 | — | 2 | 2 | no | — |
| 115 | D4⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 116 | D4⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 117 | #322 | — | 2 | 1 | no | — |
| 118 | #322 | — | 2 | 1 | no | — |
| 119 | C82 | — | 2 | 1 | no | — |
| 120 | C82 | — | 2 | 1 | no | — |
| 121 | C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 122 | C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 123 | C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 124 | C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 125 | C2×#41 | C2×Q8 | 1 | 2 | yes | — |
| 126 | #135 | — | 4 | 2 | no | — |
| 127 | C2×Q16 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 128 | C2×SD16 | C4⋊C4 | 1 | 2 | yes | — |
| 129 | C2×D8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 130 | C2×M16 | — | 2 | 2 | no | — |
| 131 | C2×M16 | — | 2 | 2 | no | — |
| 132 | C2×M16 | — | 2 | 1 | no | — |
| 133 | C2×M16 | — | 2 | 1 | no | — |
| 134 | C2×M16 | — | 2 | 2 | no | — |
| 135 | C2×M16 | — | 2 | 2 | no | — |
| 136 | C2×M16 | — | 4 | 2 | no | — |
| 137 | C22×C8 | — | 2 | 1 | no | — |
| 138 | C22×C8 | — | 4 | 2 | no | — |
| 139 | C22×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 140 | C22×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 141 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 142 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 143 | C4×Q8 | C22×C4 | 1 | 2 | yes | — |
| 144 | C4×Q8 | C2×D4 | 1 | 2 | yes | — |
| 145 | C4×D4 | — | 2 | 2 | no | — |
| 146 | C4×D4 | — | 2 | 2 | no | — |
| 147 | C4×D4 | C22×C4 | 1 | 2 | yes | — |
| 148 | C4×D4 | C2×D4 | 1 | 2 | yes | — |
| 149 | #117 | — | 2 | 2 | no | — |
| 150 | #117 | C2×Q8 | 1 | 2 | yes | — |
| 151 | C2×C42 | — | 2 | 2 | no | — |
| 152 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 153 | #108 | — | 2 | 4 | no | — |
| 154 | #107 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 155 | #107 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 156 | #106 | C4⋊C4 | 1 | 2 | yes | — |
| 157 | C4⋊C8 | — | 2 | 2 | no | — |
| 158 | C4⋊C8 | — | 2 | 2 | no | — |
| 159 | C4⋊C8 | — | 2 | 2 | no | — |
| 160 | C4⋊C8 | — | 2 | 2 | no | — |
| 161 | C4⋊C8 | — | 2 | 2 | no | — |
| 162 | C4⋊C8 | — | 2 | 2 | no | — |
| 163 | C4⋊C8 | — | 4 | 2 | no | — |
| 164 | C4⋊C8 | — | 4 | 2 | no | — |
| 165 | C4⋊C8 | C22×C4 | 1 | 2 | yes | — |
| 166 | C4⋊C8 | C22×C4 | 1 | 2 | yes | — |
| 167 | C4⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 168 | C4⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 169 | C4⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 170 | C4⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 171 | #103 | C4⋊C4 | 1 | 2 | yes | — |
| 172 | #103 | C4⋊C4 | 1 | 2 | yes | — |
| 173 | #102 | C4⋊C4 | 1 | 2 | yes | — |
| 174 | #102 | C4⋊C4 | 1 | 2 | yes | — |
| 175 | C22⋊C8 | — | 2 | 2 | no | — |
| 176 | C22⋊C8 | — | 2 | 1 | no | — |
| 177 | C22⋊C8 | — | 2 | 1 | no | — |
| 178 | C22⋊C8 | — | 2 | 2 | no | — |
| 179 | C22⋊C8 | — | 2 | 2 | no | — |
| 180 | C22⋊C8 | — | 4 | 2 | no | — |
| 181 | C22⋊C8 | — | 4 | 2 | no | — |
| 182 | #97 | — | 2 | 2 | no | — |
| 183 | #97 | — | 2 | 2 | no | — |
| 184 | #97 | — | 2 | 1 | no | — |
| 185 | #97 | — | 2 | 1 | no | — |
| 186 | #97 | — | 4 | 2 | no | — |
| 187 | C4×C8 | — | 2 | 1 | no | — |
| 188 | C4×C8 | — | 2 | 1 | no | — |
| 189 | C4×C8 | — | 2 | 2 | no | — |
| 190 | C4×C8 | — | 2 | 1 | no | — |
| 191 | C4×C8 | — | 2 | 1 | no | — |
| 192 | C4×C8 | — | 2 | 1 | no | — |
| 193 | C4×C8 | — | 4 | 2 | no | — |
| 194 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 195 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 196 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 197 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 198 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 199 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 200 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 201 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 202 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 203 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 204 | #41 | — | 4 | 2 | no | — |
| 205 | #41 | — | 4 | 2 | no | — |
| 206 | C2×Q8 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 207 | C2×D4 | — | 2 | 2 | no | — |
| 208 | C2×D4 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 209 | C22×C4 | — | 2 | 2 | no | — |
| 210 | C22×C4 | — | 2 | 2 | no | — |
| 211 | C22×C4 | — | 2 | 2 | no | — |
| 212 | C22×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 213 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 214 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 215 | Q16 | — | 2 | 2 | no | — |
| 216 | SD16 | — | 4 | 2 | no | — |
| 217 | D8 | — | 2 | 2 | no | — |
| 218 | M16 | — | 2 | 2 | no | — |
| 219 | M16 | — | 2 | 2 | no | — |
| 220 | M16 | — | 4 | 1 | no | — |
| 221 | M16 | — | 4 | 1 | no | — |
| 222 | M16 | — | 4 | 4 | no | — |
| 223 | M16 | — | 8 | 2 | no | — |
| 224 | M16 | — | 8 | 2 | no | — |
| 225 | M16 | — | 8 | 2 | no | — |
| 226 | M16 | — | 16 | 2 | no | — |
| 227 | C2×C8 | — | 2 | 2 | no | — |
| 228 | C2×C8 | — | 2 | 2 | no | — |
| 229 | C2×C8 | — | 2 | 2 | no | — |
| 230 | C2×C8 | — | 2 | 2 | no | — |
| 231 | C2×C8 | — | 2 | 1 | no | — |
| 232 | C2×C8 | — | 2 | 1 | no | — |
| 233 | C2×C8 | — | 2 | 1 | no | — |
| 234 | C2×C8 | — | 2 | 1 | no | — |
| 235 | C2×C8 | — | 2 | 1 | no | — |
| 236 | C2×C8 | — | 2 | 1 | no | — |
| 237 | C2×C8 | — | 4 | 1 | no | — |
| 238 | C2×C8 | — | 4 | 1 | no | — |
| 239 | C2×C8 | — | 4 | 1 | no | — |
| 240 | C2×C8 | — | 4 | 1 | no | — |
| 241 | C2×C8 | — | 4 | 1 | no | — |
| 242 | C2×C8 | — | 4 | 1 | no | — |
| 243 | C2×C8 | — | 4 | 2 | no | — |
| 244 | C2×C8 | — | 4 | 2 | no | — |
| 245 | C2×C8 | — | 16 | 2 | no | — |
| 246 | C2×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 247 | C2×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 248 | C2×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 249 | C2×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 250 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 251 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 252 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 253 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 254 | C4⋊C4 | — | 2 | 2 | no | — |
| 255 | C4⋊C4 | — | 2 | 2 | no | — |
| 256 | C4⋊C4 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 257 | C4⋊C4 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 258 | C22⋊C4 | — | 2 | 2 | no | — |
| 259 | C22⋊C4 | — | 2 | 2 | no | — |
| 260 | C22⋊C4 | — | 2 | 2 | no | — |
| 261 | C42 | — | 2 | 2 | no | — |
| 262 | C42 | — | 2 | 2 | no | — |
| 263 | C42 | — | 2 | 2 | no | — |
| 264 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 265 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 266 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 267 | C42 | 2−1+4 | 1 | 1 | yes | characteristic |
| 268 | C23 | — | 2 | 2 | no | — |
| 269 | C23 | #554 | 1 | 1 | yes | characteristic |
| 270 | Q8 | — | 2 | 2 | no | — |
| 271 | Q8 | — | 2 | 2 | no | — |
| 272 | D4 | — | 2 | 2 | no | — |
| 273 | D4 | — | 2 | 2 | no | — |
| 274 | D4 | — | 8 | 2 | no | — |
| 275 | C2×C4 | — | 2 | 1 | no | — |
| 276 | C2×C4 | — | 2 | 1 | no | — |
| 277 | C2×C4 | — | 2 | 1 | no | — |
| 278 | C2×C4 | — | 2 | 1 | no | — |
| 279 | C2×C4 | — | 2 | 1 | no | — |
| 280 | C2×C4 | — | 2 | 1 | no | — |
| 281 | C2×C4 | — | 2 | 2 | no | — |
| 282 | C2×C4 | — | 2 | 2 | no | — |
| 283 | C2×C4 | — | 2 | 2 | no | — |
| 284 | C2×C4 | — | 4 | 2 | no | — |
| 285 | C2×C4 | — | 4 | 2 | no | — |
| 286 | C2×C4 | — | 4 | 2 | no | — |
| 287 | C2×C4 | — | 4 | 2 | no | — |
| 288 | C2×C4 | — | 8 | 2 | no | — |
| 289 | C2×C4 | C22×C4⋊C4 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | #520 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | #547 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | D4×Q8 | 1 | 1 | yes | characteristic |
| 295 | C8 | — | 2 | 1 | no | — |
| 296 | C8 | — | 2 | 1 | no | — |
| 297 | C8 | — | 2 | 1 | no | — |
| 298 | C8 | — | 2 | 1 | no | — |
| 299 | C8 | — | 2 | 1 | no | — |
| 300 | C8 | — | 2 | 1 | no | — |
| 301 | C8 | — | 4 | 1 | no | — |
| 302 | C8 | — | 4 | 1 | no | — |
| 303 | C8 | — | 4 | 2 | no | — |
| 304 | C8 | — | 4 | 2 | no | — |
| 305 | C8 | — | 4 | 2 | no | — |
| 306 | C8 | — | 4 | 2 | no | — |
| 307 | C8 | — | 4 | 1 | no | — |
| 308 | C8 | — | 4 | 1 | no | — |
| 309 | C8 | — | 4 | 1 | no | — |
| 310 | C8 | — | 4 | 1 | no | — |
| 311 | C8 | — | 8 | 2 | no | — |
| 312 | C8 | — | 8 | 2 | no | — |
| 313 | C8 | #430 | 1 | 2 | yes | — |
| 314 | C8 | #443 | 1 | 2 | yes | — |
| 315 | C22 | — | 2 | 1 | no | — |
| 316 | C22 | — | 2 | 1 | no | — |
| 317 | C22 | — | 2 | 1 | no | — |
| 318 | C22 | — | 4 | 2 | no | — |
| 319 | C22 | — | 4 | 2 | no | — |
| 320 | C22 | — | 4 | 2 | no | — |
| 321 | C22 | D4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 322 | C4 | — | 2 | 1 | no | — |
| 323 | C4 | — | 2 | 1 | no | — |
| 324 | C4 | — | 2 | 1 | no | — |
| 325 | C4 | — | 4 | 2 | no | — |
| 326 | C4 | — | 4 | 2 | no | — |
| 327 | C4 | — | 4 | 2 | no | — |
| 328 | C4 | C2×#430 | 1 | 1 | yes | characteristic |
| 329 | C4 | #2916 | 1 | 1 | yes | characteristic |
| 330 | C4 | C2×#443 | 1 | 1 | yes | characteristic |
| 331 | C4 | #2957 | 1 | 1 | yes | characteristic |
| 332 | C4 | #3290 | 1 | 1 | yes | characteristic |
| 333 | C4 | #3290 | 1 | 1 | yes | characteristic |
| 334 | C2 | — | 4 | 1 | no | — |
| 335 | C2 | — | 8 | 2 | no | — |
| 336 | C2 | #22669 | 1 | 1 | yes | characteristic |
| 337 | C2 | #22688 | 1 | 1 | yes | characteristic |
| 338 | C2 | #22690 | 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