← #225786
Subgroups — #225786
Order 512.
533 conjugacy classes of subgroups.
The subgroup lattice has 279 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #18059 | C2 | 1 | 2 | yes | maximal |
| 2 | C8×#135 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #17835 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #9634 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #8901 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #8824 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #8670 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | C8×#108 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #2991 | C22 | 1 | 4 | yes | — |
| 10 | C8×#41 | C22 | 1 | 2 | yes | — |
| 11 | C4×#135 | C22 | 1 | 1 | yes | characteristic |
| 12 | #1837 | C22 | 1 | 2 | yes | — |
| 13 | C4×#108 | C22 | 1 | 1 | yes | characteristic |
| 14 | #1589 | — | 2 | 2 | no | — |
| 15 | #1588 | — | 2 | 4 | no | — |
| 16 | C8×Q16 | — | 2 | 2 | no | — |
| 17 | C8×Q16 | C4 | 1 | 1 | yes | characteristic |
| 18 | C8×Q16 | C4 | 1 | 1 | yes | characteristic |
| 19 | C8×SD16 | C4 | 1 | 2 | yes | — |
| 20 | C8×SD16 | C4 | 1 | 2 | yes | — |
| 21 | C8×D8 | C4 | 1 | 1 | yes | characteristic |
| 22 | C8×D8 | C4 | 1 | 1 | yes | characteristic |
| 23 | #1574 | — | 2 | 4 | no | — |
| 24 | #1574 | C22 | 1 | 4 | yes | — |
| 25 | #1567 | C22 | 1 | 2 | yes | — |
| 26 | #1566 | — | 2 | 2 | no | — |
| 27 | #1566 | C22 | 1 | 2 | yes | — |
| 28 | #1564 | — | 2 | 2 | no | — |
| 29 | #1564 | C22 | 1 | 1 | yes | characteristic |
| 30 | #1564 | C22 | 1 | 1 | yes | characteristic |
| 31 | #1489 | C22 | 1 | 4 | yes | — |
| 32 | #1476 | C22 | 1 | 2 | yes | — |
| 33 | C8×M16 | C22 | 1 | 2 | yes | — |
| 34 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 35 | #1299 | C22 | 1 | 4 | yes | — |
| 36 | #1278 | C22 | 1 | 2 | yes | — |
| 37 | C2×#135 | Q8 | 1 | 1 | yes | characteristic |
| 38 | C4×#41 | C23 | 1 | 2 | yes | — |
| 39 | C8×Q8 | — | 2 | 4 | no | — |
| 40 | C8×Q8 | C2×C4 | 1 | 2 | yes | — |
| 41 | C8×Q8 | C2×C4 | 1 | 2 | yes | — |
| 42 | C4×Q16 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | C4×Q16 | D4 | 1 | 1 | yes | characteristic |
| 44 | C4×SD16 | C2×C4 | 1 | 2 | yes | — |
| 45 | C4×SD16 | D4 | 1 | 2 | yes | — |
| 46 | C4×D8 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C4×D8 | D4 | 1 | 1 | yes | characteristic |
| 48 | #436 | — | 2 | 4 | no | — |
| 49 | #436 | — | 4 | 4 | no | — |
| 50 | C8×D4 | — | 2 | 2 | no | — |
| 51 | C8×D4 | — | 2 | 2 | no | — |
| 52 | C8×D4 | C2×C4 | 1 | 2 | yes | — |
| 53 | C8×D4 | C2×C4 | 1 | 2 | yes | — |
| 54 | #431 | — | 2 | 2 | no | — |
| 55 | #431 | — | 2 | 4 | no | — |
| 56 | #431 | C23 | 1 | 2 | yes | — |
| 57 | #431 | C23 | 1 | 2 | yes | — |
| 58 | C2×#108 | — | 2 | 1 | no | — |
| 59 | #427 | Q8 | 1 | 1 | yes | characteristic |
| 60 | #423 | — | 2 | 4 | no | — |
| 61 | #423 | C23 | 1 | 4 | yes | — |
| 62 | #416 | Q8 | 1 | 2 | yes | — |
| 63 | C4×M16 | C23 | 1 | 2 | yes | — |
| 64 | C2×C4×C8 | — | 2 | 2 | no | — |
| 65 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 66 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 67 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 68 | #341 | — | 2 | 1 | no | — |
| 69 | #341 | — | 4 | 2 | no | — |
| 70 | #335 | — | 2 | 4 | no | — |
| 71 | #335 | — | 2 | 2 | no | — |
| 72 | #335 | — | 2 | 2 | no | — |
| 73 | #335 | C2×C4 | 1 | 1 | yes | characteristic |
| 74 | #335 | C2×C4 | 1 | 1 | yes | characteristic |
| 75 | #335 | D4 | 1 | 2 | yes | — |
| 76 | #334 | — | 2 | 4 | no | — |
| 77 | #334 | — | 2 | 2 | no | — |
| 78 | #334 | — | 2 | 2 | no | — |
| 79 | #334 | C2×C4 | 1 | 2 | yes | — |
| 80 | #334 | D4 | 1 | 1 | yes | characteristic |
| 81 | #334 | D4 | 1 | 1 | yes | characteristic |
| 82 | Q8⋊C8 | — | 2 | 4 | no | — |
| 83 | Q8⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 84 | Q8⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 85 | D4⋊C8 | — | 2 | 4 | no | — |
| 86 | D4⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 87 | D4⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 88 | #322 | — | 2 | 2 | no | — |
| 89 | C82 | — | 2 | 2 | no | — |
| 90 | C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 91 | C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 92 | C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 93 | C82 | C2×C4 | 1 | 1 | yes | characteristic |
| 94 | C2×#41 | C2×Q8 | 1 | 2 | yes | — |
| 95 | #135 | — | 2 | 4 | no | — |
| 96 | C2×Q16 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 97 | C2×SD16 | C4⋊C4 | 1 | 2 | yes | — |
| 98 | C2×D8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 99 | C2×M16 | — | 2 | 2 | no | — |
| 100 | C2×M16 | — | 2 | 4 | no | — |
| 101 | C2×M16 | — | 4 | 4 | no | — |
| 102 | C22×C8 | — | 2 | 1 | no | — |
| 103 | C22×C8 | — | 2 | 2 | no | — |
| 104 | C22×C8 | — | 2 | 2 | no | — |
| 105 | C22×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 106 | C22×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 107 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 108 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 109 | C4×Q8 | C22×C4 | 1 | 2 | yes | — |
| 110 | C4×Q8 | C2×D4 | 1 | 2 | yes | — |
| 111 | C4×D4 | — | 2 | 2 | no | — |
| 112 | C4×D4 | — | 2 | 2 | no | — |
| 113 | C4×D4 | C22×C4 | 1 | 2 | yes | — |
| 114 | C4×D4 | C2×D4 | 1 | 2 | yes | — |
| 115 | #117 | — | 2 | 2 | no | — |
| 116 | #117 | C2×Q8 | 1 | 2 | yes | — |
| 117 | C2×C42 | — | 2 | 2 | no | — |
| 118 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 119 | #108 | — | 2 | 4 | no | — |
| 120 | #107 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 121 | #107 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 122 | #106 | C4⋊C4 | 1 | 2 | yes | — |
| 123 | C4⋊C8 | — | 2 | 2 | no | — |
| 124 | C4⋊C8 | — | 2 | 2 | no | — |
| 125 | C4⋊C8 | — | 2 | 4 | no | — |
| 126 | C4⋊C8 | — | 2 | 4 | no | — |
| 127 | C4⋊C8 | — | 4 | 4 | no | — |
| 128 | C4⋊C8 | C22×C4 | 1 | 2 | yes | — |
| 129 | C4⋊C8 | C22×C4 | 1 | 2 | yes | — |
| 130 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 131 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 132 | #103 | C4⋊C4 | 1 | 2 | yes | — |
| 133 | #103 | C4⋊C4 | 1 | 2 | yes | — |
| 134 | #102 | C4⋊C4 | 1 | 2 | yes | — |
| 135 | #102 | C4⋊C4 | 1 | 2 | yes | — |
| 136 | C22⋊C8 | — | 2 | 2 | no | — |
| 137 | C22⋊C8 | — | 2 | 2 | no | — |
| 138 | C22⋊C8 | — | 2 | 2 | no | — |
| 139 | C22⋊C8 | — | 2 | 2 | no | — |
| 140 | C22⋊C8 | — | 4 | 4 | no | — |
| 141 | #97 | — | 2 | 2 | no | — |
| 142 | C4×C8 | — | 2 | 2 | no | — |
| 143 | C4×C8 | — | 2 | 2 | no | — |
| 144 | C4×C8 | — | 2 | 2 | no | — |
| 145 | C4×C8 | — | 2 | 2 | no | — |
| 146 | C4×C8 | — | 2 | 2 | no | — |
| 147 | C4×C8 | — | 2 | 1 | no | — |
| 148 | C4×C8 | — | 4 | 4 | no | — |
| 149 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 150 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 151 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 152 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 153 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 154 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 155 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 156 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 157 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 158 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 159 | #41 | — | 2 | 4 | no | — |
| 160 | #41 | — | 4 | 2 | no | — |
| 161 | C2×Q8 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 162 | C2×D4 | — | 2 | 2 | no | — |
| 163 | C2×D4 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 164 | C22×C4 | — | 2 | 2 | no | — |
| 165 | C22×C4 | — | 2 | 2 | no | — |
| 166 | C22×C4 | — | 2 | 2 | no | — |
| 167 | C22×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 168 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 169 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 170 | Q16 | #108 | 1 | 4 | yes | complemented |
| 171 | SD16 | — | 2 | 4 | no | — |
| 172 | D8 | — | 2 | 2 | no | — |
| 173 | M16 | — | 2 | 4 | no | — |
| 174 | M16 | — | 4 | 2 | no | — |
| 175 | M16 | — | 8 | 4 | no | — |
| 176 | M16 | — | 16 | 4 | no | — |
| 177 | C2×C8 | — | 2 | 2 | no | — |
| 178 | C2×C8 | — | 2 | 2 | no | — |
| 179 | C2×C8 | — | 2 | 2 | no | — |
| 180 | C2×C8 | — | 2 | 2 | no | — |
| 181 | C2×C8 | — | 2 | 1 | no | — |
| 182 | C2×C8 | — | 2 | 1 | no | — |
| 183 | C2×C8 | — | 2 | 2 | no | — |
| 184 | C2×C8 | — | 2 | 2 | no | — |
| 185 | C2×C8 | — | 4 | 1 | no | — |
| 186 | C2×C8 | — | 4 | 1 | no | — |
| 187 | C2×C8 | — | 4 | 2 | no | — |
| 188 | C2×C8 | — | 4 | 2 | no | — |
| 189 | C2×C8 | — | 4 | 4 | no | — |
| 190 | C2×C8 | — | 4 | 4 | no | — |
| 191 | C2×C8 | — | 8 | 2 | no | — |
| 192 | C2×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 193 | C2×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 194 | C2×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 195 | C2×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 196 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 197 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 198 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 199 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 200 | C4⋊C4 | — | 2 | 2 | no | — |
| 201 | C4⋊C4 | — | 2 | 2 | no | — |
| 202 | C4⋊C4 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 203 | C4⋊C4 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 204 | C22⋊C4 | — | 2 | 2 | no | — |
| 205 | C22⋊C4 | — | 2 | 2 | no | — |
| 206 | C22⋊C4 | — | 2 | 2 | no | — |
| 207 | C42 | — | 2 | 2 | no | — |
| 208 | C42 | — | 2 | 2 | no | — |
| 209 | C42 | — | 2 | 2 | no | — |
| 210 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 211 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 212 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 213 | C42 | 2−1+4 | 1 | 1 | yes | characteristic |
| 214 | C23 | — | 2 | 2 | no | — |
| 215 | C23 | #554 | 1 | 1 | yes | characteristic |
| 216 | Q8 | — | 2 | 2 | no | — |
| 217 | Q8 | C2×#108 | 1 | 4 | yes | — |
| 218 | D4 | — | 2 | 2 | no | — |
| 219 | D4 | — | 2 | 2 | no | — |
| 220 | D4 | — | 4 | 4 | no | — |
| 221 | C2×C4 | — | 2 | 2 | no | — |
| 222 | C2×C4 | — | 2 | 1 | no | — |
| 223 | C2×C4 | — | 2 | 1 | no | — |
| 224 | C2×C4 | — | 2 | 1 | no | — |
| 225 | C2×C4 | — | 2 | 1 | no | — |
| 226 | C2×C4 | — | 2 | 2 | no | — |
| 227 | C2×C4 | — | 2 | 2 | no | — |
| 228 | C2×C4 | — | 2 | 2 | no | — |
| 229 | C2×C4 | — | 4 | 4 | no | — |
| 230 | C2×C4 | — | 4 | 2 | no | — |
| 231 | C2×C4 | — | 4 | 2 | no | — |
| 232 | C2×C4 | — | 4 | 4 | no | — |
| 233 | C2×C4 | C22×C4⋊C4 | 1 | 1 | yes | characteristic |
| 234 | C2×C4 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 235 | C2×C4 | #520 | 1 | 1 | yes | characteristic |
| 236 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 237 | C2×C4 | #547 | 1 | 1 | yes | characteristic |
| 238 | C2×C4 | D4×Q8 | 1 | 1 | yes | characteristic |
| 239 | C8 | — | 2 | 1 | no | — |
| 240 | C8 | — | 2 | 1 | no | — |
| 241 | C8 | — | 2 | 1 | no | — |
| 242 | C8 | — | 2 | 1 | no | — |
| 243 | C8 | — | 2 | 1 | no | — |
| 244 | C8 | — | 2 | 1 | no | — |
| 245 | C8 | — | 4 | 1 | no | — |
| 246 | C8 | — | 4 | 1 | no | — |
| 247 | C8 | — | 4 | 4 | no | — |
| 248 | C8 | — | 4 | 2 | no | — |
| 249 | C8 | — | 4 | 2 | no | — |
| 250 | C8 | — | 4 | 2 | no | — |
| 251 | C8 | — | 4 | 2 | no | — |
| 252 | C8 | — | 8 | 4 | no | — |
| 253 | C8 | C2×#108 | 1 | 2 | yes | — |
| 254 | C8 | #443 | 1 | 2 | yes | — |
| 255 | C22 | — | 2 | 1 | no | — |
| 256 | C22 | — | 2 | 1 | no | — |
| 257 | C22 | — | 2 | 1 | no | — |
| 258 | C22 | — | 4 | 2 | no | — |
| 259 | C22 | — | 4 | 2 | no | — |
| 260 | C22 | — | 4 | 2 | no | — |
| 261 | C22 | D4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 262 | C4 | — | 2 | 1 | no | — |
| 263 | C4 | — | 2 | 4 | no | — |
| 264 | C4 | — | 2 | 1 | no | — |
| 265 | C4 | — | 2 | 1 | no | — |
| 266 | C4 | — | 4 | 2 | no | — |
| 267 | C4 | — | 4 | 2 | no | — |
| 268 | C4 | C22×#108 | 1 | 1 | yes | characteristic |
| 269 | C4 | #2916 | 1 | 1 | yes | characteristic |
| 270 | C4 | C2×#443 | 1 | 1 | yes | characteristic |
| 271 | C4 | #2957 | 1 | 1 | yes | characteristic |
| 272 | C4 | #3290 | 1 | 2 | yes | — |
| 273 | C2 | — | 4 | 1 | no | — |
| 274 | C2 | — | 8 | 2 | no | — |
| 275 | C2 | D4×#108 | 1 | 1 | yes | characteristic |
| 276 | C2 | #22688 | 1 | 1 | yes | characteristic |
| 277 | 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