← #219613
Subgroups — #219613
Order 512.
476 conjugacy classes of subgroups.
The subgroup lattice has 272 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22258 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22254 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #19664 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #9778 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #9759 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #9298 | C2 | 1 | 2 | yes | maximal |
| 7 | #9297 | C2 | 1 | 2 | yes | maximal |
| 8 | #9290 | C2 | 1 | 2 | yes | maximal |
| 9 | #9288 | C2 | 1 | 2 | yes | maximal |
| 10 | #8791 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | #8142 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #3230 | C22 | 1 | 2 | yes | — |
| 13 | #3209 | C22 | 1 | 2 | yes | — |
| 14 | #3160 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#500 | C22 | 1 | 1 | yes | characteristic |
| 16 | #2961 | C22 | 1 | 2 | yes | — |
| 17 | #2331 | C22 | 1 | 2 | yes | — |
| 18 | #1995 | C22 | 1 | 2 | yes | — |
| 19 | #1993 | C22 | 1 | 2 | yes | — |
| 20 | #1922 | — | 2 | 2 | no | — |
| 21 | #1922 | — | 2 | 2 | no | — |
| 22 | #1921 | — | 2 | 2 | no | — |
| 23 | #1918 | — | 2 | 2 | no | — |
| 24 | #1837 | C22 | 1 | 1 | yes | characteristic |
| 25 | #1835 | C22 | 1 | 1 | yes | characteristic |
| 26 | #1746 | C22 | 1 | 1 | yes | characteristic |
| 27 | #1723 | C4 | 1 | 4 | yes | complemented |
| 28 | #1713 | C4 | 1 | 2 | yes | complemented |
| 29 | #1710 | C4 | 1 | 2 | yes | complemented |
| 30 | #1549 | C22 | 1 | 2 | yes | — |
| 31 | #1548 | C22 | 1 | 2 | yes | — |
| 32 | #1518 | C22 | 1 | 2 | yes | — |
| 33 | #1518 | C22 | 1 | 2 | yes | — |
| 34 | #1452 | C22 | 1 | 1 | yes | characteristic |
| 35 | #1450 | C22 | 1 | 1 | yes | characteristic |
| 36 | #1286 | C22 | 1 | 2 | yes | — |
| 37 | #1286 | C22 | 1 | 2 | yes | — |
| 38 | #1275 | C22 | 1 | 1 | yes | characteristic |
| 39 | #1275 | C22 | 1 | 1 | yes | characteristic |
| 40 | #1274 | C22 | 1 | 1 | yes | characteristic |
| 41 | #1274 | C22 | 1 | 1 | yes | characteristic |
| 42 | #533 | C23 | 1 | 2 | yes | — |
| 43 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 44 | #500 | — | 2 | 1 | no | — |
| 45 | #500 | — | 2 | 1 | no | — |
| 46 | #500 | C2×C4 | 1 | 2 | yes | — |
| 47 | #500 | C2×C4 | 1 | 2 | yes | — |
| 48 | #500 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | #500 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | #500 | C2×C4 | 1 | 4 | yes | — |
| 51 | #499 | — | 2 | 1 | no | — |
| 52 | #499 | — | 2 | 1 | no | — |
| 53 | #498 | C2×C4 | 1 | 2 | yes | — |
| 54 | #494 | C2×C4 | 1 | 4 | yes | — |
| 55 | #487 | C2×C4 | 1 | 4 | yes | — |
| 56 | #487 | C2×C4 | 1 | 4 | yes | — |
| 57 | #484 | — | 4 | 2 | no | — |
| 58 | #477 | — | 4 | 2 | no | — |
| 59 | #462 | — | 4 | 2 | no | — |
| 60 | #451 | — | 4 | 2 | no | — |
| 61 | #446 | — | 4 | 2 | no | — |
| 62 | #446 | — | 4 | 2 | no | — |
| 63 | #431 | C23 | 1 | 1 | yes | characteristic |
| 64 | #431 | C23 | 1 | 1 | yes | characteristic |
| 65 | #431 | C23 | 1 | 1 | yes | characteristic |
| 66 | #431 | C23 | 1 | 1 | yes | characteristic |
| 67 | #427 | — | 2 | 1 | no | — |
| 68 | #427 | — | 2 | 1 | no | — |
| 69 | C2×#107 | — | 2 | 1 | no | — |
| 70 | C2×#107 | — | 2 | 1 | no | — |
| 71 | #424 | — | 2 | 2 | no | — |
| 72 | #423 | C23 | 1 | 2 | yes | — |
| 73 | #405 | — | 2 | 2 | no | — |
| 74 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 75 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 76 | #391 | — | 4 | 2 | no | — |
| 77 | #389 | — | 2 | 2 | no | — |
| 78 | #387 | — | 4 | 2 | no | — |
| 79 | #382 | C23 | 1 | 2 | yes | — |
| 80 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 81 | #376 | C23 | 1 | 1 | yes | characteristic |
| 82 | #341 | — | 2 | 1 | no | — |
| 83 | #341 | — | 2 | 1 | no | — |
| 84 | #340 | — | 2 | 1 | no | — |
| 85 | #340 | — | 2 | 1 | no | — |
| 86 | #340 | — | 2 | 4 | no | — |
| 87 | #339 | — | 2 | 4 | no | — |
| 88 | #336 | D4 | 1 | 2 | yes | — |
| 89 | #336 | D4 | 1 | 2 | yes | — |
| 90 | #333 | — | 4 | 2 | no | — |
| 91 | #332 | — | 4 | 2 | no | — |
| 92 | #328 | — | 4 | 2 | no | — |
| 93 | #328 | — | 4 | 2 | no | — |
| 94 | #322 | — | 2 | 1 | no | — |
| 95 | #322 | — | 2 | 1 | no | — |
| 96 | #322 | C2×C4 | 1 | 2 | yes | — |
| 97 | C82 | C2×C4 | 1 | 2 | yes | — |
| 98 | C22×Q8 | — | 2 | 2 | no | — |
| 99 | C2×Q16 | — | 4 | 4 | no | — |
| 100 | C2×M16 | — | 2 | 2 | no | — |
| 101 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 102 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | C4⋊Q8 | — | 2 | 2 | no | — |
| 105 | C4⋊Q8 | — | 2 | 2 | no | — |
| 106 | C4⋊Q8 | — | 2 | 2 | no | — |
| 107 | C4⋊Q8 | C22×C4 | 1 | 4 | yes | — |
| 108 | C4⋊Q8 | C22×C4 | 1 | 2 | yes | — |
| 109 | C4⋊Q8 | C22×C4 | 1 | 2 | yes | — |
| 110 | #125 | — | 2 | 2 | no | — |
| 111 | C22⋊Q8 | — | 2 | 2 | no | — |
| 112 | C22⋊Q8 | — | 2 | 2 | no | — |
| 113 | C4×Q8 | — | 2 | 2 | no | — |
| 114 | C4×Q8 | — | 2 | 2 | no | — |
| 115 | #117 | — | 2 | 2 | no | — |
| 116 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 117 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 118 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 119 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 120 | C2×C42 | — | 4 | 2 | no | — |
| 121 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 122 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 123 | #107 | — | 2 | 2 | no | — |
| 124 | #107 | — | 2 | 2 | no | — |
| 125 | #107 | — | 2 | 2 | no | — |
| 126 | #107 | — | 2 | 2 | no | — |
| 127 | #107 | — | 2 | 1 | no | — |
| 128 | #107 | — | 2 | 1 | no | — |
| 129 | #107 | — | 2 | 1 | no | — |
| 130 | #107 | — | 2 | 1 | no | — |
| 131 | #107 | — | 4 | 4 | no | — |
| 132 | #106 | — | 2 | 2 | no | — |
| 133 | #106 | — | 2 | 2 | no | — |
| 134 | C4⋊C8 | — | 2 | 1 | no | — |
| 135 | C4⋊C8 | — | 2 | 1 | no | — |
| 136 | C4⋊C8 | — | 2 | 1 | no | — |
| 137 | C4⋊C8 | — | 2 | 2 | no | — |
| 138 | C4⋊C8 | — | 2 | 1 | no | — |
| 139 | C4⋊C8 | — | 2 | 2 | no | — |
| 140 | C4⋊C8 | — | 4 | 4 | no | — |
| 141 | #103 | — | 4 | 4 | no | — |
| 142 | #103 | — | 4 | 4 | no | — |
| 143 | C22⋊C8 | — | 2 | 1 | no | — |
| 144 | C22⋊C8 | — | 2 | 1 | no | — |
| 145 | C22⋊C8 | — | 2 | 1 | no | — |
| 146 | C22⋊C8 | — | 2 | 1 | no | — |
| 147 | #97 | — | 2 | 1 | no | — |
| 148 | #97 | — | 2 | 1 | no | — |
| 149 | #97 | — | 4 | 2 | no | — |
| 150 | C4×C8 | — | 2 | 1 | no | — |
| 151 | C4×C8 | — | 2 | 1 | no | — |
| 152 | C4×C8 | — | 4 | 2 | no | — |
| 153 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 154 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 155 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 156 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 157 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 158 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 159 | #95 | — | 2 | 1 | no | — |
| 160 | #95 | — | 2 | 1 | no | — |
| 161 | #95 | — | 2 | 2 | no | — |
| 162 | #95 | — | 2 | 2 | no | — |
| 163 | #95 | — | 4 | 2 | no | — |
| 164 | #95 | — | 4 | 2 | no | — |
| 165 | C2×Q8 | — | 2 | 2 | no | — |
| 166 | C2×Q8 | — | 2 | 4 | no | — |
| 167 | C2×Q8 | — | 2 | 2 | no | — |
| 168 | C2×Q8 | — | 4 | 2 | no | — |
| 169 | C2×Q8 | — | 16 | 2 | no | — |
| 170 | C22×C4 | — | 2 | 2 | no | — |
| 171 | C22×C4 | — | 4 | 2 | no | — |
| 172 | C22×C4 | — | 4 | 2 | no | — |
| 173 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 174 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 175 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 176 | Q16 | — | 16 | 4 | no | — |
| 177 | M16 | — | 4 | 2 | no | — |
| 178 | M16 | — | 4 | 2 | no | — |
| 179 | C2×C8 | — | 2 | 2 | no | — |
| 180 | C2×C8 | — | 2 | 2 | no | — |
| 181 | C2×C8 | — | 2 | 2 | no | — |
| 182 | C2×C8 | — | 2 | 2 | no | — |
| 183 | C2×C8 | — | 2 | 2 | no | — |
| 184 | C2×C8 | — | 2 | 2 | no | — |
| 185 | C2×C8 | — | 2 | 2 | no | — |
| 186 | C2×C8 | — | 4 | 2 | no | — |
| 187 | C2×C8 | — | 4 | 1 | no | — |
| 188 | C2×C8 | — | 4 | 1 | no | — |
| 189 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 190 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 191 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 192 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 193 | C4⋊C4 | — | 2 | 2 | no | — |
| 194 | C4⋊C4 | — | 2 | 2 | no | — |
| 195 | C4⋊C4 | — | 2 | 4 | no | — |
| 196 | C4⋊C4 | — | 2 | 2 | no | — |
| 197 | C4⋊C4 | — | 2 | 4 | no | — |
| 198 | C4⋊C4 | — | 2 | 2 | no | — |
| 199 | C4⋊C4 | — | 4 | 2 | no | — |
| 200 | C4⋊C4 | — | 4 | 2 | no | — |
| 201 | C4⋊C4 | — | 4 | 2 | no | — |
| 202 | C4⋊C4 | — | 4 | 2 | no | — |
| 203 | C4⋊C4 | — | 4 | 2 | no | — |
| 204 | C4⋊C4 | — | 4 | 2 | no | — |
| 205 | C4⋊C4 | — | 16 | 2 | no | — |
| 206 | C22⋊C4 | — | 4 | 2 | no | — |
| 207 | C42 | — | 2 | 2 | no | — |
| 208 | C42 | — | 2 | 2 | no | — |
| 209 | C42 | — | 4 | 2 | no | — |
| 210 | C42 | — | 16 | 2 | no | — |
| 211 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 212 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 213 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 214 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 215 | C23 | #535 | 1 | 1 | yes | characteristic |
| 216 | Q8 | — | 8 | 2 | no | — |
| 217 | Q8 | — | 8 | 4 | no | — |
| 218 | Q8 | — | 8 | 2 | no | — |
| 219 | Q8 | — | 16 | 2 | no | — |
| 220 | C2×C4 | — | 4 | 1 | no | — |
| 221 | C2×C4 | — | 4 | 1 | no | — |
| 222 | C2×C4 | — | 4 | 2 | no | — |
| 223 | C2×C4 | — | 4 | 2 | no | — |
| 224 | C2×C4 | — | 4 | 2 | no | — |
| 225 | C2×C4 | — | 4 | 2 | no | — |
| 226 | C2×C4 | — | 4 | 4 | no | — |
| 227 | C2×C4 | — | 4 | 2 | no | — |
| 228 | C2×C4 | — | 8 | 2 | no | — |
| 229 | C2×C4 | — | 8 | 2 | no | — |
| 230 | C2×C4 | — | 8 | 2 | no | — |
| 231 | C2×C4 | — | 8 | 2 | no | — |
| 232 | C2×C4 | — | 8 | 2 | no | — |
| 233 | C2×C4 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 234 | C2×C4 | #518 | 1 | 1 | yes | characteristic |
| 235 | C2×C4 | #518 | 1 | 1 | yes | characteristic |
| 236 | C2×C4 | #535 | 1 | 1 | yes | characteristic |
| 237 | C2×C4 | #540 | 1 | 1 | yes | characteristic |
| 238 | C2×C4 | #540 | 1 | 1 | yes | characteristic |
| 239 | C2×C4 | #578 | 1 | 2 | yes | — |
| 240 | C2×C4 | #578 | 1 | 2 | yes | — |
| 241 | C8 | — | 2 | 1 | no | — |
| 242 | C8 | — | 2 | 1 | no | — |
| 243 | C8 | — | 2 | 1 | no | — |
| 244 | C8 | — | 2 | 1 | no | — |
| 245 | C8 | — | 4 | 2 | no | — |
| 246 | C8 | — | 4 | 2 | no | — |
| 247 | C8 | — | 4 | 2 | no | — |
| 248 | C8 | — | 8 | 1 | no | — |
| 249 | C8 | — | 8 | 1 | no | — |
| 250 | C8 | — | 8 | 2 | no | — |
| 251 | C22 | — | 2 | 1 | no | — |
| 252 | C22 | — | 2 | 1 | no | — |
| 253 | C22 | #2328 | 1 | 1 | yes | characteristic |
| 254 | C22 | #3223 | 1 | 2 | yes | — |
| 255 | C4 | — | 2 | 1 | no | — |
| 256 | C4 | — | 2 | 1 | no | — |
| 257 | C4 | — | 2 | 1 | no | — |
| 258 | C4 | — | 2 | 1 | no | — |
| 259 | C4 | — | 8 | 2 | no | — |
| 260 | C4 | — | 8 | 2 | no | — |
| 261 | C4 | — | 8 | 4 | no | — |
| 262 | C4 | — | 8 | 2 | no | — |
| 263 | C4 | — | 16 | 2 | no | — |
| 264 | C4 | #2950 | 1 | 1 | yes | characteristic |
| 265 | C4 | #2950 | 1 | 1 | yes | characteristic |
| 266 | C4 | #3223 | 1 | 2 | yes | — |
| 267 | C2 | — | 2 | 2 | no | — |
| 268 | C2 | #22277 | 1 | 1 | yes | characteristic |
| 269 | C2 | #22285 | 1 | 1 | yes | characteristic |
| 270 | C2 | #22286 | 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