Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4125 conjugacy classes of subgroups.
The subgroup lattice has 265 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #59668 | C2 | 1 | 12 | yes | maximal |
| 2 | #51080 | C2 | 1 | 24 | yes | maximal |
| 3 | #46153 | C2 | 1 | 3 | yes | maximal |
| 4 | #46007 | C2 | 1 | 6 | yes | maximal |
| 5 | #39653 | C2 | 1 | 12 | yes | maximal |
| 6 | #36565 | C2 | 1 | 3 | yes | maximal |
| 7 | #34526 | C2 | 1 | 2 | yes | maximal |
| 8 | C4×#564 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | C2×#564 | C22 | 1 | 1 | yes | characteristic |
| 10 | #2863 | C22 | 1 | 4 | yes | — |
| 11 | #2862 | C22 | 1 | 6 | yes | — |
| 12 | #2859 | C22 | 1 | 3 | yes | — |
| 13 | #2845 | C22 | 1 | 6 | yes | — |
| 14 | #2806 | C22 | 1 | 24 | yes | — |
| 15 | #2806 | C22 | 1 | 24 | yes | — |
| 16 | #2802 | C22 | 1 | 24 | yes | — |
| 17 | #2755 | C22 | 1 | 24 | yes | — |
| 18 | #2755 | C22 | 1 | 24 | yes | — |
| 19 | #2726 | C22 | 1 | 12 | yes | — |
| 20 | #2726 | C22 | 1 | 24 | yes | — |
| 21 | #2713 | C22 | 1 | 24 | yes | — |
| 22 | #2666 | C22 | 1 | 24 | yes | — |
| 23 | #2586 | C22 | 1 | 24 | yes | — |
| 24 | #2584 | C22 | 1 | 24 | yes | — |
| 25 | #2552 | C22 | 1 | 12 | yes | — |
| 26 | #2551 | C22 | 1 | 6 | yes | — |
| 27 | #2551 | C22 | 1 | 12 | yes | — |
| 28 | #2550 | C22 | 1 | 24 | yes | — |
| 29 | #2550 | C22 | 1 | 12 | yes | — |
| 30 | #2546 | C22 | 1 | 24 | yes | — |
| 31 | #2511 | C22 | 1 | 12 | yes | — |
| 32 | #2507 | C22 | 1 | 12 | yes | — |
| 33 | #2507 | C22 | 1 | 12 | yes | — |
| 34 | #2507 | C22 | 1 | 12 | yes | — |
| 35 | #2462 | C22 | 1 | 24 | yes | — |
| 36 | #2451 | C22 | 1 | 24 | yes | — |
| 37 | #2451 | C22 | 1 | 24 | yes | — |
| 38 | #2451 | C22 | 1 | 12 | yes | — |
| 39 | #2446 | C22 | 1 | 24 | yes | — |
| 40 | #2382 | C22 | 1 | 12 | yes | — |
| 41 | #2382 | C22 | 1 | 3 | yes | — |
| 42 | #2382 | C22 | 1 | 6 | yes | — |
| 43 | #2370 | C22 | 1 | 24 | yes | — |
| 44 | #2369 | C22 | 1 | 12 | yes | — |
| 45 | #2369 | C22 | 1 | 12 | yes | — |
| 46 | #2355 | C22 | 1 | 6 | yes | — |
| 47 | #2336 | C22 | 1 | 24 | yes | — |
| 48 | C4×C4⋊Q8 | C22 | 1 | 12 | yes | — |
| 49 | C4×#125 | C22 | 1 | 12 | yes | — |
| 50 | C4×#125 | C22 | 1 | 3 | yes | — |
| 51 | C4×#125 | C22 | 1 | 6 | yes | — |
| 52 | C42×Q8 | C22 | 1 | 1 | yes | characteristic |
| 53 | #564 | Q8 | 1 | 16 | yes | complemented |
| 54 | C2×C4⋊Q8 | C23 | 1 | 12 | yes | — |
| 55 | C2×C4⋊Q8 | C23 | 1 | 12 | yes | — |
| 56 | C2×C4⋊Q8 | C23 | 1 | 24 | yes | — |
| 57 | C2×#125 | C23 | 1 | 24 | yes | — |
| 58 | C2×#125 | C23 | 1 | 6 | yes | — |
| 59 | C2×#125 | C23 | 1 | 3 | yes | — |
| 60 | C2×#125 | C23 | 1 | 12 | yes | — |
| 61 | C2×C4×Q8 | C23 | 1 | 3 | yes | — |
| 62 | C2×C4×Q8 | C23 | 1 | 24 | yes | — |
| 63 | #400 | C23 | 1 | 24 | yes | — |
| 64 | #400 | C23 | 1 | 24 | yes | — |
| 65 | #398 | C23 | 1 | 24 | yes | — |
| 66 | #398 | C23 | 1 | 24 | yes | — |
| 67 | #398 | C23 | 1 | 24 | yes | — |
| 68 | #398 | C23 | 1 | 24 | yes | — |
| 69 | #398 | C23 | 1 | 24 | yes | — |
| 70 | #398 | C23 | 1 | 24 | yes | — |
| 71 | #398 | C23 | 1 | 24 | yes | — |
| 72 | #398 | C23 | 1 | 24 | yes | — |
| 73 | #398 | C23 | 1 | 24 | yes | — |
| 74 | #395 | C23 | 1 | 24 | yes | — |
| 75 | #395 | C23 | 1 | 24 | yes | — |
| 76 | #395 | C23 | 1 | 24 | yes | — |
| 77 | #395 | C23 | 1 | 24 | yes | — |
| 78 | #391 | C23 | 1 | 24 | yes | — |
| 79 | #391 | C23 | 1 | 24 | yes | — |
| 80 | #391 | C23 | 1 | 24 | yes | — |
| 81 | #391 | C23 | 1 | 24 | yes | — |
| 82 | #391 | C23 | 1 | 12 | yes | — |
| 83 | #391 | C23 | 1 | 24 | yes | — |
| 84 | #391 | C23 | 1 | 24 | yes | — |
| 85 | #389 | C23 | 1 | 12 | yes | — |
| 86 | #389 | C23 | 1 | 12 | yes | — |
| 87 | #389 | C23 | 1 | 6 | yes | — |
| 88 | #389 | C23 | 1 | 12 | yes | — |
| 89 | #389 | C23 | 1 | 24 | yes | — |
| 90 | #389 | C23 | 1 | 12 | yes | — |
| 91 | #389 | C23 | 1 | 12 | yes | — |
| 92 | #389 | C23 | 1 | 24 | yes | — |
| 93 | #389 | C23 | 1 | 24 | yes | — |
| 94 | #389 | C23 | 1 | 24 | yes | — |
| 95 | #389 | C23 | 1 | 24 | yes | — |
| 96 | #389 | C23 | 1 | 24 | yes | — |
| 97 | #389 | C23 | 1 | 12 | yes | — |
| 98 | #389 | C23 | 1 | 24 | yes | — |
| 99 | #389 | C23 | 1 | 12 | yes | — |
| 100 | #389 | C23 | 1 | 12 | yes | — |
| 101 | #389 | C23 | 1 | 24 | yes | — |
| 102 | #387 | C23 | 1 | 12 | yes | — |
| 103 | #387 | C23 | 1 | 24 | yes | — |
| 104 | #387 | C23 | 1 | 12 | yes | — |
| 105 | #387 | C23 | 1 | 12 | yes | — |
| 106 | #387 | C23 | 1 | 12 | yes | — |
| 107 | #387 | C23 | 1 | 12 | yes | — |
| 108 | #387 | C23 | 1 | 24 | yes | — |
| 109 | #387 | C23 | 1 | 24 | yes | — |
| 110 | #387 | C23 | 1 | 24 | yes | — |
| 111 | #387 | C23 | 1 | 24 | yes | — |
| 112 | #387 | C23 | 1 | 24 | yes | — |
| 113 | #387 | C23 | 1 | 24 | yes | — |
| 114 | #387 | C23 | 1 | 24 | yes | — |
| 115 | #384 | C23 | 1 | 12 | yes | — |
| 116 | #384 | C23 | 1 | 6 | yes | — |
| 117 | #383 | C23 | 1 | 8 | yes | — |
| 118 | #382 | C23 | 1 | 6 | yes | — |
| 119 | #382 | C23 | 1 | 3 | yes | — |
| 120 | #382 | C23 | 1 | 12 | yes | — |
| 121 | #382 | C23 | 1 | 12 | yes | — |
| 122 | C4×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 123 | C4×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 124 | C4×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 125 | C4×C4⋊C4 | C23 | 1 | 24 | yes | — |
| 126 | C4×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 127 | C4×C4⋊C4 | C23 | 1 | 24 | yes | — |
| 128 | C4×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 129 | C4×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 130 | C4×C4⋊C4 | C23 | 1 | 24 | yes | — |
| 131 | C4×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 132 | C4×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 133 | C43 | C23 | 1 | 4 | yes | — |
| 134 | C43 | C23 | 1 | 1 | yes | characteristic |
| 135 | C43 | C23 | 1 | 2 | yes | — |
| 136 | C22×Q8 | C24 | 1 | 1 | yes | characteristic |
| 137 | C22×Q8 | C24 | 1 | 24 | yes | — |
| 138 | C4⋊Q8 | — | 2 | 48 | no | — |
| 139 | C4⋊Q8 | — | 8 | 24 | no | — |
| 140 | C4⋊Q8 | — | 8 | 12 | no | — |
| 141 | #125 | — | 8 | 24 | no | — |
| 142 | #125 | — | 8 | 6 | no | — |
| 143 | #125 | C2×Q8 | 1 | 24 | yes | — |
| 144 | #125 | C2×Q8 | 1 | 96 | yes | — |
| 145 | C4×Q8 | — | 2 | 12 | no | — |
| 146 | C4×Q8 | — | 8 | 24 | no | — |
| 147 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 148 | C2×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 149 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 150 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 151 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 152 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 153 | C2×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 154 | C2×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 155 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 156 | C2×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 157 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 158 | C2×C4⋊C4 | C24 | 1 | 24 | yes | — |
| 159 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 160 | C2×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 161 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 162 | C2×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 163 | C2×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 164 | C2×C4⋊C4 | C24 | 1 | 3 | yes | — |
| 165 | C2×C42 | C24 | 1 | 3 | yes | — |
| 166 | C2×C42 | C24 | 1 | 6 | yes | — |
| 167 | C2×C42 | C24 | 1 | 12 | yes | — |
| 168 | C2×C42 | C24 | 1 | 12 | yes | — |
| 169 | C2×C42 | C24 | 1 | 24 | yes | — |
| 170 | C2×C42 | C24 | 1 | 24 | yes | — |
| 171 | C2×C42 | C24 | 1 | 3 | yes | — |
| 172 | C2×C42 | C24 | 1 | 6 | yes | — |
| 173 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 174 | C2×C42 | C24 | 1 | 4 | yes | — |
| 175 | #95 | C24 | 1 | 24 | yes | — |
| 176 | #95 | C24 | 1 | 6 | yes | — |
| 177 | #95 | C24 | 1 | 12 | yes | — |
| 178 | #95 | C24 | 1 | 12 | yes | — |
| 179 | #95 | C24 | 1 | 12 | yes | — |
| 180 | #95 | C24 | 1 | 12 | yes | — |
| 181 | #95 | C24 | 1 | 24 | yes | — |
| 182 | #95 | C24 | 1 | 12 | yes | — |
| 183 | #95 | C24 | 1 | 24 | yes | — |
| 184 | #95 | C24 | 1 | 24 | yes | — |
| 185 | #95 | C24 | 1 | 24 | yes | — |
| 186 | #95 | C24 | 1 | 24 | yes | — |
| 187 | #95 | C24 | 1 | 24 | yes | — |
| 188 | #95 | C24 | 1 | 24 | yes | — |
| 189 | C2×Q8 | — | 2 | 6 | no | — |
| 190 | C2×Q8 | — | 2 | 48 | no | — |
| 191 | C2×Q8 | — | 4 | 24 | no | — |
| 192 | C2×Q8 | — | 4 | 24 | no | — |
| 193 | C22×C4 | C25 | 1 | 2 | yes | — |
| 194 | C22×C4 | C25 | 1 | 12 | yes | — |
| 195 | C22×C4 | C25 | 1 | 24 | yes | — |
| 196 | C22×C4 | C25 | 1 | 3 | yes | — |
| 197 | C22×C4 | C25 | 1 | 3 | yes | — |
| 198 | C22×C4 | C25 | 1 | 12 | yes | — |
| 199 | C22×C4 | C25 | 1 | 6 | yes | — |
| 200 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 201 | C4⋊C4 | — | 2 | 12 | no | — |
| 202 | C4⋊C4 | — | 2 | 48 | no | — |
| 203 | C4⋊C4 | — | 2 | 24 | no | — |
| 204 | C4⋊C4 | — | 4 | 24 | no | — |
| 205 | C4⋊C4 | — | 4 | 24 | no | — |
| 206 | C4⋊C4 | — | 4 | 24 | no | — |
| 207 | C4⋊C4 | — | 4 | 12 | no | — |
| 208 | C4⋊C4 | — | 4 | 12 | no | — |
| 209 | C4⋊C4 | — | 4 | 24 | no | — |
| 210 | C4⋊C4 | — | 4 | 24 | no | — |
| 211 | C4⋊C4 | — | 4 | 12 | no | — |
| 212 | C4⋊C4 | — | 4 | 24 | no | — |
| 213 | C4⋊C4 | — | 4 | 12 | no | — |
| 214 | C4⋊C4 | — | 4 | 6 | no | — |
| 215 | C4⋊C4 | C22×Q8 | 1 | 24 | yes | — |
| 216 | C4⋊C4 | C22×Q8 | 1 | 48 | yes | — |
| 217 | C4⋊C4 | C22×Q8 | 1 | 48 | yes | — |
| 218 | C4⋊C4 | 2−1+4 | 1 | 12 | yes | — |
| 219 | C42 | — | 2 | 6 | no | — |
| 220 | C42 | — | 2 | 12 | no | — |
| 221 | C42 | — | 2 | 24 | no | — |
| 222 | C42 | — | 4 | 12 | no | — |
| 223 | C42 | — | 4 | 24 | no | — |
| 224 | C42 | — | 4 | 24 | no | — |
| 225 | C42 | — | 4 | 3 | no | — |
| 226 | C42 | — | 4 | 6 | no | — |
| 227 | C42 | C22×Q8 | 1 | 4 | yes | — |
| 228 | C42 | C22×Q8 | 1 | 16 | yes | — |
| 229 | C23 | C26 | 1 | 1 | yes | characteristic |
| 230 | Q8 | — | 4 | 4 | no | — |
| 231 | Q8 | — | 8 | 48 | no | — |
| 232 | C2×C4 | — | 2 | 3 | no | — |
| 233 | C2×C4 | — | 2 | 12 | no | — |
| 234 | C2×C4 | — | 2 | 12 | no | — |
| 235 | C2×C4 | — | 2 | 3 | no | — |
| 236 | C2×C4 | — | 2 | 12 | no | — |
| 237 | C2×C4 | — | 2 | 3 | no | — |
| 238 | C2×C4 | — | 2 | 6 | no | — |
| 239 | C2×C4 | — | 2 | 12 | no | — |
| 240 | C2×C4 | — | 2 | 24 | no | — |
| 241 | C2×C4 | — | 2 | 24 | no | — |
| 242 | C2×C4 | — | 2 | 6 | no | — |
| 243 | C2×C4 | — | 2 | 24 | no | — |
| 244 | C2×C4 | — | 2 | 12 | no | — |
| 245 | C2×C4 | — | 2 | 12 | no | — |
| 246 | C2×C4 | C23×Q8 | 1 | 6 | yes | — |
| 247 | C2×C4 | C23×Q8 | 1 | 24 | yes | — |
| 248 | C2×C4 | C2×2−1+4 | 1 | 12 | yes | — |
| 249 | C2×C4 | C2×2−1+4 | 1 | 6 | yes | — |
| 250 | C22 | C24×Q8 | 1 | 1 | yes | characteristic |
| 251 | C22 | C22×2−1+4 | 1 | 3 | yes | — |
| 252 | C22 | H(7,𝔽2) | 1 | 3 | yes | — |
| 253 | C4 | — | 2 | 6 | no | — |
| 254 | C4 | — | 2 | 24 | no | — |
| 255 | C4 | — | 2 | 12 | no | — |
| 256 | C4 | — | 4 | 2 | no | — |
| 257 | C4 | — | 4 | 3 | no | — |
| 258 | C4 | — | 4 | 24 | no | — |
| 259 | C4 | — | 4 | 12 | no | — |
| 260 | C4 | #3556 | 1 | 4 | yes | — |
| 261 | C2 | C2×#3556 | 1 | 1 | yes | characteristic |
| 262 | C2 | Q8×2−1+4 | 1 | 3 | yes | — |
| 263 | C2 | #63054 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 53. #564 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #59668 | 2 |
| #51080 | 2 |
| #46153 | 2 |
| #46007 | 2 |
| #39653 | 2 |
| #36565 | 2 |
| #34526 | 2 |
| C4×#564 | 2 |