Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4747 conjugacy classes of subgroups.
The subgroup lattice has 266 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #59673 | C2 | 1 | 9 | yes | maximal |
| 2 | #59665 | C2 | 1 | 18 | yes | maximal |
| 3 | #36584 | C2 | 1 | 6 | yes | maximal |
| 4 | #36574 | C2 | 1 | 9 | yes | maximal |
| 5 | #36466 | C2 | 1 | 6 | yes | maximal |
| 6 | #36447 | C2 | 1 | 6 | yes | maximal |
| 7 | C4×Q82 | C2 | 1 | 2 | yes | maximal |
| 8 | C4×Q82 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | C4×#557 | C2 | 1 | 6 | yes | maximal |
| 10 | C2×Q82 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#557 | C22 | 1 | 6 | yes | — |
| 12 | #2859 | C22 | 1 | 9 | yes | — |
| 13 | #2859 | C22 | 1 | 18 | yes | — |
| 14 | #2799 | C22 | 1 | 36 | yes | — |
| 15 | #2584 | C22 | 1 | 36 | yes | — |
| 16 | #2581 | C22 | 1 | 18 | yes | — |
| 17 | #2571 | C22 | 1 | 18 | yes | — |
| 18 | #2571 | C22 | 1 | 9 | yes | — |
| 19 | #2552 | C22 | 1 | 18 | yes | — |
| 20 | #2551 | C22 | 1 | 9 | yes | — |
| 21 | #2550 | C22 | 1 | 36 | yes | — |
| 22 | #2509 | C22 | 1 | 36 | yes | — |
| 23 | #2509 | C22 | 1 | 36 | yes | — |
| 24 | #2508 | C22 | 1 | 36 | yes | — |
| 25 | #2506 | C22 | 1 | 36 | yes | — |
| 26 | #2453 | C22 | 1 | 36 | yes | — |
| 27 | #2370 | C22 | 1 | 18 | yes | — |
| 28 | #2370 | C22 | 1 | 18 | yes | — |
| 29 | #2369 | C22 | 1 | 9 | yes | — |
| 30 | #2369 | C22 | 1 | 18 | yes | — |
| 31 | #2365 | C22 | 1 | 18 | yes | — |
| 32 | #2355 | C22 | 1 | 18 | yes | — |
| 33 | #2351 | C22 | 1 | 12 | yes | — |
| 34 | #2351 | C22 | 1 | 2 | yes | — |
| 35 | #2351 | C22 | 1 | 6 | yes | — |
| 36 | #2351 | C22 | 1 | 6 | yes | — |
| 37 | Q8×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 38 | Q8×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 39 | Q8×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 40 | Q8×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 41 | C4×C4⋊Q8 | C22 | 1 | 18 | yes | — |
| 42 | C4×C4⋊Q8 | C22 | 1 | 18 | yes | — |
| 43 | C4×C4⋊Q8 | C22 | 1 | 9 | yes | — |
| 44 | C4×#125 | C22 | 1 | 18 | yes | — |
| 45 | C4×#125 | C22 | 1 | 18 | yes | — |
| 46 | C42×Q8 | C22 | 1 | 6 | yes | — |
| 47 | C42×Q8 | C22 | 1 | 6 | yes | — |
| 48 | C42×Q8 | C22 | 1 | 9 | yes | — |
| 49 | C42×Q8 | C22 | 1 | 6 | yes | — |
| 50 | Q82 | Q8 | 1 | 16 | yes | complemented |
| 51 | #557 | — | 2 | 48 | no | — |
| 52 | C2×C4⋊Q8 | C23 | 1 | 18 | yes | — |
| 53 | C2×C4⋊Q8 | C23 | 1 | 9 | yes | — |
| 54 | C2×#125 | C23 | 1 | 18 | yes | — |
| 55 | C2×#125 | C23 | 1 | 18 | yes | — |
| 56 | C2×#125 | C23 | 1 | 18 | yes | — |
| 57 | C2×C4×Q8 | C23 | 1 | 18 | yes | — |
| 58 | C2×C4×Q8 | C23 | 1 | 6 | yes | — |
| 59 | C2×C4×Q8 | C23 | 1 | 6 | yes | — |
| 60 | C2×C4×Q8 | C23 | 1 | 2 | yes | — |
| 61 | C2×C4×Q8 | C23 | 1 | 6 | yes | — |
| 62 | C2×C4×Q8 | C23 | 1 | 2 | yes | — |
| 63 | C2×C4×Q8 | C23 | 1 | 6 | yes | — |
| 64 | C2×C4×Q8 | C23 | 1 | 9 | yes | — |
| 65 | C2×C4×Q8 | C23 | 1 | 12 | yes | — |
| 66 | C2×C4×Q8 | C23 | 1 | 6 | yes | — |
| 67 | C2×C4×Q8 | C23 | 1 | 2 | yes | — |
| 68 | #400 | C23 | 1 | 36 | yes | — |
| 69 | #400 | C23 | 1 | 72 | yes | — |
| 70 | #398 | C23 | 1 | 72 | yes | — |
| 71 | #395 | C23 | 1 | 36 | yes | — |
| 72 | #391 | C23 | 1 | 36 | yes | — |
| 73 | #391 | C23 | 1 | 36 | yes | — |
| 74 | #391 | C23 | 1 | 18 | yes | — |
| 75 | #389 | C23 | 1 | 6 | yes | — |
| 76 | #389 | C23 | 1 | 18 | yes | — |
| 77 | #389 | C23 | 1 | 18 | yes | — |
| 78 | #389 | C23 | 1 | 18 | yes | — |
| 79 | #389 | C23 | 1 | 6 | yes | — |
| 80 | #389 | C23 | 1 | 36 | yes | — |
| 81 | #389 | C23 | 1 | 36 | yes | — |
| 82 | #389 | C23 | 1 | 36 | yes | — |
| 83 | #389 | C23 | 1 | 18 | yes | — |
| 84 | #389 | C23 | 1 | 36 | yes | — |
| 85 | #389 | C23 | 1 | 36 | yes | — |
| 86 | #389 | C23 | 1 | 36 | yes | — |
| 87 | #389 | C23 | 1 | 36 | yes | — |
| 88 | #389 | C23 | 1 | 36 | yes | — |
| 89 | #387 | C23 | 1 | 36 | yes | — |
| 90 | #387 | C23 | 1 | 36 | yes | — |
| 91 | #387 | C23 | 1 | 36 | yes | — |
| 92 | #387 | C23 | 1 | 36 | yes | — |
| 93 | #384 | C23 | 1 | 9 | yes | — |
| 94 | #384 | C23 | 1 | 18 | yes | — |
| 95 | #384 | C23 | 1 | 18 | yes | — |
| 96 | #382 | C23 | 1 | 6 | yes | — |
| 97 | #382 | C23 | 1 | 18 | yes | — |
| 98 | #382 | C23 | 1 | 36 | yes | — |
| 99 | #382 | C23 | 1 | 18 | yes | — |
| 100 | C4×C4⋊C4 | C23 | 1 | 36 | yes | — |
| 101 | C4×C4⋊C4 | C23 | 1 | 18 | yes | — |
| 102 | C4×C4⋊C4 | C23 | 1 | 18 | yes | — |
| 103 | C4×C4⋊C4 | C23 | 1 | 18 | yes | — |
| 104 | C4×C4⋊C4 | C23 | 1 | 18 | yes | — |
| 105 | C4×C4⋊C4 | C23 | 1 | 18 | yes | — |
| 106 | C4×C4⋊C4 | C23 | 1 | 18 | yes | — |
| 107 | C4×C4⋊C4 | C23 | 1 | 18 | yes | — |
| 108 | C4×C4⋊C4 | C23 | 1 | 18 | yes | — |
| 109 | C4×C4⋊C4 | C23 | 1 | 9 | yes | — |
| 110 | C4×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 111 | C4×C4⋊C4 | C23 | 1 | 18 | yes | — |
| 112 | C4×C4⋊C4 | C23 | 1 | 18 | yes | — |
| 113 | C4×C4⋊C4 | C23 | 1 | 18 | yes | — |
| 114 | C4×C4⋊C4 | C23 | 1 | 18 | yes | — |
| 115 | C43 | C23 | 1 | 9 | yes | — |
| 116 | C43 | C23 | 1 | 18 | yes | — |
| 117 | C22×Q8 | C24 | 1 | 9 | yes | — |
| 118 | C22×Q8 | C24 | 1 | 6 | yes | — |
| 119 | C22×Q8 | C24 | 1 | 2 | yes | — |
| 120 | C4⋊Q8 | — | 2 | 72 | no | — |
| 121 | C4⋊Q8 | C2×Q8 | 1 | 72 | yes | — |
| 122 | #125 | — | 2 | 72 | no | — |
| 123 | #125 | — | 2 | 72 | no | — |
| 124 | #125 | — | 8 | 18 | no | — |
| 125 | C4×Q8 | — | 2 | 24 | no | — |
| 126 | C4×Q8 | — | 2 | 24 | no | — |
| 127 | C4×Q8 | — | 2 | 24 | no | — |
| 128 | C4×Q8 | — | 2 | 8 | no | — |
| 129 | C4×Q8 | — | 2 | 8 | no | — |
| 130 | C4×Q8 | — | 2 | 72 | no | — |
| 131 | C4×Q8 | — | 4 | 12 | no | — |
| 132 | C4×Q8 | — | 4 | 18 | no | — |
| 133 | C4×Q8 | — | 8 | 12 | no | — |
| 134 | C4×Q8 | C2×Q8 | 1 | 48 | yes | — |
| 135 | C4×Q8 | #41 | 1 | 16 | yes | — |
| 136 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 137 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 138 | C2×C4⋊C4 | C24 | 1 | 18 | yes | — |
| 139 | C2×C4⋊C4 | C24 | 1 | 18 | yes | — |
| 140 | C2×C4⋊C4 | C24 | 1 | 18 | yes | — |
| 141 | C2×C4⋊C4 | C24 | 1 | 18 | yes | — |
| 142 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 143 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 144 | C2×C4⋊C4 | C24 | 1 | 18 | yes | — |
| 145 | C2×C4⋊C4 | C24 | 1 | 36 | yes | — |
| 146 | C2×C4⋊C4 | C24 | 1 | 36 | yes | — |
| 147 | C2×C4⋊C4 | C24 | 1 | 9 | yes | — |
| 148 | C2×C4⋊C4 | C24 | 1 | 18 | yes | — |
| 149 | C2×C4⋊C4 | C24 | 1 | 36 | yes | — |
| 150 | C2×C4⋊C4 | C24 | 1 | 18 | yes | — |
| 151 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 152 | C2×C42 | C24 | 1 | 9 | yes | — |
| 153 | C2×C42 | C24 | 1 | 18 | yes | — |
| 154 | C2×C42 | C24 | 1 | 18 | yes | — |
| 155 | C2×C42 | C24 | 1 | 6 | yes | — |
| 156 | C2×C42 | C24 | 1 | 6 | yes | — |
| 157 | C2×C42 | C24 | 1 | 18 | yes | — |
| 158 | C2×C42 | C24 | 1 | 18 | yes | — |
| 159 | C2×C42 | C24 | 1 | 18 | yes | — |
| 160 | C2×C42 | C24 | 1 | 9 | yes | — |
| 161 | C2×C42 | C24 | 1 | 9 | yes | — |
| 162 | C2×C42 | C24 | 1 | 6 | yes | — |
| 163 | #95 | C24 | 1 | 1 | yes | characteristic |
| 164 | #95 | C24 | 1 | 6 | yes | — |
| 165 | #95 | C24 | 1 | 9 | yes | — |
| 166 | #95 | C24 | 1 | 18 | yes | — |
| 167 | #95 | C24 | 1 | 12 | yes | — |
| 168 | #95 | C24 | 1 | 36 | yes | — |
| 169 | #95 | C24 | 1 | 36 | yes | — |
| 170 | #95 | C24 | 1 | 36 | yes | — |
| 171 | #95 | C24 | 1 | 36 | yes | — |
| 172 | #95 | C24 | 1 | 36 | yes | — |
| 173 | C2×Q8 | — | 2 | 36 | no | — |
| 174 | C2×Q8 | — | 4 | 6 | no | — |
| 175 | C2×Q8 | — | 4 | 9 | no | — |
| 176 | C2×Q8 | — | 4 | 6 | no | — |
| 177 | C2×Q8 | C22×Q8 | 1 | 24 | yes | — |
| 178 | C2×Q8 | C22×Q8 | 1 | 8 | yes | — |
| 179 | C2×Q8 | C2×#41 | 1 | 8 | yes | — |
| 180 | C2×Q8 | 2−1+4 | 1 | 8 | yes | — |
| 181 | C22×C4 | C25 | 1 | 6 | yes | — |
| 182 | C22×C4 | C25 | 1 | 6 | yes | — |
| 183 | C22×C4 | C25 | 1 | 18 | yes | — |
| 184 | C22×C4 | C25 | 1 | 9 | yes | — |
| 185 | C22×C4 | C25 | 1 | 9 | yes | — |
| 186 | C22×C4 | C25 | 1 | 6 | yes | — |
| 187 | C22×C4 | C25 | 1 | 2 | yes | — |
| 188 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 189 | C22×C4 | C25 | 1 | 6 | yes | — |
| 190 | C4⋊C4 | — | 2 | 12 | no | — |
| 191 | C4⋊C4 | — | 2 | 36 | no | — |
| 192 | C4⋊C4 | — | 2 | 36 | no | — |
| 193 | C4⋊C4 | — | 2 | 36 | no | — |
| 194 | C4⋊C4 | — | 2 | 36 | no | — |
| 195 | C4⋊C4 | — | 2 | 72 | no | — |
| 196 | C4⋊C4 | — | 2 | 12 | no | — |
| 197 | C4⋊C4 | — | 2 | 12 | no | — |
| 198 | C4⋊C4 | — | 2 | 36 | no | — |
| 199 | C4⋊C4 | — | 2 | 12 | no | — |
| 200 | C4⋊C4 | — | 4 | 18 | no | — |
| 201 | C4⋊C4 | — | 4 | 9 | no | — |
| 202 | C4⋊C4 | — | 4 | 36 | no | — |
| 203 | C4⋊C4 | — | 4 | 36 | no | — |
| 204 | C4⋊C4 | C22×Q8 | 1 | 72 | yes | — |
| 205 | C4⋊C4 | C2×#41 | 1 | 24 | yes | — |
| 206 | C42 | — | 2 | 36 | no | — |
| 207 | C42 | — | 2 | 36 | no | — |
| 208 | C42 | — | 2 | 12 | no | — |
| 209 | C42 | — | 2 | 12 | no | — |
| 210 | C42 | — | 2 | 36 | no | — |
| 211 | C42 | — | 2 | 18 | no | — |
| 212 | C42 | — | 4 | 18 | no | — |
| 213 | C42 | — | 4 | 18 | no | — |
| 214 | C42 | — | 4 | 9 | no | — |
| 215 | C42 | C22×Q8 | 1 | 36 | yes | — |
| 216 | C42 | C2×#41 | 1 | 24 | yes | — |
| 217 | C23 | C26 | 1 | 1 | yes | characteristic |
| 218 | Q8 | — | 4 | 24 | no | — |
| 219 | Q8 | — | 4 | 36 | no | — |
| 220 | Q8 | #557 | 1 | 32 | yes | complemented |
| 221 | C2×C4 | — | 2 | 6 | no | — |
| 222 | C2×C4 | — | 2 | 2 | no | — |
| 223 | C2×C4 | — | 2 | 2 | no | — |
| 224 | C2×C4 | — | 2 | 9 | no | — |
| 225 | C2×C4 | — | 2 | 6 | no | — |
| 226 | C2×C4 | — | 2 | 2 | no | — |
| 227 | C2×C4 | — | 2 | 9 | no | — |
| 228 | C2×C4 | — | 2 | 6 | no | — |
| 229 | C2×C4 | — | 2 | 18 | no | — |
| 230 | C2×C4 | — | 2 | 18 | no | — |
| 231 | C2×C4 | — | 2 | 6 | no | — |
| 232 | C2×C4 | — | 2 | 6 | no | — |
| 233 | C2×C4 | — | 2 | 18 | no | — |
| 234 | C2×C4 | — | 2 | 6 | no | — |
| 235 | C2×C4 | — | 2 | 9 | no | — |
| 236 | C2×C4 | — | 2 | 6 | no | — |
| 237 | C2×C4 | — | 2 | 6 | no | — |
| 238 | C2×C4 | — | 2 | 18 | no | — |
| 239 | C2×C4 | C23×Q8 | 1 | 18 | yes | — |
| 240 | C2×C4 | C23×Q8 | 1 | 12 | yes | — |
| 241 | C2×C4 | C22×#41 | 1 | 4 | yes | — |
| 242 | C2×C4 | C22×#41 | 1 | 12 | yes | — |
| 243 | C2×C4 | C22×#41 | 1 | 12 | yes | — |
| 244 | C2×C4 | C2×2−1+4 | 1 | 12 | yes | — |
| 245 | C2×C4 | #585 | 1 | 2 | yes | — |
| 246 | C22 | C24×Q8 | 1 | 1 | yes | characteristic |
| 247 | C22 | C23×#41 | 1 | 2 | yes | — |
| 248 | C22 | C22×2−1+4 | 1 | 2 | yes | — |
| 249 | C22 | C2×#585 | 1 | 1 | yes | characteristic |
| 250 | C22 | H(7,𝔽2) | 1 | 1 | yes | characteristic |
| 251 | C4 | — | 2 | 18 | no | — |
| 252 | C4 | — | 2 | 4 | no | — |
| 253 | C4 | — | 2 | 2 | no | — |
| 254 | C4 | — | 2 | 12 | no | — |
| 255 | C4 | — | 4 | 6 | no | — |
| 256 | C4 | — | 4 | 18 | no | — |
| 257 | C4 | — | 4 | 6 | no | — |
| 258 | C4 | — | 4 | 9 | no | — |
| 259 | C4 | C2×#557 | 1 | 24 | yes | — |
| 260 | C2 | C22×#557 | 1 | 2 | yes | — |
| 261 | C2 | #412 | 1 | 1 | yes | characteristic |
| 262 | C2 | #62989 | 1 | 1 | yes | characteristic |
| 263 | C2 | #62999 | 1 | 2 | yes | — |
| 264 | C2 | #63038 | 1 | 1 | yes | characteristic |
| 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 |
|---|---|---|
| 50. Q82 | Q8 | semidirect |
| 220. Q8 | #557 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #59673 | 2 |
| #59665 | 2 |
| #36584 | 2 |
| #36574 | 2 |
| #36466 | 2 |
| #36447 | 2 |
| C4×Q82 | 2 |
| C4×#557 | 2 |