Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5478 conjugacy classes of subgroups.
The subgroup lattice has 257 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #36641 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | #36585 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | Q8×#126 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #36458 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | #36448 | C2 | 1 | 12 | yes | maximal, complemented |
| 6 | #35160 | C2 | 1 | 18 | yes | maximal, complemented |
| 7 | #34257 | C2 | 1 | 3 | yes | maximal |
| 8 | #34249 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | C2×C42×Q8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | C22×2−1+4 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#519 | C22 | 1 | 3 | yes | — |
| 12 | C22×C4×Q8 | C22 | 1 | 3 | yes | — |
| 13 | #2863 | C22 | 1 | 18 | yes | — |
| 14 | #2863 | C22 | 1 | 6 | yes | — |
| 15 | #2860 | C22 | 1 | 18 | yes | — |
| 16 | #2860 | C22 | 1 | 6 | yes | — |
| 17 | #2852 | C22 | 1 | 4 | yes | — |
| 18 | #2852 | C22 | 1 | 36 | yes | — |
| 19 | #2643 | C22 | 1 | 24 | yes | — |
| 20 | #2514 | C22 | 1 | 36 | yes | — |
| 21 | #2513 | C22 | 1 | 36 | yes | — |
| 22 | #2512 | C22 | 1 | 36 | yes | — |
| 23 | #2511 | C22 | 1 | 36 | yes | — |
| 24 | #2497 | C22 | 1 | 36 | yes | — |
| 25 | #2496 | C22 | 1 | 36 | yes | — |
| 26 | #2495 | C22 | 1 | 36 | yes | — |
| 27 | #2494 | C22 | 1 | 36 | yes | — |
| 28 | #2351 | C22 | 1 | 12 | yes | — |
| 29 | #2351 | C22 | 1 | 12 | yes | — |
| 30 | Q8×C4⋊C4 | C22 | 1 | 12 | yes | — |
| 31 | Q8×C4⋊C4 | C22 | 1 | 12 | yes | — |
| 32 | #2341 | C22 | 1 | 12 | yes | — |
| 33 | #2341 | C22 | 1 | 12 | yes | — |
| 34 | Q8×C22⋊C4 | C22 | 1 | 12 | yes | — |
| 35 | Q8×C22⋊C4 | C22 | 1 | 12 | yes | — |
| 36 | #2317 | C22 | 1 | 18 | yes | — |
| 37 | #2311 | C22 | 1 | 18 | yes | — |
| 38 | C4×#126 | C22 | 1 | 36 | yes | — |
| 39 | C4×#126 | C22 | 1 | 12 | yes | — |
| 40 | #2297 | C22 | 1 | 18 | yes | — |
| 41 | #2296 | C22 | 1 | 18 | yes | — |
| 42 | C42×Q8 | C22 | 1 | 12 | yes | — |
| 43 | C42×Q8 | C22 | 1 | 4 | yes | — |
| 44 | C2×C4×C4⋊C4 | C22 | 1 | 9 | yes | — |
| 45 | C2×C43 | C22 | 1 | 3 | yes | — |
| 46 | C2×2−1+4 | — | 2 | 24 | no | — |
| 47 | C22×#41 | C23 | 1 | 6 | yes | — |
| 48 | C22×#41 | C23 | 1 | 4 | yes | — |
| 49 | C23×Q8 | C23 | 1 | 4 | yes | — |
| 50 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 51 | C2×#126 | C23 | 1 | 4 | yes | — |
| 52 | C2×#126 | C23 | 1 | 36 | yes | — |
| 53 | #519 | — | 2 | 24 | no | — |
| 54 | C2×C4×Q8 | — | 2 | 24 | no | — |
| 55 | C2×C4×Q8 | C23 | 1 | 12 | yes | — |
| 56 | C2×C4×Q8 | C23 | 1 | 12 | yes | — |
| 57 | C2×C4×Q8 | C23 | 1 | 12 | yes | — |
| 58 | C2×C4×Q8 | C23 | 1 | 12 | yes | — |
| 59 | C2×#117 | C23 | 1 | 18 | yes | — |
| 60 | C22×C4⋊C4 | C23 | 1 | 9 | yes | — |
| 61 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 62 | C22×C42 | C23 | 1 | 9 | yes | — |
| 63 | C22×C42 | C23 | 1 | 9 | yes | — |
| 64 | #401 | C23 | 1 | 24 | yes | — |
| 65 | #400 | C23 | 1 | 72 | yes | — |
| 66 | #397 | C23 | 1 | 72 | yes | — |
| 67 | #396 | C23 | 1 | 24 | yes | — |
| 68 | #391 | C23 | 1 | 36 | yes | — |
| 69 | #391 | C23 | 1 | 18 | yes | — |
| 70 | #391 | C23 | 1 | 18 | yes | — |
| 71 | #390 | C23 | 1 | 36 | yes | — |
| 72 | #390 | C23 | 1 | 18 | yes | — |
| 73 | #390 | C23 | 1 | 18 | yes | — |
| 74 | #389 | C23 | 1 | 36 | yes | — |
| 75 | #389 | C23 | 1 | 36 | yes | — |
| 76 | #389 | C23 | 1 | 18 | yes | — |
| 77 | #389 | C23 | 1 | 18 | yes | — |
| 78 | #389 | C23 | 1 | 18 | yes | — |
| 79 | #389 | C23 | 1 | 18 | yes | — |
| 80 | #388 | C23 | 1 | 12 | yes | — |
| 81 | #388 | C23 | 1 | 36 | yes | — |
| 82 | #388 | C23 | 1 | 36 | yes | — |
| 83 | #388 | C23 | 1 | 36 | yes | — |
| 84 | #387 | C23 | 1 | 12 | yes | — |
| 85 | #387 | C23 | 1 | 36 | yes | — |
| 86 | #387 | C23 | 1 | 36 | yes | — |
| 87 | #387 | C23 | 1 | 36 | yes | — |
| 88 | #386 | C23 | 1 | 36 | yes | — |
| 89 | #386 | C23 | 1 | 12 | yes | — |
| 90 | #385 | C23 | 1 | 36 | yes | — |
| 91 | #385 | C23 | 1 | 12 | yes | — |
| 92 | #384 | C23 | 1 | 36 | yes | — |
| 93 | #383 | C23 | 1 | 4 | yes | — |
| 94 | #383 | C23 | 1 | 36 | yes | — |
| 95 | #382 | C23 | 1 | 36 | yes | — |
| 96 | #380 | C23 | 1 | 36 | yes | — |
| 97 | #380 | C23 | 1 | 36 | yes | — |
| 98 | C4×C4⋊C4 | C23 | 1 | 36 | yes | — |
| 99 | C4×C4⋊C4 | C23 | 1 | 36 | yes | — |
| 100 | C4×C4⋊C4 | C23 | 1 | 36 | yes | — |
| 101 | C4×C4⋊C4 | C23 | 1 | 36 | yes | — |
| 102 | C4×C22⋊C4 | C23 | 1 | 36 | yes | — |
| 103 | C4×C22⋊C4 | C23 | 1 | 36 | yes | — |
| 104 | C2×#95 | C23 | 1 | 6 | yes | — |
| 105 | C43 | C23 | 1 | 18 | yes | — |
| 106 | C43 | C23 | 1 | 6 | yes | — |
| 107 | 2−1+4 | — | 4 | 64 | no | — |
| 108 | C2×#41 | — | 2 | 48 | no | — |
| 109 | C2×#41 | — | 2 | 72 | no | — |
| 110 | C22×Q8 | — | 2 | 48 | no | — |
| 111 | C22×Q8 | #41 | 1 | 24 | yes | — |
| 112 | C22×Q8 | C24 | 1 | 6 | yes | — |
| 113 | C22×Q8 | C24 | 1 | 4 | yes | — |
| 114 | C22×D4 | C24 | 1 | 4 | yes | — |
| 115 | C22×D4 | C24 | 1 | 6 | yes | — |
| 116 | C23×C4 | C24 | 1 | 12 | yes | — |
| 117 | C23×C4 | C24 | 1 | 3 | yes | — |
| 118 | C23×C4 | C24 | 1 | 9 | yes | — |
| 119 | C23×C4 | C24 | 1 | 3 | yes | — |
| 120 | #126 | — | 2 | 16 | no | — |
| 121 | #126 | — | 4 | 72 | no | — |
| 122 | C4×Q8 | — | 2 | 48 | no | — |
| 123 | C4×Q8 | — | 2 | 48 | no | — |
| 124 | C4×Q8 | — | 2 | 48 | no | — |
| 125 | C4×Q8 | — | 2 | 48 | no | — |
| 126 | #117 | — | 2 | 72 | no | — |
| 127 | C2×C4⋊C4 | — | 2 | 36 | no | — |
| 128 | C2×C4⋊C4 | C24 | 1 | 36 | yes | — |
| 129 | C2×C4⋊C4 | C24 | 1 | 18 | yes | — |
| 130 | C2×C4⋊C4 | C24 | 1 | 18 | yes | — |
| 131 | C2×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 132 | C2×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 133 | C2×C4⋊C4 | C24 | 1 | 36 | yes | — |
| 134 | C2×C4⋊C4 | C24 | 1 | 36 | yes | — |
| 135 | C2×C22⋊C4 | C24 | 1 | 12 | yes | — |
| 136 | C2×C22⋊C4 | C24 | 1 | 12 | yes | — |
| 137 | C2×C22⋊C4 | C24 | 1 | 12 | yes | — |
| 138 | C2×C22⋊C4 | C24 | 1 | 36 | yes | — |
| 139 | C2×C22⋊C4 | C24 | 1 | 36 | yes | — |
| 140 | C2×C42 | — | 2 | 36 | no | — |
| 141 | C2×C42 | — | 2 | 4 | no | — |
| 142 | C2×C42 | — | 4 | 18 | no | — |
| 143 | C2×C42 | C24 | 1 | 36 | yes | — |
| 144 | C2×C42 | C24 | 1 | 18 | yes | — |
| 145 | C2×C42 | C24 | 1 | 18 | yes | — |
| 146 | C2×C42 | C24 | 1 | 18 | yes | — |
| 147 | C2×C42 | C24 | 1 | 3 | yes | — |
| 148 | C2×C42 | C24 | 1 | 18 | yes | — |
| 149 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 150 | #95 | C24 | 1 | 24 | yes | — |
| 151 | #95 | C24 | 1 | 12 | yes | — |
| 152 | #95 | C24 | 1 | 12 | yes | — |
| 153 | #95 | C24 | 1 | 36 | yes | — |
| 154 | #95 | C24 | 1 | 36 | yes | — |
| 155 | #95 | C24 | 1 | 12 | yes | — |
| 156 | #95 | C24 | 1 | 12 | yes | — |
| 157 | #95 | C24 | 1 | 36 | yes | — |
| 158 | #95 | C24 | 1 | 36 | yes | — |
| 159 | C24 | C25 | 1 | 4 | yes | — |
| 160 | C24 | C25 | 1 | 1 | yes | characteristic |
| 161 | #41 | — | 4 | 64 | no | — |
| 162 | #41 | — | 4 | 96 | no | — |
| 163 | C2×Q8 | — | 2 | 36 | no | — |
| 164 | C2×Q8 | — | 4 | 64 | no | — |
| 165 | C2×Q8 | #126 | 1 | 64 | yes | complemented |
| 166 | C2×Q8 | C2×#41 | 1 | 48 | yes | — |
| 167 | C2×D4 | — | 2 | 24 | no | — |
| 168 | C2×D4 | — | 2 | 36 | no | — |
| 169 | C22×C4 | — | 2 | 6 | no | — |
| 170 | C22×C4 | — | 2 | 6 | no | — |
| 171 | C22×C4 | — | 2 | 6 | no | — |
| 172 | C22×C4 | — | 2 | 18 | no | — |
| 173 | C22×C4 | — | 2 | 72 | no | — |
| 174 | C22×C4 | — | 4 | 9 | no | — |
| 175 | C22×C4 | — | 4 | 9 | no | — |
| 176 | C22×C4 | C2×#41 | 1 | 36 | yes | — |
| 177 | C22×C4 | C25 | 1 | 4 | yes | — |
| 178 | C22×C4 | C25 | 1 | 12 | yes | — |
| 179 | C22×C4 | C25 | 1 | 18 | yes | — |
| 180 | C22×C4 | C25 | 1 | 12 | yes | — |
| 181 | C22×C4 | C25 | 1 | 3 | yes | — |
| 182 | C22×C4 | C25 | 1 | 3 | yes | — |
| 183 | C22×C4 | C25 | 1 | 6 | yes | — |
| 184 | C4⋊C4 | — | 2 | 72 | no | — |
| 185 | C4⋊C4 | — | 2 | 36 | no | — |
| 186 | C4⋊C4 | — | 2 | 36 | no | — |
| 187 | C4⋊C4 | — | 2 | 24 | no | — |
| 188 | C4⋊C4 | — | 2 | 24 | no | — |
| 189 | C4⋊C4 | — | 4 | 36 | no | — |
| 190 | C4⋊C4 | — | 4 | 36 | no | — |
| 191 | C22⋊C4 | — | 2 | 24 | no | — |
| 192 | C22⋊C4 | — | 2 | 24 | no | — |
| 193 | C22⋊C4 | — | 2 | 24 | no | — |
| 194 | C22⋊C4 | — | 4 | 36 | no | — |
| 195 | C22⋊C4 | — | 4 | 36 | no | — |
| 196 | C42 | — | 2 | 36 | no | — |
| 197 | C42 | — | 2 | 36 | no | — |
| 198 | C42 | — | 2 | 72 | no | — |
| 199 | C42 | — | 2 | 6 | no | — |
| 200 | C42 | — | 2 | 36 | no | — |
| 201 | C42 | — | 4 | 18 | no | — |
| 202 | C42 | 2−1+4 | 1 | 4 | yes | complemented |
| 203 | C23 | — | 2 | 1 | no | — |
| 204 | C23 | — | 2 | 3 | no | — |
| 205 | C23 | — | 2 | 12 | no | — |
| 206 | C23 | — | 2 | 4 | no | — |
| 207 | C23 | — | 2 | 12 | no | — |
| 208 | C23 | C22×#41 | 1 | 6 | yes | — |
| 209 | C23 | C26 | 1 | 1 | yes | characteristic |
| 210 | Q8 | — | 4 | 24 | no | — |
| 211 | Q8 | C2×#126 | 1 | 64 | yes | — |
| 212 | D4 | — | 4 | 16 | no | — |
| 213 | D4 | — | 4 | 24 | no | — |
| 214 | C2×C4 | — | 2 | 24 | no | — |
| 215 | C2×C4 | — | 2 | 3 | no | — |
| 216 | C2×C4 | — | 2 | 3 | no | — |
| 217 | C2×C4 | — | 2 | 3 | no | — |
| 218 | C2×C4 | — | 2 | 3 | no | — |
| 219 | C2×C4 | — | 2 | 12 | no | — |
| 220 | C2×C4 | — | 2 | 12 | no | — |
| 221 | C2×C4 | — | 2 | 18 | no | — |
| 222 | C2×C4 | — | 2 | 12 | no | — |
| 223 | C2×C4 | — | 2 | 12 | no | — |
| 224 | C2×C4 | — | 2 | 18 | no | — |
| 225 | C2×C4 | — | 2 | 18 | no | — |
| 226 | C2×C4 | — | 2 | 12 | no | — |
| 227 | C2×C4 | — | 2 | 12 | no | — |
| 228 | C2×C4 | — | 2 | 12 | no | — |
| 229 | C2×C4 | — | 4 | 36 | no | — |
| 230 | C2×C4 | — | 4 | 48 | no | — |
| 231 | C2×C4 | C2×#126 | 1 | 48 | yes | — |
| 232 | C2×C4 | C22×#41 | 1 | 36 | yes | — |
| 233 | C2×C4 | C2×2−1+4 | 1 | 6 | yes | — |
| 234 | C2×C4 | #585 | 1 | 6 | yes | — |
| 235 | C22 | — | 2 | 6 | no | — |
| 236 | C22 | — | 2 | 6 | no | — |
| 237 | C22 | — | 4 | 12 | no | — |
| 238 | C22 | — | 4 | 12 | no | — |
| 239 | C22 | — | 4 | 4 | no | — |
| 240 | C22 | C22×#126 | 1 | 4 | yes | — |
| 241 | C22 | C23×#41 | 1 | 3 | yes | — |
| 242 | C22 | C22×2−1+4 | 1 | 1 | yes | characteristic |
| 243 | C22 | C2×#585 | 1 | 3 | yes | — |
| 244 | C4 | — | 2 | 6 | no | — |
| 245 | C4 | — | 2 | 6 | no | — |
| 246 | C4 | — | 4 | 12 | no | — |
| 247 | C4 | — | 4 | 12 | no | — |
| 248 | C4 | — | 4 | 18 | no | — |
| 249 | C4 | — | 4 | 4 | no | — |
| 250 | C4 | C22×#126 | 1 | 24 | yes | — |
| 251 | C2 | — | 2 | 4 | no | — |
| 252 | C2 | — | 8 | 4 | no | — |
| 253 | C2 | C23×#126 | 1 | 1 | yes | characteristic |
| 254 | C2 | #62788 | 1 | 3 | yes | — |
| 255 | C2 | #62793 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×2−1+4 | C22 |
| 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 |
|---|---|---|
| 1. #36641 | C2 | semidirect |
| 2. #36585 | C2 | semidirect |
| 3. Q8×#126 | C2 | semidirect |
| 4. #36458 | C2 | semidirect |
| 5. #36448 | C2 | semidirect |
| 6. #35160 | C2 | semidirect |
| 8. #34249 | C2 | semidirect |
| 9. C2×C42×Q8 | C2 | semidirect |
| 165. C2×Q8 | #126 | semidirect |
| 202. C42 | 2−1+4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #36641 | 2 |
| #36585 | 2 |
| Q8×#126 | 2 |
| #36458 | 2 |
| #36448 | 2 |
| #35160 | 2 |
| #34257 | 2 |
| #34249 | 2 |
| C2×C42×Q8 | 2 |