Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4388 conjugacy classes of subgroups.
The subgroup lattice has 204 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #37565 | C2 | 1 | 12 | yes | maximal, complemented |
| 2 | #37481 | C2 | 1 | 6 | yes | maximal |
| 3 | #37480 | C2 | 1 | 6 | yes | maximal |
| 4 | #35294 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | #35213 | C2 | 1 | 24 | yes | maximal, complemented |
| 6 | #34012 | C2 | 1 | 2 | yes | maximal |
| 7 | #34012 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C23×C42 | C22 | 1 | 1 | yes | characteristic |
| 9 | #2868 | C22 | 1 | 4 | yes | complemented |
| 10 | #2863 | C22 | 1 | 32 | yes | complemented |
| 11 | #2859 | C22 | 1 | 12 | yes | complemented |
| 12 | #2843 | C22 | 1 | 16 | yes | complemented |
| 13 | #2842 | C22 | 1 | 12 | yes | — |
| 14 | #2840 | C22 | 1 | 4 | yes | — |
| 15 | #2667 | C22 | 1 | 48 | yes | — |
| 16 | #2665 | C22 | 1 | 48 | yes | — |
| 17 | #2629 | C22 | 1 | 24 | yes | — |
| 18 | #2628 | C22 | 1 | 24 | yes | — |
| 19 | #2612 | C22 | 1 | 48 | yes | complemented |
| 20 | #2608 | C22 | 1 | 24 | yes | — |
| 21 | #2607 | C22 | 1 | 24 | yes | — |
| 22 | #2570 | C22 | 1 | 48 | yes | complemented |
| 23 | #2550 | C22 | 1 | 48 | yes | complemented |
| 24 | #2441 | C22 | 1 | 48 | yes | — |
| 25 | #2439 | C22 | 1 | 48 | yes | — |
| 26 | #2331 | C22 | 1 | 24 | yes | complemented |
| 27 | #2330 | C22 | 1 | 24 | yes | complemented |
| 28 | #2285 | C22 | 1 | 6 | yes | — |
| 29 | #2285 | C22 | 1 | 12 | yes | — |
| 30 | #2283 | C22 | 1 | 12 | yes | — |
| 31 | #2283 | C22 | 1 | 12 | yes | — |
| 32 | #2283 | C22 | 1 | 12 | yes | — |
| 33 | #2280 | C22 | 1 | 3 | yes | — |
| 34 | #2280 | C22 | 1 | 12 | yes | — |
| 35 | #2280 | C22 | 1 | 3 | yes | — |
| 36 | C4×#117 | C22 | 1 | 12 | yes | — |
| 37 | C4×#117 | C22 | 1 | 6 | yes | — |
| 38 | C24×C4 | C23 | 1 | 3 | yes | — |
| 39 | C2×C4⋊Q8 | C23 | 1 | 24 | yes | — |
| 40 | C2×#125 | C23 | 1 | 24 | yes | — |
| 41 | C2×#117 | C23 | 1 | 6 | yes | — |
| 42 | C2×#117 | C23 | 1 | 6 | yes | — |
| 43 | C2×#117 | C23 | 1 | 24 | yes | — |
| 44 | C22×C42 | Q8 | 1 | 16 | yes | complemented |
| 45 | C22×C42 | C23 | 1 | 6 | yes | — |
| 46 | C22×C42 | C23 | 1 | 3 | yes | — |
| 47 | C22×C42 | C23 | 1 | 3 | yes | — |
| 48 | #400 | C23 | 1 | 48 | yes | — |
| 49 | #399 | C23 | 1 | 48 | yes | — |
| 50 | #398 | C23 | 1 | 96 | yes | — |
| 51 | #396 | C23 | 1 | 96 | yes | — |
| 52 | #395 | C23 | 1 | 48 | yes | — |
| 53 | #393 | C23 | 1 | 48 | yes | — |
| 54 | #391 | C23 | 1 | 96 | yes | — |
| 55 | #389 | C23 | 1 | 48 | yes | — |
| 56 | #389 | C23 | 1 | 48 | yes | — |
| 57 | #389 | C23 | 1 | 96 | yes | — |
| 58 | #387 | C23 | 1 | 96 | yes | — |
| 59 | #387 | C23 | 1 | 96 | yes | — |
| 60 | #384 | C23 | 1 | 24 | yes | — |
| 61 | #384 | C23 | 1 | 4 | yes | — |
| 62 | #384 | C23 | 1 | 8 | yes | — |
| 63 | #383 | C23 | 1 | 8 | yes | — |
| 64 | #383 | C23 | 1 | 8 | yes | — |
| 65 | #383 | C23 | 1 | 8 | yes | — |
| 66 | #382 | C23 | 1 | 24 | yes | — |
| 67 | #382 | C23 | 1 | 24 | yes | — |
| 68 | #382 | C23 | 1 | 12 | yes | — |
| 69 | #381 | C23 | 1 | 6 | yes | — |
| 70 | #381 | C23 | 1 | 6 | yes | — |
| 71 | #381 | C23 | 1 | 24 | yes | — |
| 72 | #380 | C23 | 1 | 24 | yes | — |
| 73 | #380 | C23 | 1 | 12 | yes | — |
| 74 | #380 | C23 | 1 | 12 | yes | — |
| 75 | #380 | C23 | 1 | 24 | yes | — |
| 76 | #379 | C23 | 1 | 1 | yes | characteristic |
| 77 | #379 | C23 | 1 | 8 | yes | — |
| 78 | #379 | C23 | 1 | 3 | yes | — |
| 79 | C4×C4⋊C4 | C23 | 1 | 24 | yes | — |
| 80 | C4×C4⋊C4 | C23 | 1 | 24 | yes | — |
| 81 | C4×C4⋊C4 | C23 | 1 | 24 | yes | — |
| 82 | C4×C22⋊C4 | C23 | 1 | 24 | yes | — |
| 83 | C4×C22⋊C4 | C23 | 1 | 24 | yes | — |
| 84 | C4×C22⋊C4 | C23 | 1 | 12 | yes | — |
| 85 | C4×C22⋊C4 | C23 | 1 | 12 | yes | — |
| 86 | #376 | C23 | 1 | 12 | yes | — |
| 87 | #376 | C23 | 1 | 24 | yes | — |
| 88 | C43 | C23 | 1 | 4 | yes | — |
| 89 | C43 | C23 | 1 | 8 | yes | — |
| 90 | C25 | C24 | 1 | 1 | yes | characteristic |
| 91 | C22×Q8 | C24 | 1 | 16 | yes | — |
| 92 | C23×C4 | — | 4 | 12 | no | — |
| 93 | C23×C4 | C2×Q8 | 1 | 24 | yes | — |
| 94 | C23×C4 | C24 | 1 | 6 | yes | — |
| 95 | C23×C4 | C24 | 1 | 3 | yes | — |
| 96 | C23×C4 | C24 | 1 | 3 | yes | — |
| 97 | C23×C4 | C24 | 1 | 6 | yes | — |
| 98 | C4⋊Q8 | — | 4 | 48 | no | — |
| 99 | #125 | — | 4 | 48 | no | — |
| 100 | #117 | — | 4 | 12 | no | — |
| 101 | #117 | — | 4 | 12 | no | — |
| 102 | #117 | — | 4 | 48 | no | — |
| 103 | C2×C4⋊C4 | C24 | 1 | 48 | yes | — |
| 104 | C2×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 105 | C2×C4⋊C4 | C24 | 1 | 12 | yes | — |
| 106 | C2×C4⋊C4 | C24 | 1 | 96 | yes | — |
| 107 | C2×C4⋊C4 | C24 | 1 | 48 | yes | — |
| 108 | C2×C22⋊C4 | C24 | 1 | 6 | yes | — |
| 109 | C2×C22⋊C4 | C24 | 1 | 48 | yes | — |
| 110 | C2×C22⋊C4 | C24 | 1 | 12 | yes | — |
| 111 | C2×C22⋊C4 | C24 | 1 | 6 | yes | — |
| 112 | C2×C42 | C2×Q8 | 1 | 24 | yes | — |
| 113 | C2×C42 | C2×Q8 | 1 | 48 | yes | — |
| 114 | C2×C42 | C2×Q8 | 1 | 24 | yes | — |
| 115 | C2×C42 | C24 | 1 | 12 | yes | — |
| 116 | C2×C42 | C24 | 1 | 12 | yes | — |
| 117 | C2×C42 | C24 | 1 | 48 | yes | — |
| 118 | C2×C42 | C24 | 1 | 2 | yes | — |
| 119 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 120 | C2×C42 | C24 | 1 | 3 | yes | — |
| 121 | C2×C42 | C24 | 1 | 3 | yes | — |
| 122 | C2×C42 | C24 | 1 | 6 | yes | — |
| 123 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 124 | #95 | C24 | 1 | 24 | yes | — |
| 125 | #95 | C24 | 1 | 48 | yes | — |
| 126 | #95 | C24 | 1 | 24 | yes | — |
| 127 | #95 | C24 | 1 | 48 | yes | — |
| 128 | #95 | C24 | 1 | 48 | yes | — |
| 129 | #95 | C24 | 1 | 48 | yes | — |
| 130 | C24 | — | 4 | 6 | no | — |
| 131 | C24 | C22×Q8 | 1 | 4 | yes | — |
| 132 | C24 | C25 | 1 | 3 | yes | — |
| 133 | C2×Q8 | — | 4 | 48 | no | — |
| 134 | C22×C4 | — | 2 | 24 | no | — |
| 135 | C22×C4 | — | 2 | 12 | no | — |
| 136 | C22×C4 | — | 4 | 48 | no | — |
| 137 | C22×C4 | — | 4 | 6 | no | — |
| 138 | C22×C4 | — | 4 | 12 | no | — |
| 139 | C22×C4 | C22×Q8 | 1 | 24 | yes | — |
| 140 | C22×C4 | C22×Q8 | 1 | 12 | yes | — |
| 141 | C22×C4 | C22×Q8 | 1 | 12 | yes | — |
| 142 | C22×C4 | C22×Q8 | 1 | 24 | yes | — |
| 143 | C22×C4 | C25 | 1 | 4 | yes | — |
| 144 | C22×C4 | C25 | 1 | 32 | yes | — |
| 145 | C22×C4 | C25 | 1 | 12 | yes | — |
| 146 | C22×C4 | C25 | 1 | 6 | yes | — |
| 147 | C22×C4 | C25 | 1 | 3 | yes | — |
| 148 | C22×C4 | C25 | 1 | 3 | yes | — |
| 149 | C4⋊C4 | — | 2 | 96 | no | — |
| 150 | C4⋊C4 | — | 4 | 12 | no | — |
| 151 | C4⋊C4 | — | 4 | 12 | no | — |
| 152 | C4⋊C4 | — | 4 | 96 | no | — |
| 153 | C4⋊C4 | — | 4 | 48 | no | — |
| 154 | C22⋊C4 | — | 4 | 12 | no | — |
| 155 | C22⋊C4 | — | 4 | 6 | no | — |
| 156 | C22⋊C4 | — | 4 | 6 | no | — |
| 157 | C22⋊C4 | — | 4 | 48 | no | — |
| 158 | C42 | — | 2 | 24 | no | — |
| 159 | C42 | — | 2 | 24 | no | — |
| 160 | C42 | — | 4 | 48 | no | — |
| 161 | C42 | C22×Q8 | 1 | 8 | yes | — |
| 162 | C42 | C22×Q8 | 1 | 12 | yes | — |
| 163 | C42 | C22×Q8 | 1 | 4 | yes | — |
| 164 | C42 | C22×Q8 | 1 | 12 | yes | — |
| 165 | C42 | C22×Q8 | 1 | 24 | yes | — |
| 166 | C42 | C22×Q8 | 1 | 4 | yes | — |
| 167 | C23 | — | 2 | 6 | no | — |
| 168 | C23 | — | 2 | 12 | no | — |
| 169 | C23 | — | 4 | 12 | no | — |
| 170 | C23 | — | 4 | 16 | no | — |
| 171 | C23 | C23×Q8 | 1 | 6 | yes | — |
| 172 | C23 | C26 | 1 | 1 | yes | characteristic |
| 173 | Q8 | — | 16 | 16 | no | — |
| 174 | C2×C4 | — | 2 | 12 | no | — |
| 175 | C2×C4 | — | 2 | 6 | no | — |
| 176 | C2×C4 | — | 2 | 48 | no | — |
| 177 | C2×C4 | — | 2 | 24 | no | — |
| 178 | C2×C4 | — | 2 | 24 | no | — |
| 179 | C2×C4 | — | 2 | 12 | no | — |
| 180 | C2×C4 | — | 2 | 12 | no | — |
| 181 | C2×C4 | — | 2 | 96 | no | — |
| 182 | C2×C4 | — | 4 | 12 | no | — |
| 183 | C2×C4 | — | 4 | 24 | no | — |
| 184 | C2×C4 | C23×Q8 | 1 | 12 | yes | — |
| 185 | C2×C4 | C23×Q8 | 1 | 6 | yes | — |
| 186 | C2×C4 | C23×Q8 | 1 | 6 | yes | — |
| 187 | C2×C4 | #585 | 1 | 12 | yes | — |
| 188 | C22 | — | 2 | 12 | no | — |
| 189 | C22 | — | 2 | 6 | no | — |
| 190 | C22 | — | 4 | 16 | no | — |
| 191 | C22 | — | 4 | 12 | no | — |
| 192 | C22 | C24×Q8 | 1 | 1 | yes | characteristic |
| 193 | C22 | C2×#585 | 1 | 6 | yes | — |
| 194 | C4 | — | 2 | 12 | no | — |
| 195 | C4 | — | 2 | 6 | no | — |
| 196 | C4 | — | 4 | 12 | no | — |
| 197 | C4 | — | 4 | 32 | no | — |
| 198 | C4 | — | 4 | 4 | no | — |
| 199 | C4 | #3458 | 1 | 12 | yes | — |
| 200 | C2 | — | 4 | 6 | no | — |
| 201 | C2 | C2×#3458 | 1 | 3 | yes | — |
| 202 | C2 | #62839 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| 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. #37565 | C2 | semidirect |
| 4. #35294 | C2 | semidirect |
| 5. #35213 | C2 | semidirect |
| 9. #2868 | C22 | semidirect |
| 10. #2863 | C22 | semidirect |
| 11. #2859 | C22 | semidirect |
| 12. #2843 | C22 | semidirect |
| 19. #2612 | C22 | semidirect |
| 22. #2570 | C22 | semidirect |
| 23. #2550 | C22 | semidirect |
| 26. #2331 | C22 | semidirect |
| 27. #2330 | C22 | semidirect |
| 44. C22×C42 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #37565 | 2 |
| #37481 | 2 |
| #37480 | 2 |
| #35294 | 2 |
| #35213 | 2 |
| #34012 | 2 |