Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
28878 conjugacy classes of subgroups; only the 1194 normal subgroups are shown.
This group has 28878 conjugacy classes of subgroups — too many to draw. Only its 1194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C26×C4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#1779 | C2 | 1 | 14 | yes | maximal, complemented |
| 3 | C27 | C4 | 1 | 1 | yes | characteristic |
| 4 | C25×C4 | — | 2 | 56 | no | — |
| 5 | C25×C4 | C4 | 1 | 7 | yes | — |
| 6 | C25×C4 | C22 | 1 | 7 | yes | — |
| 7 | C22×C22⋊C8 | C22 | 1 | 28 | yes | complemented |
| 8 | #1779 | — | 2 | 56 | no | — |
| 9 | C26 | — | 2 | 28 | no | — |
| 10 | C26 | — | 2 | 28 | no | — |
| 11 | C26 | C8 | 1 | 8 | yes | complemented |
| 12 | C26 | C2×C4 | 1 | 7 | yes | — |
| 13 | C24×C4 | — | 2 | 896 | no | — |
| 14 | C24×C4 | — | 2 | 28 | no | — |
| 15 | C24×C4 | — | 2 | 168 | no | — |
| 16 | C24×C4 | — | 2 | 168 | no | — |
| 17 | C24×C4 | C2×C4 | 1 | 21 | yes | — |
| 18 | C24×C4 | D4 | 1 | 56 | yes | — |
| 19 | C24×C4 | C23 | 1 | 7 | yes | — |
| 20 | C23×C8 | C23 | 1 | 8 | yes | complemented |
| 21 | C2×C22⋊C8 | — | 2 | 168 | no | — |
| 22 | C25 | — | 2 | 224 | no | — |
| 23 | C25 | — | 2 | 672 | no | — |
| 24 | C25 | — | 2 | 224 | no | — |
| 25 | C25 | — | 2 | 84 | no | — |
| 26 | C25 | — | 2 | 84 | no | — |
| 27 | C25 | C22⋊C4 | 1 | 28 | yes | — |
| 28 | C25 | C2×C8 | 1 | 28 | yes | — |
| 29 | C25 | M16 | 1 | 28 | yes | — |
| 30 | C25 | C22×C4 | 1 | 7 | yes | — |
| 31 | C23×C4 | — | 2 | 2048 | no | — |
| 32 | C23×C4 | — | 2 | 672 | no | — |
| 33 | C23×C4 | — | 2 | 896 | no | — |
| 34 | C23×C4 | — | 2 | 896 | no | — |
| 35 | C23×C4 | — | 2 | 84 | no | — |
| 36 | C23×C4 | — | 2 | 168 | no | — |
| 37 | C23×C4 | — | 2 | 672 | no | — |
| 38 | C23×C4 | — | 2 | 56 | no | — |
| 39 | C23×C4 | C22⋊C4 | 1 | 84 | yes | — |
| 40 | C23×C4 | C22×C4 | 1 | 7 | yes | — |
| 41 | C23×C4 | C2×D4 | 1 | 84 | yes | — |
| 42 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 43 | C22×C8 | — | 2 | 56 | no | — |
| 44 | C22⋊C8 | — | 4 | 112 | no | — |
| 45 | C24 | — | 2 | 256 | no | — |
| 46 | C24 | — | 2 | 224 | no | — |
| 47 | C24 | — | 2 | 1792 | no | — |
| 48 | C24 | — | 2 | 672 | no | — |
| 49 | C24 | — | 2 | 1792 | no | — |
| 50 | C24 | — | 2 | 28 | no | — |
| 51 | C24 | — | 2 | 28 | no | — |
| 52 | C24 | — | 2 | 168 | no | — |
| 53 | C24 | — | 2 | 336 | no | — |
| 54 | C24 | — | 2 | 168 | no | — |
| 55 | C24 | — | 2 | 336 | no | — |
| 56 | C24 | C22⋊C8 | 1 | 112 | yes | complemented |
| 57 | C24 | C2×C22⋊C4 | 1 | 42 | yes | — |
| 58 | C24 | C22×C8 | 1 | 14 | yes | — |
| 59 | C24 | C2×M16 | 1 | 42 | yes | — |
| 60 | C24 | C23×C4 | 1 | 1 | yes | characteristic |
| 61 | C22×C4 | — | 2 | 1792 | no | — |
| 62 | C22×C4 | — | 2 | 672 | no | — |
| 63 | C22×C4 | — | 2 | 1792 | no | — |
| 64 | C22×C4 | — | 2 | 28 | no | — |
| 65 | C22×C4 | — | 2 | 168 | no | — |
| 66 | C22×C4 | — | 2 | 336 | no | — |
| 67 | C22×C4 | — | 2 | 336 | no | — |
| 68 | C22×C4 | C2×C22⋊C4 | 1 | 42 | yes | — |
| 69 | C22×C4 | #120 | 1 | 112 | yes | — |
| 70 | C22×C4 | C22×D4 | 1 | 14 | yes | — |
| 71 | C2×C8 | — | 4 | 56 | no | — |
| 72 | C23 | — | 2 | 224 | no | — |
| 73 | C23 | — | 2 | 2048 | no | — |
| 74 | C23 | — | 2 | 896 | no | — |
| 75 | C23 | — | 2 | 672 | no | — |
| 76 | C23 | — | 2 | 896 | no | — |
| 77 | C23 | — | 2 | 56 | no | — |
| 78 | C23 | — | 2 | 84 | no | — |
| 79 | C23 | — | 2 | 168 | no | — |
| 80 | C23 | — | 2 | 84 | no | — |
| 81 | C23 | — | 2 | 672 | no | — |
| 82 | C23 | #379 | 1 | 28 | yes | — |
| 83 | C23 | C2×C22⋊C8 | 1 | 84 | yes | — |
| 84 | C23 | #407 | 1 | 84 | yes | — |
| 85 | C23 | C22×C22⋊C4 | 1 | 7 | yes | — |
| 86 | C23 | C23×C8 | 1 | 1 | yes | characteristic |
| 87 | C23 | C22×M16 | 1 | 7 | yes | — |
| 88 | C2×C4 | — | 2 | 672 | no | — |
| 89 | C2×C4 | — | 2 | 84 | no | — |
| 90 | C2×C4 | — | 2 | 224 | no | — |
| 91 | C2×C4 | #379 | 1 | 28 | yes | — |
| 92 | C2×C4 | C2×#120 | 1 | 28 | yes | — |
| 93 | C8 | — | 8 | 8 | no | — |
| 94 | C22 | — | 2 | 896 | no | — |
| 95 | C22 | — | 2 | 28 | no | — |
| 96 | C22 | — | 2 | 168 | no | — |
| 97 | C22 | — | 2 | 28 | no | — |
| 98 | C22 | — | 2 | 168 | no | — |
| 99 | C22 | #1779 | 1 | 56 | yes | complemented |
| 100 | C22 | C2×#379 | 1 | 7 | yes | — |
| 101 | C22 | C22×C22⋊C8 | 1 | 7 | yes | — |
| 102 | C22 | C2×#407 | 1 | 21 | yes | — |
| 103 | C4 | — | 2 | 28 | no | — |
| 104 | C4 | #2846 | 1 | 8 | yes | — |
| 105 | C2 | — | 2 | 56 | no | — |
| 106 | C2 | #14679 | 1 | 1 | yes | characteristic |
| 107 | C2 | C2×#1779 | 1 | 7 | yes | — |
| 108 | C2 | #20579 | 1 | 7 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator | C23 | C23×C8 |
| socle | C24 | C23×C4 |
| prime radical | C27 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1779 | C2 | semidirect |
| 7. C22×C22⋊C8 | C22 | semidirect |
| 11. C26 | C8 | semidirect |
| 20. C23×C8 | C23 | semidirect |
| 56. C24 | C22⋊C8 | semidirect |
| 99. C22 | #1779 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C26×C4 | 2 |
| C2×#1779 | 2 |