Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4026 conjugacy classes of subgroups; only the 1858 normal subgroups are shown.
This group has 4026 conjugacy classes of subgroups — too many to draw. Only its 1858 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13829 | C2 | direct (N × H) |
| 7. C2×#382 | C4 | semidirect |
| 22. #382 | C2×C4 | semidirect |
| 45. C2×C42 | C42 | semidirect |
| 73. C42 | C2×C42 | semidirect |
| 109. C2 | #13829 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#382 | 2 |
| #13829 | 2 |
| C2×#1436 | 2 |