Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
556 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21737 | C2 | semidirect |
| 3. #13542 | C2 | semidirect |
| 4. #13487 | C2 | semidirect |
| 6. #13367 | C2 | semidirect |
| 7. #12647 | C2 | semidirect |
| 8. #12596 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21737 | 2 |
| #21697 | 2 |
| #13542 | 2 |
| #13487 | 2 |
| #13413 | 2 |
| #13367 | 2 |
| #12647 | 2 |
| #12596 | 2 |