Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
732 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 732 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#366 | C2 | semidirect |
| 4. #13493 | C2 | semidirect |
| 5. #13492 | C2 | semidirect |
| 6. #12557 | C2 | semidirect |
| 7. #12556 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#366 | 2 |
| C2×#2113 | 2 |
| #20663 | 2 |
| #13493 | 2 |
| #13492 | 2 |
| #12557 | 2 |
| #12556 | 2 |