Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
599 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 599 conjugacy classes of subgroups — too many to draw. Only its 146 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 | #2537 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #28176 | C2 | semidirect |
| 4. C2×C8×Q16 | C2 | semidirect |
| 5. #10459 | C2 | semidirect |
| 6. #8824 | C2 | semidirect |
| 7. #8688 | C2 | semidirect |
| 8. #8617 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28176 | 2 |
| #19261 | 2 |
| #17956 | 2 |
| C2×C8×Q16 | 2 |
| #10459 | 2 |
| #8824 | 2 |
| #8688 | 2 |
| #8617 | 2 |