Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
517 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 517 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3227 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #28202 |
| prime radical | #3150 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ 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. #33637 | C2 | semidirect |
| 2. #21733 | C2 | semidirect |
| 4. #13511 | C2 | semidirect |
| 6. #13140 | C2 | semidirect |
| 7. #12881 | C2 | semidirect |
| 8. #12825 | C2 | semidirect |
| 43. #499 | D4 | semidirect |
| 44. #498 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33637 | 2 |
| #21733 | 2 |
| #21708 | 2 |
| #13511 | 2 |
| #13416 | 2 |
| #13140 | 2 |
| #12881 | 2 |
| #12825 | 2 |