Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
576 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 144 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 | C2×#459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #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. #33627 | C2 | semidirect |
| 2. #21733 | C2 | semidirect |
| 4. #13509 | C2 | semidirect |
| 6. #13136 | C2 | semidirect |
| 7. #12884 | C2 | semidirect |
| 8. #12818 | C2 | semidirect |
| 42. #500 | D4 | semidirect |
| 44. #499 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33627 | 2 |
| #21733 | 2 |
| #21697 | 2 |
| #13509 | 2 |
| #13413 | 2 |
| #13136 | 2 |
| #12884 | 2 |
| #12818 | 2 |