Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
440 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 150 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 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#2246 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33726 | C2 | semidirect |
| 3. #13723 | C2 | semidirect |
| 4. #13719 | C2 | semidirect |
| 5. #13617 | C2 | semidirect |
| 6. C4×M64 | C2 | semidirect |
| 14. #2253 | C22 | semidirect |
| 15. #2252 | C22 | semidirect |
| 23. #1415 | C22 | semidirect |
| 24. #1397 | C22 | semidirect |
| 25. C4×C32 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33726 | 2 |
| C2×#2246 | 2 |
| #13723 | 2 |
| #13719 | 2 |
| #13617 | 2 |
| C4×M64 | 2 |