Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
470 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 470 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×#466 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×D4 | D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33734 | C2 | semidirect |
| 4. #12469 | C2 | semidirect |
| 5. #12452 | C2 | semidirect |
| 6. #12431 | C2 | semidirect |
| 7. #12022 | C2 | semidirect |
| 8. #11983 | C2 | semidirect |
| 13. #2252 | C22 | semidirect |
| 14. #2247 | C22 | semidirect |
| 15. #2241 | C22 | semidirect |
| 32. #1371 | C22 | semidirect |
| 33. #1371 | C22 | semidirect |
| 34. #1365 | C22 | semidirect |
| 35. #1362 | C22 | semidirect |
| 145. C2×D4 | Q32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33734 | 2 |
| C2×D4×C16 | 2 |
| #19256 | 2 |
| #12469 | 2 |
| #12452 | 2 |
| #12431 | 2 |
| #12022 | 2 |
| #11983 | 2 |