Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
464 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 130 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 | #19101 | C2 |
| 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. #33776 | C2 | semidirect |
| 2. #33486 | C2 | semidirect |
| 3. #19259 | C2 | semidirect |
| 5. #12470 | C2 | semidirect |
| 6. #12467 | C2 | semidirect |
| 7. #12432 | C2 | semidirect |
| 8. #12431 | C2 | semidirect |
| 9. #12034 | C2 | semidirect |
| 10. #11963 | C2 | semidirect |
| 11. #11950 | C2 | semidirect |
| 17. #2248 | C22 | semidirect |
| 18. #2247 | C22 | semidirect |
| 19. #2241 | C22 | semidirect |
| 20. #2240 | C22 | semidirect |
| 39. #1372 | C22 | semidirect |
| 40. #1371 | C22 | semidirect |
| 41. #1364 | C22 | semidirect |
| 42. #1362 | C22 | semidirect |
| 43. #1362 | C22 | semidirect |
| 44. #1361 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33776 | 2 |
| #33486 | 2 |
| #19259 | 2 |
| #19101 | 2 |
| #12470 | 2 |
| #12467 | 2 |
| #12432 | 2 |
| #12431 | 2 |
| #12034 | 2 |
| #11963 | 2 |
| #11950 | 2 |