Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
459 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 459 conjugacy classes of subgroups — too many to draw. Only its 126 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 | #19187 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ 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. #33774 | C2 | semidirect |
| 2. #33495 | C2 | semidirect |
| 3. #19330 | C2 | semidirect |
| 5. #12472 | C2 | semidirect |
| 6. #12468 | C2 | semidirect |
| 7. #12435 | C2 | semidirect |
| 8. #12432 | C2 | semidirect |
| 9. #12034 | C2 | semidirect |
| 10. #11964 | C2 | semidirect |
| 11. #11953 | C2 | semidirect |
| 23. #2169 | C22 | semidirect |
| 25. D4×C16 | C22 | semidirect |
| 31. #1675 | C22 | semidirect |
| 32. #1669 | C22 | semidirect |
| 39. #1372 | C22 | semidirect |
| 40. #1371 | C22 | semidirect |
| 41. #1365 | C22 | semidirect |
| 42. #1362 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33774 | 2 |
| #33495 | 2 |
| #19330 | 2 |
| #19187 | 2 |
| #12472 | 2 |
| #12468 | 2 |
| #12435 | 2 |
| #12432 | 2 |
| #12034 | 2 |
| #11964 | 2 |
| #11953 | 2 |