Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
491 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 491 conjugacy classes of subgroups — too many to draw. Only its 140 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 | C4×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ 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. #33628 | C2 | semidirect |
| 4. #17897 | C2 | semidirect |
| 5. #12486 | C2 | semidirect |
| 6. #12484 | C2 | semidirect |
| 7. #12005 | C2 | semidirect |
| 8. #11939 | C2 | semidirect |
| 18. #2227 | C22 | semidirect |
| 19. #2225 | C22 | semidirect |
| 31. #1591 | C22 | semidirect |
| 35. #1370 | C22 | semidirect |
| 36. #1365 | C22 | semidirect |
| 37. #1364 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33628 | 2 |
| C2×#2207 | 2 |
| #19652 | 2 |
| #17897 | 2 |
| #12486 | 2 |
| #12484 | 2 |
| #12005 | 2 |
| #11939 | 2 |
| #11931 | 2 |