Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
457 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 457 conjugacy classes of subgroups — too many to draw. Only its 128 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 | #33780 | C2 |
| 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. #33793 | C2 | semidirect |
| 3. #19655 | C2 | semidirect |
| 4. #17970 | C2 | semidirect |
| 5. #12487 | C2 | semidirect |
| 6. #12485 | C2 | semidirect |
| 7. #12420 | C2 | semidirect |
| 8. #12418 | C2 | semidirect |
| 9. #12014 | C2 | semidirect |
| 10. #11975 | C2 | semidirect |
| 11. #11963 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33793 | 2 |
| #33780 | 2 |
| #19655 | 2 |
| #17970 | 2 |
| #12487 | 2 |
| #12485 | 2 |
| #12420 | 2 |
| #12418 | 2 |
| #12014 | 2 |
| #11975 | 2 |
| #11963 | 2 |