Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
424 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 424 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×#459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #3169 | 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 |
|---|---|---|
| 3. #19668 | C2 | semidirect |
| 5. #12489 | C2 | semidirect |
| 6. #12487 | C2 | semidirect |
| 7. #12449 | C2 | semidirect |
| 8. #12426 | C2 | semidirect |
| 9. #12005 | C2 | semidirect |
| 10. #11987 | C2 | semidirect |
| 11. #11984 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33820 | 2 |
| #33812 | 2 |
| #19668 | 2 |
| #17970 | 2 |
| #12489 | 2 |
| #12487 | 2 |
| #12449 | 2 |
| #12426 | 2 |
| #12005 | 2 |
| #11987 | 2 |
| #11984 | 2 |