Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
443 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 138 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 | #1505 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 10. #2166 | C4 | semidirect |
| 14. #1846 | C4 | semidirect |
| 15. #1844 | C4 | semidirect |
| 19. #1776 | C4 | semidirect |
| 22. #1747 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2166 | 2 |
| #20820 | 2 |
| C2×#1747 | 2 |
| #7255 | 2 |
| #7254 | 2 |