Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
377 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 377 conjugacy classes of subgroups — too many to draw. Only its 88 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 | #1893 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 18. #1979 | C4 | semidirect |
| 19. #1979 | C4 | semidirect |
| 20. #1952 | C4 | semidirect |
| 21. #1952 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33422 | 2 |
| #33124 | 2 |
| C2×#1350 | 2 |
| #11706 | 2 |
| #10828 | 2 |
| #7265 | 2 |
| #7260 | 2 |