Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
334 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #354 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | C2×#384 | C22 |
| prime radical | C24 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G | |
| LECS | G ▷ C2×#384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. #1325 | C4 | semidirect |
| 19. C2×#97 | C8 | semidirect |
| 26. #382 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1325 | 2 |
| #10843 | 2 |