Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
435 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 98 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 | #1987 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C2×#498 | C4 | semidirect |
| 14. #1941 | C4 | semidirect |
| 15. #1844 | C4 | semidirect |
| 54. #341 | Q8 | semidirect |
| 58. #340 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33207 | 2 |
| #12788 | 2 |
| #10706 | 2 |
| #7263 | 2 |
| #7262 | 2 |