Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
350 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 350 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 | #1982 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13278 | C2 | semidirect |
| 4. #12004 | C2 | semidirect |
| 5. #11936 | C2 | semidirect |
| 7. #8283 | C2 | semidirect |
| 18. #1707 | C4 | semidirect |
| 19. #1707 | C4 | semidirect |
| 20. #1706 | C4 | semidirect |
| 21. #1706 | C4 | semidirect |
| 260. C8 | #360 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #19596 | 2 |
| #13278 | 2 |
| #13255 | 2 |
| #12004 | 2 |
| #11936 | 2 |
| #11925 | 2 |
| #8283 | 2 |