Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
496 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 150 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 | C2×#493 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33797 | C2 | semidirect |
| 3. #19604 | C2 | semidirect |
| 5. #12535 | C2 | semidirect |
| 6. #12534 | C2 | semidirect |
| 7. #12532 | C2 | semidirect |
| 8. #12531 | C2 | semidirect |
| 9. #12522 | C2 | semidirect |
| 10. #12511 | C2 | semidirect |
| 11. #11410 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33797 | 2 |
| #33794 | 2 |
| #19604 | 2 |
| #19600 | 2 |
| #12535 | 2 |
| #12534 | 2 |
| #12532 | 2 |
| #12531 | 2 |
| #12522 | 2 |
| #12511 | 2 |
| #11410 | 2 |