Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
456 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 118 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 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ 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. #33905 | C2 | semidirect |
| 2. #33853 | C2 | semidirect |
| 3. #19937 | C2 | semidirect |
| 4. #12503 | C2 | semidirect |
| 5. #12486 | C2 | semidirect |
| 6. #12452 | C2 | semidirect |
| 7. #12430 | C2 | semidirect |
| 8. #12373 | C2 | semidirect |
| 9. #12342 | C2 | semidirect |
| 10. #12083 | C2 | semidirect |
| 11. C16×D8 | C2 | semidirect |
| 232. D4 | #508 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33905 | 2 |
| #33853 | 2 |
| #19937 | 2 |
| #12503 | 2 |
| #12486 | 2 |
| #12452 | 2 |
| #12430 | 2 |
| #12373 | 2 |
| #12342 | 2 |
| #12083 | 2 |
| C16×D8 | 2 |