Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
423 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 423 conjugacy classes of subgroups — too many to draw. Only its 128 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×#466 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #19332 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ 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. #33792 | C2 | semidirect |
| 2. #33493 | C2 | semidirect |
| 4. #19193 | C2 | semidirect |
| 5. #12472 | C2 | semidirect |
| 6. #12468 | C2 | semidirect |
| 7. #12455 | C2 | semidirect |
| 8. #12452 | C2 | semidirect |
| 9. #12036 | C2 | semidirect |
| 10. #11972 | C2 | semidirect |
| 11. #11948 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33792 | 2 |
| #33493 | 2 |
| #19332 | 2 |
| #19193 | 2 |
| #12472 | 2 |
| #12468 | 2 |
| #12455 | 2 |
| #12452 | 2 |
| #12036 | 2 |
| #11972 | 2 |
| #11948 | 2 |