Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
400 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 126 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 | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33499 | C2 | semidirect |
| 4. #19208 | C2 | semidirect |
| 5. #12479 | C2 | semidirect |
| 6. #12475 | C2 | semidirect |
| 7. #12460 | C2 | semidirect |
| 8. #12457 | C2 | semidirect |
| 9. #12034 | C2 | semidirect |
| 10. #11974 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33792 | 2 |
| #33499 | 2 |
| #19341 | 2 |
| #19208 | 2 |
| #12479 | 2 |
| #12475 | 2 |
| #12460 | 2 |
| #12457 | 2 |
| #12034 | 2 |
| #11974 | 2 |
| #11951 | 2 |