Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
467 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 467 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 |
| 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. #33733 | C2 | semidirect |
| 2. #33434 | C2 | semidirect |
| 3. #19329 | C2 | semidirect |
| 4. #12467 | C2 | semidirect |
| 5. #12454 | C2 | semidirect |
| 6. #12433 | C2 | semidirect |
| 7. #12024 | C2 | semidirect |
| 8. #11985 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33733 | 2 |
| #33434 | 2 |
| #19329 | 2 |
| #12467 | 2 |
| #12454 | 2 |
| #12433 | 2 |
| #12024 | 2 |
| #11985 | 2 |