Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
144 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12430 |
| commutator, Frattini, cosocle | C4×C16 | C23 |
| prime radical | #12436 | C2 |
| LCS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2170 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13727 | C2 | semidirect |
| 2. #13647 | C2 | semidirect |
| 3. #12437 | C2 | semidirect |
| 5. #7462 | C2 | semidirect |
| 6. #7449 | C2 | semidirect |
| 7. #7448 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13727 | 2 |
| #13647 | 2 |
| #12437 | 2 |
| #12436 | 2 |
| #7462 | 2 |
| #7449 | 2 |
| #7448 | 2 |