Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
354 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 122 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 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×D4 | D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #13720 | C2 | semidirect |
| 4. #13715 | C2 | semidirect |
| 5. #13706 | C2 | semidirect |
| 6. #13695 | C2 | semidirect |
| 7. #13688 | C2 | semidirect |
| 8. #13665 | C2 | semidirect |
| 9. C4×Q64 | C2 | semidirect |
| 11. C2×#1416 | C2 | semidirect |
| 188. D4 | Q64 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33905 | 2 |
| D4×Q32 | 2 |
| #13720 | 2 |
| #13715 | 2 |
| #13706 | 2 |
| #13695 | 2 |
| #13688 | 2 |
| #13665 | 2 |
| C4×Q64 | 2 |
| D4×C32 | 2 |
| C2×#1416 | 2 |