Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
311 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 311 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×D32 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ Q8×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 |
|---|---|---|
| 1. #33947 | C2 | semidirect |
| 2. Q8×D16 | C2 | semidirect |
| 3. #13734 | C2 | semidirect |
| 4. #13733 | C2 | semidirect |
| 5. #13707 | C2 | semidirect |
| 6. #13705 | C2 | semidirect |
| 7. #13705 | C2 | semidirect |
| 8. #13703 | C2 | semidirect |
| 9. #13660 | C2 | semidirect |
| 58. D32 | Q8 | semidirect |
| 98. D16 | C2×Q8 | semidirect |
| 120. C32 | C2×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33947 | 2 |
| Q8×D16 | 2 |
| #13734 | 2 |
| #13733 | 2 |
| #13707 | 2 |
| #13705 | 2 |
| #13703 | 2 |
| #13660 | 2 |
| #13654 | 2 |
| C4×D32 | 2 |