Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
377 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 377 conjugacy classes of subgroups — too many to draw. Only its 116 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 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×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. #33733 | C2 | semidirect |
| 2. C4×#508 | C2 | semidirect |
| 3. #13713 | C2 | semidirect |
| 4. #13704 | C2 | semidirect |
| 5. #13703 | C2 | semidirect |
| 6. #13686 | C2 | semidirect |
| 7. #13685 | C2 | semidirect |
| 8. #13663 | C2 | semidirect |
| 9. #13654 | C2 | semidirect |
| 10. #13652 | C2 | semidirect |
| 11. #13640 | C2 | semidirect |
| 119. Q32 | #41 | semidirect |
| 125. D16 | #41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33733 | 2 |
| C4×#508 | 2 |
| #13713 | 2 |
| #13704 | 2 |
| #13703 | 2 |
| #13686 | 2 |
| #13685 | 2 |
| #13663 | 2 |
| #13654 | 2 |
| #13652 | 2 |
| #13640 | 2 |