Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
398 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 146 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 | C22×D16 | C22 |
| 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. #33726 | C2 | semidirect |
| 3. #13722 | C2 | semidirect |
| 4. #13721 | C2 | semidirect |
| 5. #13618 | C2 | semidirect |
| 7. C4×M64 | C2 | semidirect |
| 16. #2253 | C22 | semidirect |
| 17. #2252 | C22 | semidirect |
| 25. #1416 | C22 | semidirect |
| 27. #1397 | C22 | semidirect |
| 28. C4×C32 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33726 | 2 |
| C2×#2249 | 2 |
| #13722 | 2 |
| #13721 | 2 |
| #13618 | 2 |
| #13617 | 2 |
| C4×M64 | 2 |