Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
635 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 635 conjugacy classes of subgroups — too many to draw. Only its 104 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 | #12731 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#3415 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13625 | C2 | semidirect |
| 3. C2×#1419 | C2 | semidirect |
| 4. #13622 | C2 | semidirect |
| 5. #13617 | C2 | semidirect |
| 6. #13558 | C2 | semidirect |
| 7. #12841 | C2 | semidirect |
| 12. #3415 | C4 | semidirect |
| 20. C2×M64 | C22 | semidirect |
| 21. C2×M64 | C22 | semidirect |
| 22. C2×#368 | C22 | semidirect |
| 23. #2158 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3415 | 2 |
| #13625 | 2 |
| C2×#1419 | 2 |
| #13622 | 2 |
| #13617 | 2 |
| #13558 | 2 |
| #12841 | 2 |