Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
426 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D16 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C22 | C4×D16 |
| prime radical | C2×#2262 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ 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 |
|---|---|---|
| 2. C2×#2178 | C2 | semidirect |
| 3. #13655 | C2 | direct (N × H) |
| 4. C2×#1425 | C2 | semidirect |
| 5. C2×#1417 | C2 | semidirect |
| 6. C2×C4×C32 | C2 | semidirect |
| 9. #2262 | C4 | semidirect |
| 11. C2×Q64 | C4 | semidirect |
| 12. C2×SD64 | C4 | semidirect |
| 13. C2×D32 | C4 | semidirect |
| 17. #2178 | C22 | semidirect |
| 21. #1425 | C22 | semidirect |
| 22. #1417 | C22 | semidirect |
| 23. C4×C32 | C22 | semidirect |
| 40. Q64 | C2×C4 | semidirect |
| 41. SD64 | C2×C4 | semidirect |
| 42. D32 | C2×C4 | semidirect |
| 173. C2 | #13655 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2262 | 2 |
| C2×#2178 | 2 |
| #13655 | 2 |
| C2×#1425 | 2 |
| C2×#1417 | 2 |
| C2×C4×C32 | 2 |