Finite Group DatabasebetaFinite Group DatabasebetaOrder 16448.
234 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D257 |
| commutator, socle, cosocle | C22×C257 | C24 |
| Fitting | #123×C257 | C2 |
| prime radical | #22551564 | C2 |
| hypercenter | #123 | D257 |
| hypocenter | C257 | #529 |
| LCS | G ▷ C22×C257 ▷ C257 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22551587 | C2 | semidirect |
| 2. #22551585 | C2 | semidirect |
| 3. #22551582 | C2 | semidirect |
| 4. #22551578 | C2 | semidirect |
| 5. #22551575 | C2 | semidirect |
| 6. #22551569 | C2 | semidirect |
| 8. #22551557 | C2 | semidirect |
| 9. D4×Dic257 | C2 | semidirect |
| 10. #22551552 | C2 | semidirect |
| 11. #22551552 | C2 | semidirect |
| 12. #22551550 | C2 | semidirect |
| 13. #22551548 | C2 | semidirect |
| 14. C2×C4×Dic257 | C2 | semidirect |
| 15. #123×C257 | C2 | semidirect |
| 103. C257 | #529 | semidirect |
| 110. #123 | D257 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22551587 | 2 |
| #22551585 | 2 |
| #22551582 | 2 |
| #22551578 | 2 |
| #22551575 | 2 |
| #22551569 | 2 |
| #22551564 | 2 |
| #22551557 | 2 |
| D4×Dic257 | 2 |
| #22551552 | 2 |
| #22551550 | 2 |
| #22551548 | 2 |
| C2×C4×Dic257 | 2 |
| #123×C257 | 2 |
| #529 | 257 |