Finite Group DatabasebetaFinite Group DatabasebetaOrder 16448.
292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 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 |
| hypercenter | #123 | D257 |
| hypocenter | C257 | #547 |
| 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. C2×#16861618 | C2 | semidirect |
| 2. C2×#16861617 | C2 | semidirect |
| 3. #22551591 | C2 | semidirect |
| 4. #22551577 | C2 | semidirect |
| 5. #22551576 | C2 | semidirect |
| 6. #22551574 | C2 | semidirect |
| 7. #22551570 | C2 | semidirect |
| 8. #22551569 | C2 | semidirect |
| 9. #22551565 | C2 | semidirect |
| 10. #22551564 | C2 | semidirect |
| 11. #22551552 | C2 | semidirect |
| 12. #22551552 | C2 | semidirect |
| 13. C2×#16861597 | C2 | semidirect |
| 14. C4⋊C4×D257 | C2 | semidirect |
| 15. #123×C257 | C2 | semidirect |
| 61. C22⋊D257 | D4 | semidirect |
| 107. Dic257 | #41 | semidirect |
| 115. C257 | #547 | semidirect |
| 123. #123 | D257 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16861618 | 2 |
| C2×#16861617 | 2 |
| #22551591 | 2 |
| #22551577 | 2 |
| #22551576 | 2 |
| #22551574 | 2 |
| #22551570 | 2 |
| #22551569 | 2 |
| #22551565 | 2 |
| #22551564 | 2 |
| #22551552 | 2 |
| C2×#16861597 | 2 |
| C4⋊C4×D257 | 2 |
| #123×C257 | 2 |
| #547 | 257 |