Finite Group DatabasebetaFinite Group DatabasebetaOrder 16448.
212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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 | #124×C257 | C2 |
| prime radical | C2×D4×C257 | C22 |
| hypercenter | #124 | D257 |
| hypocenter | C257 | #556 |
| LCS | G ▷ C22×C257 ▷ C257 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22551590 | C2 | semidirect |
| 2. #22551585 | C2 | semidirect |
| 3. #22551585 | C2 | semidirect |
| 5. #22551569 | C2 | semidirect |
| 6. Q8×Dic257 | C2 | semidirect |
| 7. C4×Dic514 | C2 | semidirect |
| 8. C4×Dic514 | C2 | semidirect |
| 10. #22551548 | C2 | semidirect |
| 69. C257 | #556 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22551590 | 2 |
| #22551585 | 2 |
| #22551579 | 2 |
| #22551569 | 2 |
| Q8×Dic257 | 2 |
| C4×Dic514 | 2 |
| D4×Dic257 | 2 |
| #22551548 | 2 |
| #124×C257 | 2 |
| #556 | 257 |