Finite Group DatabasebetaFinite Group DatabasebetaOrder 16448.
310 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 109 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 | #121×C257 | C2 |
| hypercenter | #121 | D257 |
| hypocenter | C257 | #532 |
| LCS | G ▷ C22×C257 ▷ C257 | |
| UCS | C1 ◁ C22 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16861617 | C2 | semidirect |
| 2. #22551591 | C2 | semidirect |
| 3. #22551588 | C2 | semidirect |
| 4. #22551578 | C2 | semidirect |
| 5. #22551566 | C2 | semidirect |
| 6. #22551564 | C2 | semidirect |
| 7. D4×Dic257 | C2 | semidirect |
| 8. #22551554 | C2 | semidirect |
| 9. #22551552 | C2 | semidirect |
| 10. C2×C4×Dic257 | C2 | semidirect |
| 11. #121×C257 | C2 | semidirect |
| 44. C2×Dic257 | D4 | semidirect |
| 71. C4×C257 | #41 | semidirect |
| 76. Dic257 | #41 | semidirect |
| 83. C257 | #532 | semidirect |
| 91. #121 | D257 | semidirect |
| 160. C4 | #16861617 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16861617 | 2 |
| #22551591 | 2 |
| #22551588 | 2 |
| #22551578 | 2 |
| #22551566 | 2 |
| #22551564 | 2 |
| D4×Dic257 | 2 |
| #22551554 | 2 |
| #22551552 | 2 |
| C2×C4×Dic257 | 2 |
| #121×C257 | 2 |
| #532 | 257 |