Finite Group DatabasebetaFinite Group DatabasebetaOrder 28096.
220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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×D439 |
| commutator, socle, cosocle | C22×C439 | C24 |
| Fitting | C22⋊Q8×C439 | C2 |
| hypercenter | C22⋊Q8 | D439 |
| hypocenter | C439 | #562 |
| LCS | G ▷ C22×C439 ▷ C439 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #28380733 | C2 | semidirect |
| 2. #28380732 | C2 | semidirect |
| 3. #28380729 | C2 | semidirect |
| 4. #28380728 | C2 | semidirect |
| 5. #28380728 | C2 | semidirect |
| 6. #28380726 | C2 | semidirect |
| 7. #28380725 | C2 | semidirect |
| 8. #28380723 | C2 | semidirect |
| 9. #28380721 | C2 | semidirect |
| 10. #28380719 | C2 | semidirect |
| 11. #28380718 | C2 | semidirect |
| 12. #28380717 | C2 | semidirect |
| 13. #28380716 | C2 | semidirect |
| 14. #28380716 | C2 | semidirect |
| 15. C22⋊Q8×C439 | C2 | semidirect |
| 109. C439 | #562 | semidirect |
| 121. C22⋊Q8 | D439 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28380733 | 2 |
| #28380732 | 2 |
| #28380729 | 2 |
| #28380728 | 2 |
| #28380726 | 2 |
| #28380725 | 2 |
| #28380723 | 2 |
| #28380721 | 2 |
| #28380719 | 2 |
| #28380718 | 2 |
| #28380717 | 2 |
| #28380716 | 2 |
| C22⋊Q8×C439 | 2 |
| #562 | 439 |