Finite Group DatabasebetaFinite Group DatabasebetaOrder 15424.
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×D241 |
| commutator, socle, cosocle | C22×C241 | C24 |
| Fitting | C22⋊Q8×C241 | C2 |
| hypercenter | C22⋊Q8 | D241 |
| hypocenter | C241 | #562 |
| LCS | G ▷ C22×C241 ▷ C241 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21440632 | C2 | semidirect |
| 2. #21440631 | C2 | semidirect |
| 3. #21440628 | C2 | semidirect |
| 4. #21440627 | C2 | semidirect |
| 5. #21440627 | C2 | semidirect |
| 6. #21440625 | C2 | semidirect |
| 7. #21440624 | C2 | semidirect |
| 8. #21440622 | C2 | semidirect |
| 9. #21440620 | C2 | semidirect |
| 10. #21440618 | C2 | semidirect |
| 11. #21440617 | C2 | semidirect |
| 12. #21440616 | C2 | semidirect |
| 13. #21440615 | C2 | semidirect |
| 14. #21440615 | C2 | semidirect |
| 15. C22⋊Q8×C241 | C2 | semidirect |
| 109. C241 | #562 | semidirect |
| 121. C22⋊Q8 | D241 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21440632 | 2 |
| #21440631 | 2 |
| #21440628 | 2 |
| #21440627 | 2 |
| #21440625 | 2 |
| #21440624 | 2 |
| #21440622 | 2 |
| #21440620 | 2 |
| #21440618 | 2 |
| #21440617 | 2 |
| #21440616 | 2 |
| #21440615 | 2 |
| C22⋊Q8×C241 | 2 |
| #562 | 241 |