Finite Group DatabasebetaFinite Group DatabasebetaOrder 26944.
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×D421 |
| commutator, socle, cosocle | C22×C421 | C24 |
| Fitting | C22⋊Q8×C421 | C2 |
| hypercenter | C22⋊Q8 | D421 |
| hypocenter | C421 | #562 |
| LCS | G ▷ C22×C421 ▷ C421 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27212356 | C2 | semidirect |
| 2. #27212355 | C2 | semidirect |
| 3. #27212352 | C2 | semidirect |
| 4. #27212351 | C2 | semidirect |
| 5. #27212351 | C2 | semidirect |
| 6. #27212349 | C2 | semidirect |
| 7. #27212348 | C2 | semidirect |
| 8. #27212346 | C2 | semidirect |
| 9. #27212344 | C2 | semidirect |
| 10. #27212342 | C2 | semidirect |
| 11. #27212341 | C2 | semidirect |
| 12. #27212340 | C2 | semidirect |
| 13. #27212339 | C2 | semidirect |
| 14. #27212339 | C2 | semidirect |
| 15. C22⋊Q8×C421 | C2 | semidirect |
| 109. C421 | #562 | semidirect |
| 121. C22⋊Q8 | D421 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27212356 | 2 |
| #27212355 | 2 |
| #27212352 | 2 |
| #27212351 | 2 |
| #27212349 | 2 |
| #27212348 | 2 |
| #27212346 | 2 |
| #27212344 | 2 |
| #27212342 | 2 |
| #27212341 | 2 |
| #27212340 | 2 |
| #27212339 | 2 |
| C22⋊Q8×C421 | 2 |
| #562 | 421 |