Finite Group DatabasebetaFinite Group DatabasebetaOrder 12608.
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×D197 |
| commutator, socle, cosocle | C22×C197 | C24 |
| Fitting | C22⋊Q8×C197 | C2 |
| hypercenter | C22⋊Q8 | D197 |
| hypocenter | C197 | #562 |
| LCS | G ▷ C22×C197 ▷ C197 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20263316 | C2 | semidirect |
| 2. #20263315 | C2 | semidirect |
| 3. #20263312 | C2 | semidirect |
| 4. #20263311 | C2 | semidirect |
| 5. #20263311 | C2 | semidirect |
| 6. #20263309 | C2 | semidirect |
| 7. #20263308 | C2 | semidirect |
| 8. #20263306 | C2 | semidirect |
| 9. #20263304 | C2 | semidirect |
| 10. #20263302 | C2 | semidirect |
| 11. #20263301 | C2 | semidirect |
| 12. #20263300 | C2 | semidirect |
| 13. #20263299 | C2 | semidirect |
| 14. #20263299 | C2 | semidirect |
| 15. C22⋊Q8×C197 | C2 | semidirect |
| 109. C197 | #562 | semidirect |
| 121. C22⋊Q8 | D197 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20263316 | 2 |
| #20263315 | 2 |
| #20263312 | 2 |
| #20263311 | 2 |
| #20263309 | 2 |
| #20263308 | 2 |
| #20263306 | 2 |
| #20263304 | 2 |
| #20263302 | 2 |
| #20263301 | 2 |
| #20263300 | 2 |
| #20263299 | 2 |
| C22⋊Q8×C197 | 2 |
| #562 | 197 |