Finite Group DatabasebetaFinite Group DatabasebetaOrder 12352.
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×D193 |
| commutator, socle, cosocle | C22×C193 | C24 |
| Fitting | C22⋊Q8×C193 | C2 |
| hypercenter | C22⋊Q8 | D193 |
| hypocenter | C193 | #562 |
| LCS | G ▷ C22×C193 ▷ C193 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20259889 | C2 | semidirect |
| 2. #20259888 | C2 | semidirect |
| 3. #20259885 | C2 | semidirect |
| 4. #20259884 | C2 | semidirect |
| 5. #20259884 | C2 | semidirect |
| 6. #20259882 | C2 | semidirect |
| 7. #20259881 | C2 | semidirect |
| 8. #20259879 | C2 | semidirect |
| 9. #20259877 | C2 | semidirect |
| 10. #20259875 | C2 | semidirect |
| 11. #20259874 | C2 | semidirect |
| 12. #20259873 | C2 | semidirect |
| 13. #20259872 | C2 | semidirect |
| 14. #20259872 | C2 | semidirect |
| 15. C22⋊Q8×C193 | C2 | semidirect |
| 109. C193 | #562 | semidirect |
| 121. C22⋊Q8 | D193 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20259889 | 2 |
| #20259888 | 2 |
| #20259885 | 2 |
| #20259884 | 2 |
| #20259882 | 2 |
| #20259881 | 2 |
| #20259879 | 2 |
| #20259877 | 2 |
| #20259875 | 2 |
| #20259874 | 2 |
| #20259873 | 2 |
| #20259872 | 2 |
| C22⋊Q8×C193 | 2 |
| #562 | 193 |