Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
294 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 103 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×D199 |
| commutator, socle, cosocle | C22×C199 | C24 |
| Fitting | C22⋊Q8×C199 | C2 |
| hypercenter | C22⋊Q8 | D199 |
| hypocenter | C199 | #537 |
| LCS | G ▷ C22×C199 ▷ C199 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15691887 | C2 | semidirect |
| 2. C2×#15691886 | C2 | semidirect |
| 3. #20263933 | C2 | semidirect |
| 4. #20263931 | C2 | semidirect |
| 5. #20263930 | C2 | semidirect |
| 6. #20263930 | C2 | semidirect |
| 7. #20263929 | C2 | semidirect |
| 8. #20263928 | C2 | semidirect |
| 9. #20263928 | C2 | semidirect |
| 10. C4⋊C4×D199 | C2 | semidirect |
| 11. C22⋊Q8×C199 | C2 | semidirect |
| 54. C4×D199 | D4 | semidirect |
| 89. C199 | #537 | semidirect |
| 95. C22⋊Q8 | D199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15691887 | 2 |
| C2×#15691886 | 2 |
| #20263933 | 2 |
| #20263931 | 2 |
| #20263930 | 2 |
| #20263929 | 2 |
| #20263928 | 2 |
| C4⋊C4×D199 | 2 |
| C22⋊Q8×C199 | 2 |
| #537 | 199 |