Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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 | C4×D4×C199 | C2 |
| hypercenter | C4×D4 | D199 |
| hypocenter | C199 | #550 |
| LCS | G ▷ C22×C199 ▷ C199 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15691886 | C2 | semidirect |
| 2. #20263952 | C2 | semidirect |
| 3. #20263951 | C2 | semidirect |
| 4. #20263929 | C2 | semidirect |
| 5. #20263928 | C2 | semidirect |
| 6. C4×Dic398 | C2 | semidirect |
| 7. #20263917 | C2 | semidirect |
| 8. C4×D4×C199 | C2 | semidirect |
| 27. Dic398 | D4 | semidirect |
| 53. C4×C199 | #41 | semidirect |
| 61. C199 | #550 | semidirect |
| 70. C4×D4 | D199 | semidirect |
| 117. C4 | #15691886 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15691886 | 2 |
| #20263952 | 2 |
| #20263951 | 2 |
| #20263929 | 2 |
| #20263928 | 2 |
| C4×Dic398 | 2 |
| #20263917 | 2 |
| C4×D4×C199 | 2 |
| #550 | 199 |