Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
290 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 113 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×Q8×C199 | C2 |
| hypercenter | C4×Q8 | D199 |
| hypocenter | C199 | #548 |
| LCS | G ▷ C22×C199 ▷ C199 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15691887 | C2 | semidirect |
| 2. C2×Q8×D199 | C2 | semidirect |
| 3. #20263943 | C2 | semidirect |
| 4. #20263935 | C2 | semidirect |
| 5. #20263930 | C2 | semidirect |
| 6. C4×D796 | C2 | semidirect |
| 7. C4×Q8×C199 | C2 | semidirect |
| 31. Q8×C199 | D4 | semidirect |
| 41. C2×D199 | #41 | semidirect |
| 52. C199 | #548 | semidirect |
| 59. C4×Q8 | D199 | semidirect |
| 78. Q8 | D796 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15691887 | 2 |
| C2×Q8×D199 | 2 |
| #20263943 | 2 |
| #20263935 | 2 |
| #20263930 | 2 |
| C4×D796 | 2 |
| C4×Q8×C199 | 2 |
| #548 | 199 |