Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
280 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 115 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 |
| prime radical | C2×D796 | C22 |
| hypercenter | C4⋊Q8 | D199 |
| hypocenter | C199 | D4×Q8 |
| 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×Q8×D199 | C2 | semidirect |
| 2. #20263954 | C2 | semidirect |
| 3. #20263933 | C2 | semidirect |
| 4. #20263931 | C2 | semidirect |
| 5. C4×Dic398 | C2 | semidirect |
| 8. C4⋊Q8×C199 | C2 | semidirect |
| 28. Dic398 | D4 | semidirect |
| 29. D796 | Q8 | semidirect |
| 43. C2×D199 | C2×Q8 | semidirect |
| 47. C4×C199 | C2×Q8 | semidirect |
| 55. C199 | D4×Q8 | semidirect |
| 59. C4⋊Q8 | D199 | semidirect |
| 106. C4 | Q8×D199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×D199 | 2 |
| #20263954 | 2 |
| #20263933 | 2 |
| #20263931 | 2 |
| C4×Dic398 | 2 |
| C4×D796 | 2 |
| #20263917 | 2 |
| C4⋊Q8×C199 | 2 |
| D4×Q8 | 199 |