Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
260 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 260 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×D199 |
| commutator, socle, cosocle | C22×C199 | C24 |
| Fitting | #124×C199 | C2 |
| hypercenter | #124 | D199 |
| hypocenter | C199 | H(3,𝔽4) |
| LCS | G ▷ C22×C199 ▷ C199 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20263944 | C2 | semidirect |
| 2. #20263943 | C2 | semidirect |
| 3. #20263941 | C2 | semidirect |
| 4. #20263925 | C2 | semidirect |
| 5. #20263923 | C2 | semidirect |
| 6. #124×C199 | C2 | semidirect |
| 7. C23×D199 | C22 | semidirect |
| 8. C2×Dic398 | C22 | semidirect |
| 12. C22⋊Dic199 | C22 | semidirect |
| 13. (C2×D199)⋊C4 | C22 | semidirect |
| 14. (C2×D199)⋊C4 | C22 | semidirect |
| 15. (C2×D199)⋊C4 | C22 | semidirect |
| 16. C4×Dic199 | C22 | semidirect |
| 17. C2×Q8×C199 | C22 | semidirect |
| 19. C22⋊C4×C199 | C22 | semidirect |
| 20. C42×C199 | C22 | semidirect |
| 50. C199 | H(3,𝔽4) | semidirect |
| 55. #124 | D199 | semidirect |
| 60. C2×Q8 | C2×D199 | semidirect |
| 69. C22⋊C4 | C2×D199 | semidirect |
| 71. C42 | C2×D199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20263944 | 2 |
| #20263943 | 2 |
| #20263941 | 2 |
| #20263925 | 2 |
| #20263923 | 2 |
| #124×C199 | 2 |
| H(3,𝔽4) | 199 |