Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 111 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 |
| prime radical | C2×D4×D199 | C2 |
| hypercenter | #124 | D199 |
| hypocenter | C199 | C2×#124 |
| 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. C2×Q8×D199 | C2 | semidirect |
| 3. #20263944 | C2 | semidirect |
| 4. #20263943 | C2 | semidirect |
| 5. #20263942 | C2 | semidirect |
| 6. #20263941 | C2 | semidirect |
| 7. C22⋊C4×D199 | C2 | semidirect |
| 8. C42×D199 | C2 | semidirect |
| 9. #124×C199 | C2 | semidirect |
| 11. C2×Dic398 | C22 | semidirect |
| 12. C2×Dic398 | C22 | semidirect |
| 20. C22⋊Dic199 | C22 | semidirect |
| 21. (C2×D199)⋊C4 | C22 | semidirect |
| 22. (C2×D199)⋊C4 | C22 | semidirect |
| 23. C4×Dic199 | C22 | semidirect |
| 24. C4×Dic199 | C22 | semidirect |
| 25. C2×Q8×C199 | C22 | semidirect |
| 27. C22⋊C4×C199 | C22 | semidirect |
| 28. C42×C199 | C22 | semidirect |
| 67. D199 | #124 | direct (N × H) |
| 68. C199 | C2×#124 | semidirect |
| 76. #124 | D199 | direct (N × H) |
| 83. C2×Q8 | C2×D199 | semidirect |
| 94. C22⋊C4 | C2×D199 | semidirect |
| 97. C42 | C2×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 |
| C2×D4×D199 | 2 |
| #20263944 | 2 |
| #20263943 | 2 |
| #20263942 | 2 |
| #20263941 | 2 |
| C22⋊C4×D199 | 2 |
| C42×D199 | 2 |
| #124×C199 | 2 |
| C2×#124 | 199 |