Finite Group DatabasebetaFinite Group DatabasebetaOrder 2624.
252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 252 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×D41 |
| commutator, socle, cosocle | C22×C41 | C24 |
| Fitting | #126×C41 | C2 |
| hypercenter | #126 | D41 |
| hypocenter | C41 | #560 |
| LCS | G ▷ C22×C41 ▷ C41 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13050161 | C2 | semidirect |
| 2. #13050158 | C2 | semidirect |
| 3. #13050149 | C2 | semidirect |
| 4. #13050139 | C2 | semidirect |
| 5. #13050137 | C2 | semidirect |
| 6. #13050134 | C2 | semidirect |
| 7. #126×C41 | C2 | semidirect |
| 12. C2×D164 | C22 | semidirect |
| 13. C2×C4×D41 | C22 | semidirect |
| 14. C4⋊C4×C41 | C22 | semidirect |
| 16. C42×C41 | C22 | semidirect |
| 19. #10620281 | C22 | semidirect |
| 20. #10620279 | C22 | semidirect |
| 51. C41 | #560 | semidirect |
| 54. #126 | D41 | semidirect |
| 66. C4⋊C4 | C2×D41 | semidirect |
| 71. C42 | C2×D41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13050161 | 2 |
| #13050158 | 2 |
| #13050149 | 2 |
| #13050139 | 2 |
| #13050137 | 2 |
| #13050134 | 2 |
| #126×C41 | 2 |
| #560 | 41 |