Finite Group DatabasebetaFinite Group DatabasebetaOrder 2624.
234 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 97 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 |
| prime radical | #13050137 | C2 |
| hypercenter | #126 | D41 |
| hypocenter | C41 | #529 |
| 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. #13050158 | C2 | semidirect |
| 2. #13050154 | C2 | semidirect |
| 3. #13050150 | C2 | semidirect |
| 4. #13050149 | C2 | semidirect |
| 5. #13050146 | C2 | semidirect |
| 6. #13050142 | C2 | semidirect |
| 8. #13050129 | C2 | semidirect |
| 9. #13050126 | C2 | semidirect |
| 10. #13050124 | C2 | semidirect |
| 11. #13050124 | C2 | semidirect |
| 12. #13050122 | C2 | semidirect |
| 13. #13050120 | C2 | semidirect |
| 14. C42×D41 | C2 | semidirect |
| 15. #126×C41 | C2 | semidirect |
| 104. C41 | #529 | semidirect |
| 106. #126 | D41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13050158 | 2 |
| #13050154 | 2 |
| #13050150 | 2 |
| #13050149 | 2 |
| #13050146 | 2 |
| #13050142 | 2 |
| #13050137 | 2 |
| #13050129 | 2 |
| #13050126 | 2 |
| #13050124 | 2 |
| #13050122 | 2 |
| #13050120 | 2 |
| C42×D41 | 2 |
| #126×C41 | 2 |
| #529 | 41 |