Finite Group DatabasebetaFinite Group DatabasebetaOrder 2624.
292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 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 | #123×C41 | C2 |
| hypercenter | #123 | D41 |
| hypocenter | C41 | #547 |
| LCS | G ▷ C22×C41 ▷ C41 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#10620306 | C2 | semidirect |
| 2. C2×#10620308 | C2 | semidirect |
| 3. #13050163 | C2 | semidirect |
| 4. #13050149 | C2 | semidirect |
| 5. #13050148 | C2 | semidirect |
| 6. #13050146 | C2 | semidirect |
| 7. #13050142 | C2 | semidirect |
| 8. #13050141 | C2 | semidirect |
| 9. #13050137 | C2 | semidirect |
| 10. #13050136 | C2 | semidirect |
| 11. #13050124 | C2 | semidirect |
| 12. #13050124 | C2 | semidirect |
| 13. C2×#10620279 | C2 | semidirect |
| 14. C4⋊C4×D41 | C2 | semidirect |
| 15. #123×C41 | C2 | semidirect |
| 67. C22⋊D41 | D4 | semidirect |
| 107. Dic41 | #41 | semidirect |
| 116. C41 | #547 | semidirect |
| 123. #123 | D41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#10620306 | 2 |
| C2×#10620308 | 2 |
| #13050163 | 2 |
| #13050149 | 2 |
| #13050148 | 2 |
| #13050146 | 2 |
| #13050142 | 2 |
| #13050141 | 2 |
| #13050137 | 2 |
| #13050136 | 2 |
| #13050124 | 2 |
| C2×#10620279 | 2 |
| C4⋊C4×D41 | 2 |
| #123×C41 | 2 |
| #547 | 41 |