Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
930 conjugacy classes of subgroups; only the 451 normal subgroups are shown.
This group has 930 conjugacy classes of subgroups — too many to draw. Only its 451 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 | C24×D97 |
| commutator, socle, cosocle | C22×C97 | C25 |
| Fitting | #564×C97 | C2 |
| hypercenter | #564 | D97 |
| hypocenter | C97 | #3531 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C22 ◁ #564 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20261489 | C2 | semidirect |
| 2. #20261483 | C2 | semidirect |
| 3. #20261291 | C2 | semidirect |
| 4. #20261270 | C2 | semidirect |
| 5. #564×C97 | C2 | semidirect |
| 54. C97 | #3531 | semidirect |
| 55. #564 | D97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20261489 | 2 |
| #20261483 | 2 |
| #20261291 | 2 |
| #20261270 | 2 |
| #564×C97 | 2 |
| #3531 | 97 |