Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
516 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D97 |
| commutator, socle, cosocle | C23×C97 | C24 |
| Fitting | C43×C97 | C2 |
| prime radical | C22×D388 | C22 |
| hypercenter | C43 | D97 |
| hypocenter | C97 | #2860 |
| LCS | G ▷ C23×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ C43 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690804 | C2 | semidirect |
| 3. #20260633 | C2 | semidirect |
| 4. #20260626 | C2 | semidirect |
| 5. C43×C97 | C2 | semidirect |
| 28. C42×C97 | D4 | semidirect |
| 52. C4×C97 | #126 | semidirect |
| 59. C97 | #2860 | semidirect |
| 64. C43 | D97 | semidirect |
| 87. C42 | D388 | semidirect |
| 112. C4 | #15690804 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690804 | 2 |
| #20260637 | 2 |
| #20260633 | 2 |
| #20260626 | 2 |
| C43×C97 | 2 |
| #2860 | 97 |