Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1260 conjugacy classes of subgroups; only the 485 normal subgroups are shown.
This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 485 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D97 |
| commutator, cosocle | C22×C97 | C25 |
| Frattini | C22 | C24×D97 |
| Fitting | C2×#117×C97 | C2 |
| socle | C23×C97 | C24 |
| hypercenter | C2×#117 | D97 |
| hypocenter | C97 | C2×#540 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ C2×#117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20261340 | C2 | direct (N × H) |
| 2. C2×#15690807 | C2 | semidirect |
| 3. C2×#15690802 | C2 | semidirect |
| 4. C2×#15690795 | C2 | semidirect |
| 5. C2×#15690784 | C2 | semidirect |
| 6. C2×#15690783 | C2 | semidirect |
| 7. C2×C4×D388 | C2 | semidirect |
| 8. C2×#117×C97 | C2 | semidirect |
| 13. #15690807 | C22 | semidirect |
| 14. #15690802 | C22 | semidirect |
| 15. #15690795 | C22 | semidirect |
| 16. #15690784 | C22 | semidirect |
| 17. #15690783 | C22 | semidirect |
| 18. C4×D388 | C22 | semidirect |
| 24. #117×C97 | C22 | semidirect |
| 93. C2×C97 | #540 | semidirect |
| 98. C97 | C2×#540 | semidirect |
| 106. C2×#117 | D97 | semidirect |
| 118. #117 | C2×D97 | semidirect |
| 192. C2 | #20261340 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20261340 | 2 |
| C2×#15690807 | 2 |
| C2×#15690802 | 2 |
| C2×#15690795 | 2 |
| C2×#15690784 | 2 |
| C2×#15690783 | 2 |
| C2×C4×D388 | 2 |
| C2×#117×C97 | 2 |
| C2×#540 | 97 |