Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
422 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 205 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D97 |
| commutator, socle | C22×C97 | C23×C4 |
| Fitting, prime radical | #525×C97 | C2 |
| cosocle | C2×C4×C97 | C24 |
| hypercenter | #525 | D97 |
| hypocenter | C97 | #2977 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C2×C4 ◁ #525 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20260807 | C2 | semidirect |
| 2. D4×#11715895 | C2 | semidirect |
| 3. #20260796 | C2 | semidirect |
| 4. #20260702 | C2 | semidirect |
| 5. C2×C22⋊#11715895 | C2 | semidirect |
| 7. C2×#13095495 | C22 | semidirect |
| 9. C22×#11715895 | C22 | semidirect |
| 10. C4⋊#11715895 | C22 | semidirect |
| 14. C22⋊#11715895 | C22 | semidirect |
| 15. #15690721 | C22 | semidirect |
| 16. C4×#11715895 | C22 | semidirect |
| 90. C97 | #2977 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20260807 | 2 |
| D4×#11715895 | 2 |
| #20260796 | 2 |
| #20260702 | 2 |
| C2×C22⋊#11715895 | 2 |
| #525×C97 | 2 |
| #2977 | 97 |