Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1160 conjugacy classes of subgroups; only the 309 normal subgroups are shown.
This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 309 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 | C22×C97 | C23×C4 |
| Fitting, prime radical | C23×D4×C97 | C2 |
| socle, cosocle | C23×C97 | C24 |
| hypercenter | C23×D4 | D97 |
| hypocenter | C97 | #2308 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ C23×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C22⋊Dic97 | C2 | semidirect |
| 2. #20260615 | C2 | semidirect |
| 3. #20260598 | C2 | semidirect |
| 4. #20260591 | C2 | semidirect |
| 6. C23×Dic97 | C22 | semidirect |
| 9. C2×C22⋊Dic97 | C22 | semidirect |
| 10. C2×C22⋊Dic97 | C22 | semidirect |
| 11. #15690717 | C22 | semidirect |
| 15. C22×D4×C97 | C4 | semidirect |
| 28. C24×C97 | C2×C4 | semidirect |
| 36. C2×D4×C97 | C2×C4 | semidirect |
| 38. C22×C4×C97 | C2×C4 | semidirect |
| 49. C23×C97 | C22⋊C4 | semidirect |
| 88. C97 | #2308 | semidirect |
| 97. C22×D4 | Dic97 | semidirect |
| 110. C24 | C2×Dic97 | semidirect |
| 118. C2×D4 | C2×Dic97 | semidirect |
| 123. C22×C4 | C2×Dic97 | semidirect |
| 137. C23 | C22⋊Dic97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C22⋊Dic97 | 2 |
| #20260615 | 2 |
| #20260598 | 2 |
| #20260591 | 2 |
| C23×D4×C97 | 2 |
| #2308 | 97 |