Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
6520 conjugacy classes of subgroups; only the 3301 normal subgroups are shown.
This group has 6520 conjugacy classes of subgroups — too many to draw. Only its 3301 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D97 |
| commutator, cosocle | C2×C97 | C26 |
| Frattini | C2 | C25×D97 |
| Fitting | C24×C4×C97 | C2 |
| socle | C24×C97 | C23 |
| hypercenter | C24×C4 | D97 |
| hypocenter | C97 | C23×#41 |
| LCS | G ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#13095511 | C2 | direct (N × H) |
| 2. C23×Dic194 | C2 | semidirect |
| 3. C23×D388 | C2 | semidirect |
| 4. C23×C22⋊D97 | C2 | semidirect |
| 5. C23×C4×D97 | C2 | semidirect |
| 6. C24×C4×C97 | C2 | semidirect |
| 8. C2×#13095511 | C22 | direct (N × H) |
| 9. C22×Dic194 | C22 | semidirect |
| 10. C22×D388 | C22 | semidirect |
| 11. C22×C22⋊D97 | C22 | semidirect |
| 13. C22×C4×D97 | C22 | semidirect |
| 17. C23×C4×C97 | C22 | semidirect |
| 20. #13095511 | C23 | direct (N × H) |
| 21. C2×Dic194 | C23 | semidirect |
| 22. C2×D388 | C23 | semidirect |
| 23. C2×C22⋊D97 | C23 | semidirect |
| 25. C2×C4×D97 | C23 | semidirect |
| 31. C22×C4×C97 | C23 | semidirect |
| 33. C23×C97 | #41 | semidirect |
| 40. C2×C4×C97 | C24 | semidirect |
| 41. C22⋊D97 | C24 | semidirect |
| 43. D388 | C24 | semidirect |
| 44. C4×D97 | C24 | semidirect |
| 45. Dic194 | C24 | semidirect |
| 47. C22×C97 | C2×#41 | semidirect |
| 56. C2×C97 | C22×#41 | semidirect |
| 60. C97 | C23×#41 | semidirect |
| 66. C24×C4 | D97 | semidirect |
| 77. C23×C4 | C2×D97 | semidirect |
| 91. C22×C4 | C22×D97 | semidirect |
| 95. C23 | #13095511 | direct (N × H) |
| 105. C2×C4 | C23×D97 | semidirect |
| 109. C22 | C2×#13095511 | direct (N × H) |
| 117. C2 | C22×#13095511 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#13095511 | 2 |
| C23×Dic194 | 2 |
| C23×D388 | 2 |
| C23×C22⋊D97 | 2 |
| C23×C4×D97 | 2 |
| C24×C4×C97 | 2 |
| C23×#41 | 97 |