Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1296 conjugacy classes of subgroups; only the 517 normal subgroups are shown.
This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 517 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D97 |
| commutator, socle, cosocle | C22×C97 | C25 |
| Fitting | #543×C97 | C2 |
| prime radical | C24×D97 | C22 |
| hypercenter | #543 | D97 |
| hypocenter | C97 | C2×#543 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C22 ◁ #543 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20261361 | C2 | semidirect |
| 2. #20261360 | C2 | semidirect |
| 3. C22⋊Q8×D97 | C2 | semidirect |
| 5. #543×C97 | C2 | semidirect |
| 7. C2×Q8×D97 | C22 | semidirect |
| 9. #15690790 | C22 | semidirect |
| 10. #15690789 | C22 | semidirect |
| 11. #15690788 | C22 | semidirect |
| 12. #15690787 | C22 | semidirect |
| 13. #15690786 | C22 | semidirect |
| 14. C4⋊C4×D97 | C22 | semidirect |
| 19. C22⋊Q8×C97 | C22 | semidirect |
| 22. C23×D97 | Q8 | semidirect |
| 25. C2×Dic194 | C23 | semidirect |
| 31. C4⋊Dic97 | C23 | semidirect |
| 32. #13095490 | C23 | semidirect |
| 36. C2×Q8×C97 | C23 | semidirect |
| 38. C4⋊C4×C97 | C23 | semidirect |
| 41. C23×C97 | C2×Q8 | semidirect |
| 47. C22×D97 | C2×Q8 | semidirect |
| 74. D97 | #543 | direct (N × H) |
| 76. C97 | C2×#543 | semidirect |
| 79. #543 | D97 | direct (N × H) |
| 90. C22⋊Q8 | C2×D97 | semidirect |
| 104. C2×Q8 | C22×D97 | semidirect |
| 112. C4⋊C4 | C22×D97 | semidirect |
| 124. C23 | Q8×D97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20261361 | 2 |
| #20261360 | 2 |
| C22⋊Q8×D97 | 2 |
| C2×C22⋊C4×D97 | 2 |
| #543×C97 | 2 |
| C2×#543 | 97 |