Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
5872 conjugacy classes of subgroups; only the 3213 normal subgroups are shown.
This group has 5872 conjugacy classes of subgroups — too many to draw. Only its 3213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C24×D97 |
| commutator, cosocle | C2×C97 | C26 |
| Frattini | C2 | C25×D97 |
| Fitting | C2×2−1+4×C97 | C2 |
| socle | C22×C97 | C25 |
| hypercenter | C2×2−1+4 | D97 |
| hypocenter | C97 | C22×2−1+4 |
| LCS | G ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C22 ◁ C2×2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690858 | C2 | semidirect |
| 2. C2×#15690857 | C2 | semidirect |
| 3. 2−1+4×D97 | C2 | direct (N × H) |
| 4. C2×#41×D97 | C2 | semidirect |
| 5. C22×Q8×D97 | C2 | semidirect |
| 6. C2×2−1+4×C97 | C2 | semidirect |
| 7. #15690858 | C22 | semidirect |
| 8. #15690857 | C22 | semidirect |
| 9. C2×#13095512 | C22 | semidirect |
| 10. C2×#13095511 | C22 | semidirect |
| 11. C2×#13095510 | C22 | semidirect |
| 12. #41×D97 | C22 | semidirect |
| 13. C22×Dic194 | C22 | semidirect |
| 15. C2×Q8×D97 | C22 | semidirect |
| 16. C2×Q8×D97 | C22 | semidirect |
| 19. 2−1+4×C97 | C22 | semidirect |
| 20. C2×#41×C97 | C22 | semidirect |
| 21. C22×Q8×C97 | C22 | semidirect |
| 23. #13095512 | C23 | semidirect |
| 24. #13095511 | C23 | semidirect |
| 25. #13095510 | C23 | semidirect |
| 27. C2×Dic194 | C23 | semidirect |
| 29. Q8×D97 | C23 | semidirect |
| 37. #41×C97 | C23 | semidirect |
| 39. C2×Q8×C97 | C23 | semidirect |
| 63. C2×D97 | 2−1+4 | direct (N × H) |
| 69. C2×C97 | C2×2−1+4 | semidirect |
| 72. D97 | C2×2−1+4 | direct (N × H) |
| 74. C97 | C22×2−1+4 | semidirect |
| 78. C2×2−1+4 | D97 | direct (N × H) |
| 83. 2−1+4 | C2×D97 | direct (N × H) |
| 88. C2×#41 | C2×D97 | semidirect |
| 93. C22×Q8 | C2×D97 | semidirect |
| 100. #41 | C22×D97 | semidirect |
| 106. C2×Q8 | C22×D97 | semidirect |
| 145. C2 | 2−1+4×D97 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690858 | 2 |
| C2×#15690857 | 2 |
| 2−1+4×D97 | 2 |
| C2×#41×D97 | 2 |
| C22×Q8×D97 | 2 |
| C2×2−1+4×C97 | 2 |
| C22×2−1+4 | 97 |