Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.
This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D97 |
| commutator, cosocle | C2×C97 | C26 |
| Frattini | C2 | C25×D97 |
| Fitting | C22×#41×C97 | C2 |
| socle | C23×C97 | C24 |
| hypercenter | C22×#41 | D97 |
| hypocenter | C97 | C22×H(5,𝔽2) |
| LCS | G ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ C22×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690856 | C2 | direct (N × H) |
| 2. C22×#13095512 | C2 | semidirect |
| 3. C22×#13095511 | C2 | semidirect |
| 4. C23×D388 | C2 | semidirect |
| 5. C22×D4×D97 | C2 | semidirect |
| 6. C22×#41×C97 | C2 | semidirect |
| 7. C24×D97 | C22 | semidirect |
| 8. #15690856 | C22 | direct (N × H) |
| 9. C2×#13095512 | C22 | semidirect |
| 10. C2×#13095511 | C22 | semidirect |
| 11. C22×Dic194 | C22 | semidirect |
| 12. C22×D388 | C22 | semidirect |
| 13. C22×D388 | C22 | semidirect |
| 14. C22×D388 | C22 | semidirect |
| 15. C22×C22⋊D97 | C22 | semidirect |
| 16. C2×D4×D97 | C22 | semidirect |
| 17. C22×C4×D97 | C22 | semidirect |
| 18. C2×#41×C97 | C22 | semidirect |
| 19. C22×Q8×C97 | C22 | semidirect |
| 20. C22×D4×C97 | C22 | semidirect |
| 21. C23×C4×C97 | C22 | semidirect |
| 24. C23×D97 | C23 | semidirect |
| 25. #13095512 | C23 | semidirect |
| 26. #13095511 | C23 | semidirect |
| 27. C2×Dic194 | C23 | semidirect |
| 28. C2×D388 | C23 | semidirect |
| 29. C2×D388 | C23 | semidirect |
| 30. C2×D388 | C23 | semidirect |
| 31. C2×C22⋊D97 | C23 | semidirect |
| 32. D4×D97 | C23 | semidirect |
| 34. C2×C4×D97 | C23 | semidirect |
| 36. #41×C97 | C23 | semidirect |
| 37. C2×Q8×C97 | C23 | semidirect |
| 38. C2×D4×C97 | C23 | semidirect |
| 41. C22×C4×C97 | C23 | semidirect |
| 48. C22×D97 | C24 | semidirect |
| 49. Q8×C97 | C24 | semidirect |
| 50. D4×C97 | C24 | semidirect |
| 53. C2×C4×C97 | C24 | semidirect |
| 54. C22⋊D97 | C24 | semidirect |
| 56. D388 | C24 | semidirect |
| 57. D388 | C24 | semidirect |
| 58. C4×D97 | C24 | semidirect |
| 59. Dic194 | C24 | semidirect |
| 61. C22×C97 | H(5,𝔽2) | semidirect |
| 71. C2×C97 | C2×H(5,𝔽2) | semidirect |
| 75. C97 | C22×H(5,𝔽2) | semidirect |
| 79. C22×#41 | D97 | semidirect |
| 86. C2×#41 | C2×D97 | semidirect |
| 88. C22×Q8 | C2×D97 | semidirect |
| 94. C22×D4 | C2×D97 | semidirect |
| 96. C23×C4 | C2×D97 | semidirect |
| 103. #41 | C22×D97 | semidirect |
| 105. C2×Q8 | C22×D97 | semidirect |
| 111. C2×D4 | C22×D97 | semidirect |
| 116. C22×C4 | C22×D97 | semidirect |
| 125. Q8 | C23×D97 | semidirect |
| 129. D4 | C23×D97 | semidirect |
| 134. C2×C4 | C23×D97 | semidirect |
| 139. C22 | #15690856 | direct (N × H) |
| 147. C2 | C2×#15690856 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690856 | 2 |
| C22×#13095512 | 2 |
| C22×#13095511 | 2 |
| C23×D388 | 2 |
| C22×D4×D97 | 2 |
| C22×#41×C97 | 2 |
| C22×H(5,𝔽2) | 97 |