Finite Group DatabasebetaFinite Group DatabasebetaOrder 26816.
334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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 | C23×D419 |
| commutator, socle, cosocle | C22×C419 | C24 |
| Fitting | #121×C419 | C2 |
| hypercenter | #121 | D419 |
| hypocenter | C419 | #546 |
| LCS | G ▷ C22×C419 ▷ C419 | |
| UCS | C1 ◁ C22 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#20269410 | C2 | semidirect |
| 2. C2×D4×D419 | C2 | semidirect |
| 3. #27211704 | C2 | semidirect |
| 4. #27211703 | C2 | semidirect |
| 5. #27211699 | C2 | semidirect |
| 6. #27211696 | C2 | semidirect |
| 7. #27211691 | C2 | semidirect |
| 8. #27211690 | C2 | semidirect |
| 9. #27211686 | C2 | semidirect |
| 10. #27211686 | C2 | semidirect |
| 11. C4×C22⋊D419 | C2 | semidirect |
| 12. (C22⋊D419)⋊C4 | C2 | semidirect |
| 13. C2×(C2×D419)⋊C4 | C2 | semidirect |
| 14. C22⋊C4×D419 | C2 | semidirect |
| 15. #121×C419 | C2 | semidirect |
| 19. C2×Dic838 | C22 | semidirect |
| 24. C2×C22⋊D419 | C22 | semidirect |
| 25. C2×C22⋊D419 | C22 | semidirect |
| 27. C22×Dic419 | C22 | semidirect |
| 28. C22×Dic419 | C22 | semidirect |
| 30. C2×C4×D419 | C22 | semidirect |
| 31. C2×C4×D419 | C22 | semidirect |
| 32. Dic419⋊C4 | C22 | semidirect |
| 33. Dic419⋊C4 | C22 | semidirect |
| 34. Dic419⋊C4 | C22 | semidirect |
| 35. C22⋊Dic419 | C22 | semidirect |
| 36. C22⋊Dic419 | C22 | semidirect |
| 37. C22⋊Dic419 | C22 | semidirect |
| 41. (C2×D419)⋊C4 | C22 | semidirect |
| 42. (C2×D419)⋊C4 | C22 | semidirect |
| 43. (C2×D419)⋊C4 | C22 | semidirect |
| 44. (C2×D419)⋊C4 | C22 | semidirect |
| 45. C4×Dic419 | C22 | semidirect |
| 47. C2×D4×C419 | C22 | semidirect |
| 48. C2×D4×C419 | C22 | semidirect |
| 49. C22×C4×C419 | C22 | semidirect |
| 50. C4⋊C4×C419 | C22 | semidirect |
| 51. C22⋊C4×C419 | C22 | semidirect |
| 52. C22⋊C4×C419 | C22 | semidirect |
| 68. C22⋊D419 | D4 | semidirect |
| 110. C2×D419 | #41 | semidirect |
| 129. C419 | #546 | semidirect |
| 139. #121 | D419 | semidirect |
| 158. C2×D4 | C2×D419 | semidirect |
| 159. C2×D4 | C2×D419 | semidirect |
| 165. C22×C4 | C2×D419 | semidirect |
| 169. C4⋊C4 | C2×D419 | semidirect |
| 180. C22⋊C4 | C2×D419 | semidirect |
| 181. C22⋊C4 | C2×D419 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#20269410 | 2 |
| C2×D4×D419 | 2 |
| #27211704 | 2 |
| #27211703 | 2 |
| #27211699 | 2 |
| #27211696 | 2 |
| #27211691 | 2 |
| #27211690 | 2 |
| #27211686 | 2 |
| C4×C22⋊D419 | 2 |
| (C22⋊D419)⋊C4 | 2 |
| C2×(C2×D419)⋊C4 | 2 |
| C22⋊C4×D419 | 2 |
| #121×C419 | 2 |
| #546 | 419 |