Finite Group DatabasebetaFinite Group DatabasebetaOrder 1472.
250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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×D23 |
| commutator, socle, cosocle | C22×C23 | C24 |
| Fitting | C23×#123 | C2 |
| hypercenter | #123 | D23 |
| hypocenter | C23 | #538 |
| LCS | G ▷ C22×C23 ▷ C23 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#123 | C2 | semidirect |
| 2. #10625109 | C2 | semidirect |
| 3. #10625108 | C2 | semidirect |
| 4. #10625101 | C2 | semidirect |
| 5. C4×C22⋊D23 | C2 | semidirect |
| 6. #10625069 | C2 | semidirect |
| 7. #10625068 | C2 | semidirect |
| 8. #10625065 | C2 | semidirect |
| 9. #10625064 | C2 | semidirect |
| 10. #10625056 | C2 | semidirect |
| 11. #10625056 | C2 | semidirect |
| 12. #10625055 | C2 | semidirect |
| 13. #10625053 | C2 | semidirect |
| 14. C22⋊C4×D23 | C2 | semidirect |
| 15. #10625050 | C2 | semidirect |
| 18. C22×C4×C23 | C22 | semidirect |
| 22. C2×C22⋊D23 | C22 | semidirect |
| 27. C2×C4×D23 | C22 | semidirect |
| 28. C2×Dic46 | C22 | semidirect |
| 29. C4⋊C4×C23 | C22 | semidirect |
| 30. C4⋊C4×C23 | C22 | semidirect |
| 33. C22⋊C4×C23 | C22 | semidirect |
| 36. C22⋊Dic23 | C22 | semidirect |
| 43. (C2×D23)⋊C4 | C22 | semidirect |
| 44. (C2×D23)⋊C4 | C22 | semidirect |
| 45. C4⋊Dic23 | C22 | semidirect |
| 46. C4⋊Dic23 | C22 | semidirect |
| 47. Dic23⋊C4 | C22 | semidirect |
| 48. Dic23⋊C4 | C22 | semidirect |
| 49. C4×Dic23 | C22 | semidirect |
| 50. C4×Dic23 | C22 | semidirect |
| 122. #123 | D23 | semidirect |
| 132. C23 | #538 | semidirect |
| 145. C22×C4 | C2×D23 | semidirect |
| 150. C4⋊C4 | C2×D23 | semidirect |
| 151. C4⋊C4 | C2×D23 | semidirect |
| 165. C22⋊C4 | C2×D23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#123 | 2 |
| #10625109 | 2 |
| #10625108 | 2 |
| #10625101 | 2 |
| C4×C22⋊D23 | 2 |
| #10625069 | 2 |
| #10625068 | 2 |
| #10625065 | 2 |
| #10625064 | 2 |
| #10625056 | 2 |
| #10625055 | 2 |
| #10625053 | 2 |
| C22⋊C4×D23 | 2 |
| #10625050 | 2 |
| #538 | 23 |