Finite Group DatabasebetaFinite Group DatabasebetaOrder 12224.
248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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×D191 |
| commutator, socle, cosocle | C22×C191 | C24 |
| Fitting | #123×C191 | C2 |
| prime radical | #20259290 | C2 |
| hypercenter | #123 | D191 |
| hypocenter | C191 | #551 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20259317 | C2 | semidirect |
| 2. #20259317 | C2 | semidirect |
| 3. #20259308 | C2 | semidirect |
| 4. #20259306 | C2 | semidirect |
| 5. #20259304 | C2 | semidirect |
| 6. #20259302 | C2 | semidirect |
| 7. #20259299 | C2 | semidirect |
| 9. D4×Dic191 | C2 | semidirect |
| 10. (C22⋊D191)⋊C4 | C2 | semidirect |
| 11. #20259281 | C2 | semidirect |
| 12. #20259280 | C2 | semidirect |
| 13. C2×(C2×D191)⋊C4 | C2 | semidirect |
| 14. C22⋊C4×D191 | C2 | semidirect |
| 15. #123×C191 | C2 | semidirect |
| 17. C2×Dic382 | C22 | semidirect |
| 21. C22×Dic191 | C22 | semidirect |
| 24. C4⋊Dic191 | C22 | semidirect |
| 25. C4⋊Dic191 | C22 | semidirect |
| 26. C4⋊Dic191 | C22 | semidirect |
| 27. C4⋊Dic191 | C22 | semidirect |
| 28. Dic191⋊C4 | C22 | semidirect |
| 29. Dic191⋊C4 | C22 | semidirect |
| 32. C22⋊Dic191 | C22 | semidirect |
| 41. C4×Dic191 | C22 | semidirect |
| 42. C4×Dic191 | C22 | semidirect |
| 43. C4×Dic191 | C22 | semidirect |
| 46. C4⋊C4×C191 | C22 | semidirect |
| 47. C4⋊C4×C191 | C22 | semidirect |
| 49. C22⋊C4×C191 | C22 | semidirect |
| 50. C22⋊C4×C191 | C22 | semidirect |
| 110. C191 | #551 | semidirect |
| 117. #123 | D191 | semidirect |
| 143. C4⋊C4 | C2×D191 | semidirect |
| 144. C4⋊C4 | C2×D191 | semidirect |
| 157. C22⋊C4 | C2×D191 | semidirect |
| 158. C22⋊C4 | C2×D191 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20259317 | 2 |
| #20259308 | 2 |
| #20259306 | 2 |
| #20259304 | 2 |
| #20259302 | 2 |
| #20259299 | 2 |
| #20259290 | 2 |
| D4×Dic191 | 2 |
| (C22⋊D191)⋊C4 | 2 |
| #20259281 | 2 |
| #20259280 | 2 |
| C2×(C2×D191)⋊C4 | 2 |
| C22⋊C4×D191 | 2 |
| #123×C191 | 2 |
| #551 | 191 |