Finite Group DatabasebetaFinite Group DatabasebetaOrder 12608.
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×D197 |
| commutator, socle, cosocle | C22×C197 | C24 |
| Fitting | C4×D4×C197 | C2 |
| prime radical | #20263294 | C2 |
| hypercenter | C4×D4 | D197 |
| hypocenter | C197 | #551 |
| LCS | G ▷ C22×C197 ▷ C197 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20263321 | C2 | semidirect |
| 2. #20263320 | C2 | semidirect |
| 3. #20263312 | C2 | semidirect |
| 4. #20263309 | C2 | semidirect |
| 5. #20263308 | C2 | semidirect |
| 6. #20263307 | C2 | semidirect |
| 7. #20263305 | C2 | semidirect |
| 8. #20263304 | C2 | semidirect |
| 10. C4×C22⋊D197 | C2 | semidirect |
| 11. #20263283 | C2 | semidirect |
| 12. #20263282 | C2 | semidirect |
| 13. C2×#15691335 | C2 | semidirect |
| 14. C22⋊C4×D197 | C2 | semidirect |
| 15. C4×D4×C197 | C2 | semidirect |
| 17. C2×Dic394 | C22 | semidirect |
| 23. C4⋊Dic197 | C22 | semidirect |
| 24. #15691337 | C22 | semidirect |
| 25. #15691337 | C22 | semidirect |
| 26. #15691337 | C22 | semidirect |
| 27. #15691337 | C22 | semidirect |
| 28. #15691337 | C22 | semidirect |
| 29. #15691337 | C22 | semidirect |
| 32. C22⋊Dic197 | C22 | semidirect |
| 33. C22⋊Dic197 | C22 | semidirect |
| 42. C4×Dic197 | C22 | semidirect |
| 43. C4×Dic197 | C22 | semidirect |
| 46. C22×C4×C197 | C22 | semidirect |
| 47. C4⋊C4×C197 | C22 | semidirect |
| 49. C22⋊C4×C197 | C22 | semidirect |
| 50. C42×C197 | C22 | semidirect |
| 110. C197 | #551 | semidirect |
| 122. C4×D4 | D197 | semidirect |
| 136. C22×C4 | C2×D197 | semidirect |
| 144. C4⋊C4 | C2×D197 | semidirect |
| 158. C22⋊C4 | C2×D197 | semidirect |
| 161. C42 | C2×D197 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20263321 | 2 |
| #20263320 | 2 |
| #20263312 | 2 |
| #20263309 | 2 |
| #20263308 | 2 |
| #20263307 | 2 |
| #20263305 | 2 |
| #20263304 | 2 |
| #20263294 | 2 |
| C4×C22⋊D197 | 2 |
| #20263283 | 2 |
| #20263282 | 2 |
| C2×#15691335 | 2 |
| C22⋊C4×D197 | 2 |
| C4×D4×C197 | 2 |
| #551 | 197 |