Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
1058 conjugacy classes of subgroups; only the 451 normal subgroups are shown.
This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 451 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 | C24×D251 |
| commutator, socle, cosocle | C22×C251 | C25 |
| Fitting | H(3,𝔽4)×C251 | C2 |
| hypercenter | H(3,𝔽4) | D251 |
| hypocenter | C251 | #3529 |
| LCS | G ▷ C22×C251 ▷ C251 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30632437 | C2 | semidirect |
| 2. #30632431 | C2 | semidirect |
| 3. #30632426 | C2 | semidirect |
| 4. #30632236 | C2 | semidirect |
| 5. #30632216 | C2 | semidirect |
| 6. #30632193 | C2 | semidirect |
| 7. H(3,𝔽4)×C251 | C2 | semidirect |
| 8. C2×#16860811 | C22 | semidirect |
| 10. C23×Dic251 | C22 | semidirect |
| 11. #22548713 | C22 | semidirect |
| 13. #22548708 | C22 | semidirect |
| 14. #22548707 | C22 | semidirect |
| 15. #22548703 | C22 | semidirect |
| 16. #22548699 | C22 | semidirect |
| 17. #22548697 | C22 | semidirect |
| 19. #22548692 | C22 | semidirect |
| 20. #22548691 | C22 | semidirect |
| 21. #22548688 | C22 | semidirect |
| 24. #22548684 | C22 | semidirect |
| 25. Q8×Dic251 | C22 | semidirect |
| 27. (C22⋊D251)⋊C4 | C22 | semidirect |
| 28. #22548671 | C22 | semidirect |
| 29. C42×D251 | C22 | semidirect |
| 97. C251 | #3529 | semidirect |
| 101. H(3,𝔽4) | D251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30632437 | 2 |
| #30632431 | 2 |
| #30632426 | 2 |
| #30632236 | 2 |
| #30632216 | 2 |
| #30632193 | 2 |
| H(3,𝔽4)×C251 | 2 |
| #3529 | 251 |