Finite Group DatabasebetaFinite Group DatabasebetaOrder 34432.
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×D269 |
| commutator, socle, cosocle | C22×C269 | C25 |
| Fitting | H(3,𝔽4)×C269 | C2 |
| hypercenter | H(3,𝔽4) | D269 |
| hypocenter | C269 | #3529 |
| LCS | G ▷ C22×C269 ▷ C269 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #31771355 | C2 | semidirect |
| 2. #31771349 | C2 | semidirect |
| 3. #31771344 | C2 | semidirect |
| 4. #31771154 | C2 | semidirect |
| 5. #31771134 | C2 | semidirect |
| 6. #31771111 | C2 | semidirect |
| 7. H(3,𝔽4)×C269 | C2 | semidirect |
| 8. C2×#16864403 | C22 | semidirect |
| 10. C23×Dic269 | C22 | semidirect |
| 11. #22575451 | C22 | semidirect |
| 13. #22575446 | C22 | semidirect |
| 14. #22575445 | C22 | semidirect |
| 15. #22575441 | C22 | semidirect |
| 16. #22575437 | C22 | semidirect |
| 17. #22575435 | C22 | semidirect |
| 19. #22575430 | C22 | semidirect |
| 20. #22575429 | C22 | semidirect |
| 21. #22575426 | C22 | semidirect |
| 24. #22575422 | C22 | semidirect |
| 25. Q8×Dic269 | C22 | semidirect |
| 27. #22575413 | C22 | semidirect |
| 28. #22575409 | C22 | semidirect |
| 29. C42×D269 | C22 | semidirect |
| 97. C269 | #3529 | semidirect |
| 101. H(3,𝔽4) | D269 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #31771355 | 2 |
| #31771349 | 2 |
| #31771344 | 2 |
| #31771154 | 2 |
| #31771134 | 2 |
| #31771111 | 2 |
| H(3,𝔽4)×C269 | 2 |
| #3529 | 269 |