Finite Group DatabasebetaFinite Group DatabasebetaOrder 12928.
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×D101 |
| commutator, socle, cosocle | C22×C101 | C25 |
| Fitting | H(3,𝔽4)×C101 | C2 |
| hypercenter | H(3,𝔽4) | D101 |
| hypocenter | C101 | #3529 |
| LCS | G ▷ C22×C101 ▷ C101 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20265927 | C2 | semidirect |
| 2. #20265921 | C2 | semidirect |
| 3. #20265916 | C2 | semidirect |
| 4. #20265726 | C2 | semidirect |
| 5. #20265706 | C2 | semidirect |
| 6. #20265683 | C2 | semidirect |
| 7. H(3,𝔽4)×C101 | C2 | semidirect |
| 8. C2×#13108075 | C22 | semidirect |
| 10. C23×Dic101 | C22 | semidirect |
| 11. #15692253 | C22 | semidirect |
| 13. #15692248 | C22 | semidirect |
| 14. #15692247 | C22 | semidirect |
| 15. #15692243 | C22 | semidirect |
| 16. #15692239 | C22 | semidirect |
| 17. #15692237 | C22 | semidirect |
| 19. #15692232 | C22 | semidirect |
| 20. #15692231 | C22 | semidirect |
| 21. #15692228 | C22 | semidirect |
| 24. #15692224 | C22 | semidirect |
| 25. Q8×Dic101 | C22 | semidirect |
| 27. #15692215 | C22 | semidirect |
| 28. #15692211 | C22 | semidirect |
| 29. C42×D101 | C22 | semidirect |
| 98. C101 | #3529 | semidirect |
| 101. H(3,𝔽4) | D101 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20265927 | 2 |
| #20265921 | 2 |
| #20265916 | 2 |
| #20265726 | 2 |
| #20265706 | 2 |
| #20265683 | 2 |
| H(3,𝔽4)×C101 | 2 |
| #3529 | 101 |