Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
3472 conjugacy classes of subgroups; only the 723 normal subgroups are shown.
This group has 3472 conjugacy classes of subgroups — too many to draw. Only its 723 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×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting, prime radical | C33×#120 | C2 |
| hypercenter | #120 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #551 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ #120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#88864 | C2 | semidirect |
| 3. #11723977 | C2 | semidirect |
| 4. #11723976 | C2 | semidirect |
| 5. #11723976 | C2 | semidirect |
| 6. D4×C3⋊(C3⋊Dic3) | C2 | semidirect |
| 7. #11723920 | C2 | semidirect |
| 8. #11723919 | C2 | semidirect |
| 9. #11723918 | C2 | semidirect |
| 16. C22×C3⋊(C3⋊Dic3) | C22 | semidirect |
| 17. C2×#6308 | C22 | semidirect |
| 24. #88864 | C22 | semidirect |
| 25. #88864 | C22 | semidirect |
| 26. C4⋊(C3⋊(C3⋊Dic3)) | C22 | semidirect |
| 27. #88857 | C22 | semidirect |
| 28. #88857 | C22 | semidirect |
| 29. #88857 | C22 | semidirect |
| 30. C4×C3⋊(C3⋊Dic3) | C22 | semidirect |
| 31. C4×C3⋊(C3⋊Dic3) | C22 | semidirect |
| 172. C33 | #551 | semidirect |
| 231. C32 | #5587 | semidirect |
| 269. C3 | #10596058 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×#120 | 2 |
| C2×#88864 | 2 |
| #11723977 | 2 |
| #11723976 | 2 |
| D4×C3⋊(C3⋊Dic3) | 2 |
| #11723920 | 2 |
| #11723919 | 2 |
| #11723918 | 2 |
| #10596058 | 3 |