Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1796 conjugacy classes of subgroups; only the 332 normal subgroups are shown.
This group has 1796 conjugacy classes of subgroups — too many to draw. Only its 332 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#6290 |
| commutator, cosocle | C33×C4 | C24 |
| Frattini | C4 | C22×D3×C3⋊D3 |
| Fitting | C33×#41 | C22 |
| socle | C2×C33 | C22×D4 |
| hypercenter | #41 | D3×C3⋊D3 |
| hypocenter | C33 | #577 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#64367 | C2 | semidirect |
| 2. C32×#1009 | C2 | semidirect |
| 3. #11723989 | C2 | semidirect |
| 4. #11723373 | C2 | semidirect |
| 5. C2×#88792 | C2 | semidirect |
| 6. #11723364 | C2 | semidirect |
| 7. #11723363 | C2 | semidirect |
| 13. C3×#3828 | C22 | semidirect |
| 14. C2×C3×#634 | C22 | semidirect |
| 19. #88855 | C22 | semidirect |
| 20. C2×C3⋊(C3⋊(C3⋊C8)) | C22 | semidirect |
| 21. #88798 | C22 | semidirect |
| 22. #88793 | C22 | semidirect |
| 23. #88792 | C22 | semidirect |
| 24. #88792 | C22 | semidirect |
| 25. #88789 | C22 | semidirect |
| 27. #64367 | D3 | semidirect |
| 29. C3×#1009 | D3 | semidirect |
| 69. #3828 | C2×D3 | semidirect |
| 70. C2×#634 | C2×D3 | semidirect |
| 106. #1009 | C3⋊D3 | semidirect |
| 223. C33 | #577 | semidirect |
| 309. C32 | #5836 | semidirect |
| 310. C32 | #5837 | semidirect |
| 347. C3 | #10595446 | semidirect |
| 348. C3 | #10596307 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#64367 | 2 |
| C32×#1009 | 2 |
| #11723989 | 2 |
| #11723373 | 2 |
| C2×#88792 | 2 |
| #11723364 | 2 |
| #11723363 | 2 |
| #10596307 | 3 |
| #10595446 | 3 |