Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
611 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 611 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×A4×D9 |
| commutator, cosocle, hypocenter | Q8×C9 | C23×C3 |
| Frattini, socle | C2×C3 | C22×D3×A4 |
| Fitting | C2×C9×#41 | C2×C3 |
| hypercenter | C2×C4 | A4×D9 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C9×#41⋊C3 | C2 | semidirect |
| 2. #11722416 | C2 | semidirect |
| 3. D9×#41⋊C3 | C2 | semidirect |
| 4. #11722412 | C2 | semidirect |
| 5. #11722410 | C2 | semidirect |
| 7. #10593139 | C3 | semidirect |
| 9. C9×#41⋊C3 | C22 | semidirect |
| 10. C9×#41⋊C3 | C22 | semidirect |
| 11. C2×C9×SL(2,𝔽3) | C22 | semidirect |
| 12. D9×SL(2,𝔽3) | C22 | semidirect |
| 13. #88615 | C22 | semidirect |
| 19. C2×C9×#41 | C2×C3 | semidirect |
| 21. #63716 | C2×C3 | semidirect |
| 23. #63715 | C2×C3 | semidirect |
| 25. #41×D9 | C2×C3 | semidirect |
| 26. #63713 | C2×C3 | semidirect |
| 103. C2×#41⋊C3 | D9 | semidirect |
| 188. #41⋊C3 | C2×D9 | semidirect |
| 189. #41⋊C3 | C2×D9 | semidirect |
| 192. C2×SL(2,𝔽3) | C2×D9 | semidirect |
| 225. C2×#41 | C3×D9 | semidirect |
| 338. C9 | #5945 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C9×#41⋊C3 | 2 |
| #11722416 | 2 |
| D9×#41⋊C3 | 2 |
| #11722412 | 2 |
| #11722410 | 2 |
| #5964⋊C3 | 3 |
| #10593139 | 3 |
| C3×#3697 | 4 |