Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
729 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 729 conjugacy classes of subgroups — too many to draw. Only its 211 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | C22×A4 |
| commutator, hypocenter | Q8 | C23×C3×C9 |
| Frattini, socle | C2×C3 | C23×C3×A4 |
| Fitting | C9×#585 | C3 |
| prime radical | C3×#5945 | C3 |
| cosocle | C3×Q8 | C23×C32 |
| hypercenter | C9×#41 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C4×C9 ◁ C9×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×2−1+4⋊C3 | C2 | semidirect |
| 2. C9×#994 | C2 | semidirect |
| 3. C2×C9×#41⋊C3 | C2 | semidirect |
| 5. C9×#585 | C3 | semidirect |
| 6. #10593120 | C3 | semidirect |
| 8. C9×#41⋊C3 | C22 | semidirect |
| 9. C9×#41⋊C3 | C22 | semidirect |
| 10. C2×C9×SL(2,𝔽3) | C22 | semidirect |
| 15. C9×2−1+4 | C2×C3 | semidirect |
| 17. C9×H(5,𝔽2) | C2×C3 | semidirect |
| 20. C2×C9×#41 | C2×C3 | semidirect |
| 21. 2−1+4⋊C9 | C2×C3 | semidirect |
| 22. #63700 | C2×C3 | semidirect |
| 23. C2×#41⋊C9 | C2×C3 | semidirect |
| 29. #5945 | C9 | direct (N × H) |
| 67. 2−1+4⋊C3 | C2×C9 | semidirect |
| 68. #994 | C2×C9 | semidirect |
| 69. C2×#41⋊C3 | C2×C9 | semidirect |
| 97. #585 | C3×C9 | semidirect |
| 120. #41⋊C3 | C22×C9 | semidirect |
| 121. #41⋊C3 | C22×C9 | semidirect |
| 122. C2×SL(2,𝔽3) | C22×C9 | semidirect |
| 139. 2−1+4 | C2×C3×C9 | semidirect |
| 141. H(5,𝔽2) | C2×C3×C9 | semidirect |
| 144. C2×#41 | C2×C3×C9 | semidirect |
| 210. C9 | #5945 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×2−1+4⋊C3 | 2 |
| C9×#994 | 2 |
| C2×C9×#41⋊C3 | 2 |
| C3×#5945 | 3 |
| C9×#585 | 3 |
| #10593120 | 3 |
| C3×C9×#41 | 4 |