Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
227 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | #120⋊C3 |
| commutator, hypocenter | C22×Q8 | C2×C3×C9 |
| Frattini, socle | C23×C3 | C2×C3×A4 |
| Fitting | C9×#393 | C3 |
| prime radical | C3×#393⋊C3 | C3 |
| cosocle | C22×C3×Q8 | C2×C32 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×C22⋊SL(2,𝔽3) | C2 | semidirect |
| 3. C9×#393 | C3 | semidirect |
| 4. #393⋊C9 | C3 | semidirect |
| 7. C22×Q8×C9 | C2×C3 | semidirect |
| 8. C22⋊(Q8⋊C9) | C2×C3 | semidirect |
| 15. #393⋊C3 | C9 | direct (N × H) |
| 26. C22⋊SL(2,𝔽3) | C2×C9 | semidirect |
| 34. C23×C9 | SL(2,𝔽3) | semidirect |
| 45. #393 | C3×C9 | semidirect |
| 66. C22×Q8 | C2×C3×C9 | semidirect |
| 108. C9 | #393⋊C3 | direct (N × H) |
| 111. C23 | C9×SL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×C22⋊SL(2,𝔽3) | 2 |
| C3×#393⋊C3 | 3 |
| C9×#393 | 3 |
| #393⋊C9 | 3 |
| C22×C9×A4 | 4 |