Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
292 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C9 | C22⋊A4 |
| commutator, hypocenter | #543 | C3×C9 |
| Frattini, socle | C22×C3 | C3×C22⋊A4 |
| Fitting | C9×#543 | C3 |
| prime radical | C3×#543⋊C3 | C3 |
| cosocle | C3×#543 | C32 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C22×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C9×#543 | C3 | semidirect |
| 3. #543⋊C9 | C3 | semidirect |
| 12. #543⋊C3 | C9 | direct (N × H) |
| 17. C2×Q8×C9 | A4 | semidirect |
| 32. C23×C9 | SL(2,𝔽3) | semidirect |
| 39. #543 | C3×C9 | semidirect |
| 82. C2×Q8 | C9×A4 | semidirect |
| 97. C9 | #543⋊C3 | direct (N × H) |
| 101. 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 |
|---|---|
| C3×#543⋊C3 | 3 |
| C9×#543 | 3 |
| #543⋊C9 | 3 |
| C22×C9×A4 | 4 |
| C2×C9×SL(2,𝔽3) | 4 |