Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
370 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 45 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 | C24 | C3×C4×C9 |
| Frattini, socle | C23×C3 | C2×C3×A4 |
| Fitting | C9×#379 | C3 |
| prime radical | C2×C3×C22⋊A4 | C2×C3 |
| cosocle | C25×C3 | C2×C32 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C2×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C9×#379 | C3 | semidirect |
| 4. #379⋊C9 | C3 | semidirect |
| 5. C9×C22⋊A4 | C4 | semidirect |
| 15. #379⋊C3 | C9 | direct (N × H) |
| 22. C24×C9 | C3×C4 | semidirect |
| 23. C22⋊(C22⋊C9) | C3×C4 | semidirect |
| 25. C22×C4×C9 | A4 | semidirect |
| 58. #379 | C3×C9 | semidirect |
| 65. C22⋊A4 | C4×C9 | semidirect |
| 129. C24 | C3×C4×C9 | semidirect |
| 131. C22×C4 | C9×A4 | semidirect |
| 156. C9 | #379⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C9×C22⋊A4 | 2 |
| C3×#379⋊C3 | 3 |
| C9×#379 | 3 |
| #379⋊C9 | 3 |
| C4×C9×A4 | 4 |