Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
870 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 870 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C9 | C22⋊A4 |
| commutator, hypocenter | C24 | C3×C4×C9 |
| Frattini | C2×C3 | C2×C3×C22⋊A4 |
| Fitting | C24×C4×C9 | C3 |
| socle, cosocle | C25×C3 | C2×C32 |
| prime radical | C2×C3×C22⋊A4 | C2×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C4×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C24×C4×C9 | C3 | semidirect |
| 4. C4×C22⋊(C22⋊C9) | C3 | semidirect |
| 5. C9×C22⋊A4 | C4 | direct (N × H) |
| 14. C4×C22⋊A4 | C9 | direct (N × H) |
| 19. C24×C9 | C3×C4 | semidirect |
| 20. C22⋊(C22⋊C9) | C3×C4 | semidirect |
| 24. C22×C4×C9 | A4 | semidirect |
| 42. C24×C4 | C3×C9 | semidirect |
| 47. C22⋊A4 | C4×C9 | direct (N × H) |
| 59. C22×C9 | C4×A4 | semidirect |
| 63. C4×C9 | C22⋊A4 | direct (N × H) |
| 84. C24 | C3×C4×C9 | semidirect |
| 88. C22×C4 | C9×A4 | semidirect |
| 100. C9 | C4×C22⋊A4 | direct (N × H) |
| 117. C22 | C4×C9×A4 | semidirect |
| 119. C4 | C9×C22⋊A4 | 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×C4×C22⋊A4 | 3 |
| C24×C4×C9 | 3 |
| C4×C22⋊(C22⋊C9) | 3 |
| C4×C9×A4 | 4 |