Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
519 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 519 conjugacy classes of subgroups — too many to draw. Only its 195 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42×C9 | A4 |
| commutator, hypocenter | C22 | C3×C42×C9 |
| Frattini | C22×C3 | C22×C3×A4 |
| Fitting | C22×C42×C9 | C3 |
| socle, cosocle | C24×C3 | C22×C32 |
| prime radical | C22×C3×A4 | C22×C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C42×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C22×C42×C9 | C3 | semidirect |
| 4. C42×C22⋊C9 | C3 | semidirect |
| 6. C4×C9×A4 | C4 | direct (N × H) |
| 16. C42×A4 | C9 | direct (N × H) |
| 25. C22×C4×C9 | C3×C4 | semidirect |
| 26. C4×C22⋊C9 | C3×C4 | semidirect |
| 30. C42×C9 | A4 | direct (N × H) |
| 32. C9×A4 | C42 | direct (N × H) |
| 51. C22×C42 | C3×C9 | semidirect |
| 59. C4×A4 | C4×C9 | direct (N × H) |
| 72. C22×C9 | C3×C42 | semidirect |
| 74. C22⋊C9 | C3×C42 | semidirect |
| 77. C4×C9 | C4×A4 | direct (N × H) |
| 102. C22×C4 | C3×C4×C9 | semidirect |
| 105. C42 | C9×A4 | direct (N × H) |
| 113. A4 | C42×C9 | direct (N × H) |
| 119. C9 | C42×A4 | direct (N × H) |
| 137. C22 | C3×C42×C9 | semidirect |
| 140. C4 | C4×C9×A4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C9×A4 | 2 |
| C3×C42×A4 | 3 |
| C22×C42×C9 | 3 |
| C42×C22⋊C9 | 3 |
| C3×C42×C9 | 4 |