Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
210 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 48 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 | C42⋊C3 |
| commutator, hypocenter | C42 | C3×C4×C9 |
| Frattini, socle | C23×C3 | C2×C3×A4 |
| Fitting | C43×C9 | C3 |
| prime radical | C2×C3×C42⋊C3 | C2×C3 |
| cosocle | C2×C3×C42 | C2×C32 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C4×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C43×C9 | C3 | semidirect |
| 4. C4×C42⋊C9 | C3 | semidirect |
| 6. C9×C42⋊C3 | C4 | direct (N × H) |
| 14. C4×C42⋊C3 | C9 | direct (N × H) |
| 25. C42×C9 | C3×C4 | semidirect |
| 26. C42⋊C9 | C3×C4 | semidirect |
| 44. C43 | C3×C9 | semidirect |
| 55. C42⋊C3 | C4×C9 | direct (N × H) |
| 67. C4×C9 | C42⋊C3 | direct (N × H) |
| 94. C42 | C3×C4×C9 | semidirect |
| 107. C9 | C4×C42⋊C3 | direct (N × H) |
| 128. C4 | C9×C42⋊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×C42⋊C3 | 2 |
| C3×C4×C42⋊C3 | 3 |
| C43×C9 | 3 |
| C4×C42⋊C9 | 3 |
| C4×C9×A4 | 4 |