Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
290 conjugacy classes of subgroups; only the 227 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 227 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×C8 | C3⋊D9 |
| commutator, hypocenter | C3×C9 | C4×C16 |
| Frattini | C2×C3×C8 | C2×C3⋊D3 |
| Fitting | C3×C4×C8×C9 | C2 |
| socle, prime radical | C22×C32 | C2×C3⋊C8 |
| cosocle | C2×C3×C8×C9 | C22 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C4×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. C3⋊(C9⋊C16) | C4 | direct (N × H) |
| 28. C3×C4×C9 | C16 | semidirect |
| 59. C4×C9 | C3⋊C16 | semidirect |
| 64. C3×C9 | C4×C16 | semidirect |
| 83. C3×C4 | C9⋊C16 | semidirect |
| 90. C9 | C4×C3⋊C16 | semidirect |
| 100. C4 | C3⋊(C9⋊C16) | direct (N × H) |
| 103. C3 | C4×C9⋊C16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C4×C8×C9 | 2 |
| C2×C3⋊(C9⋊C16) | 2 |
| C4×C3⋊(C3⋊C16) | 3 |
| C4×C9⋊C16 | 3 |