Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
290 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C9 | C22⋊C4 |
| commutator | C22 | C2×C3×C8×C9 |
| Frattini, cosocle | C2×C3×C8 | C22×C32 |
| socle | C2×C32 | C3×C22⋊C8 |
| prime radical | C23×C32 | C3×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C9 ◁ C2×C3×C8×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C9×M32 | C2 | semidirect |
| 4. C9×#349 | C3 | direct (N × H) |
| 13. C9×M32 | C2×C3 | semidirect |
| 21. C3×#349 | C9 | direct (N × H) |
| 41. C3×M32 | C2×C9 | semidirect |
| 57. #349 | C3×C9 | direct (N × H) |
| 84. M32 | C2×C3×C9 | semidirect |
| 85. C3×C9 | #349 | direct (N × H) |
| 124. C9 | C3×#349 | direct (N × H) |
| 144. C3 | C9×#349 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C9×M16 | 2 |
| C3×C9×M32 | 2 |
| C32×#349 | 3 |
| C9×#349 | 3 |