Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
812 conjugacy classes of subgroups; only the 476 normal subgroups are shown.
This group has 812 conjugacy classes of subgroups — too many to draw. Only its 476 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C33×C4 | C22⋊C4 |
| commutator | C22 | C2×C33×C8 |
| Frattini, cosocle | C2×C8 | C22×C33 |
| socle | C2×C33 | C22⋊C8 |
| prime radical | C23×C33 | C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C33×C4 ◁ C2×C33×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C33×M32 | C2 | semidirect |
| 3. C32×#349 | C3 | direct (N × H) |
| 10. C32×M32 | C2×C3 | semidirect |
| 18. C3×#349 | C32 | direct (N × H) |
| 31. C3×M32 | C2×C32 | semidirect |
| 39. #349 | C33 | direct (N × H) |
| 55. M32 | C2×C33 | semidirect |
| 56. C33 | #349 | direct (N × H) |
| 78. C32 | C3×#349 | direct (N × H) |
| 95. C3 | C32×#349 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C33×M16 | 2 |
| C33×M32 | 2 |
| C32×#349 | 3 |