Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
202 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D3×AGL(1,𝔽9) |
| commutator | C2×C33 | C2×C16 |
| Fitting, socle | C22×C33 | C2×C8 |
| prime radical | C22×C32×D3 | C8 |
| cosocle | C2×C3×#622 | C22 |
| hypocenter | C33 | C22⋊C16 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3⋊#3771 | C2 | semidirect |
| 3. C2×C3×#3771 | C2 | semidirect |
| 14. C2×#3771 | D3 | semidirect |
| 35. C2×C32×D3 | C16 | semidirect |
| 88. C33 | C22⋊C16 | semidirect |
| 129. C2×D3 | #3771 | semidirect |
| 139. C32 | (C2×D3)⋊C16 | semidirect |
| 168. C3 | C22⋊#3771 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D3×#622 | 2 |
| C2×C3⋊#3771 | 2 |
| C2×C3×#3771 | 2 |
| C22⋊#3771 | 3 |
| (C2×D3)⋊C16 | 9 |