Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
216 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C8 | D3×A4 |
| commutator, hypocenter | C3×Q8 | C32×C8 |
| Frattini | C8 | C3×D3×A4 |
| Fitting | C32×#131 | C2×C3 |
| socle | C2×C32 | C8×A4 |
| prime radical | C32×#41⋊C3 | C4 |
| cosocle | C3×#131 | C2×C32 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C3×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#5140 | C3 | semidirect |
| 4. C3⋊(#504⋊C3) | C3 | direct (N × H) |
| 17. #5140 | C32 | semidirect |
| 127. C32 | #504⋊C3 | semidirect |
| 147. C3 | C3⋊(#504⋊C3) | direct (N × H) |
| 148. C3 | C3×#504⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#131⋊C3 | 2 |
| C3×#5140 | 3 |
| C3×#504⋊C3 | 3 |
| C3⋊(#504⋊C3) | 3 |
| C32×C3⋊C16 | 4 |