Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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 | C16 | #1113 |
| commutator, hypocenter | C33 | C2×C32 |
| Fitting, cosocle | C33×C16 | C22 |
| socle | C2×C33 | C2×C16 |
| prime radical | C2×C3×C3⋊D3 | C16 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C3⋊(C3⋊C32) | C2 | semidirect |
| 11. C3⋊(C3⋊C32) | D3 | semidirect |
| 54. C3×C3⋊D3 | C32 | semidirect |
| 79. C33 | C2×C32 | semidirect |
| 96. C3⋊D3 | C3⋊C32 | semidirect |
| 116. C32 | D3×C32 | semidirect |
| 117. C32 | C2×C3⋊C32 | semidirect |
| 135. C3 | D3×C3⋊C32 | semidirect |
| 136. C3 | #10589067 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C16×C3⋊D3 | 2 |
| C3×C3⋊(C3⋊C32) | 2 |
| #10589067 | 3 |
| D3×C3⋊C32 | 3 |