Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
304 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 76 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 | D3×C3⋊D3 |
| commutator, socle | C2×C33 | C2×C16 |
| Fitting, cosocle | C33×C16 | C22 |
| prime radical | C2×C3⋊(C3⋊D3) | C16 |
| hypocenter | C33 | M64 |
| LCS | G ▷ C2×C33 ▷ 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 |
| 3. C32×C3⋊C32 | C2 | semidirect |
| 13. C3⋊(C3⋊C32) | D3 | semidirect |
| 15. C3×C3⋊C32 | D3 | semidirect |
| 39. C3⋊C32 | C3⋊D3 | semidirect |
| 73. C33 | M64 | semidirect |
| 102. C32 | #4447 | semidirect |
| 103. C32 | #4447 | semidirect |
| 117. C3 | #10589069 | semidirect |
| 118. C3 | #10589313 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C16×C3⋊(C3⋊D3) | 2 |
| C3×C3⋊(C3⋊C32) | 2 |
| C32×C3⋊C32 | 2 |
| #10589313 | 3 |
| #10589069 | 3 |