Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
296 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | A4×D9 |
| commutator, hypocenter | Q8×C9 | C2×C3×C4 |
| Frattini, socle | C22×C3 | C2×D3×A4 |
| Fitting | C2×C9×#41 | C2×C3 |
| prime radical | C2×C3×#41⋊C3 | D3 |
| cosocle | C2×Q8×C9 | C22×C3 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. SL(2,𝔽3)×Dic9 | C2 | semidirect |
| 4. #41×Dic9 | C3 | semidirect |
| 5. C9×#41⋊C3 | C4 | semidirect |
| 18. C9×SL(2,𝔽3) | C2×C4 | semidirect |
| 25. C9×#41 | C3×C4 | semidirect |
| 75. #41⋊C3 | Dic9 | direct (N × H) |
| 96. Dic9 | #41⋊C3 | direct (N × H) |
| 119. SL(2,𝔽3) | C2×Dic9 | semidirect |
| 128. #41 | C3×Dic9 | semidirect |
| 153. C9 | C4×#41⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C9×#41⋊C3 | 2 |
| SL(2,𝔽3)×Dic9 | 2 |
| Dic3×#41⋊C3 | 3 |
| #41×Dic9 | 3 |
| C3×C4×Dic9 | 4 |