Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
362 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42×C9 | A4 |
| commutator, hypocenter | Q8 | C2×C3×C4×C9 |
| Frattini, socle | C22×C3 | C22×C3×A4 |
| Fitting | C4×C9×#41 | C3 |
| prime radical | C2×C3×#41⋊C3 | C2×C3 |
| cosocle | C2×C3×Q8 | C22×C32 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C42×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×C9×SL(2,𝔽3) | C2 | semidirect |
| 4. C4×C9×#41 | C3 | semidirect |
| 5. C4×#41⋊C9 | C3 | semidirect |
| 6. C9×#41⋊C3 | C4 | direct (N × H) |
| 20. C9×SL(2,𝔽3) | C2×C4 | semidirect |
| 22. C4×#41⋊C3 | C9 | direct (N × H) |
| 27. C9×#41 | C3×C4 | semidirect |
| 32. #41⋊C9 | C3×C4 | semidirect |
| 49. C4×SL(2,𝔽3) | C2×C9 | semidirect |
| 68. C4×#41 | C3×C9 | semidirect |
| 77. #41⋊C3 | C4×C9 | direct (N × H) |
| 98. C4×C9 | #41⋊C3 | direct (N × H) |
| 119. SL(2,𝔽3) | C2×C4×C9 | semidirect |
| 128. #41 | C3×C4×C9 | semidirect |
| 151. C9 | C4×#41⋊C3 | direct (N × H) |
| 176. C4 | C9×#41⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C9×#41⋊C3 | 2 |
| C4×C9×SL(2,𝔽3) | 2 |
| C3×C4×#41⋊C3 | 3 |
| C4×C9×#41 | 3 |
| C4×#41⋊C9 | 3 |
| C3×C42×C9 | 4 |