Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
470 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 470 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | #99 |
| commutator | C23 | C2×C3×C4×C9 |
| Frattini, cosocle | C2×C3×D4 | C22×C32 |
| socle | C2×C32 | C3×#99 |
| prime radical | C32×#120 | C2×C3 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ C22×C3×C9 ◁ C2×C3×D4×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C9×#100 | C2 | semidirect |
| 3. C3×C9×#99 | C2 | semidirect |
| 5. C9×AΣL(1,𝔽16) | C3 | direct (N × H) |
| 6. C24×C3×C9 | C4 | semidirect |
| 12. C3×C9×C22⋊C4 | C4 | semidirect |
| 17. C9×#100 | C2×C3 | semidirect |
| 18. C9×#99 | C2×C3 | semidirect |
| 30. C3×AΣL(1,𝔽16) | C9 | direct (N × H) |
| 35. C24×C9 | C3×C4 | semidirect |
| 45. C9×C22⋊C4 | C3×C4 | semidirect |
| 61. C3×#100 | C2×C9 | semidirect |
| 63. C3×#99 | C2×C9 | semidirect |
| 87. AΣL(1,𝔽16) | C3×C9 | direct (N × H) |
| 93. C24×C3 | C4×C9 | semidirect |
| 105. C3×C22⋊C4 | C4×C9 | semidirect |
| 134. #100 | C2×C3×C9 | semidirect |
| 135. #99 | C2×C3×C9 | semidirect |
| 136. C3×C9 | AΣL(1,𝔽16) | direct (N × H) |
| 169. C24 | C3×C4×C9 | semidirect |
| 175. C22⋊C4 | C3×C4×C9 | semidirect |
| 203. C9 | C3×AΣL(1,𝔽16) | direct (N × H) |
| 238. C3 | C9×AΣL(1,𝔽16) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C9×#120 | 2 |
| C3×C9×#100 | 2 |
| C3×C9×#99 | 2 |
| C32×AΣL(1,𝔽16) | 3 |
| C9×AΣL(1,𝔽16) | 3 |