Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
800 conjugacy classes of subgroups; only the 270 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 270 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 | #120 |
| commutator | C23 | C23×C3×C9 |
| Frattini, cosocle | C23×C3 | C23×C32 |
| socle | C2×C32 | C3×#120 |
| prime radical | C32×H(5,𝔽2) | C2×C3 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ C23×C3×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C9×#123 | C2 | semidirect |
| 3. C3×C9×#99 | C2 | semidirect |
| 5. C9×#458 | C3 | direct (N × H) |
| 9. C22×C3×C4×C9 | C22 | semidirect |
| 12. C3×C9×C22⋊C4 | C22 | semidirect |
| 17. C9×#123 | C2×C3 | semidirect |
| 18. C9×#99 | C2×C3 | semidirect |
| 29. C3×#458 | C9 | direct (N × H) |
| 41. C22×C4×C9 | C22×C3 | semidirect |
| 44. C9×C22⋊C4 | C22×C3 | semidirect |
| 55. C3×#123 | C2×C9 | semidirect |
| 57. C3×#99 | C2×C9 | semidirect |
| 79. #458 | C3×C9 | direct (N × H) |
| 89. C22×C3×C4 | C22×C9 | semidirect |
| 95. C3×C22⋊C4 | C22×C9 | semidirect |
| 114. #123 | C2×C3×C9 | semidirect |
| 115. #99 | C2×C3×C9 | semidirect |
| 116. C3×C9 | #458 | direct (N × H) |
| 146. C22×C4 | C22×C3×C9 | semidirect |
| 149. C22⋊C4 | C22×C3×C9 | semidirect |
| 167. C9 | C3×#458 | direct (N × H) |
| 192. C3 | C9×#458 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C9×H(5,𝔽2) | 2 |
| C3×C9×#123 | 2 |
| C3×C9×#99 | 2 |
| C32×#458 | 3 |
| C9×#458 | 3 |