Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
320 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 320 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 | C2×C4 | C2×C3×C4×C9 |
| Frattini, cosocle | C2×C3×Q8 | C22×C32 |
| socle | C2×C32 | C3×#99 |
| prime radical | C2×C32×D4 | C3×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ C22×C3×C9 ◁ C2×C3×Q8×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C9×#101 | C2 | semidirect |
| 4. C9×#355 | C3 | direct (N × H) |
| 13. C9×#101 | C2×C3 | semidirect |
| 21. C3×#355 | C9 | direct (N × H) |
| 41. C3×#101 | C2×C9 | semidirect |
| 57. #355 | C3×C9 | direct (N × H) |
| 84. #101 | C2×C3×C9 | semidirect |
| 85. C3×C9 | #355 | direct (N × H) |
| 124. C9 | C3×#355 | direct (N × H) |
| 144. C3 | C9×#355 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C9×#124 | 2 |
| C3×C9×#101 | 2 |
| C32×#355 | 3 |
| C9×#355 | 3 |