Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
450 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 250 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C9 | C2×D4 |
| commutator | C2×C4 | C23×C3×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle, prime radical | C22×C32 | C2×C3×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C9 ◁ C3×C42×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C9×#491 | C3 | direct (N × H) |
| 27. C3×#491 | C9 | direct (N × H) |
| 66. #491 | C3×C9 | direct (N × H) |
| 95. C3×C9 | #491 | direct (N × H) |
| 129. C9 | C3×#491 | direct (N × H) |
| 145. C3 | C9×#491 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C9×C4⋊Q8 | 2 |
| C3×C9×#125 | 2 |
| C3×C9×#103 | 2 |
| C3×C9×#97 | 2 |
| C32×#491 | 3 |
| C9×#491 | 3 |