Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
230 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 170 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 | D8 |
| commutator | C8 | C2×C3×C4×C9 |
| Frattini, cosocle | C2×C3×C8 | C22×C32 |
| socle, prime radical | C22×C32 | C3×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C9 ◁ C2×C3×C4×C9 ◁ C2×C3×C8×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C9×#367 | C3 | direct (N × H) |
| 7. C3×C9×C16 | C4 | semidirect |
| 16. C3×#367 | C9 | direct (N × H) |
| 23. C9×C16 | C3×C4 | semidirect |
| 40. #367 | C3×C9 | direct (N × H) |
| 47. C3×C16 | C4×C9 | semidirect |
| 59. C3×C9 | #367 | direct (N × H) |
| 74. C16 | C3×C4×C9 | semidirect |
| 84. C9 | C3×#367 | direct (N × H) |
| 97. C3 | C9×#367 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C9×C16 | 2 |
| C3×C9×#107 | 2 |
| C32×#367 | 3 |
| C9×#367 | 3 |