Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
426 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C42 | #1113 |
| commutator | C2×C33 | C4×C8 |
| Fitting, cosocle | C33×C42 | C22 |
| socle, prime radical | C22×C33 | C42 |
| hypocenter | C33 | #322 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 18. C3×C3⋊(C3⋊C8) | C8 | semidirect |
| 56. C3⋊(C3⋊C8) | C3⋊C8 | semidirect |
| 96. C33 | #322 | semidirect |
| 151. C32 | #4454 | semidirect |
| 152. C32 | #4455 | semidirect |
| 175. C3 | #10589078 | semidirect |
| 176. C3 | #10589079 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C4×C3⋊(C3⋊C8) | 2 |
| #10589079 | 3 |
| #10589078 | 3 |