Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
354 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 123 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 | C2×C8 | #1113 |
| commutator, hypocenter | C33 | C4×C16 |
| Fitting, cosocle | C2×C33×C8 | C22 |
| socle, prime radical | C22×C33 | C2×C8 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. C3×C3⋊(C3⋊C16) | C4 | semidirect |
| 33. C3⋊(C3⋊C16) | Dic3 | semidirect |
| 40. C3×C3⋊Dic3 | C16 | semidirect |
| 99. C3⋊Dic3 | C3⋊C16 | semidirect |
| 104. C33 | C4×C16 | semidirect |
| 158. C32 | C4×C3⋊C16 | semidirect |
| 159. C32 | Dic3×C16 | semidirect |
| 181. C3 | Dic3×C3⋊C16 | semidirect |
| 182. C3 | #10589102 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C8×C3⋊Dic3 | 2 |
| C2×C3×C3⋊(C3⋊C16) | 2 |
| #10589102 | 3 |
| Dic3×C3⋊C16 | 3 |