Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
3332 conjugacy classes of subgroups; only the 741 normal subgroups are shown.
This group has 3332 conjugacy classes of subgroups — too many to draw. Only its 741 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C23×C33 | C23 |
| Fitting, prime radical | C22×C33×D4 | C2 |
| hypercenter | C22×D4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #394 |
| LCS | G ▷ C23×C33 ▷ C33 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#88864 | C2 | semidirect |
| 3. C2×#88864 | C2 | semidirect |
| 4. C2×#88857 | C2 | semidirect |
| 5. #11722201 | C2 | semidirect |
| 40. C2×C3⋊(C3⋊Dic3) | D4 | semidirect |
| 130. C33 | #394 | semidirect |
| 182. C32 | #5222 | semidirect |
| 224. C3 | #10592921 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C33×D4 | 2 |
| C2×#88864 | 2 |
| C2×#88857 | 2 |
| #11722201 | 2 |
| #10592921 | 3 |