Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
3360 conjugacy classes of subgroups; only the 667 normal subgroups are shown.
This group has 3360 conjugacy classes of subgroups — too many to draw. Only its 667 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C33×#117 | C2 |
| hypercenter | #117 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #540 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C33×#117 | C2 | semidirect |
| 2. #11723971 | C2 | semidirect |
| 3. #11723935 | C2 | semidirect |
| 4. #11723925 | C2 | semidirect |
| 5. #11723915 | C2 | semidirect |
| 6. C4×C4⋊(C3⋊(C3⋊D3)) | C2 | semidirect |
| 7. #11723911 | C2 | semidirect |
| 22. C32×#117 | D3 | semidirect |
| 66. C3×#117 | C3⋊D3 | semidirect |
| 120. #117 | C3⋊(C3⋊D3) | semidirect |
| 121. C33 | #540 | semidirect |
| 162. C32 | #5535 | semidirect |
| 190. C3 | #10596006 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×#117 | 2 |
| #11723971 | 2 |
| #11723935 | 2 |
| #11723925 | 2 |
| #11723915 | 2 |
| C4×C4⋊(C3⋊(C3⋊D3)) | 2 |
| #11723911 | 2 |
| #10596006 | 3 |