Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
3052 conjugacy classes of subgroups; only the 667 normal subgroups are shown.
This group has 3052 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, prime radical | C33×#121 | C2 |
| hypercenter | #121 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #539 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11723976 | C2 | semidirect |
| 3. #11723976 | C2 | semidirect |
| 4. D4×C3⋊(C3⋊Dic3) | C2 | semidirect |
| 5. #11723965 | C2 | semidirect |
| 6. C2×#88857 | C2 | semidirect |
| 7. #11723930 | C2 | semidirect |
| 8. #11723920 | C2 | semidirect |
| 9. #11723919 | C2 | semidirect |
| 168. C33 | #539 | semidirect |
| 225. C32 | #5602 | semidirect |
| 261. C3 | #10596073 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×#121 | 2 |
| #11723976 | 2 |
| D4×C3⋊(C3⋊Dic3) | 2 |
| #11723965 | 2 |
| C2×#88857 | 2 |
| #11723930 | 2 |
| #11723920 | 2 |
| #11723919 | 2 |
| #10596073 | 3 |