Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
154 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 50 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 | C22 | D3×AGL(1,𝔽9) |
| commutator | C2×C33 | C4×C8 |
| Fitting, socle, prime radical | C22×C33 | C2×C8 |
| cosocle | C2×C3×#622 | C22 |
| hypocenter | C33 | #346 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C3⋊#3771 | C4 | semidirect |
| 8. C3×#3771 | C4 | semidirect |
| 23. #3771 | Dic3 | semidirect |
| 69. C33 | #346 | semidirect |
| 108. C32 | #4502 | semidirect |
| 131. C3 | #10593263 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3⋊#3771 | 2 |
| C2×C3×#3771 | 2 |
| Dic3×#622 | 2 |
| #10593263 | 3 |
| #4502 | 9 |