Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
162 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C3⋊AGL(1,𝔽9) |
| commutator, hypocenter | C33 | C4×C16 |
| Frattini | C22 | C2×C3⋊AGL(1,𝔽9) |
| Fitting | C2×C33×C4 | C8 |
| socle, prime radical | C22×C33 | C2×C8 |
| cosocle | C2×C3×#622 | C22 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C3⋊#3771 | C4 | direct (N × H) |
| 22. C33×C4 | C16 | semidirect |
| 23. C3×C3⋊Dic3 | C16 | semidirect |
| 51. C32×C4 | C3⋊C16 | semidirect |
| 52. C3⋊Dic3 | C3⋊C16 | semidirect |
| 60. C33 | C4×C16 | semidirect |
| 91. C3×C4 | #3771 | semidirect |
| 97. C32 | C4×C3⋊C16 | semidirect |
| 116. C4 | C3⋊#3771 | direct (N × H) |
| 119. C3 | C4×#3771 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3⋊#3771 | 2 |
| C3×C4×#622 | 2 |
| C4×#3771 | 3 |
| C4×C3⋊C16 | 9 |