Finite Group DatabasebetaFinite Group Databasebeta← C3×C4×C22⋊(C22⋊C9) #13164876
Order 1728.
1530 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1530 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C4 | C22⋊A4 |
| commutator, hypocenter | C24 | C3×C4×C9 |
| Frattini | C2×C3 | C2×C3×C22⋊A4 |
| Fitting | C24×C32×C4 | C3 |
| socle, prime radical | C25×C32 | C2×C3 |
| cosocle | C25×C3 | C2×C32 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C32×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×C22⋊(C22⋊C9) | C3 | direct (N × H) |
| 4. C3×C22⋊(C22⋊C9) | C4 | direct (N × H) |
| 11. C24×C3×C4 | C9 | semidirect |
| 20. C22⋊(C22⋊C9) | C3×C4 | direct (N × H) |
| 39. C24×C4 | C3×C9 | semidirect |
| 45. C24×C3 | C4×C9 | semidirect |
| 53. C22×C3×C4 | C22⋊C9 | semidirect |
| 90. C24 | C3×C4×C9 | semidirect |
| 94. C22×C4 | C3×C22⋊C9 | semidirect |
| 103. C22×C3 | C4×C22⋊C9 | semidirect |
| 107. C3×C4 | C22⋊(C22⋊C9) | direct (N × H) |
| 129. C22 | C3×C4×C22⋊C9 | semidirect |
| 131. C4 | C3×C22⋊(C22⋊C9) | direct (N × H) |
| 132. C3 | C4×C22⋊(C22⋊C9) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C22⋊(C22⋊C9) | 2 |
| C24×C32×C4 | 3 |
| C4×C22⋊(C22⋊C9) | 3 |
| C3×C4×C22⋊C9 | 4 |