Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1041 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 1041 conjugacy classes of subgroups — too many to draw. Only its 61 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 | C3 | A4×C22⋊A4 |
| commutator, hypocenter | C26 | C3×C9 |
| Fitting, socle, cosocle | C26×C3 | C32 |
| prime radical | C24×C3×A4 | C3 |
| LCS | G ▷ C26 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22⋊(C22⋊(C22⋊C9)) | C3 | semidirect |
| 3. C22×C22⋊(C22⋊C9) | C3 | semidirect |
| 9. C24×A4 | C9 | semidirect |
| 16. C22⋊(C22⋊C9) | A4 | direct (N × H) |
| 27. C26 | C3×C9 | semidirect |
| 39. C22×A4 | C22⋊C9 | semidirect |
| 73. C24 | C9×A4 | semidirect |
| 74. C24 | C3×C22⋊C9 | semidirect |
| 86. A4 | C22⋊(C22⋊C9) | direct (N × H) |
| 111. C22 | A4×C22⋊C9 | semidirect |
| 112. C22 | C3×C22⋊(C22⋊C9) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×A4 | 3 |
| C22⋊(C22⋊(C22⋊C9)) | 3 |
| C22×C22⋊(C22⋊C9) | 3 |
| C3×C22⋊(C22⋊C9) | 4 |
| A4×C22⋊C9 | 4 |