Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
951 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 951 conjugacy classes of subgroups — too many to draw. Only its 43 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 | C3×C22⋊(C22⋊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 |
| 4. C24×C22⋊C9 | C3 | semidirect |
| 10. C22⋊(C22⋊A4) | C9 | semidirect |
| 18. C22⋊(C22⋊C9) | A4 | semidirect |
| 21. C22×C22⋊C9 | A4 | semidirect |
| 30. C26 | C3×C9 | semidirect |
| 54. C22⋊C9 | C22⋊A4 | semidirect |
| 80. C24 | C9×A4 | semidirect |
| 81. C24 | C9×A4 | semidirect |
| 121. C22 | (C22⋊A4)⋊C9 | semidirect |
| 122. C22 | C9×C22⋊A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C22⋊(C22⋊A4) | 3 |
| C22⋊(C22⋊(C22⋊C9)) | 3 |
| C22×C22⋊(C22⋊C9) | 3 |
| C24×C22⋊C9 | 3 |
| C9×C22⋊A4 | 4 |
| (C22⋊A4)⋊C9 | 4 |