Finite Group DatabasebetaFinite Group DatabasebetaOrder 6144.
At least 10039 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has at least 10039 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #72209713 |
| commutator, cosocle, hypocenter | #72219811 | C2×C3 |
| Frattini | C26 | C2×C22⋊A4 |
| Fitting | #72226613 | C3 |
| socle | C24 | #85645 |
| LCS | G ▷ #72219811 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72226613 | C3 | semidirect |
| 2. #72226611 | C2 | semidirect |
| 3. #72219811 | C2×C3 | semidirect |
| 5. C22×#2029 | A4 | semidirect |
| 6. #123533 | A4 | semidirect |
| 7. C28 | SL(2,𝔽3) | semidirect |
| 9. C22×#543 | C2×A4 | semidirect |
| 10. #13794 | C2×A4 | semidirect |
| 22. C24 | #2029⋊C3 | semidirect |
| 23. C24 | #73292 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72226613 | 3 |
| #72226611 | 2 |
| C23×C22⋊(C22⋊A4) | 4 |
| #72209914 | 4 |
| #72209873 | 4 |