Finite Group DatabasebetaFinite Group DatabasebetaOrder 12288.
At least 21039 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has at least 21039 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #72209719 |
| commutator, cosocle, hypocenter | #72267604 | C22×C3 |
| Frattini | C25×Q8 | C22×A4 |
| Fitting | #72267906 | C3 |
| hypercenter | C24 | #11709314 |
| LCS | G ▷ #72267604 | |
| UCS | C1 ◁ C23 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72267915 | C2 | semidirect |
| 2. #72267906 | C3 | semidirect |
| 3. #72267904 | C2×C3 | semidirect |
| 4. #72267894 | C2 | semidirect |
| 5. #72267893 | C22 | semidirect |
| 6. #72267605 | C2×C3 | semidirect |
| 7. #72267604 | C22×C3 | semidirect |
| 9. C24×2−1+4 | SL(2,𝔽3) | semidirect |
| 34. 2−1+4 | #2029⋊C3 | semidirect |
| 38. C24 | #13378⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72267915 | 2 |
| #72267906 | 3 |
| C23×#87545 | 4 |
| #72267894 | 2 |