Finite Group DatabasebetaFinite Group DatabasebetaOrder 12288.
At least 13471 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has at least 13471 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #72209720 |
| commutator, cosocle, hypocenter | #72268378 | C22×C3 |
| Frattini | C22×Q82 | C22×A4 |
| Fitting | #72271474 | C3 |
| hypercenter | C24 | #11709314 |
| LCS | G ▷ #72268378 | |
| UCS | C1 ◁ C22 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72271474 | C3 | semidirect |
| 2. #72271440 | C2 | semidirect |
| 3. #72271430 | C2×C3 | semidirect |
| 4. #72271387 | C2 | semidirect |
| 5. #72271377 | C2×C3 | semidirect |
| 6. #72268379 | C22 | semidirect |
| 7. #72268378 | C22×C3 | semidirect |
| 8. 2−1+42 | A4 | semidirect |
| 21. 2−1+4 | #2202⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72271474 | 3 |
| #72271440 | 2 |
| #72271387 | 2 |
| #72268200 | 4 |