Finite Group DatabasebetaFinite Group DatabasebetaOrder 12288.
At least 14708 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has at least 14708 conjugacy classes of subgroups — too many to draw. Only its 46 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 | #72268220 | C22×C3 |
| Frattini | C22×#543 | C22×A4 |
| Fitting | #72268229 | C3 |
| hypercenter | C24 | #11709314 |
| LCS | G ▷ #72268220 | |
| UCS | C1 ◁ C22 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72271289 | C2 | semidirect |
| 2. #72268240 | C2 | semidirect |
| 3. #72268238 | C22 | semidirect |
| 4. #72268229 | C3 | semidirect |
| 5. #72268227 | C2×C3 | semidirect |
| 6. #72268224 | C2×C3 | semidirect |
| 7. #72268220 | C22×C3 | semidirect |
| 8. #72268062 | A4 | semidirect |
| 16. C2×2−1+4 | B(4,𝔽2)⋊C3 | semidirect |
| 20. #519 | B(4,𝔽2)⋊C3 | semidirect |
| 21. C25 | #2202⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72271289 | 2 |
| #72268240 | 2 |
| #72268229 | 3 |
| #72268063 | 4 |