Finite Group DatabasebetaFinite Group DatabasebetaOrder 12288.
At least 12343 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has at least 12343 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #72260581 |
| commutator, cosocle, hypocenter | #72260911 | C22 |
| Frattini | C24 | C2×#87588 |
| Fitting | #72260850 | D3 |
| socle | C23 | C2×#11715850 |
| hypercenter | C22 | #72260462 |
| LCS | G ▷ #72260911 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72260977 | C2 | semidirect |
| 2. #72260916 | C2 | semidirect |
| 3. #72260911 | C22 | semidirect |
| 4. #72260850 | D3 | semidirect |
| 6. C2×#35924 | S4 | semidirect |
| 7. C23×#392 | S4 | semidirect |
| 8. #4131228 | S4 | semidirect |
| 9. #2032498 | S4 | semidirect |
| 10. #609922 | S4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72260977 | 2 |
| #72260916 | 2 |
| #72260882 | 3 |
| #72260680 | 4 |
| #72260674 | 4 |
| #72260665 | 4 |
| #72242868 | 4 |
| #72234221 | 4 |