Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
14334 conjugacy classes of subgroups; only the 4427 normal subgroups are shown.
This group has 14334 conjugacy classes of subgroups — too many to draw. Only its 4427 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C25 | C22×D3 |
| commutator, cosocle | C22×C3 | C26 |
| Frattini | C22 | C25×D3 |
| Fitting, prime radical | C24×C3×D4 | C2 |
| socle | C25×C3 | C23 |
| hypercenter | C24×D4 | D3 |
| hypocenter | C3 | C23×#124 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C25 ◁ C24×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C24×Dic6 | C2 | semidirect |
| 3. C23×#217 | C2 | semidirect |
| 4. C22×#936 | C2 | direct (N × H) |
| 5. C23×C4×Dic3 | C2 | semidirect |
| 14. C23×Dic6 | C22 | semidirect |
| 15. C22×#217 | C22 | semidirect |
| 16. C2×#936 | C22 | direct (N × H) |
| 17. C22×C4×Dic3 | C22 | semidirect |
| 36. C22×Dic6 | C23 | semidirect |
| 37. C2×#217 | C23 | semidirect |
| 38. #936 | C23 | direct (N × H) |
| 39. C2×C4×Dic3 | C23 | semidirect |
| 67. C2×Dic6 | C24 | semidirect |
| 68. #217 | C24 | semidirect |
| 69. C4×Dic3 | C24 | semidirect |
| 90. C23×C3 | #124 | semidirect |
| 125. C22×C3 | C2×#124 | semidirect |
| 135. C23 | #936 | direct (N × H) |
| 149. C2×C3 | C22×#124 | semidirect |
| 155. C22 | C2×#936 | direct (N × H) |
| 162. C3 | C23×#124 | semidirect |
| 164. C2 | C22×#936 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×D4 | 2 |
| C24×Dic6 | 2 |
| C23×#217 | 2 |
| C22×#936 | 2 |
| C23×C4×Dic3 | 2 |
| C23×#124 | 3 |