Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
464 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 98 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×C3 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C3×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#3237 | C2 | semidirect |
| 4. C3×#1995 | C2 | semidirect |
| 5. C3×#1506 | C2 | semidirect |
| 7. C3×#1307 | C2 | semidirect |
| 8. #10215 | C3 | direct (N × H) |
| 35. #3237 | C2×C3 | semidirect |
| 37. #1995 | C2×C3 | semidirect |
| 38. #1506 | C2×C3 | semidirect |
| 40. #1307 | C2×C3 | semidirect |
| 377. C3 | #10215 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3254 | 2 |
| C3×#3237 | 2 |
| C3×#3216 | 2 |
| C3×#1995 | 2 |
| C3×#1506 | 2 |
| C3×#1505 | 2 |
| C3×#1307 | 2 |
| #10215 | 3 |