Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
564 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 114 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×#3205 | C2 |
| 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 |
|---|---|---|
| 1. C3×#3208 | C2 | semidirect |
| 3. C3×#3135 | C2 | semidirect |
| 4. C3×#1961 | C2 | semidirect |
| 5. C3×#1518 | C2 | semidirect |
| 6. C3×#1517 | C2 | semidirect |
| 7. C3×#1281 | C2 | semidirect |
| 8. #10219 | C3 | direct (N × H) |
| 32. #3208 | C2×C3 | semidirect |
| 34. #3135 | C2×C3 | semidirect |
| 35. #1961 | C2×C3 | semidirect |
| 36. #1518 | C2×C3 | semidirect |
| 37. #1517 | C2×C3 | semidirect |
| 38. #1281 | C2×C3 | semidirect |
| 385. C3 | #10219 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3208 | 2 |
| C3×#3205 | 2 |
| C3×#3135 | 2 |
| C3×#1961 | 2 |
| C3×#1518 | 2 |
| C3×#1517 | 2 |
| C3×#1281 | 2 |
| #10219 | 3 |