Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
286 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 78 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 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23×C3 |
| prime radical | C3×#3355 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#436 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#2255 | C2 | semidirect |
| 3. C3×#2224 | C2 | semidirect |
| 4. C3×#1695 | C2 | semidirect |
| 5. C3×#1363 | C2 | semidirect |
| 6. C3×#1333 | C2 | semidirect |
| 7. C3×#1330 | C2 | semidirect |
| 8. #12400 | C3 | direct (N × H) |
| 29. #2255 | C2×C3 | semidirect |
| 30. #2224 | C2×C3 | semidirect |
| 31. #1695 | C2×C3 | semidirect |
| 32. #1363 | C2×C3 | semidirect |
| 33. #1333 | C2×C3 | semidirect |
| 34. #1330 | C2×C3 | semidirect |
| 275. C3 | #12400 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3355 | 2 |
| C3×#2255 | 2 |
| C3×#2224 | 2 |
| C3×#1695 | 2 |
| C3×#1363 | 2 |
| C3×#1333 | 2 |
| C3×#1330 | 2 |
| #12400 | 3 |