Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
412 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #127 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | #390 |
| prime radical | C3×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#1837 | C2 | semidirect |
| 3. C3×#1493 | C2 | semidirect |
| 5. C2×C3×#322 | C2 | semidirect |
| 6. #9863 | C3 | direct (N × H) |
| 21. #1837 | C2×C3 | semidirect |
| 22. #1493 | C2×C3 | semidirect |
| 24. C2×#322 | C2×C3 | semidirect |
| 198. C3 | #9863 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3131 | 2 |
| C3×#1837 | 2 |
| C3×#1493 | 2 |
| C3×#1492 | 2 |
| C2×C3×#322 | 2 |
| #9863 | 3 |