Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
728 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 728 conjugacy classes of subgroups — too many to draw. Only its 238 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#376 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#2276 | C2 | semidirect |
| 4. C3×#1920 | C2 | semidirect |
| 6. C3×#1865 | C2 | semidirect |
| 7. C3×#1840 | C2 | semidirect |
| 8. #33089 | C3 | direct (N × H) |
| 27. #2276 | C2×C3 | semidirect |
| 29. #1920 | C2×C3 | semidirect |
| 31. #1865 | C2×C3 | semidirect |
| 32. #1840 | C2×C3 | semidirect |
| 232. C3 | #33089 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#2644 | 2 |
| C3×#2276 | 2 |
| C3×#1926 | 2 |
| C3×#1920 | 2 |
| C3×#1880 | 2 |
| C3×#1865 | 2 |
| C3×#1840 | 2 |
| #33089 | 3 |