Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1284 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 1284 conjugacy classes of subgroups — too many to draw. Only its 242 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 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C3×#3459 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#2409 | C2 | semidirect |
| 3. C3×#2062 | C2 | semidirect |
| 4. C3×#2044 | C2 | semidirect |
| 5. C3×#1799 | C2 | semidirect |
| 6. #29709 | C3 | direct (N × H) |
| 25. #2409 | C2×C3 | semidirect |
| 26. #2062 | C2×C3 | semidirect |
| 27. #2044 | C2×C3 | semidirect |
| 28. #1799 | C2×C3 | semidirect |
| 328. C3 | #29709 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3459 | 2 |
| C3×#2409 | 2 |
| C3×#2062 | 2 |
| C3×#2044 | 2 |
| C3×#1799 | 2 |
| #29709 | 3 |