Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
406 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | #124 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C2×C3 | #1955 |
| prime radical | C3×#575 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C22×C3×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#2151 | C2 | semidirect |
| 3. C2×C3×#350 | C2 | semidirect |
| 4. C3×#2115 | C2 | semidirect |
| 5. C3×#2110 | C2 | semidirect |
| 6. #13199 | C3 | direct (N × H) |
| 22. #2151 | C2×C3 | semidirect |
| 23. C2×#350 | C2×C3 | semidirect |
| 24. #2115 | C2×C3 | semidirect |
| 25. #2110 | C2×C3 | semidirect |
| 218. C3 | #13199 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#2955 | 2 |
| C3×#2151 | 2 |
| C2×C3×#350 | 2 |
| C3×#2115 | 2 |
| C3×#2110 | 2 |
| #13199 | 3 |