Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
366 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 110 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 | #329 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | #97 | C23×C3 |
| prime radical | C2×C3×#127 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×#347 | C2 | semidirect |
| 3. C3×#1355 | C2 | direct (N × H) |
| 4. C2×#1355 | C3 | direct (N × H) |
| 14. C3×#347 | C22 | semidirect |
| 16. C2×#347 | C2×C3 | semidirect |
| 17. #1355 | C2×C3 | direct (N × H) |
| 47. #347 | C22×C3 | semidirect |
| 170. C2×C3 | #1355 | direct (N × H) |
| 185. C3 | C2×#1355 | direct (N × H) |
| 189. C2 | C3×#1355 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#490 | 2 |
| C2×C3×#347 | 2 |
| C3×#1355 | 2 |
| C2×#1355 | 3 |