Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
404 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | #2571 |
| prime radical | C23×C3 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. #29231 | C3 | direct (N × H) |
| 137. C3 | #29231 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#2102 | 2 |
| C3×#1953 | 2 |
| C3×#1849 | 2 |
| C3×#1833 | 2 |
| C3×#1773 | 2 |
| #29231 | 3 |