Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
324 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 122 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 | #476 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C23×C3 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. #13458 | C3 | direct (N × H) |
| 176. C3 | #13458 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#499 | 2 |
| C2×C3×#365 | 2 |
| C3×#1945 | 2 |
| C3×#1386 | 2 |
| C3×#1376 | 2 |
| #13458 | 3 |