Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1162 conjugacy classes of subgroups; only the 366 normal subgroups are shown.
This group has 1162 conjugacy classes of subgroups — too many to draw. Only its 366 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #120 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| socle | C22×C3 | C2×#120 |
| prime radical | C24×C3 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. #23564 | C3 | direct (N × H) |
| 29. C3×#117 | Q8 | semidirect |
| 81. #117 | C3×Q8 | semidirect |
| 176. C3 | #23564 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3458 | 2 |
| C3×#2296 | 2 |
| C3×#2062 | 2 |
| C3×#1740 | 2 |
| #23564 | 3 |