Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
260 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 116 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 | C2×D4 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C22×C3 | C2×C22⋊C4 |
| prime radical | C23×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. #1986 | C3 | direct (N × H) |
| 7. C3×C4⋊Q8 | C4 | semidirect |
| 8. C3×C4⋊Q8 | C4 | semidirect |
| 9. C3×#125 | C4 | semidirect |
| 32. C3×M16 | Q8 | semidirect |
| 49. C4⋊Q8 | C3×C4 | semidirect |
| 50. C4⋊Q8 | C3×C4 | semidirect |
| 51. #125 | C3×C4 | semidirect |
| 89. M16 | C3×Q8 | semidirect |
| 157. C3 | #1986 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#533 | 2 |
| C3×C4×M16 | 2 |
| C3×C4×C4⋊C4 | 2 |
| C3×#339 | 2 |
| #1986 | 3 |