Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
262 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 100 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 | #117 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C2×C3 | #381 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C3×#117 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C3×#339 | C2 | semidirect |
| 5. C3×H(3,ℤ/4) | C2 | semidirect |
| 6. #1827 | C3 | direct (N × H) |
| 12. C3×#117 | C4 | semidirect |
| 15. C2×C3×C42 | C4 | semidirect |
| 23. #339 | C2×C3 | semidirect |
| 24. H(3,ℤ/4) | C2×C3 | semidirect |
| 54. #117 | C3×C4 | semidirect |
| 57. C2×C42 | C3×C4 | semidirect |
| 154. C3 | #1827 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C4×#41 | 2 |
| C3×#413 | 2 |
| C3×#410 | 2 |
| C3×#339 | 2 |
| C3×H(3,ℤ/4) | 2 |
| #1827 | 3 |