Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
190 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 72 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 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C2×C3 | #397 |
| prime radical | C2×C3×#41 | C22 |
| 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 |
|---|---|---|
| 1. C3×#424 | C2 | semidirect |
| 5. C3×#344 | C2 | semidirect |
| 6. C3×#341 | C2 | semidirect |
| 7. C3×H(3,ℤ/4) | C2 | semidirect |
| 8. #2080 | C3 | direct (N × H) |
| 22. #424 | C2×C3 | semidirect |
| 26. #344 | C2×C3 | semidirect |
| 27. #341 | C2×C3 | semidirect |
| 28. H(3,ℤ/4) | C2×C3 | semidirect |
| 173. C3 | #2080 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#424 | 2 |
| C3×#421 | 2 |
| C3×#416 | 2 |
| C3×#413 | 2 |
| C3×#344 | 2 |
| C3×#341 | 2 |
| C3×H(3,ℤ/4) | 2 |
| #2080 | 3 |