Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
332 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 108 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 | #121 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C2×C3 | #386 |
| prime radical | C3×#575 | C2 |
| LCS | G ▷ C2×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 |
|---|---|---|
| 2. C3×C4×#41 | C2 | semidirect |
| 3. C3×#421 | C2 | semidirect |
| 4. C2×C3×#104 | C2 | semidirect |
| 5. C3×#410 | C2 | semidirect |
| 6. C3×#344 | C2 | semidirect |
| 7. C3×#339 | C2 | semidirect |
| 8. #1906 | C3 | direct (N × H) |
| 12. C3×#137 | C4 | semidirect |
| 14. C3×AGL(1,ℤ/8) | C4 | semidirect |
| 31. C4×#41 | C2×C3 | semidirect |
| 32. #421 | C2×C3 | semidirect |
| 33. C2×#104 | C2×C3 | semidirect |
| 34. #410 | C2×C3 | semidirect |
| 35. #344 | C2×C3 | semidirect |
| 36. #339 | C2×C3 | semidirect |
| 45. C2×C3×Q8 | C2×C4 | semidirect |
| 49. C2×C3×D4 | C2×C4 | semidirect |
| 63. C3×M16 | C2×C4 | semidirect |
| 81. #137 | C3×C4 | semidirect |
| 83. AGL(1,ℤ/8) | C3×C4 | semidirect |
| 142. C2×Q8 | C2×C3×C4 | semidirect |
| 146. C2×D4 | C2×C3×C4 | semidirect |
| 160. M16 | C2×C3×C4 | semidirect |
| 252. C3 | #1906 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#575 | 2 |
| C3×C4×#41 | 2 |
| C3×#421 | 2 |
| C2×C3×#104 | 2 |
| C3×#410 | 2 |
| C3×#344 | 2 |
| C3×#339 | 2 |
| #1906 | 3 |