Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
154 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 154 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, socle | C2×C3 | #442 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C3×#509 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#444 | C2 | semidirect |
| 3. C3×#442 | C2 | semidirect |
| 4. C3×#365 | C2 | semidirect |
| 5. C3×#361 | C2 | semidirect |
| 6. C3×#360 | C2 | semidirect |
| 7. C3×#347 | C2 | semidirect |
| 8. AGL(1,ℤ/16) | C3 | direct (N × H) |
| 13. C3×SD32 | C4 | semidirect |
| 14. C3×D16 | C4 | semidirect |
| 23. #444 | C2×C3 | semidirect |
| 24. #442 | C2×C3 | semidirect |
| 25. #365 | C2×C3 | semidirect |
| 26. #361 | C2×C3 | semidirect |
| 27. #360 | C2×C3 | semidirect |
| 28. #347 | C2×C3 | semidirect |
| 32. C3×Q16 | C2×C4 | semidirect |
| 36. C3×D8 | C2×C4 | semidirect |
| 46. C3×C16 | C2×C4 | semidirect |
| 51. SD32 | C3×C4 | semidirect |
| 52. D16 | C3×C4 | semidirect |
| 78. Q16 | C2×C3×C4 | semidirect |
| 82. D8 | C2×C3×C4 | semidirect |
| 92. C16 | C2×C3×C4 | semidirect |
| 132. C3 | AGL(1,ℤ/16) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#509 | 2 |
| C3×#444 | 2 |
| C3×#442 | 2 |
| C3×#365 | 2 |
| C3×#361 | 2 |
| C3×#360 | 2 |
| C3×#347 | 2 |
| AGL(1,ℤ/16) | 3 |