Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
516 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 296 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C23 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C23 | C24×C3 |
| socle | C23×C3 | C24 |
| prime radical | C24×C3 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×#387 | C2 | semidirect |
| 4. C3×#383 | C2 | semidirect |
| 5. C3×C4×C4⋊C4 | C2 | semidirect |
| 7. C3×C43 | C2 | semidirect |
| 8. C4×#126 | C3 | direct (N × H) |
| 10. C3×#126 | C4 | direct (N × H) |
| 25. #387 | C2×C3 | semidirect |
| 26. #383 | C2×C3 | semidirect |
| 27. C4×C4⋊C4 | C2×C3 | semidirect |
| 29. C43 | C2×C3 | semidirect |
| 39. C3×C4⋊C4 | C2×C4 | semidirect |
| 45. C3×C42 | C2×C4 | semidirect |
| 47. #126 | C3×C4 | direct (N × H) |
| 88. C4⋊C4 | C2×C3×C4 | semidirect |
| 94. C42 | C2×C3×C4 | semidirect |
| 106. C3×C4 | #126 | direct (N × H) |
| 141. C4 | C3×#126 | direct (N × H) |
| 142. C3 | C4×#126 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#126 | 2 |
| C3×#388 | 2 |
| C3×#387 | 2 |
| C3×#383 | 2 |
| C3×C4×C4⋊C4 | 2 |
| C3×C4×C22⋊C4 | 2 |
| C3×C43 | 2 |
| C4×#126 | 3 |