Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
2152 conjugacy classes of subgroups; only the 354 normal subgroups are shown.
This group has 2152 conjugacy classes of subgroups — too many to draw. Only its 354 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4 | A4 |
| commutator, hypocenter | C22 | C23×C3×C4 |
| Frattini | C2 | C24×A4 |
| Fitting | C25×C4 | C3 |
| socle | C26 | C2×C3 |
| prime radical | C24×A4 | C2 |
| cosocle | C23 | C24×C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C23×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C4×A4 | C2 | direct (N × H) |
| 3. C25×C4 | C3 | semidirect |
| 4. C23×A4 | C4 | direct (N × H) |
| 7. C2×C4×A4 | C22 | direct (N × H) |
| 12. C24×C4 | C2×C3 | semidirect |
| 14. C22×A4 | C2×C4 | direct (N × H) |
| 17. C4×A4 | C23 | direct (N × H) |
| 21. C25 | C3×C4 | semidirect |
| 30. C23×C4 | A4 | direct (N × H) |
| 31. C23×C4 | C22×C3 | semidirect |
| 34. C2×A4 | C22×C4 | direct (N × H) |
| 46. C24 | C2×C3×C4 | semidirect |
| 56. C22×C4 | C2×A4 | direct (N × H) |
| 57. C22×C4 | C23×C3 | semidirect |
| 60. A4 | C23×C4 | direct (N × H) |
| 73. C23 | C4×A4 | direct (N × H) |
| 74. C23 | C22×C3×C4 | semidirect |
| 82. C2×C4 | C22×A4 | direct (N × H) |
| 94. C22 | C2×C4×A4 | direct (N × H) |
| 95. C22 | C23×C3×C4 | semidirect |
| 98. C4 | C23×A4 | direct (N × H) |
| 103. C2 | C22×C4×A4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×A4 | 2 |
| C22×C4×A4 | 2 |
| C25×C4 | 3 |
| C23×C3×C4 | 4 |