Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
216 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C16 | D3 |
| commutator, hypocenter | C3 | C2×C4×C16 |
| Frattini | C2×C8 | C22×D3 |
| Fitting | C3×C4×C16 | C2 |
| socle | C22×C3 | C22×C8 |
| prime radical | C22×D3 | C2×C8 |
| cosocle | C2×C3×C8 | C23 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C4×C16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×C4×C16 | C2 | semidirect |
| 4. Dic3×C16 | C2 | semidirect |
| 5. C4×C3⋊C16 | C2 | semidirect |
| 16. D3×C16 | C4 | direct (N × H) |
| 21. C4×C16 | D3 | direct (N × H) |
| 33. C3×C16 | C2×C4 | semidirect |
| 34. C3⋊C16 | C2×C4 | semidirect |
| 50. C4×D3 | C16 | direct (N × H) |
| 69. C16 | C4×D3 | direct (N × H) |
| 73. C3×C4 | C2×C16 | semidirect |
| 77. Dic3 | C2×C16 | semidirect |
| 92. D3 | C4×C16 | direct (N × H) |
| 98. C4 | D3×C16 | direct (N × H) |
| 101. C3 | C2×C4×C16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D3×C16 | 2 |
| C4×D3×C8 | 2 |
| C3×C4×C16 | 2 |
| Dic3×C16 | 2 |
| C4×C3⋊C16 | 2 |
| C2×C4×C16 | 3 |