Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
924 conjugacy classes of subgroups; only the 381 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 381 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D3 |
| commutator | C2×C3 | C22×C42 |
| Frattini | C23 | C23×D3 |
| Fitting | C3×C43 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C22×D12 | C22 |
| hypercenter | C43 | D3 |
| hypocenter | C3 | C42×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C42 ◁ C43 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C42×D3 | C2 | semidirect |
| 3. C3×C43 | C2 | semidirect |
| 4. C4×(C2×D3)⋊C4 | C2 | semidirect |
| 5. C4×C4⋊Dic3 | C2 | semidirect |
| 15. C4×D12 | C4 | direct (N × H) |
| 22. C43 | D3 | semidirect |
| 31. C2×C4×D3 | C2×C4 | semidirect |
| 33. C3×C42 | C2×C4 | semidirect |
| 34. C3×C42 | D4 | semidirect |
| 35. (C2×D3)⋊C4 | C2×C4 | semidirect |
| 36. C4⋊Dic3 | C2×C4 | semidirect |
| 62. D12 | C42 | direct (N × H) |
| 77. C42 | C4×D3 | semidirect |
| 78. C42 | D12 | direct (N × H) |
| 84. C2×D3 | C2×C42 | semidirect |
| 86. C3×C4 | C2×C42 | semidirect |
| 87. C3×C4 | C4×D4 | semidirect |
| 116. C4 | C42×D3 | semidirect |
| 117. C4 | C4×D12 | direct (N × H) |
| 118. C3 | C42×D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D12 | 2 |
| C2×C42×D3 | 2 |
| C3×C43 | 2 |
| C4×(C2×D3)⋊C4 | 2 |
| C4×C4⋊Dic3 | 2 |
| C42×D4 | 3 |