Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1880 conjugacy classes of subgroups; only the 645 normal subgroups are shown.
This group has 1880 conjugacy classes of subgroups — too many to draw. Only its 645 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting, prime radical | C23×C3×D4 | C2 |
| socle | C24×C3 | C23 |
| hypercenter | C23×D4 | D3 |
| hypocenter | C3 | C22×#124 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C23×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×Dic6 | C2 | semidirect |
| 3. C22×#217 | C2 | semidirect |
| 4. C2×#936 | C2 | direct (N × H) |
| 5. C22×C4×Dic3 | C2 | semidirect |
| 14. C22×Dic6 | C22 | semidirect |
| 15. C2×#217 | C22 | semidirect |
| 16. #936 | C22 | direct (N × H) |
| 17. C2×C4×Dic3 | C22 | semidirect |
| 36. C2×Dic6 | C23 | semidirect |
| 37. #217 | C23 | semidirect |
| 38. C4×Dic3 | C23 | semidirect |
| 84. C22×C3 | #124 | semidirect |
| 107. C2×C3 | C2×#124 | semidirect |
| 113. C22 | #936 | direct (N × H) |
| 120. C3 | C22×#124 | semidirect |
| 122. C2 | C2×#936 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×D4 | 2 |
| C23×Dic6 | 2 |
| C22×#217 | 2 |
| C2×#936 | 2 |
| C22×C4×Dic3 | 2 |
| C22×#124 | 3 |