Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1240 conjugacy classes of subgroups; only the 501 normal subgroups are shown.
This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 501 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting, prime radical | C2×C3×#127 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C2×#127 | D3 |
| hypocenter | C3 | C2×#559 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#936 | C2 | semidirect |
| 3. C2×#935 | C2 | semidirect |
| 4. C2×D4×Dic3 | C2 | semidirect |
| 5. #5713 | C2 | direct (N × H) |
| 6. C2×C4×Dic6 | C2 | semidirect |
| 16. #936 | C22 | semidirect |
| 17. #935 | C22 | semidirect |
| 18. D4×Dic3 | C22 | semidirect |
| 22. C4×Dic6 | C22 | semidirect |
| 125. C2×C3 | #559 | semidirect |
| 138. C3 | C2×#559 | semidirect |
| 140. C2 | #5713 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#127 | 2 |
| C2×#936 | 2 |
| C2×#935 | 2 |
| C2×D4×Dic3 | 2 |
| #5713 | 2 |
| C2×C4×Dic6 | 2 |
| C2×#559 | 3 |