Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
644 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 209 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#384 | C2 |
| prime radical | C23×D3 | C23 |
| hypercenter | #384 | D3 |
| hypocenter | C3 | #2857 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#869 | C2 | semidirect |
| 4. C3×#384 | C2 | semidirect |
| 6. #4984 | C2 | semidirect |
| 7. #4973 | C2 | semidirect |
| 29. #384 | D3 | semidirect |
| 43. C2×C4×D3 | Q8 | semidirect |
| 129. C2×D3 | C4⋊Q8 | semidirect |
| 132. C3×C4 | C22⋊Q8 | semidirect |
| 183. C4 | #897 | semidirect |
| 184. C3 | #2857 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#897 | 2 |
| C2×C42×D3 | 2 |
| C2×#869 | 2 |
| C3×#384 | 2 |
| #4987 | 2 |
| #4984 | 2 |
| #4973 | 2 |
| #2857 | 3 |