Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.
This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 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×#380 | C2 |
| prime radical | C24×D3 | C22 |
| hypercenter | #380 | D3 |
| hypocenter | C3 | #2848 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #380 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#924 | C2 | semidirect |
| 3. C2×#897 | C2 | semidirect |
| 4. C3×#380 | C2 | semidirect |
| 5. #4987 | C2 | semidirect |
| 7. #4950 | C2 | semidirect |
| 8. #4668 | C2 | semidirect |
| 17. C22×Dic6 | C22 | semidirect |
| 18. C2×C3×C4⋊C4 | C22 | semidirect |
| 23. C2×C4⋊Dic3 | C22 | semidirect |
| 24. C2×C4⋊Dic3 | C22 | semidirect |
| 26. C2×Dic3⋊C4 | C22 | semidirect |
| 28. C3×#95 | C22 | semidirect |
| 29. C3⋊#95 | C22 | semidirect |
| 36. #380 | D3 | semidirect |
| 43. C23×D3 | Q8 | semidirect |
| 85. C2×C4⋊C4 | C2×D3 | semidirect |
| 91. #95 | C2×D3 | semidirect |
| 167. C22×C3 | C22⋊Q8 | semidirect |
| 180. C2×D3 | C22⋊Q8 | semidirect |
| 243. C22 | #897 | semidirect |
| 255. C3 | #2848 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×D3 | 2 |
| C2×#924 | 2 |
| C2×#897 | 2 |
| C3×#380 | 2 |
| #4987 | 2 |
| #4955 | 2 |
| #4950 | 2 |
| #4668 | 2 |
| #2848 | 3 |