Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
456 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 213 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 | C22×C3 | C23×C4 |
| Fitting | C2×C3×#126 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C24×C3 | C23 |
| hypercenter | C2×#126 | D3 |
| hypocenter | C3 | #2383 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #4984 | C2 | semidirect |
| 3. #4980 | C2 | semidirect |
| 6. #4937 | C2 | semidirect |
| 7. #4935 | C2 | semidirect |
| 10. C3×#126 | C4 | semidirect |
| 38. C3×C4⋊C4 | C2×C4 | semidirect |
| 40. C3×C42 | C2×C4 | semidirect |
| 47. #126 | Dic3 | semidirect |
| 89. C4⋊C4 | C2×Dic3 | semidirect |
| 93. C42 | C2×Dic3 | semidirect |
| 140. C3 | #2383 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#126 | 2 |
| #4984 | 2 |
| #4980 | 2 |
| #4951 | 2 |
| #4950 | 2 |
| #4937 | 2 |
| #4935 | 2 |
| #2383 | 3 |