Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
832 conjugacy classes of subgroups; only the 261 normal subgroups are shown.
This group has 832 conjugacy classes of subgroups — too many to draw. Only its 261 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, prime radical | C2×C3×#127 | C2 |
| socle, cosocle | C23×C3 | C24 |
| hypercenter | C2×#127 | D3 |
| hypocenter | C3 | #2366 |
| 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×D4×Dic3 | C2 | semidirect |
| 3. #5221 | C2 | semidirect |
| 4. #5220 | C2 | semidirect |
| 5. C4×C4⋊Dic3 | C2 | semidirect |
| 10. C3×#127 | C4 | semidirect |
| 29. C2×C3×D4 | C2×C4 | semidirect |
| 30. C2×C3×D4 | C2×C4 | semidirect |
| 37. C3×C42 | C2×C4 | semidirect |
| 41. C4⋊Dic3 | D4 | semidirect |
| 46. #127 | Dic3 | semidirect |
| 81. C2×D4 | C2×Dic3 | semidirect |
| 82. C2×D4 | C2×Dic3 | semidirect |
| 94. C42 | C2×Dic3 | semidirect |
| 106. C3×C4 | C4×D4 | semidirect |
| 149. C4 | D4×Dic3 | semidirect |
| 150. C3 | #2366 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#127 | 2 |
| C2×D4×Dic3 | 2 |
| #5221 | 2 |
| #5220 | 2 |
| C4×C4⋊Dic3 | 2 |
| #2366 | 3 |