Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
174 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | D8 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C22×C3 | C2×D8 |
| prime radical | C2×C3×D16 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×C42 ◁ C3×C4×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C4×D8 | C2 | semidirect |
| 3. C3×#366 | C2 | semidirect |
| 4. C3×#357 | C2 | semidirect |
| 5. C3×C4×C16 | C2 | semidirect |
| 6. C4×D16 | C3 | direct (N × H) |
| 9. C3×D16 | C4 | direct (N × H) |
| 16. C4×D8 | C2×C3 | semidirect |
| 17. #366 | C2×C3 | semidirect |
| 18. #357 | C2×C3 | semidirect |
| 19. C4×C16 | C2×C3 | semidirect |
| 23. C3×D8 | C2×C4 | semidirect |
| 30. C3×C16 | C2×C4 | semidirect |
| 33. D16 | C3×C4 | direct (N × H) |
| 53. D8 | C2×C3×C4 | semidirect |
| 60. C16 | C2×C3×C4 | semidirect |
| 67. C3×C4 | D16 | direct (N × H) |
| 91. C4 | C3×D16 | direct (N × H) |
| 94. C3 | C4×D16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×D16 | 2 |
| C3×C4×D8 | 2 |
| C3×#366 | 2 |
| C3×#357 | 2 |
| C3×C4×C16 | 2 |
| C4×D16 | 3 |