Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
360 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C42 | D4 |
| commutator | C4 | C2×C3×C42 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C22×C3 | C2×C22⋊C4 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C42 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×C4×M16 | C2 | semidirect |
| 4. C3×C43 | C2 | semidirect |
| 5. C3×#339 | C2 | semidirect |
| 6. C4×#104 | C3 | direct (N × H) |
| 9. C3×C4×Q8 | C4 | semidirect |
| 11. C3×C4×D4 | C4 | semidirect |
| 18. C3×#104 | C4 | direct (N × H) |
| 23. C4×M16 | C2×C3 | semidirect |
| 24. C43 | C2×C3 | semidirect |
| 25. #339 | C2×C3 | semidirect |
| 37. C3×M16 | C2×C4 | semidirect |
| 48. C3×C42 | C2×C4 | semidirect |
| 53. C4×Q8 | C3×C4 | semidirect |
| 55. C4×D4 | C3×C4 | semidirect |
| 62. #104 | C3×C4 | direct (N × H) |
| 68. C3×Q8 | C42 | semidirect |
| 71. C3×D4 | C42 | semidirect |
| 97. M16 | C2×C3×C4 | semidirect |
| 108. C42 | C2×C3×C4 | semidirect |
| 120. C3×C4 | #104 | direct (N × H) |
| 127. Q8 | C3×C42 | semidirect |
| 130. D4 | C3×C42 | semidirect |
| 158. C4 | C3×#104 | direct (N × H) |
| 162. C3 | C4×#104 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C4×#41 | 2 |
| C2×C3×#104 | 2 |
| C3×C4×M16 | 2 |
| C3×C43 | 2 |
| C3×#339 | 2 |
| C4×#104 | 3 |