Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
160 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×D8 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C2×C3×D8 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C4×Q16 | C2 | semidirect |
| 4. C3×#366 | C2 | semidirect |
| 5. C3×#358 | C2 | semidirect |
| 7. C3×#346 | C2 | semidirect |
| 8. #2175 | C3 | direct (N × H) |
| 13. C3×SD32 | C4 | semidirect |
| 22. C4×Q16 | C2×C3 | semidirect |
| 24. #366 | C2×C3 | semidirect |
| 25. #358 | C2×C3 | semidirect |
| 27. #346 | C2×C3 | semidirect |
| 32. C3×Q16 | C2×C4 | semidirect |
| 44. C3×C16 | C2×C4 | semidirect |
| 49. SD32 | C3×C4 | semidirect |
| 76. Q16 | C2×C3×C4 | semidirect |
| 88. C16 | C2×C3×C4 | semidirect |
| 130. C3 | #2175 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×SD32 | 2 |
| C3×C4×Q16 | 2 |
| C3×C4×D8 | 2 |
| C3×#366 | 2 |
| C3×#358 | 2 |
| C3×#357 | 2 |
| C3×#346 | 2 |
| #2175 | 3 |