Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
368 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 88 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 | #99 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×D4 | C23×C3 |
| prime radical | C2×C3×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×D4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×#99 | C2 | semidirect |
| 3. C3×#353 | C2 | direct (N × H) |
| 4. C2×#353 | C3 | direct (N × H) |
| 6. C22×C3×D4 | C4 | semidirect |
| 9. C3×#127 | C4 | semidirect |
| 12. C2×C3×C42 | C4 | semidirect |
| 14. C3×#99 | C22 | semidirect |
| 16. C2×#99 | C2×C3 | semidirect |
| 17. #353 | C2×C3 | direct (N × H) |
| 29. C2×C3×D4 | C2×C4 | semidirect |
| 38. C3×C42 | C2×C4 | semidirect |
| 40. C22×D4 | C3×C4 | semidirect |
| 43. #127 | C3×C4 | semidirect |
| 46. C2×C42 | C3×C4 | semidirect |
| 48. #99 | C22×C3 | semidirect |
| 85. C2×D4 | C2×C3×C4 | semidirect |
| 94. C42 | C2×C3×C4 | semidirect |
| 143. C2×C3 | #353 | direct (N × H) |
| 163. C3 | C2×#353 | direct (N × H) |
| 169. C2 | C3×#353 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#127 | 2 |
| C2×C3×#99 | 2 |
| C3×#353 | 2 |
| C2×#353 | 3 |