Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
708 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 708 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| prime radical | C2×C3×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#123 | C2 | semidirect |
| 3. C22×C3×C4⋊C4 | C2 | semidirect |
| 4. C3×#399 | C2 | semidirect |
| 5. C3×#390 | C2 | semidirect |
| 6. C3×#384 | C2 | semidirect |
| 7. C3×C4×C22⋊C4 | C2 | semidirect |
| 8. #2501 | C3 | direct (N × H) |
| 25. C2×#123 | C2×C3 | semidirect |
| 27. C22×C4⋊C4 | C2×C3 | semidirect |
| 28. #399 | C2×C3 | semidirect |
| 29. #390 | C2×C3 | semidirect |
| 30. #384 | C2×C3 | semidirect |
| 31. C4×C22⋊C4 | C2×C3 | semidirect |
| 53. C3×C22⋊C4 | D4 | semidirect |
| 116. C22⋊C4 | C3×D4 | semidirect |
| 134. C3×C4 | #123 | semidirect |
| 181. C4 | C3×#123 | semidirect |
| 182. C3 | #2501 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#123 | 2 |
| C2×C3×#121 | 2 |
| C22×C3×C4⋊C4 | 2 |
| C3×#399 | 2 |
| C3×#390 | 2 |
| C3×#384 | 2 |
| C3×C4×C22⋊C4 | 2 |
| #2501 | 3 |