Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
456 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 264 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 | C22 | C23×C3×C4 |
| Frattini, cosocle | C23 | C24×C3 |
| prime radical | C24×C3 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#389 | C2 | semidirect |
| 4. C3×#387 | C2 | semidirect |
| 6. C3×#384 | C2 | semidirect |
| 7. C3×#383 | C2 | semidirect |
| 8. #2383 | C3 | direct (N × H) |
| 10. C3×#126 | C4 | semidirect |
| 24. #389 | C2×C3 | semidirect |
| 26. #387 | C2×C3 | semidirect |
| 28. #384 | C2×C3 | semidirect |
| 29. #383 | C2×C3 | semidirect |
| 41. C3×C4⋊C4 | C2×C4 | semidirect |
| 45. C3×C42 | C2×C4 | semidirect |
| 47. #126 | C3×C4 | semidirect |
| 89. C4⋊C4 | C2×C3×C4 | semidirect |
| 93. C42 | C2×C3×C4 | semidirect |
| 140. C3 | #2383 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#126 | 2 |
| C3×#389 | 2 |
| C3×#388 | 2 |
| C3×#387 | 2 |
| C3×#385 | 2 |
| C3×#384 | 2 |
| C3×#383 | 2 |
| #2383 | 3 |