Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
520 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 520 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 | C22×C3×D4 | C22 |
| 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×#391 | C2 | semidirect |
| 3. C3×#388 | C2 | semidirect |
| 5. C3×#385 | C2 | semidirect |
| 6. C3×#382 | C2 | semidirect |
| 7. #2379 | C3 | direct (N × H) |
| 11. C3×#124 | C4 | semidirect |
| 24. #391 | C2×C3 | semidirect |
| 25. #388 | C2×C3 | semidirect |
| 27. #385 | C2×C3 | semidirect |
| 28. #382 | C2×C3 | semidirect |
| 32. C2×C3×Q8 | C2×C4 | semidirect |
| 48. C3×C22⋊C4 | C2×C4 | semidirect |
| 50. C3×C42 | C2×C4 | semidirect |
| 54. #124 | C3×C4 | semidirect |
| 93. C2×Q8 | C2×C3×C4 | semidirect |
| 109. C22⋊C4 | C2×C3×C4 | semidirect |
| 111. C42 | C2×C3×C4 | semidirect |
| 172. C3 | #2379 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#124 | 2 |
| C3×#391 | 2 |
| C3×#388 | 2 |
| C3×#386 | 2 |
| C3×#385 | 2 |
| C3×#382 | 2 |
| #2379 | 3 |