Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
756 conjugacy classes of subgroups; only the 340 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 340 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×C22⋊C4 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C2×C3×H(5,𝔽2) | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×#41 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#568 | C2 | semidirect |
| 3. C3×#413 | C2 | semidirect |
| 4. C2×C3×#100 | C2 | semidirect |
| 5. #2888 | C3 | direct (N × H) |
| 13. C2×C3×M16 | C22 | semidirect |
| 14. C3×#101 | C22 | semidirect |
| 15. C3×#100 | C22 | semidirect |
| 17. #568 | C2×C3 | semidirect |
| 18. #413 | C2×C3 | semidirect |
| 19. C2×#100 | C2×C3 | semidirect |
| 44. C2×M16 | C22×C3 | semidirect |
| 45. #101 | C22×C3 | semidirect |
| 46. #100 | C22×C3 | semidirect |
| 118. C3 | #2888 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×H(5,𝔽2) | 2 |
| C3×#568 | 2 |
| C3×#413 | 2 |
| C2×C3×#100 | 2 |
| #2888 | 3 |