Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1241 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 1241 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C24×C3 |
| Frattini | C2 | C24×A4 |
| Fitting | C2×#585 | C3 |
| socle | C22 | C23×A4 |
| hypercenter | C2×#41 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5945 | C2 | direct (N × H) |
| 2. C2×2−1+4⋊C3 | C2 | semidirect |
| 3. C2×#994 | C2 | semidirect |
| 4. C22×#41⋊C3 | C2 | semidirect |
| 5. C2×#585 | C3 | semidirect |
| 7. 2−1+4⋊C3 | C22 | semidirect |
| 8. #994 | C22 | semidirect |
| 9. C2×#41⋊C3 | C22 | semidirect |
| 10. C2×#41⋊C3 | C22 | semidirect |
| 11. C2×#41⋊C3 | C22 | semidirect |
| 12. C22×SL(2,𝔽3) | C22 | semidirect |
| 14. #585 | C2×C3 | semidirect |
| 16. C2×2−1+4 | C2×C3 | semidirect |
| 18. C2×H(5,𝔽2) | C2×C3 | semidirect |
| 21. C22×#41 | C2×C3 | semidirect |
| 26. #41⋊C3 | C23 | semidirect |
| 27. #41⋊C3 | C23 | semidirect |
| 29. C2×SL(2,𝔽3) | C23 | semidirect |
| 33. 2−1+4 | C22×C3 | semidirect |
| 37. H(5,𝔽2) | C22×C3 | semidirect |
| 53. C2×#41 | C22×C3 | semidirect |
| 181. C2 | #5945 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5945 | 2 |
| C2×2−1+4⋊C3 | 2 |
| C2×#994 | 2 |
| C22×#41⋊C3 | 2 |
| C2×#585 | 3 |
| C2×C3×#41 | 4 |