Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
2744 conjugacy classes of subgroups; only the 538 normal subgroups are shown.
This group has 2744 conjugacy classes of subgroups — too many to draw. Only its 538 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 | C24×A4 |
| commutator, cosocle, hypocenter | Q8 | C24×C32 |
| Frattini | C2 | C24×C3×A4 |
| Fitting | C3×H(7,𝔽2) | C3 |
| hypercenter | C3×2−1+4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C3×2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#5945 | C2 | semidirect |
| 2. C2×C3×#994 | C2 | semidirect |
| 3. C3×H(7,𝔽2) | C3 | semidirect |
| 4. #87536 | C3 | direct (N × H) |
| 6. C3×2−1+4⋊C3 | C22 | semidirect |
| 7. C3×#994 | C22 | semidirect |
| 8. C2×C3×#41⋊C3 | C22 | semidirect |
| 11. C3×#585 | C2×C3 | semidirect |
| 13. C2×C3×H(5,𝔽2) | C2×C3 | semidirect |
| 14. #5945 | C2×C3 | semidirect |
| 15. C2×#994 | C2×C3 | semidirect |
| 18. C3×#41⋊C3 | C23 | semidirect |
| 19. C2×C3×SL(2,𝔽3) | C23 | semidirect |
| 20. H(7,𝔽2) | C32 | semidirect |
| 38. 2−1+4⋊C3 | C22×C3 | semidirect |
| 39. #994 | C22×C3 | semidirect |
| 40. C2×#41⋊C3 | C22×C3 | semidirect |
| 47. #585 | C2×C32 | semidirect |
| 49. C2×H(5,𝔽2) | C2×C32 | semidirect |
| 74. #41⋊C3 | C23×C3 | semidirect |
| 75. C2×SL(2,𝔽3) | C23×C3 | semidirect |
| 179. C3 | #87536 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#5945 | 2 |
| C2×C3×#994 | 2 |
| C3×H(7,𝔽2) | 3 |
| #87536 | 3 |
| C32×2−1+4 | 4 |