Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1808 conjugacy classes of subgroups; only the 562 normal subgroups are shown.
This group has 1808 conjugacy classes of subgroups — too many to draw. Only its 562 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C23×C33 |
| Frattini | C2 | C23×C32×A4 |
| Fitting | C2×C32×H(5,𝔽2) | C3 |
| hypercenter | C2×C32×Q8 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C32 ◁ C2×C32×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#994 | C2 | direct (N × H) |
| 2. C2×C32×#41⋊C3 | C2 | semidirect |
| 3. C2×C32×H(5,𝔽2) | C3 | semidirect |
| 4. C2×C3×#994 | C3 | direct (N × H) |
| 6. C32×#41⋊C3 | C22 | semidirect |
| 7. C2×C32×SL(2,𝔽3) | C22 | semidirect |
| 9. C32×H(5,𝔽2) | C2×C3 | semidirect |
| 13. C3×#994 | C2×C3 | direct (N × H) |
| 14. C2×C3×#41⋊C3 | C2×C3 | semidirect |
| 17. C32×SL(2,𝔽3) | C23 | semidirect |
| 18. C2×C3×H(5,𝔽2) | C32 | semidirect |
| 19. C2×#994 | C32 | direct (N × H) |
| 33. C3×#41⋊C3 | C22×C3 | semidirect |
| 34. C2×C3×SL(2,𝔽3) | C22×C3 | semidirect |
| 38. C3×H(5,𝔽2) | C2×C32 | semidirect |
| 42. #994 | C2×C32 | direct (N × H) |
| 43. C2×#41⋊C3 | C2×C32 | semidirect |
| 59. C3×SL(2,𝔽3) | C23×C3 | semidirect |
| 60. C2×H(5,𝔽2) | C33 | semidirect |
| 76. #41⋊C3 | C22×C32 | semidirect |
| 77. C2×SL(2,𝔽3) | C22×C32 | semidirect |
| 88. H(5,𝔽2) | C2×C33 | semidirect |
| 108. SL(2,𝔽3) | C23×C32 | semidirect |
| 112. C2×C32 | #994 | direct (N × H) |
| 136. C32 | C2×#994 | direct (N × H) |
| 153. C2×C3 | C3×#994 | direct (N × H) |
| 163. C3 | C2×C3×#994 | direct (N × H) |
| 165. C2 | C32×#994 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#994 | 2 |
| C2×C32×#41⋊C3 | 2 |
| C2×C32×H(5,𝔽2) | 3 |
| C2×C3×#994 | 3 |
| C2×C33×Q8 | 4 |