Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
502 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32×C4 | A4 |
| commutator, cosocle, hypocenter | Q8 | C22×C33 |
| Frattini | C2 | C22×C32×A4 |
| Fitting | C2×C32×#41 | C3 |
| socle | C22×C32 | C2×A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C32×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#41⋊C3 | C2 | direct (N × H) |
| 2. C2×C32×SL(2,𝔽3) | C2 | semidirect |
| 3. C2×C32×#41 | C3 | semidirect |
| 4. C2×C3×#41⋊C3 | C3 | direct (N × H) |
| 6. C32×SL(2,𝔽3) | C22 | semidirect |
| 8. C32×#41 | C2×C3 | semidirect |
| 12. C3×#41⋊C3 | C2×C3 | direct (N × H) |
| 13. C2×C3×SL(2,𝔽3) | C2×C3 | semidirect |
| 16. C2×C3×#41 | C32 | semidirect |
| 17. C2×#41⋊C3 | C32 | direct (N × H) |
| 28. C3×SL(2,𝔽3) | C22×C3 | semidirect |
| 32. C3×#41 | C2×C32 | semidirect |
| 36. #41⋊C3 | C2×C32 | direct (N × H) |
| 37. C2×SL(2,𝔽3) | C2×C32 | semidirect |
| 46. C2×#41 | C33 | semidirect |
| 58. SL(2,𝔽3) | C22×C32 | semidirect |
| 62. C2×C32 | #41⋊C3 | direct (N × H) |
| 65. #41 | C2×C33 | semidirect |
| 78. C32 | C2×#41⋊C3 | direct (N × H) |
| 91. C2×C3 | C3×#41⋊C3 | direct (N × H) |
| 100. C3 | C2×C3×#41⋊C3 | direct (N × H) |
| 102. C2 | C32×#41⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#41⋊C3 | 2 |
| C2×C32×SL(2,𝔽3) | 2 |
| C2×C32×#41 | 3 |
| C2×C3×#41⋊C3 | 3 |
| C2×C33×C4 | 4 |