Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
556 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 54 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 | C3⋊S4 |
| commutator, cosocle, hypocenter | C22⋊H(3,𝔽3) | C22×C3 |
| Frattini | C3 | C2×C3×C3⋊S4 |
| Fitting | C23×C33 | D3 |
| socle | C23×C32 | C3⋊D3 |
| LCS | G ▷ C22⋊H(3,𝔽3) | |
| UCS | C1 ◁ C2×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×C22⋊H(3,𝔽3) | C2 | semidirect |
| 2. C3×#6258 | C2 | direct (N × H) |
| 5. C2×#6258 | C3 | direct (N × H) |
| 7. C3×C22⋊H(3,𝔽3) | C22 | semidirect |
| 8. C23×C33 | D3 | semidirect |
| 10. C2×C32×A4 | D3 | semidirect |
| 17. C2×C22⋊H(3,𝔽3) | C2×C3 | semidirect |
| 18. #6258 | C2×C3 | direct (N × H) |
| 28. C22×C33 | C2×D3 | semidirect |
| 33. C32×A4 | C2×D3 | semidirect |
| 38. C22⋊H(3,𝔽3) | C22×C3 | semidirect |
| 43. C23×C32 | C3×D3 | semidirect |
| 49. C2×C3×A4 | C3×D3 | semidirect |
| 64. C2×C33 | S4 | semidirect |
| 90. C22×C32 | C2×C3×D3 | semidirect |
| 102. C3×A4 | C2×C3×D3 | semidirect |
| 107. C33 | C2×S4 | semidirect |
| 115. C23×C3 | #272 | semidirect |
| 146. C2×C32 | C3×S4 | semidirect |
| 171. C22×C3 | C2×#272 | semidirect |
| 187. C32 | C2×C3×S4 | semidirect |
| 190. C23 | C3×#272 | semidirect |
| 210. C2×C3 | #6258 | direct (N × H) |
| 219. C22 | C2×C3×#272 | semidirect |
| 225. C3 | C2×#6258 | direct (N × H) |
| 230. C2 | C3×#6258 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C22⋊H(3,𝔽3) | 2 |
| C3×#6258 | 2 |
| C2×C32×S4 | 3 |
| C2×C32×#67 | 3 |
| C2×#6258 | 3 |
| C2×C3×#272 | 4 |