Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
928 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 928 conjugacy classes of subgroups — too many to draw. Only its 162 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×C32 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23×C32 |
| Frattini | C2 | C22×C32×S4 |
| Fitting | C32×H(5,𝔽2) | D3 |
| hypercenter | C32×Q8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C32 ◁ C32×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#994 | C2 | semidirect |
| 2. C32×#985 | C2 | semidirect |
| 3. C32×#984 | C2 | semidirect |
| 4. C3×#5924 | C3 | direct (N × H) |
| 8. C32×GL(2,𝔽3) | C22 | semidirect |
| 9. C32×#226 | C22 | semidirect |
| 11. C32×H(5,𝔽2) | D3 | semidirect |
| 13. C3×#994 | C2×C3 | semidirect |
| 14. C3×#985 | C2×C3 | semidirect |
| 15. C3×#984 | C2×C3 | semidirect |
| 25. #5924 | C32 | direct (N × H) |
| 37. C3×GL(2,𝔽3) | C22×C3 | semidirect |
| 38. C3×#226 | C22×C3 | semidirect |
| 50. C3×H(5,𝔽2) | C3×D3 | semidirect |
| 52. #994 | C2×C32 | semidirect |
| 53. #985 | C2×C32 | semidirect |
| 54. #984 | C2×C32 | semidirect |
| 93. GL(2,𝔽3) | C22×C32 | semidirect |
| 94. #226 | C22×C32 | semidirect |
| 113. H(5,𝔽2) | C32×D3 | semidirect |
| 172. C32 | #5924 | direct (N × H) |
| 201. C3 | C3×#5924 | 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 |
| C32×#985 | 2 |
| C32×#984 | 2 |
| C3×#5924 | 3 |
| C32×#577 | 3 |
| C32×D3×Q8 | 4 |