Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
944 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 944 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 | C32×2−1+4 | D3 |
| hypercenter | C32×D4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C32 ◁ C32×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×2−1+4⋊C3 | C2 | semidirect |
| 2. C32×#985 | C2 | semidirect |
| 3. C32×#984 | C2 | semidirect |
| 4. C32×#983 | C2 | semidirect |
| 5. C2×C32×#226 | C2 | semidirect |
| 6. C3×#5927 | C3 | direct (N × H) |
| 15. C32×2−1+4 | D3 | semidirect |
| 17. C3×2−1+4⋊C3 | C2×C3 | semidirect |
| 18. C3×#985 | C2×C3 | semidirect |
| 19. C3×#984 | C2×C3 | semidirect |
| 20. C3×#983 | C2×C3 | semidirect |
| 21. C2×C3×#226 | C2×C3 | semidirect |
| 37. #5927 | C32 | direct (N × H) |
| 79. C3×2−1+4 | C3×D3 | semidirect |
| 81. 2−1+4⋊C3 | C2×C32 | semidirect |
| 82. #985 | C2×C32 | semidirect |
| 83. #984 | C2×C32 | semidirect |
| 84. #983 | C2×C32 | semidirect |
| 85. C2×#226 | C2×C32 | semidirect |
| 180. 2−1+4 | C32×D3 | semidirect |
| 274. C32 | #5927 | direct (N × H) |
| 317. C3 | C3×#5927 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×2−1+4⋊C3 | 2 |
| C32×#985 | 2 |
| C32×#984 | 2 |
| C32×#983 | 2 |
| C2×C32×#226 | 2 |
| C3×#5927 | 3 |
| C32×#578 | 3 |
| C32×#237 | 4 |