Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
882 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 882 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | A4×D9 |
| commutator, cosocle, hypocenter | Q8×C9 | C23×C3 |
| Frattini | C2×C3 | C22×D3×A4 |
| Fitting | C22×Q8×C9 | C2×C3 |
| socle | C23×C3 | D3×A4 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C9×SL(2,𝔽3) | C2 | semidirect |
| 2. C2×D9×SL(2,𝔽3) | C2 | direct (N × H) |
| 4. C22×Q8×D9 | C3 | semidirect |
| 6. C2×C9×SL(2,𝔽3) | C22 | semidirect |
| 7. D9×SL(2,𝔽3) | C22 | direct (N × H) |
| 10. C22×Q8×C9 | C2×C3 | semidirect |
| 12. C2×Q8×D9 | C2×C3 | semidirect |
| 19. C9×SL(2,𝔽3) | C23 | semidirect |
| 27. C2×Q8×C9 | C22×C3 | semidirect |
| 32. Q8×D9 | C22×C3 | semidirect |
| 49. C22×SL(2,𝔽3) | D9 | direct (N × H) |
| 56. C22×D9 | SL(2,𝔽3) | direct (N × H) |
| 60. Q8×C9 | C23×C3 | semidirect |
| 88. C2×SL(2,𝔽3) | C2×D9 | direct (N × H) |
| 95. C22×C9 | C2×SL(2,𝔽3) | semidirect |
| 97. C2×D9 | C2×SL(2,𝔽3) | direct (N × H) |
| 105. C22×Q8 | C3×D9 | semidirect |
| 126. SL(2,𝔽3) | C22×D9 | direct (N × H) |
| 132. C2×C9 | C22×SL(2,𝔽3) | semidirect |
| 134. D9 | C22×SL(2,𝔽3) | direct (N × H) |
| 142. C2×Q8 | C2×C3×D9 | semidirect |
| 162. C9 | C23×SL(2,𝔽3) | semidirect |
| 170. Q8 | C22×C3×D9 | semidirect |
| 186. C22 | D9×SL(2,𝔽3) | direct (N × H) |
| 195. C2 | C2×D9×SL(2,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C9×SL(2,𝔽3) | 2 |
| C2×D9×SL(2,𝔽3) | 2 |
| C22×D3×SL(2,𝔽3) | 3 |
| C22×Q8×D9 | 3 |
| C23×C3×D9 | 4 |