Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
346 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 71 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×C4 | A4×D9 |
| commutator, hypocenter | Q8×C9 | C2×C3×C4 |
| Frattini, socle | C22×C3 | C2×D3×A4 |
| Fitting | C4×Q8×C9 | C2×C3 |
| prime radical | C2×D9×SL(2,𝔽3) | C2 |
| cosocle | C2×Q8×C9 | C22×C3 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×C9×SL(2,𝔽3) | C2 | semidirect |
| 3. SL(2,𝔽3)×Dic9 | C2 | semidirect |
| 5. C4×Q8×D9 | C3 | semidirect |
| 8. D9×SL(2,𝔽3) | C4 | direct (N × H) |
| 13. C4×Q8×C9 | C2×C3 | semidirect |
| 14. Q8×Dic9 | C2×C3 | semidirect |
| 24. C9×SL(2,𝔽3) | C2×C4 | semidirect |
| 32. Q8×D9 | C3×C4 | semidirect |
| 61. C4×SL(2,𝔽3) | D9 | direct (N × H) |
| 69. Q8×C9 | C2×C3×C4 | semidirect |
| 80. C4×D9 | SL(2,𝔽3) | direct (N × H) |
| 122. C4×C9 | C2×SL(2,𝔽3) | semidirect |
| 125. Dic9 | C2×SL(2,𝔽3) | semidirect |
| 130. C4×Q8 | C3×D9 | semidirect |
| 155. SL(2,𝔽3) | C4×D9 | direct (N × H) |
| 166. D9 | C4×SL(2,𝔽3) | direct (N × H) |
| 202. C9 | C2×C4×SL(2,𝔽3) | semidirect |
| 207. Q8 | C3×C4×D9 | semidirect |
| 239. C4 | 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 |
|---|---|
| C2×D9×SL(2,𝔽3) | 2 |
| C4×C9×SL(2,𝔽3) | 2 |
| SL(2,𝔽3)×Dic9 | 2 |
| C4×D3×SL(2,𝔽3) | 3 |
| C4×Q8×D9 | 3 |
| C2×C3×C4×D9 | 4 |