Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
337 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×A4×D9 |
| commutator, cosocle | C2×Q8×C9 | C22×C3 |
| Frattini, socle | C22×C3 | C2×D3×A4 |
| Fitting | C4×Q8×C9 | C2×C3 |
| hypercenter | C2×C4 | A4×D9 |
| hypocenter | Q8×C9 | C3×D4 |
| LCS | G ▷ C2×Q8×C9 ▷ Q8×C9 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D9×SL(2,𝔽3) | C2 | semidirect |
| 2. C4×C9×SL(2,𝔽3) | C2 | semidirect |
| 4. Q8×D36 | C3 | semidirect |
| 10. C2×Q8×D9 | C2×C3 | semidirect |
| 11. C4×Q8×C9 | C2×C3 | semidirect |
| 19. C9×SL(2,𝔽3) | D4 | semidirect |
| 49. C4×SL(2,𝔽3) | D9 | semidirect |
| 56. Q8×C9 | C3×D4 | semidirect |
| 62. D36 | SL(2,𝔽3) | direct (N × H) |
| 97. C2×D9 | C2×SL(2,𝔽3) | semidirect |
| 102. C4×C9 | C2×SL(2,𝔽3) | semidirect |
| 107. C4×Q8 | C3×D9 | semidirect |
| 127. SL(2,𝔽3) | D36 | direct (N × H) |
| 168. C9 | D4×SL(2,𝔽3) | semidirect |
| 173. Q8 | C3×D36 | semidirect |
| 200. C4 | D9×SL(2,𝔽3) | semidirect |
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)×D12 | 3 |
| Q8×D36 | 3 |
| C2×C3×D36 | 4 |