Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1099 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 1099 conjugacy classes of subgroups — too many to draw. Only its 82 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×(C3⋊D3)⋊A4 |
| commutator, cosocle, hypocenter | C32×Q8 | C23×C3 |
| Frattini | C2×C3 | C22×D3×A4 |
| Fitting | C2×C32×#41 | C2×C3 |
| socle | C22×C3 | C2×D3×A4 |
| hypercenter | C2×C4 | (C3⋊D3)⋊A4 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#41⋊H(3,𝔽3) | C2 | semidirect |
| 2. #11722436 | C2 | direct (N × H) |
| 3. C2×#88622 | C2 | semidirect |
| 4. C2×#64366 | C3 | semidirect |
| 7. #41⋊H(3,𝔽3) | C22 | semidirect |
| 8. C2×Q8⋊H(3,𝔽3) | C22 | semidirect |
| 9. #88622 | C22 | semidirect |
| 12. #64366 | C2×C3 | semidirect |
| 18. C2×C3×#41⋊C3 | D3 | semidirect |
| 24. Q8⋊H(3,𝔽3) | C23 | semidirect |
| 49. C3×#41⋊C3 | C2×D3 | semidirect |
| 51. C2×C3×SL(2,𝔽3) | C2×D3 | semidirect |
| 99. C3×SL(2,𝔽3) | C22×D3 | semidirect |
| 221. C2×C32 | #994 | semidirect |
| 270. C32 | C2×#994 | semidirect |
| 319. C2 | #11722436 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41⋊H(3,𝔽3) | 2 |
| #11722436 | 2 |
| C2×#88622 | 2 |
| C2×#64366 | 3 |
| C2×C3⋊#994 | 3 |
| C2×#634⋊C3 | 4 |