Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
637 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 637 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×D9⋊A4 |
| commutator, cosocle, hypocenter | Q8×C9 | C23×C3 |
| Frattini | C2×C3 | C22×D3×A4 |
| Fitting | C2×C9×#41 | C2×C3 |
| socle | C22×C3 | C2×D3×A4 |
| hypercenter | C2×C4 | D9⋊A4 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#88683 | C2 | semidirect |
| 2. #11722411 | C2 | direct (N × H) |
| 3. C2×#88614 | C2 | semidirect |
| 5. C2×#63715 | C3 | semidirect |
| 7. #88683 | C22 | semidirect |
| 8. C2×C9⋊SL(2,𝔽3) | C22 | semidirect |
| 9. #88614 | C22 | semidirect |
| 15. #63715 | C2×C3 | semidirect |
| 24. C9⋊SL(2,𝔽3) | C23 | semidirect |
| 169. C2×C9 | #994 | semidirect |
| 206. C9 | C2×#994 | semidirect |
| 247. C2 | #11722411 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#88683 | 2 |
| #11722411 | 2 |
| C2×#88614 | 2 |
| C2×C3⋊#994 | 3 |
| C2×#63715 | 3 |
| C2×3−1+2⋊Q8 | 4 |