Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
164 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2×C3 | C22×C3×A4 |
| commutator, cosocle | C3×Q8 | C22×C32 |
| Fitting | C9×2−1+4 | C3 |
| prime radical | C3×2−1+4⋊C3 | C3 |
| hypercenter | D4×C9 | A4 |
| hypocenter | Q8 | C22×3−1+2 |
| LCS | G ▷ C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C3 ◁ D4×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #88683 | C2 | semidirect |
| 2. C2×C9⋊SL(2,𝔽3) | C2 | semidirect |
| 4. C9×2−1+4 | C3 | semidirect |
| 5. 2−1+4⋊C9 | C3 | semidirect |
| 12. C9×#41 | C2×C3 | semidirect |
| 14. C2×Q8×C9 | C2×C3 | semidirect |
| 15. #41⋊C9 | C2×C3 | semidirect |
| 16. C2×Q8⋊C9 | C2×C3 | semidirect |
| 52. 2−1+4 | 3−1+2 | semidirect |
| 74. #41 | C2×3−1+2 | semidirect |
| 76. C2×Q8 | C2×3−1+2 | semidirect |
| 86. C9 | 2−1+4⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #88683 | 2 |
| C2×C9⋊SL(2,𝔽3) | 2 |
| C3×2−1+4⋊C3 | 3 |
| C9×2−1+4 | 3 |
| 2−1+4⋊C9 | 3 |
| D4×3−1+2 | 4 |