Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
818 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 818 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D33 |
| commutator, cosocle | C2×C33 | C24 |
| Frattini | C2 | C2×D33 |
| Fitting, socle | C22×C33 | C23 |
| prime radical | C22×C3⋊(C3⋊D3) | C22 |
| hypocenter | C33 | C22×Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#6296 | C2 | semidirect |
| 3. C2×C3×#627 | C2 | semidirect |
| 4. #88958 | C2 | direct (N × H) |
| 9. #6296 | C22 | semidirect |
| 11. C3×#627 | C22 | semidirect |
| 15. C2×#627 | D3 | semidirect |
| 20. C2×C3⋊(C3⋊D3) | Q8 | semidirect |
| 37. #627 | C2×D3 | semidirect |
| 40. C2×C33 | C2×Q8 | semidirect |
| 42. C3⋊(C3⋊D3) | C2×Q8 | semidirect |
| 66. C33 | C22×Q8 | semidirect |
| 86. C2×C32 | D3×Q8 | semidirect |
| 112. C32 | C2×D3×Q8 | semidirect |
| 123. C2×C3 | #3797 | semidirect |
| 135. C3 | C2×#3797 | semidirect |
| 137. C2 | #88958 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6296 | 2 |
| C2×#6289 | 2 |
| C2×C3×#627 | 2 |
| #88958 | 2 |
| C2×#3797 | 3 |