Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
612 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 197 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×H(3,𝔽3) | C2 | semidirect |
| 2. #88722 | C2 | direct (N × H) |
| 3. C2×C4⋊((C3⋊D3)⋊C3) | C2 | semidirect |
| 4. C2×C4×(C3⋊D3)⋊C3 | C2 | semidirect |
| 5. C2×#3832 | C3 | semidirect |
| 8. Q8×H(3,𝔽3) | C22 | semidirect |
| 11. C4⋊((C3⋊D3)⋊C3) | C22 | semidirect |
| 12. C4×(C3⋊D3)⋊C3 | C22 | semidirect |
| 14. C2×C32×Q8 | D3 | semidirect |
| 15. C2×C32×Q8 | C2×C3 | semidirect |
| 16. #3832 | C2×C3 | semidirect |
| 17. C2×C4⋊(C3⋊D3) | C2×C3 | semidirect |
| 18. C2×C4×C3⋊D3 | C2×C3 | semidirect |
| 33. C32×Q8 | C2×D3 | semidirect |
| 34. C32×Q8 | C22×C3 | semidirect |
| 39. C4⋊(C3⋊D3) | C22×C3 | semidirect |
| 40. C4×C3⋊D3 | C22×C3 | semidirect |
| 44. C2×H(3,𝔽3) | #41 | semidirect |
| 73. H(3,𝔽3) | C2×#41 | semidirect |
| 96. C2×C32 | #239 | semidirect |
| 97. C2×C32 | C3×#41 | semidirect |
| 103. C2×Q8 | (C3⋊D3)⋊C3 | semidirect |
| 124. C32 | C2×#239 | semidirect |
| 125. C32 | C2×C3×#41 | semidirect |
| 127. Q8 | C2×(C3⋊D3)⋊C3 | semidirect |
| 153. C2 | #88722 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×H(3,𝔽3) | 2 |
| #88722 | 2 |
| C2×C4⋊((C3⋊D3)⋊C3) | 2 |
| C2×C4×(C3⋊D3)⋊C3 | 2 |
| C2×#3832 | 3 |
| C2×C3×#239 | 3 |