Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
477 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 477 conjugacy classes of subgroups — too many to draw. Only its 56 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×#88623 | C2 | semidirect |
| 2. C4×Q8⋊H(3,𝔽3) | C2 | semidirect |
| 3. Q8×C4⋊(C3⋊D3) | C3 | semidirect |
| 8. C2×Q8×C3⋊D3 | C2×C3 | semidirect |
| 10. C32×C4×Q8 | C2×C3 | semidirect |
| 15. C3×C4×SL(2,𝔽3) | D3 | semidirect |
| 19. Q8⋊H(3,𝔽3) | D4 | semidirect |
| 59. C32×Q8 | C3×D4 | semidirect |
| 66. C4⋊(C3⋊D3) | SL(2,𝔽3) | semidirect |
| 73. C3×SL(2,𝔽3) | D12 | semidirect |
| 110. C2×C3⋊D3 | C2×SL(2,𝔽3) | semidirect |
| 119. C32×C4 | C2×SL(2,𝔽3) | semidirect |
| 124. C4×Q8 | (C3⋊D3)⋊C3 | semidirect |
| 206. C32 | D4×SL(2,𝔽3) | semidirect |
| 211. Q8 | C4⋊((C3⋊D3)⋊C3) | semidirect |
| 242. C4 | #88623 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#88623 | 2 |
| C4×Q8⋊H(3,𝔽3) | 2 |
| Q8×C4⋊(C3⋊D3) | 3 |
| SL(2,𝔽3)×D12 | 3 |
| C2×C4⋊((C3⋊D3)⋊C3) | 4 |