Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1639 conjugacy classes of subgroups; only the 357 normal subgroups are shown.
This group has 1639 conjugacy classes of subgroups — too many to draw. Only its 357 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 |
|---|---|---|
| 2. C22×C8×H(3,𝔽3) | C2 | semidirect |
| 3. C22×#6180 | C2 | semidirect |
| 4. C2×#88527 | C2 | direct (N × H) |
| 11. C2×C8×H(3,𝔽3) | C22 | semidirect |
| 12. C2×#6180 | C22 | semidirect |
| 13. #88527 | C22 | direct (N × H) |
| 15. C22×C32×C8 | D3 | semidirect |
| 26. C8×H(3,𝔽3) | C23 | semidirect |
| 27. #6180 | C23 | semidirect |
| 35. C2×C32×C8 | C2×D3 | semidirect |
| 38. C22×H(3,𝔽3) | M16 | semidirect |
| 61. C32×C8 | C22×D3 | semidirect |
| 64. C2×H(3,𝔽3) | C2×M16 | semidirect |
| 83. C22×C32 | ΓL(1,𝔽25) | semidirect |
| 93. C22×C8 | #272 | semidirect |
| 94. H(3,𝔽3) | C22×M16 | semidirect |
| 115. C2×C32 | C2×ΓL(1,𝔽25) | semidirect |
| 125. C2×C8 | C2×#272 | semidirect |
| 140. C32 | C22×ΓL(1,𝔽25) | semidirect |
| 150. C8 | C22×#272 | semidirect |
| 159. C22 | #88527 | direct (N × H) |
| 167. C2 | C2×#88527 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×#272 | 2 |
| C22×C8×H(3,𝔽3) | 2 |
| C22×#6180 | 2 |
| C2×#88527 | 2 |
| C22×C3×ΓL(1,𝔽25) | 3 |