Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
5003 conjugacy classes of subgroups; only the 795 normal subgroups are shown.
This group has 5003 conjugacy classes of subgroups — too many to draw. Only its 795 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, cosocle | C2×H(3,𝔽3) | C25 |
| Frattini | C2×C3 | C23×D32 |
| Fitting | C24×H(3,𝔽3) | C22 |
| socle | C24×C3 | D32 |
| hypocenter | H(3,𝔽3) | C23×D4 |
| LCS | G ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×(C3⋊D3)⋊C3 | C2 | semidirect |
| 2. C23×#1084 | C2 | semidirect |
| 3. C22×#6198 | C2 | direct (N × H) |
| 7. C23×(C3⋊D3)⋊C3 | C22 | semidirect |
| 8. C22×#1084 | C22 | semidirect |
| 9. C2×#6198 | C22 | direct (N × H) |
| 10. C24×C3⋊D3 | D3 | semidirect |
| 15. C23×H(3,𝔽3) | D4 | semidirect |
| 18. C22×(C3⋊D3)⋊C3 | C23 | semidirect |
| 19. C2×#1084 | C23 | semidirect |
| 20. #6198 | C23 | direct (N × H) |
| 26. C23×C3⋊D3 | C2×D3 | semidirect |
| 32. C22×H(3,𝔽3) | C2×D4 | semidirect |
| 35. C2×(C3⋊D3)⋊C3 | C24 | semidirect |
| 36. #1084 | C24 | semidirect |
| 47. C23×C32 | #67 | semidirect |
| 50. C22×C3⋊D3 | C22×D3 | semidirect |
| 57. C2×H(3,𝔽3) | C22×D4 | semidirect |
| 76. C22×C32 | C2×#67 | semidirect |
| 79. C2×C3⋊D3 | C23×D3 | semidirect |
| 87. H(3,𝔽3) | C23×D4 | semidirect |
| 107. C2×C32 | C22×#67 | semidirect |
| 132. C32 | C23×#67 | semidirect |
| 135. C23 | #6198 | direct (N × H) |
| 150. C22 | C2×#6198 | direct (N × H) |
| 157. C2 | C22×#6198 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×(C3⋊D3)⋊C3 | 2 |
| C23×#1084 | 2 |
| C22×#6198 | 2 |
| C23×#626 | 3 |