Finite Group DatabasebetaFinite Group DatabasebetaOrder 648.
410 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 136 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×C3 | (C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | C32 | C23×C32 |
| Frattini | C3 | C22×C32×D3 |
| Fitting | C22×C3×H(3,𝔽3) | C2 |
| socle | C22×C32 | C3×D3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×H(3,𝔽3) | C2 | semidirect |
| 2. C2×C3×(C3⋊D3)⋊C3 | C2 | direct (N × H) |
| 3. C22×C3×C3⋊D3 | C3 | semidirect |
| 5. C22×(C3⋊D3)⋊C3 | C3 | direct (N × H) |
| 6. C2×C3×H(3,𝔽3) | C22 | semidirect |
| 7. C3×(C3⋊D3)⋊C3 | C22 | direct (N × H) |
| 9. C22×C33 | D3 | semidirect |
| 10. C22×C33 | C2×C3 | semidirect |
| 11. C2×C3×C3⋊D3 | C2×C3 | semidirect |
| 14. C22×H(3,𝔽3) | C2×C3 | semidirect |
| 15. C2×(C3⋊D3)⋊C3 | C2×C3 | direct (N × H) |
| 16. C3×H(3,𝔽3) | C23 | semidirect |
| 18. C22×C3⋊D3 | C32 | semidirect |
| 23. C2×C33 | C2×D3 | semidirect |
| 24. C2×C33 | C22×C3 | semidirect |
| 25. C3×C3⋊D3 | C22×C3 | semidirect |
| 28. C2×H(3,𝔽3) | C22×C3 | semidirect |
| 29. (C3⋊D3)⋊C3 | C22×C3 | direct (N × H) |
| 40. C22×C32 | C3×D3 | semidirect |
| 41. C22×C32 | C2×C32 | semidirect |
| 42. C2×C3⋊D3 | C2×C32 | semidirect |
| 47. C33 | C22×D3 | semidirect |
| 48. C33 | C23×C3 | semidirect |
| 50. H(3,𝔽3) | C23×C3 | semidirect |
| 65. C2×C32 | C2×C3×D3 | semidirect |
| 66. C2×C32 | C22×C32 | semidirect |
| 67. C3⋊D3 | C22×C32 | semidirect |
| 78. C22×C3 | (C3⋊D3)⋊C3 | direct (N × H) |
| 91. C32 | C22×C3×D3 | semidirect |
| 92. C32 | C23×C32 | semidirect |
| 101. C2×C3 | C2×(C3⋊D3)⋊C3 | direct (N × H) |
| 106. C22 | C3×(C3⋊D3)⋊C3 | direct (N × H) |
| 112. C3 | C22×(C3⋊D3)⋊C3 | direct (N × H) |
| 115. C2 | C2×C3×(C3⋊D3)⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×H(3,𝔽3) | 2 |
| C2×C3×(C3⋊D3)⋊C3 | 2 |
| C22×C3×C3⋊D3 | 3 |
| C22×C32×D3 | 3 |
| C22×(C3⋊D3)⋊C3 | 3 |