Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
6856 conjugacy classes of subgroups; only the 670 normal subgroups are shown.
This group has 6856 conjugacy classes of subgroups — too many to draw. Only its 670 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24×C3 | C3×A4 |
| commutator, cosocle | C22×C3 | C24×C32 |
| Frattini | C3 | C24×C3×A4 |
| Fitting | C26×C32 | C3 |
| socle | C26×C3 | C32 |
| hypercenter | C24×C32 | A4 |
| hypocenter | C22 | C24×H(3,𝔽3) |
| LCS | G ▷ C22×C3 ▷ C22 | |
| UCS | C1 ◁ C24×C3 ◁ C24×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C22⋊H(3,𝔽3) | C2 | direct (N × H) |
| 2. C26×C32 | C3 | semidirect |
| 3. C24×C3×A4 | C3 | semidirect |
| 5. C22×C22⋊H(3,𝔽3) | C22 | direct (N × H) |
| 8. C25×C32 | C2×C3 | semidirect |
| 9. C23×C3×A4 | C2×C3 | semidirect |
| 11. C2×C22⋊H(3,𝔽3) | C23 | direct (N × H) |
| 20. C24×C32 | A4 | semidirect |
| 21. C24×C32 | C22×C3 | semidirect |
| 22. C22×C3×A4 | C22×C3 | semidirect |
| 24. C22⋊H(3,𝔽3) | C24 | direct (N × H) |
| 37. C23×C32 | C2×A4 | semidirect |
| 38. C23×C32 | C23×C3 | semidirect |
| 39. C2×C3×A4 | C23×C3 | semidirect |
| 40. C26 | H(3,𝔽3) | semidirect |
| 61. C22×C32 | C22×A4 | semidirect |
| 62. C22×C32 | C24×C3 | semidirect |
| 63. C3×A4 | C24×C3 | semidirect |
| 66. C25 | C2×H(3,𝔽3) | semidirect |
| 85. C2×C32 | C23×A4 | semidirect |
| 90. C24 | C22⋊H(3,𝔽3) | direct (N × H) |
| 91. C24 | C22×H(3,𝔽3) | semidirect |
| 107. C32 | C24×A4 | semidirect |
| 112. C23 | C2×C22⋊H(3,𝔽3) | direct (N × H) |
| 113. C23 | C23×H(3,𝔽3) | semidirect |
| 125. C22 | C22×C22⋊H(3,𝔽3) | direct (N × H) |
| 126. C22 | C24×H(3,𝔽3) | semidirect |
| 132. C2 | C23×C22⋊H(3,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C22⋊H(3,𝔽3) | 2 |
| C26×C32 | 3 |
| C24×C3×A4 | 3 |
| C24×H(3,𝔽3) | 4 |