Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
756 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C22×H(3,𝔽3) |
| commutator, Frattini, cosocle | C2×C32 | C22×C33 |
| prime radical | C2×C3×AΣL(1,𝔽27) | C22 |
| LCS | G ▷ C2×C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C33×Q8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C34×Q8 | C3 | semidirect |
| 3. C3×Q8×3−1+2 | C3 | semidirect |
| 4. C3×Q8×H(3,𝔽3) | C3 | semidirect |
| 5. Q8×AΣL(1,𝔽27) | C3 | direct (N × H) |
| 11. C3×AΣL(1,𝔽27) | Q8 | direct (N × H) |
| 17. C33×Q8 | C32 | semidirect |
| 19. Q8×3−1+2 | C32 | semidirect |
| 21. Q8×H(3,𝔽3) | C32 | semidirect |
| 38. C34 | C3×Q8 | semidirect |
| 39. C3×3−1+2 | C3×Q8 | semidirect |
| 40. C3×H(3,𝔽3) | C3×Q8 | semidirect |
| 41. AΣL(1,𝔽27) | C3×Q8 | direct (N × H) |
| 80. C33 | C32×Q8 | semidirect |
| 82. 3−1+2 | C32×Q8 | semidirect |
| 84. H(3,𝔽3) | C32×Q8 | semidirect |
| 90. C3×Q8 | AΣL(1,𝔽27) | direct (N × H) |
| 120. Q8 | C3×AΣL(1,𝔽27) | direct (N × H) |
| 132. C3 | Q8×AΣL(1,𝔽27) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C4×AΣL(1,𝔽27) | 2 |
| C34×Q8 | 3 |
| C3×Q8×3−1+2 | 3 |
| C3×Q8×H(3,𝔽3) | 3 |
| Q8×AΣL(1,𝔽27) | 3 |