Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
200 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 56 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×C3 | D4×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32×C4 | C22×C32 |
| prime radical | D4×AΣL(1,𝔽27) | C2 |
| LCS | G ▷ C32×C4 ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C32×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×AΣL(1,𝔽27) | C2 | semidirect |
| 3. C8×AΣL(1,𝔽27) | C2 | semidirect |
| 4. C33×SD16 | C3 | semidirect |
| 5. SD16×3−1+2 | C3 | semidirect |
| 6. SD16×H(3,𝔽3) | C3 | semidirect |
| 10. C33×Q8 | C2×C3 | semidirect |
| 12. C33×C8 | C2×C3 | semidirect |
| 13. Q8×3−1+2 | C2×C3 | semidirect |
| 14. Q8×H(3,𝔽3) | C2×C3 | semidirect |
| 17. C8×3−1+2 | C2×C3 | semidirect |
| 18. C8×H(3,𝔽3) | C2×C3 | semidirect |
| 35. AΣL(1,𝔽27) | SD16 | direct (N × H) |
| 76. C33 | C3×SD16 | semidirect |
| 77. 3−1+2 | C3×SD16 | semidirect |
| 78. H(3,𝔽3) | C3×SD16 | semidirect |
| 101. SD16 | AΣL(1,𝔽27) | direct (N × H) |
| 119. Q8 | C2×AΣL(1,𝔽27) | semidirect |
| 121. C8 | C2×AΣL(1,𝔽27) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×AΣL(1,𝔽27) | 2 |
| D4×AΣL(1,𝔽27) | 2 |
| C8×AΣL(1,𝔽27) | 2 |
| C33×SD16 | 3 |
| SD16×3−1+2 | 3 |
| SD16×H(3,𝔽3) | 3 |