Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
140 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 140 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×H(3,𝔽3) | C2×C3 |
| 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×#731 | C2 | semidirect |
| 3. C8×#731 | C2 | semidirect |
| 4. SD16×3−1+2 | C3 | semidirect |
| 6. C3×C9×SD16 | C3 | semidirect |
| 10. Q8×3−1+2 | C2×C3 | semidirect |
| 12. C3×Q8×C9 | C2×C3 | semidirect |
| 16. C8×3−1+2 | C2×C3 | semidirect |
| 18. C3×C8×C9 | C2×C3 | semidirect |
| 34. #731 | SD16 | direct (N × H) |
| 68. 3−1+2 | C3×SD16 | semidirect |
| 70. C3×C9 | C3×SD16 | semidirect |
| 88. SD16 | #731 | direct (N × H) |
| 102. Q8 | C2×#731 | semidirect |
| 104. C8 | C2×#731 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×#731 | 2 |
| D4×#731 | 2 |
| C8×#731 | 2 |
| SD16×3−1+2 | 3 |
| SD16×H(3,𝔽3) | 3 |
| C3×C9×SD16 | 3 |