Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
402 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | C22×H(3,𝔽3) |
| commutator, Frattini, cosocle | C2×C32 | C22×C33 |
| socle | C2×C3 | C22×C3×H(3,𝔽3) |
| prime radical | C2×#6908 | C22 |
| LCS | G ▷ C2×C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C3×Q8×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×#737 | C3 | semidirect |
| 3. C32×Q8×C9 | C3 | semidirect |
| 4. Q8×#733 | C3 | semidirect |
| 5. Q8×#732 | C3 | semidirect |
| 6. Q8×#731 | C3 | semidirect |
| 7. Q8×AΣL(1,𝔽27) | C3 | semidirect |
| 15. #6908 | Q8 | direct (N × H) |
| 42. #737 | C3×Q8 | semidirect |
| 43. C32×C9 | C3×Q8 | semidirect |
| 44. #733 | C3×Q8 | semidirect |
| 45. #732 | C3×Q8 | semidirect |
| 46. #731 | C3×Q8 | semidirect |
| 47. AΣL(1,𝔽27) | C3×Q8 | semidirect |
| 112. Q8 | #6908 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#6908 | 2 |
| Q8×#737 | 3 |
| C32×Q8×C9 | 3 |
| Q8×#733 | 3 |
| Q8×#732 | 3 |
| Q8×#731 | 3 |
| Q8×AΣL(1,𝔽27) | 3 |