Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
595 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 595 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22×C3 | C22×C3×A4 |
| commutator, cosocle | C2×C3×Q8 | C22×C32 |
| Fitting | C32×D4×Q8 | C3 |
| hypercenter | C2×C32×D4 | A4 |
| hypocenter | Q8 | D4×H(3,𝔽3) |
| LCS | G ▷ C2×C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C32×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q8⋊H(3,𝔽3) | C2 | semidirect |
| 2. C4×Q8⋊H(3,𝔽3) | C2 | semidirect |
| 3. C32×D4×Q8 | C3 | semidirect |
| 4. C3×D4×SL(2,𝔽3) | C3 | semidirect |
| 8. C22×C32×Q8 | C2×C3 | semidirect |
| 9. C22×C3×SL(2,𝔽3) | C2×C3 | semidirect |
| 12. C32×C4×Q8 | C2×C3 | semidirect |
| 14. C3×C4×SL(2,𝔽3) | C2×C3 | semidirect |
| 18. Q8⋊H(3,𝔽3) | D4 | direct (N × H) |
| 58. C32×Q8 | C3×D4 | semidirect |
| 60. C32×D4 | SL(2,𝔽3) | semidirect |
| 68. C3×SL(2,𝔽3) | C3×D4 | semidirect |
| 69. D4×Q8 | H(3,𝔽3) | semidirect |
| 105. C22×C32 | C2×SL(2,𝔽3) | semidirect |
| 112. C32×C4 | C2×SL(2,𝔽3) | semidirect |
| 113. C22×Q8 | C2×H(3,𝔽3) | semidirect |
| 116. C4×Q8 | C2×H(3,𝔽3) | semidirect |
| 190. C32 | D4×SL(2,𝔽3) | semidirect |
| 195. Q8 | D4×H(3,𝔽3) | semidirect |
| 196. D4 | Q8⋊H(3,𝔽3) | direct (N × H) |
| 217. C22 | C2×Q8⋊H(3,𝔽3) | semidirect |
| 223. C4 | C2×Q8⋊H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8⋊H(3,𝔽3) | 2 |
| C4×Q8⋊H(3,𝔽3) | 2 |
| C32×D4×Q8 | 3 |
| C3×D4×SL(2,𝔽3) | 3 |
| C2×D4×H(3,𝔽3) | 4 |