Finite Group DatabasebetaFinite Group Databasebeta← (C3×SL(2,𝔽3))⋊H(3,𝔽3) #14517246
Order 1944.
623 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 623 conjugacy classes of subgroups — too many to draw. Only its 73 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 | C2×C32 | C32×A4 |
| commutator, cosocle | C32×Q8 | C33 |
| Fitting | C3×Q8×H(3,𝔽3) | C3 |
| hypercenter | C2×C3×H(3,𝔽3) | A4 |
| hypocenter | Q8 | C32⋊H(3,𝔽3) |
| LCS | G ▷ C32×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C3×H(3,𝔽3) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C33×SL(2,𝔽3) | C3 | semidirect |
| 2. C3×Q8×H(3,𝔽3) | C3 | semidirect |
| 3. C3×Q8⋊H(3,𝔽3) | C3 | semidirect |
| 4. C3×Q8⋊H(3,𝔽3) | C3 | semidirect |
| 9. C33×Q8 | C32 | semidirect |
| 13. C32×SL(2,𝔽3) | C32 | semidirect |
| 27. C3×H(3,𝔽3) | SL(2,𝔽3) | semidirect |
| 31. C32×Q8 | H(3,𝔽3) | semidirect |
| 38. C3×SL(2,𝔽3) | H(3,𝔽3) | semidirect |
| 58. C33 | C3×SL(2,𝔽3) | semidirect |
| 92. C32 | Q8⋊H(3,𝔽3) | semidirect |
| 94. Q8 | C32⋊H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×SL(2,𝔽3) | 3 |
| C3×Q8×H(3,𝔽3) | 3 |
| C3×Q8⋊H(3,𝔽3) | 3 |
| C2×C32⋊H(3,𝔽3) | 4 |