Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
803 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 803 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C3 | C32×D32 |
| commutator, socle, cosocle | C2×C33 | C22×C32 |
| Fitting | C2×C32×H(3,𝔽3) | C22 |
| hypercenter | C2×H(3,𝔽3) | D32 |
| hypocenter | C32 | D4×H(3,𝔽3) |
| LCS | G ▷ C2×C33 ▷ C32 | |
| UCS | C1 ◁ C2×C3 ◁ C2×H(3,𝔽3) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3⋊D3×H(3,𝔽3) | C2 | semidirect |
| 2. C2×C3×D3×H(3,𝔽3) | C2 | semidirect |
| 3. C3×Dic3×H(3,𝔽3) | C2 | semidirect |
| 4. C32×#626 | C3 | semidirect |
| 10. C2×C32×C3⋊D3 | C2×C3 | semidirect |
| 11. C2×C33×D3 | C2×C3 | semidirect |
| 12. C33×Dic3 | C2×C3 | semidirect |
| 18. C2×D3×H(3,𝔽3) | D3 | semidirect |
| 21. Dic3×H(3,𝔽3) | D3 | semidirect |
| 22. C32×H(3,𝔽3) | D4 | semidirect |
| 52. C2×C32×D3 | C3×D3 | semidirect |
| 58. C32×Dic3 | C3×D3 | semidirect |
| 65. C34 | C3×D4 | semidirect |
| 70. C3×H(3,𝔽3) | D12 | semidirect |
| 71. C3×H(3,𝔽3) | #67 | semidirect |
| 77. #626 | H(3,𝔽3) | direct (N × H) |
| 114. C2×C3⋊D3 | C2×H(3,𝔽3) | semidirect |
| 124. C2×C3×D3 | C2×H(3,𝔽3) | semidirect |
| 134. C3×Dic3 | C2×H(3,𝔽3) | semidirect |
| 144. C33 | C3×D12 | semidirect |
| 145. C33 | C3×#67 | semidirect |
| 151. H(3,𝔽3) | #626 | direct (N × H) |
| 207. C2×D3 | D3×H(3,𝔽3) | semidirect |
| 213. Dic3 | D3×H(3,𝔽3) | semidirect |
| 233. C32 | D4×H(3,𝔽3) | semidirect |
| 234. C32 | C3×#626 | semidirect |
| 271. C3 | D12×H(3,𝔽3) | semidirect |
| 272. C3 | #67×H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3⋊D3×H(3,𝔽3) | 2 |
| C2×C3×D3×H(3,𝔽3) | 2 |
| C3×Dic3×H(3,𝔽3) | 2 |
| C32×#626 | 3 |
| #67×H(3,𝔽3) | 3 |
| D12×H(3,𝔽3) | 3 |