Finite Group DatabasebetaFinite Group DatabasebetaOrder 1998.
44 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | H(3,𝔽3)×C37 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C3×D111 | — | 3 | 4 | no | maximal |
| 3 | C32×C37 | D3 | 1 | 4 | yes | complemented |
| 4 | D111 | — | 3 | 4 | no | — |
| 5 | C3×D37 | — | 9 | 1 | no | — |
| 6 | C3×C37 | — | 3 | 4 | no | — |
| 7 | C3×C37 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 8 | D37 | — | 9 | 1 | no | Hall |
| 9 | #272 | — | 37 | 1 | no | maximal, Hall |
| 10 | C37 | #272 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | H(3,𝔽3) | D37 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C3×D3 | — | 111 | 4 | no | — |
| 13 | C32 | D111 | 1 | 4 | yes | complemented |
| 14 | C2×C3 | — | 333 | 1 | no | — |
| 15 | D3 | — | 111 | 4 | no | — |
| 16 | C3 | — | 3 | 4 | no | — |
| 17 | C3 | C3⋊D111 | 1 | 1 | yes | characteristic |
| 18 | C2 | — | 333 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C3⋊D111 |
| commutator, Fitting, cosocle, hypocenter | H(3,𝔽3)×C37 | C2 |
| socle | C3×C37 | C3⋊D3 |
| LCS | G ▷ H(3,𝔽3)×C37 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(3,𝔽3)×C37 | C2 | semidirect |
| 3. C32×C37 | D3 | semidirect |
| 10. C37 | #272 | semidirect |
| 11. H(3,𝔽3) | D37 | semidirect |
| 13. C32 | D111 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3)×C37 | 2 |
| C3×D111 | 3 |
| #272 | 37 |