Finite Group DatabasebetaFinite Group DatabasebetaOrder 378.
44 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C32×D7 | C3 | 1 | 4 | yes | maximal, complemented |
| 3 | C32×C7 | C2×C3 | 1 | 4 | yes | complemented |
| 4 | C2×H(3,𝔽3) | — | 7 | 1 | no | maximal, Hall |
| 5 | C3×D7 | — | 3 | 4 | no | — |
| 6 | C3×D7 | C32 | 1 | 1 | yes | characteristic |
| 7 | H(3,𝔽3) | D7 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 8 | C3×C7 | — | 3 | 4 | no | — |
| 9 | C3×C7 | C2×C32 | 1 | 1 | yes | characteristic |
| 10 | C2×C32 | — | 7 | 4 | no | — |
| 11 | D7 | H(3,𝔽3) | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 12 | C32 | C3×D7 | 1 | 4 | yes | complemented |
| 13 | C7 | C2×H(3,𝔽3) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C2×C3 | — | 7 | 1 | no | — |
| 15 | C2×C3 | — | 21 | 4 | no | — |
| 16 | C3 | — | 3 | 4 | no | — |
| 17 | C3 | C32×D7 | 1 | 1 | yes | characteristic |
| 18 | C2 | — | 7 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C32×D7 |
| commutator, socle, cosocle | C3×C7 | C2×C32 |
| Fitting | C7×H(3,𝔽3) | C2 |
| hypercenter | H(3,𝔽3) | D7 |
| hypocenter | C7 | C2×H(3,𝔽3) |
| LCS | G ▷ C3×C7 ▷ C7 | |
| UCS | C1 ◁ C3 ◁ H(3,𝔽3) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×H(3,𝔽3) | C2 | semidirect |
| 2. C32×D7 | C3 | semidirect |
| 3. C32×C7 | C2×C3 | semidirect |
| 7. H(3,𝔽3) | D7 | direct (N × H) |
| 11. D7 | H(3,𝔽3) | direct (N × H) |
| 12. C32 | C3×D7 | semidirect |
| 13. C7 | C2×H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×H(3,𝔽3) | 2 |
| C32×D7 | 3 |
| C2×H(3,𝔽3) | 7 |