Finite Group DatabasebetaFinite Group DatabasebetaOrder 324.
52 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×AΣL(1,𝔽27) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C32×Dic3 | — | 3 | 1 | no | maximal |
| 3 | #1084 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | AΣL(1,𝔽27) | C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 5 | C2×C33 | D3 | 1 | 1 | yes | characteristic |
| 6 | C2×3−1+2 | — | 2 | 1 | no | — |
| 7 | C2×H(3,𝔽3) | C2×C3 | 1 | 1 | yes | characteristic |
| 8 | C32×C4 | — | 9 | 1 | no | — |
| 9 | C3×Dic3 | — | 3 | 3 | no | — |
| 10 | C3×Dic3 | — | 3 | 1 | no | — |
| 11 | C3×Dic3 | — | 9 | 1 | no | — |
| 12 | C33 | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 13 | 3−1+2 | — | 2 | 1 | no | — |
| 14 | H(3,𝔽3) | C3×C4 | 1 | 1 | yes | characteristic, complemented |
| 15 | C2×C32 | — | 3 | 1 | no | — |
| 16 | C2×C32 | — | 3 | 3 | no | — |
| 17 | C2×C32 | — | 3 | 1 | no | — |
| 18 | C2×C32 | C3×D3 | 1 | 1 | yes | characteristic |
| 19 | C2×C9 | — | 6 | 1 | no | — |
| 20 | C3×C4 | — | 9 | 3 | no | — |
| 21 | C3×C4 | — | 9 | 1 | no | — |
| 22 | Dic3 | — | 3 | 1 | no | — |
| 23 | Dic3 | — | 9 | 1 | no | — |
| 24 | C32 | — | 3 | 3 | no | — |
| 25 | C32 | — | 3 | 1 | no | — |
| 26 | C32 | — | 3 | 1 | no | — |
| 27 | C32 | C3×Dic3 | 1 | 1 | yes | characteristic |
| 28 | C9 | — | 6 | 1 | no | — |
| 29 | C2×C3 | — | 3 | 1 | no | — |
| 30 | C2×C3 | — | 3 | 3 | no | — |
| 31 | C2×C3 | — | 9 | 1 | no | — |
| 32 | C2×C3 | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 33 | C4 | — | 9 | 1 | no | Sylow |
| 34 | C3 | — | 3 | 3 | no | — |
| 35 | C3 | — | 3 | 1 | no | — |
| 36 | C3 | — | 9 | 1 | no | — |
| 37 | C3 | (C3⋊Dic3)⋊C3 | 1 | 1 | yes | characteristic |
| 38 | C2 | #4174 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C3 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | H(3,𝔽3) | C3×C4 |
| Frattini | C2×C32 | C3×D3 |
| Fitting, prime radical | C2×AΣL(1,𝔽27) | C2 |
| cosocle | C2×H(3,𝔽3) | C2×C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #1084 | C3 | semidirect |
| 4. AΣL(1,𝔽27) | C4 | semidirect |
| 12. C33 | Dic3 | semidirect |
| 14. H(3,𝔽3) | C3×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×AΣL(1,𝔽27) | 2 |
| C32×Dic3 | 3 |
| #1084 | 3 |