Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
101 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×Q8⋊C9 | C2 | 1 | 3 | yes | maximal, direct product factor |
| 2 | C22×C3×Q8 | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C23×C9 | — | 4 | 1 | no | maximal |
| 4 | Q8⋊C9 | C22 | 1 | 1 | yes | characteristic, direct product factor |
| 5 | C2×C3×Q8 | — | 3 | 3 | no | — |
| 6 | C2×C3×Q8 | C2×C3 | 1 | 3 | yes | — |
| 7 | C22×C3×C4 | — | 3 | 1 | no | — |
| 8 | C22×C9 | — | 4 | 4 | no | — |
| 9 | C22×C9 | — | 4 | 3 | no | — |
| 10 | C22×Q8 | C9 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | C23×C3 | A4 | 1 | 1 | yes | characteristic |
| 12 | C3×Q8 | — | 3 | 3 | no | — |
| 13 | C3×Q8 | — | 3 | 2 | no | — |
| 14 | C3×Q8 | C22×C3 | 1 | 1 | yes | characteristic |
| 15 | C2×C3×C4 | — | 3 | 3 | no | — |
| 16 | C2×C3×C4 | — | 3 | 3 | no | — |
| 17 | C2×C9 | — | 4 | 6 | no | — |
| 18 | C2×C9 | — | 4 | 1 | no | — |
| 19 | C2×Q8 | — | 3 | 3 | no | — |
| 20 | C2×Q8 | C2×C9 | 1 | 3 | yes | complemented |
| 21 | C22×C4 | — | 3 | 1 | no | — |
| 22 | C22×C3 | SL(2,𝔽3) | 1 | 4 | yes | — |
| 23 | C22×C3 | C2×A4 | 1 | 3 | yes | — |
| 24 | C3×C4 | — | 3 | 3 | no | — |
| 25 | C3×C4 | — | 3 | 1 | no | — |
| 26 | C9 | — | 4 | 1 | no | Sylow |
| 27 | C23 | C22⋊C9 | 1 | 1 | yes | characteristic |
| 28 | Q8 | — | 3 | 3 | no | — |
| 29 | Q8 | — | 3 | 2 | no | — |
| 30 | Q8 | C22×C9 | 1 | 1 | yes | characteristic, complemented |
| 31 | C2×C4 | — | 3 | 3 | no | — |
| 32 | C2×C4 | — | 3 | 3 | no | — |
| 33 | C2×C3 | C2×SL(2,𝔽3) | 1 | 6 | yes | — |
| 34 | C2×C3 | C22×A4 | 1 | 1 | yes | characteristic |
| 35 | C22 | Q8⋊C9 | 1 | 4 | yes | direct product factor |
| 36 | C22 | C2×C22⋊C9 | 1 | 3 | yes | — |
| 37 | C4 | — | 3 | 1 | no | — |
| 38 | C4 | — | 3 | 3 | no | — |
| 39 | C3 | C22×SL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 40 | C2 | C2×Q8⋊C9 | 1 | 6 | yes | direct product factor |
| 41 | C2 | C22×C22⋊C9 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical, hypercenter | C23×C3 | A4 |
| commutator, hypocenter | Q8 | C22×C9 |
| Frattini | C2×C3 | C22×A4 |
| Fitting | C22×C3×Q8 | C3 |
| cosocle | C3×Q8 | C22×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C23×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8⋊C9 | C2 | direct (N × H) |
| 4. Q8⋊C9 | C22 | direct (N × H) |
| 10. C22×Q8 | C9 | semidirect |
| 20. C2×Q8 | C2×C9 | semidirect |
| 30. Q8 | C22×C9 | semidirect |
| 35. C22 | Q8⋊C9 | direct (N × H) |
| 40. C2 | C2×Q8⋊C9 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8⋊C9 | 2 |
| C22×C3×Q8 | 3 |
| C23×C9 | 4 |