Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
114 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×H(5,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #997 | C3 | 1 | 3 | yes | maximal, direct product factor |
| 3 | C2×C3×A4 | — | 4 | 3 | no | maximal |
| 4 | C3×SL(2,𝔽3) | — | 4 | 2 | no | maximal |
| 5 | C3×#41 | — | 3 | 2 | no | — |
| 6 | C2×C3×D4 | — | 3 | 3 | no | — |
| 7 | C3×A4 | — | 4 | 3 | no | — |
| 8 | H(5,𝔽2) | C32 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 9 | C23×C3 | — | 3 | 1 | no | — |
| 10 | C23×C3 | A4 | 1 | 3 | yes | complemented |
| 11 | C2×A4 | — | 4 | 9 | no | — |
| 12 | C3×Q8 | A4 | 1 | 2 | yes | complemented |
| 13 | C3×D4 | — | 3 | 6 | no | — |
| 14 | C2×C3×C4 | — | 3 | 3 | no | — |
| 15 | SL(2,𝔽3) | — | 4 | 6 | no | — |
| 16 | C2×C32 | — | 16 | 1 | no | — |
| 17 | #41 | — | 3 | 2 | no | — |
| 18 | C2×D4 | — | 3 | 3 | no | — |
| 19 | C22×C3 | — | 3 | 3 | no | — |
| 20 | C22×C3 | — | 4 | 3 | no | — |
| 21 | C22×C3 | — | 12 | 1 | no | — |
| 22 | A4 | — | 4 | 9 | no | — |
| 23 | C3×C4 | — | 3 | 2 | no | — |
| 24 | C32 | — | 16 | 1 | no | Sylow |
| 25 | C23 | — | 3 | 1 | no | — |
| 26 | C23 | C3×A4 | 1 | 3 | yes | complemented |
| 27 | Q8 | C3×A4 | 1 | 2 | yes | complemented |
| 28 | D4 | — | 3 | 6 | no | — |
| 29 | C2×C4 | — | 3 | 3 | no | — |
| 30 | C2×C3 | — | 6 | 3 | no | — |
| 31 | C2×C3 | — | 16 | 3 | no | — |
| 32 | C2×C3 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 33 | C22 | — | 3 | 3 | no | — |
| 34 | C22 | — | 4 | 3 | no | — |
| 35 | C22 | — | 12 | 1 | no | — |
| 36 | C4 | — | 3 | 2 | no | — |
| 37 | C3 | — | 16 | 3 | no | — |
| 38 | C3 | #997 | 1 | 1 | yes | characteristic, direct product factor |
| 39 | C2 | — | 6 | 3 | no | — |
| 40 | C2 | C3×C22⋊A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C3 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C32 |
| Frattini | C2 | C3×C22⋊A4 |
| Fitting | C3×H(5,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×H(5,𝔽2) | C3 | semidirect |
| 2. #997 | C3 | direct (N × H) |
| 8. H(5,𝔽2) | C32 | semidirect |
| 10. C23×C3 | A4 | semidirect |
| 12. C3×Q8 | A4 | semidirect |
| 26. C23 | C3×A4 | semidirect |
| 27. Q8 | C3×A4 | semidirect |
| 38. C3 | #997 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×H(5,𝔽2) | 3 |
| #997 | 3 |
| C2×C3×A4 | 4 |
| C3×SL(2,𝔽3) | 4 |