Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
86 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | A42 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×S4 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | D4×A4 | — | 3 | 1 | no | maximal |
| 4 | D3×A4 | — | 4 | 1 | no | maximal |
| 5 | C3×S4 | — | 4 | 1 | no | maximal |
| 6 | C22⋊A4 | — | 2 | 1 | no | — |
| 7 | C22×A4 | — | 3 | 1 | no | — |
| 8 | C22×A4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 9 | C22×A4 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×S4 | — | 3 | 1 | no | — |
| 11 | C2×S4 | — | 3 | 1 | no | — |
| 12 | C4×A4 | — | 3 | 1 | no | — |
| 13 | C3×A4 | — | 4 | 1 | no | — |
| 14 | C3×A4 | — | 4 | 1 | no | — |
| 15 | C22×D4 | — | 3 | 1 | no | Sylow |
| 16 | C22×D3 | — | 4 | 1 | no | — |
| 17 | C2×A4 | — | 3 | 1 | no | — |
| 18 | C2×A4 | — | 3 | 1 | no | — |
| 19 | C2×A4 | — | 6 | 1 | no | — |
| 20 | S4 | — | 3 | 1 | no | — |
| 21 | S4 | A4 | 1 | 1 | yes | characteristic, direct product factor |
| 22 | C3×D4 | — | 12 | 1 | no | — |
| 23 | C3×D3 | — | 16 | 1 | no | — |
| 24 | C24 | — | 3 | 1 | no | — |
| 25 | C24 | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 26 | C2×D4 | — | 9 | 1 | no | — |
| 27 | C2×D4 | — | 9 | 1 | no | — |
| 28 | C2×D4 | — | 9 | 1 | no | — |
| 29 | C2×D4 | — | 9 | 1 | no | — |
| 30 | C22×C4 | — | 3 | 1 | no | — |
| 31 | C22×C3 | — | 4 | 1 | no | — |
| 32 | C22×C3 | — | 4 | 1 | no | — |
| 33 | C22×C3 | — | 12 | 1 | no | — |
| 34 | C2×D3 | — | 12 | 1 | no | — |
| 35 | C2×D3 | — | 12 | 1 | no | — |
| 36 | A4 | — | 8 | 1 | no | — |
| 37 | A4 | — | 8 | 1 | no | — |
| 38 | A4 | — | 24 | 1 | no | — |
| 39 | A4 | S4 | 1 | 1 | yes | characteristic, direct product factor |
| 40 | A4 | C2×A4 | 1 | 1 | yes | characteristic, complemented |
| 41 | C3×C4 | — | 12 | 1 | no | — |
| 42 | C32 | — | 16 | 1 | no | Sylow |
| 43 | C23 | — | 3 | 1 | no | — |
| 44 | C23 | — | 3 | 1 | no | — |
| 45 | C23 | — | 6 | 1 | no | — |
| 46 | C23 | — | 9 | 1 | no | — |
| 47 | C23 | — | 9 | 1 | no | — |
| 48 | C23 | — | 9 | 1 | no | — |
| 49 | C23 | — | 18 | 1 | no | — |
| 50 | D4 | — | 3 | 1 | no | — |
| 51 | D4 | — | 9 | 1 | no | — |
| 52 | D4 | — | 9 | 1 | no | — |
| 53 | D4 | — | 9 | 2 | no | — |
| 54 | D4 | — | 9 | 1 | no | — |
| 55 | C2×C4 | — | 9 | 1 | no | — |
| 56 | C2×C4 | — | 9 | 1 | no | — |
| 57 | C2×C3 | — | 12 | 1 | no | — |
| 58 | C2×C3 | — | 12 | 1 | no | — |
| 59 | C2×C3 | — | 24 | 1 | no | — |
| 60 | D3 | — | 4 | 1 | no | — |
| 61 | D3 | — | 12 | 1 | no | — |
| 62 | C22 | — | 3 | 1 | no | — |
| 63 | C22 | — | 6 | 1 | no | — |
| 64 | C22 | — | 9 | 1 | no | — |
| 65 | C22 | — | 9 | 1 | no | — |
| 66 | C22 | — | 9 | 1 | no | — |
| 67 | C22 | — | 9 | 1 | no | — |
| 68 | C22 | — | 18 | 1 | no | — |
| 69 | C22 | — | 18 | 1 | no | — |
| 70 | C22 | — | 18 | 1 | no | — |
| 71 | C22 | — | 18 | 1 | no | — |
| 72 | C22 | C3×S4 | 1 | 1 | yes | characteristic, complemented |
| 73 | C22 | D3×A4 | 1 | 1 | yes | characteristic, complemented |
| 74 | C4 | — | 3 | 1 | no | — |
| 75 | C4 | — | 9 | 1 | no | — |
| 76 | C3 | — | 4 | 1 | no | — |
| 77 | C3 | — | 4 | 1 | no | — |
| 78 | C3 | — | 32 | 1 | no | — |
| 79 | C2 | — | 3 | 1 | no | — |
| 80 | C2 | — | 3 | 1 | no | — |
| 81 | C2 | — | 6 | 1 | no | — |
| 82 | C2 | — | 9 | 1 | no | — |
| 83 | C2 | — | 18 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×A4 | C2×C3 |
| Fitting, socle | C24 | C3×D3 |
| LCS | G ▷ C22×A4 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A42 | C2 | semidirect |
| 2. C22×S4 | C3 | semidirect |
| 8. C22×A4 | D3 | semidirect |
| 9. C22×A4 | C2×C3 | semidirect |
| 21. S4 | A4 | direct (N × H) |
| 25. C24 | C3×D3 | semidirect |
| 39. A4 | S4 | direct (N × H) |
| 40. A4 | C2×A4 | semidirect |
| 72. C22 | C3×S4 | semidirect |
| 73. C22 | D3×A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A42 | 2 |
| C22×S4 | 3 |
| D4×A4 | 3 |
| D3×A4 | 4 |
| C3×S4 | 4 |