Finite Group DatabasebetaFinite Group DatabasebetaOrder 96.
56 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×S4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | SL(2,ℤ/4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C4×D4 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C2×A4 | C22 | 1 | 1 | yes | characteristic |
| 6 | S4 | C4 | 1 | 2 | yes | direct product factor |
| 7 | C4×D3 | — | 4 | 1 | no | maximal |
| 8 | C2×D4 | — | 3 | 1 | no | — |
| 9 | C22×C4 | — | 3 | 1 | no | — |
| 10 | C22×C4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C4⋊C4 | — | 3 | 1 | no | — |
| 12 | C22⋊C4 | — | 3 | 1 | no | — |
| 13 | C22⋊C4 | — | 3 | 1 | no | — |
| 14 | C42 | — | 3 | 1 | no | — |
| 15 | C2×D3 | — | 4 | 1 | no | — |
| 16 | A4 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | C3×C4 | — | 4 | 1 | no | — |
| 18 | Dic3 | — | 4 | 1 | no | — |
| 19 | C23 | — | 3 | 1 | no | — |
| 20 | C23 | C2×D3 | 1 | 1 | yes | characteristic |
| 21 | D4 | — | 3 | 2 | no | — |
| 22 | D4 | — | 3 | 2 | no | — |
| 23 | C2×C4 | — | 3 | 1 | no | — |
| 24 | C2×C4 | — | 3 | 1 | no | — |
| 25 | C2×C4 | — | 3 | 1 | no | — |
| 26 | C2×C4 | — | 3 | 1 | no | — |
| 27 | C2×C4 | — | 3 | 1 | no | — |
| 28 | C2×C4 | — | 6 | 1 | no | — |
| 29 | C2×C4 | — | 6 | 1 | no | — |
| 30 | C2×C3 | — | 4 | 1 | no | — |
| 31 | D3 | — | 4 | 2 | no | — |
| 32 | C22 | — | 3 | 1 | no | — |
| 33 | C22 | — | 3 | 1 | no | — |
| 34 | C22 | — | 3 | 2 | no | — |
| 35 | C22 | — | 6 | 1 | no | — |
| 36 | C22 | — | 6 | 1 | no | — |
| 37 | C22 | C4×D3 | 1 | 1 | yes | characteristic, complemented |
| 38 | C4 | — | 3 | 2 | no | — |
| 39 | C4 | — | 3 | 1 | no | — |
| 40 | C4 | — | 6 | 1 | no | — |
| 41 | C4 | — | 6 | 1 | no | — |
| 42 | C4 | S4 | 1 | 1 | yes | characteristic, direct product factor |
| 43 | C3 | — | 4 | 1 | no | Sylow |
| 44 | C2 | — | 3 | 1 | no | — |
| 45 | C2 | — | 3 | 1 | no | — |
| 46 | C2 | — | 6 | 2 | no | — |
| 47 | C2 | C2×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | S4 |
| commutator, hypocenter | A4 | C2×C4 |
| Frattini | C2 | C2×S4 |
| Fitting | C22×C4 | D3 |
| socle | C23 | C2×D3 |
| prime radical | C2×S4 | C2 |
| cosocle | C2×A4 | C22 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×A4 | C2 | semidirect |
| 3. SL(2,ℤ/4) | C2 | semidirect |
| 6. S4 | C4 | direct (N × H) |
| 10. C22×C4 | D3 | semidirect |
| 16. A4 | C2×C4 | semidirect |
| 37. C22 | C4×D3 | semidirect |
| 42. C4 | S4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×S4 | 2 |
| C4×A4 | 2 |
| SL(2,ℤ/4) | 2 |
| C4×D4 | 3 |
| C4×D3 | 4 |