Finite Group DatabasebetaFinite Group DatabasebetaOrder 192.
72 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C4⋊S4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | A4⋊Q8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C8×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #465 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C2×S4 | — | 2 | 1 | no | — |
| 6 | C4×A4 | C22 | 1 | 1 | yes | characteristic |
| 7 | SL(2,ℤ/4) | — | 2 | 1 | no | — |
| 8 | #204 | — | 4 | 1 | no | maximal |
| 9 | C2×SD16 | — | 3 | 1 | no | — |
| 10 | C22×C8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C22⋊Q8 | — | 3 | 1 | no | — |
| 12 | #121 | — | 3 | 1 | no | — |
| 13 | #106 | — | 3 | 1 | no | — |
| 14 | #103 | — | 3 | 1 | no | — |
| 15 | #102 | — | 3 | 1 | no | — |
| 16 | C2×A4 | D4 | 1 | 1 | yes | characteristic |
| 17 | S4 | — | 4 | 1 | no | — |
| 18 | D12 | — | 4 | 1 | no | — |
| 19 | Dic6 | — | 4 | 1 | no | — |
| 20 | C3×C8 | — | 4 | 1 | no | — |
| 21 | C2×Q8 | — | 3 | 1 | no | — |
| 22 | C2×D4 | — | 3 | 1 | no | — |
| 23 | C2×D4 | — | 6 | 1 | no | — |
| 24 | C22×C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 25 | SD16 | — | 6 | 1 | no | — |
| 26 | SD16 | — | 6 | 1 | no | — |
| 27 | C2×C8 | — | 3 | 1 | no | — |
| 28 | C2×C8 | — | 3 | 1 | no | — |
| 29 | C4⋊C4 | — | 3 | 1 | no | — |
| 30 | C4⋊C4 | — | 3 | 1 | no | — |
| 31 | C4⋊C4 | — | 6 | 1 | no | — |
| 32 | C22⋊C4 | — | 6 | 1 | no | — |
| 33 | C22⋊C4 | — | 6 | 1 | no | — |
| 34 | C2×D3 | — | 8 | 1 | no | — |
| 35 | A4 | SD16 | 1 | 1 | yes | characteristic, complemented |
| 36 | C3×C4 | — | 4 | 1 | no | — |
| 37 | Dic3 | — | 8 | 1 | no | — |
| 38 | C23 | — | 6 | 1 | no | — |
| 39 | C23 | D12 | 1 | 1 | yes | characteristic |
| 40 | Q8 | — | 6 | 1 | no | — |
| 41 | Q8 | — | 6 | 1 | no | — |
| 42 | D4 | — | 6 | 1 | no | — |
| 43 | D4 | — | 6 | 1 | no | — |
| 44 | D4 | — | 12 | 1 | no | — |
| 45 | D4 | — | 12 | 1 | no | — |
| 46 | C2×C4 | — | 3 | 1 | no | — |
| 47 | C2×C4 | — | 3 | 1 | no | — |
| 48 | C2×C4 | — | 6 | 1 | no | — |
| 49 | C2×C4 | — | 6 | 1 | no | — |
| 50 | C2×C4 | — | 6 | 1 | no | — |
| 51 | C8 | — | 3 | 1 | no | — |
| 52 | C8 | S4 | 1 | 1 | yes | characteristic, complemented |
| 53 | C2×C3 | — | 4 | 1 | no | — |
| 54 | D3 | — | 16 | 1 | no | — |
| 55 | C22 | — | 3 | 1 | no | — |
| 56 | C22 | — | 3 | 1 | no | — |
| 57 | C22 | — | 12 | 1 | no | — |
| 58 | C22 | — | 12 | 1 | no | — |
| 59 | C22 | — | 12 | 1 | no | — |
| 60 | C22 | #204 | 1 | 1 | yes | characteristic, complemented |
| 61 | C4 | — | 3 | 1 | no | — |
| 62 | C4 | — | 12 | 1 | no | — |
| 63 | C4 | — | 12 | 1 | no | — |
| 64 | C4 | — | 12 | 1 | no | — |
| 65 | C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 66 | C3 | — | 4 | 1 | no | Sylow |
| 67 | C2 | — | 3 | 1 | no | — |
| 68 | C2 | — | 3 | 1 | no | — |
| 69 | C2 | — | 24 | 1 | no | — |
| 70 | C2 | C4⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C4⋊S4 |
| commutator, cosocle | C4×A4 | C22 |
| Frattini | C4 | C2×S4 |
| Fitting | C22×C8 | D3 |
| socle | C23 | D12 |
| prime radical | C4⋊S4 | C2 |
| hypercenter | C8 | S4 |
| hypocenter | A4 | SD16 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. A4⋊Q8 | C2 | semidirect |
| 3. C8×A4 | C2 | semidirect |
| 10. C22×C8 | D3 | semidirect |
| 35. A4 | SD16 | semidirect |
| 52. C8 | S4 | semidirect |
| 60. C22 | #204 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4⋊S4 | 2 |
| A4⋊Q8 | 2 |
| C8×A4 | 2 |
| #465 | 3 |
| #204 | 4 |