Finite Group DatabasebetaFinite Group DatabasebetaOrder 192.
163 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Q8×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C4⋊S4 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C4×S4 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #548 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C2×S4 | C22 | 1 | 3 | yes | — |
| 6 | #239 | — | 4 | 1 | no | maximal |
| 7 | C4×A4 | C22 | 1 | 3 | yes | — |
| 8 | SL(2,ℤ/4) | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#41 | — | 3 | 1 | no | — |
| 10 | C22×Q8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | #124 | — | 3 | 3 | no | — |
| 12 | C22⋊Q8 | — | 3 | 3 | no | — |
| 13 | #121 | — | 3 | 3 | no | — |
| 14 | C4×Q8 | — | 3 | 1 | no | — |
| 15 | C4×D4 | — | 3 | 3 | no | — |
| 16 | C2×A4 | C23 | 1 | 1 | yes | characteristic |
| 17 | S4 | — | 2 | 3 | no | — |
| 18 | C3×Q8 | — | 4 | 1 | no | — |
| 19 | D12 | — | 4 | 3 | no | — |
| 20 | C4×D3 | — | 4 | 3 | no | — |
| 21 | #41 | — | 6 | 3 | no | — |
| 22 | #41 | — | 6 | 1 | no | — |
| 23 | C2×Q8 | — | 3 | 1 | no | — |
| 24 | C2×Q8 | — | 3 | 3 | no | — |
| 25 | C2×D4 | — | 3 | 3 | no | — |
| 26 | C2×D4 | — | 3 | 3 | no | — |
| 27 | C22×C4 | — | 3 | 3 | no | — |
| 28 | C22×C4 | C2×D3 | 1 | 3 | yes | — |
| 29 | C4⋊C4 | — | 3 | 3 | no | — |
| 30 | C4⋊C4 | — | 3 | 3 | no | — |
| 31 | C22⋊C4 | — | 3 | 3 | no | — |
| 32 | C22⋊C4 | — | 3 | 6 | no | — |
| 33 | C22⋊C4 | — | 3 | 1 | no | — |
| 34 | C42 | — | 3 | 3 | no | — |
| 35 | C2×D3 | — | 4 | 3 | no | — |
| 36 | A4 | #41 | 1 | 1 | yes | characteristic, complemented |
| 37 | C3×C4 | — | 4 | 3 | no | — |
| 38 | Dic3 | — | 4 | 1 | no | — |
| 39 | C23 | — | 3 | 3 | no | — |
| 40 | C23 | C22×D3 | 1 | 1 | yes | characteristic |
| 41 | Q8 | — | 3 | 3 | no | — |
| 42 | Q8 | — | 6 | 1 | no | — |
| 43 | Q8 | S4 | 1 | 1 | yes | characteristic, complemented |
| 44 | D4 | — | 6 | 3 | no | — |
| 45 | D4 | — | 6 | 3 | no | — |
| 46 | D4 | — | 6 | 3 | no | — |
| 47 | D4 | — | 6 | 3 | no | — |
| 48 | C2×C4 | — | 3 | 3 | no | — |
| 49 | C2×C4 | — | 3 | 3 | no | — |
| 50 | C2×C4 | — | 3 | 1 | no | — |
| 51 | C2×C4 | — | 3 | 3 | no | — |
| 52 | C2×C4 | — | 3 | 1 | no | — |
| 53 | C2×C4 | — | 6 | 3 | no | — |
| 54 | C2×C4 | — | 6 | 3 | no | — |
| 55 | C2×C3 | — | 4 | 1 | no | — |
| 56 | D3 | — | 8 | 3 | no | — |
| 57 | C22 | — | 3 | 1 | no | — |
| 58 | C22 | — | 3 | 1 | no | — |
| 59 | C22 | — | 6 | 3 | no | — |
| 60 | C22 | — | 6 | 3 | no | — |
| 61 | C22 | — | 6 | 3 | no | — |
| 62 | C22 | #239 | 1 | 1 | yes | characteristic, complemented |
| 63 | C4 | — | 3 | 3 | no | — |
| 64 | C4 | — | 6 | 1 | no | — |
| 65 | C4 | — | 6 | 3 | no | — |
| 66 | C4 | — | 6 | 1 | no | — |
| 67 | C4 | C2×S4 | 1 | 3 | yes | — |
| 68 | C3 | — | 4 | 1 | no | Sylow |
| 69 | C2 | — | 3 | 1 | no | — |
| 70 | C2 | — | 3 | 1 | no | — |
| 71 | C2 | — | 12 | 3 | no | — |
| 72 | C2 | C22×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×S4 |
| commutator, cosocle | C2×A4 | C23 |
| Fitting | C22×Q8 | D3 |
| socle | C23 | C22×D3 |
| hypercenter | Q8 | S4 |
| hypocenter | A4 | #41 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×A4 | C2 | semidirect |
| 2. C4⋊S4 | C2 | semidirect |
| 3. C4×S4 | C2 | semidirect |
| 10. C22×Q8 | D3 | semidirect |
| 36. A4 | #41 | semidirect |
| 43. Q8 | S4 | semidirect |
| 62. C22 | #239 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×A4 | 2 |
| C4⋊S4 | 2 |
| C4×S4 | 2 |
| #548 | 3 |
| #239 | 4 |