Finite Group DatabasebetaFinite Group DatabasebetaOrder 192.
70 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Q8×A4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | A4⋊Q8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | A4⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #451 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C4×A4 | — | 2 | 1 | no | — |
| 6 | C4×A4 | C22 | 1 | 1 | yes | characteristic |
| 7 | SL(2,ℤ/4) | — | 2 | 1 | no | — |
| 8 | #216 | — | 4 | 1 | no | maximal |
| 9 | C22×Q8 | D3 | 1 | 1 | yes | characteristic |
| 10 | C2×Q16 | — | 3 | 1 | no | — |
| 11 | C2×Q16 | — | 3 | 1 | no | — |
| 12 | C22⋊Q8 | — | 3 | 1 | no | — |
| 13 | #103 | — | 3 | 1 | no | — |
| 14 | #103 | — | 3 | 1 | no | — |
| 15 | C22⋊C8 | — | 3 | 1 | no | — |
| 16 | C2×A4 | D4 | 1 | 1 | yes | characteristic |
| 17 | C3×Q8 | — | 4 | 1 | no | — |
| 18 | Dic6 | — | 4 | 1 | no | — |
| 19 | C3⋊C8 | — | 4 | 1 | no | — |
| 20 | C2×Q8 | — | 3 | 1 | no | — |
| 21 | C2×Q8 | — | 3 | 1 | no | — |
| 22 | C2×Q8 | — | 3 | 1 | no | — |
| 23 | C2×Q8 | — | 6 | 1 | no | — |
| 24 | C22×C4 | — | 2 | 1 | no | — |
| 25 | C22×C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 26 | Q16 | — | 6 | 2 | no | — |
| 27 | Q16 | — | 6 | 1 | no | — |
| 28 | Q16 | — | 6 | 1 | no | — |
| 29 | C2×C8 | — | 3 | 1 | no | — |
| 30 | C2×C8 | — | 3 | 1 | no | — |
| 31 | C4⋊C4 | — | 3 | 1 | no | — |
| 32 | C4⋊C4 | — | 6 | 1 | no | — |
| 33 | C22⋊C4 | — | 6 | 1 | no | — |
| 34 | A4 | Q16 | 1 | 1 | yes | characteristic, complemented |
| 35 | C3×C4 | — | 4 | 1 | no | — |
| 36 | C3×C4 | — | 8 | 1 | no | — |
| 37 | Dic3 | — | 8 | 1 | no | — |
| 38 | C23 | #67 | 1 | 1 | yes | characteristic |
| 39 | Q8 | — | 3 | 2 | no | — |
| 40 | Q8 | — | 3 | 1 | no | — |
| 41 | Q8 | — | 6 | 1 | no | — |
| 42 | Q8 | — | 6 | 1 | no | — |
| 43 | Q8 | — | 6 | 1 | no | — |
| 44 | Q8 | S4 | 1 | 1 | yes | characteristic |
| 45 | C2×C4 | — | 3 | 1 | no | — |
| 46 | C2×C4 | — | 3 | 1 | no | — |
| 47 | C2×C4 | — | 6 | 1 | no | — |
| 48 | C2×C4 | — | 6 | 1 | no | — |
| 49 | C2×C4 | — | 6 | 1 | no | — |
| 50 | C2×C4 | — | 6 | 1 | no | — |
| 51 | C8 | — | 6 | 1 | no | — |
| 52 | C8 | — | 6 | 1 | no | — |
| 53 | C2×C3 | — | 4 | 1 | no | — |
| 54 | C22 | — | 3 | 1 | no | — |
| 55 | C22 | — | 3 | 1 | no | — |
| 56 | C22 | #216 | 1 | 1 | yes | characteristic, complemented |
| 57 | C4 | — | 2 | 1 | no | — |
| 58 | C4 | — | 3 | 1 | no | — |
| 59 | C4 | — | 6 | 1 | no | — |
| 60 | C4 | — | 12 | 1 | no | — |
| 61 | C4 | — | 12 | 1 | no | — |
| 62 | C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 63 | C3 | — | 4 | 1 | no | Sylow |
| 64 | C2 | — | 3 | 1 | no | — |
| 65 | C2 | — | 3 | 1 | no | — |
| 66 | C2 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | GL(2,ℤ/4) |
| commutator, cosocle | C4×A4 | C22 |
| Frattini | C4 | C2×S4 |
| Fitting | C22×Q8 | D3 |
| socle | C23 | #67 |
| prime radical | C2×A4 | D4 |
| hypercenter | Q8 | S4 |
| hypocenter | A4 | Q16 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 34. A4 | Q16 | semidirect |
| 56. C22 | #216 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×A4 | 2 |
| A4⋊Q8 | 2 |
| A4⋊C8 | 2 |
| #451 | 3 |
| #216 | 4 |