Finite Group DatabasebetaFinite Group DatabasebetaOrder 192.
65 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C4×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×SL(2,𝔽3) | C2 | 1 | 2 | yes | maximal |
| 3 | #557 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 4 | C2×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 5 | C3×C4⋊C4 | — | 4 | 1 | no | maximal |
| 6 | C4⋊Q8 | — | 3 | 1 | no | — |
| 7 | #125 | — | 3 | 2 | no | — |
| 8 | C4×Q8 | — | 3 | 1 | no | — |
| 9 | C4×Q8 | C2×C3 | 1 | 2 | yes | — |
| 10 | C4×Q8 | C2×C3 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×C4 | — | 4 | 1 | no | — |
| 12 | C2×C3×C4 | — | 4 | 2 | no | — |
| 13 | SL(2,𝔽3) | Q8 | 1 | 1 | yes | characteristic, complemented |
| 14 | C2×Q8 | — | 3 | 1 | no | — |
| 15 | C2×Q8 | C22×C3 | 1 | 1 | yes | characteristic |
| 16 | C4⋊C4 | — | 3 | 1 | no | — |
| 17 | C4⋊C4 | — | 3 | 2 | no | — |
| 18 | C4⋊C4 | — | 3 | 2 | no | — |
| 19 | C4⋊C4 | — | 3 | 2 | no | — |
| 20 | C4⋊C4 | A4 | 1 | 1 | yes | characteristic |
| 21 | C42 | — | 3 | 2 | no | — |
| 22 | C42 | — | 3 | 1 | no | — |
| 23 | C22×C3 | — | 4 | 1 | no | — |
| 24 | C3×C4 | — | 4 | 2 | no | — |
| 25 | C3×C4 | — | 8 | 2 | no | — |
| 26 | Q8 | — | 3 | 1 | no | — |
| 27 | Q8 | — | 6 | 2 | no | — |
| 28 | Q8 | C3×Q8 | 1 | 1 | yes | characteristic |
| 29 | C2×C4 | — | 3 | 1 | no | — |
| 30 | C2×C4 | — | 3 | 1 | no | — |
| 31 | C2×C4 | — | 3 | 2 | no | — |
| 32 | C2×C4 | C2×A4 | 1 | 2 | yes | — |
| 33 | C2×C4 | C2×A4 | 1 | 1 | yes | characteristic |
| 34 | C2×C3 | — | 4 | 1 | no | — |
| 35 | C2×C3 | — | 4 | 1 | no | — |
| 36 | C2×C3 | — | 4 | 1 | no | — |
| 37 | C22 | C22×A4 | 1 | 1 | yes | characteristic |
| 38 | C4 | — | 2 | 2 | no | — |
| 39 | C4 | — | 3 | 1 | no | — |
| 40 | C4 | — | 3 | 1 | no | — |
| 41 | C4 | — | 6 | 2 | no | — |
| 42 | C4 | — | 6 | 1 | no | — |
| 43 | C4 | #41⋊C3 | 1 | 2 | yes | — |
| 44 | C3 | — | 4 | 1 | no | Sylow |
| 45 | C2 | Q8×A4 | 1 | 1 | yes | characteristic |
| 46 | C2 | C2×#41⋊C3 | 1 | 1 | yes | characteristic |
| 47 | C2 | 2−1+4⋊C3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×A4 |
| commutator, cosocle | C2×Q8 | C22×C3 |
| Fitting | #557 | C3 |
| prime radical | C2×SL(2,𝔽3) | C22 |
| hypercenter | C4⋊C4 | A4 |
| hypocenter | Q8 | C3×Q8 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #557 | C3 | semidirect |
| 13. SL(2,𝔽3) | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×SL(2,𝔽3) | 2 |
| #557 | 3 |
| C3×C4⋊C4 | 4 |