Finite Group DatabasebetaFinite Group DatabasebetaOrder 192.
81 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #859 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #419 | — | 3 | 1 | no | maximal, Sylow |
| 4 | C2×SL(2,𝔽3) | C4 | 1 | 2 | yes | complemented |
| 5 | C2×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 6 | #217 | — | 4 | 1 | no | maximal |
| 7 | C22×Q8 | D3 | 1 | 1 | yes | characteristic |
| 8 | C2×M16 | — | 3 | 1 | no | — |
| 9 | #117 | — | 3 | 1 | no | — |
| 10 | #103 | — | 3 | 2 | no | — |
| 11 | #103 | — | 3 | 2 | no | — |
| 12 | C23×C3 | — | 4 | 1 | no | — |
| 13 | C2×Dic3 | — | 4 | 2 | no | — |
| 14 | SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 15 | C2×Q8 | — | 3 | 1 | no | — |
| 16 | C2×Q8 | — | 3 | 2 | no | — |
| 17 | C2×Q8 | Dic3 | 1 | 2 | yes | complemented |
| 18 | C2×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 19 | C22×C4 | — | 3 | 1 | no | — |
| 20 | M16 | — | 6 | 2 | no | — |
| 21 | C2×C8 | — | 3 | 2 | no | — |
| 22 | C4⋊C4 | — | 3 | 2 | no | — |
| 23 | C22⋊C4 | — | 6 | 1 | no | — |
| 24 | C42 | — | 6 | 1 | no | — |
| 25 | C22×C3 | — | 4 | 2 | no | — |
| 26 | C22×C3 | — | 4 | 1 | no | — |
| 27 | C22×C3 | — | 8 | 2 | no | — |
| 28 | Dic3 | — | 8 | 2 | no | — |
| 29 | C23 | S4 | 1 | 1 | yes | characteristic |
| 30 | Q8 | — | 3 | 2 | no | — |
| 31 | Q8 | — | 3 | 1 | no | — |
| 32 | Q8 | — | 6 | 1 | no | — |
| 33 | Q8 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 34 | C2×C4 | — | 3 | 2 | no | — |
| 35 | C2×C4 | — | 3 | 2 | no | — |
| 36 | C2×C4 | — | 3 | 1 | no | — |
| 37 | C2×C4 | — | 3 | 1 | no | — |
| 38 | C2×C4 | — | 6 | 2 | no | — |
| 39 | C8 | — | 6 | 2 | no | — |
| 40 | C2×C3 | — | 4 | 2 | no | — |
| 41 | C2×C3 | — | 4 | 1 | no | — |
| 42 | C2×C3 | — | 8 | 2 | no | — |
| 43 | C22 | — | 2 | 2 | no | — |
| 44 | C22 | SL(2,ℤ/4) | 1 | 2 | yes | — |
| 45 | C22 | C2×S4 | 1 | 1 | yes | characteristic |
| 46 | C4 | — | 3 | 1 | no | — |
| 47 | C4 | — | 3 | 1 | no | — |
| 48 | C4 | — | 3 | 2 | no | — |
| 49 | C4 | — | 12 | 2 | no | — |
| 50 | C3 | — | 4 | 1 | no | Sylow |
| 51 | C2 | — | 2 | 2 | no | — |
| 52 | C2 | #983 | 1 | 2 | yes | — |
| 53 | C2 | C2×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C2×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C4 |
| Fitting | C22×Q8 | D3 |
| prime radical | C22×SL(2,𝔽3) | C2 |
| cosocle | C2×SL(2,𝔽3) | C22 |
| hypercenter | C23 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #859 | C2 | semidirect |
| 4. C2×SL(2,𝔽3) | C4 | semidirect |
| 17. C2×Q8 | Dic3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×SL(2,𝔽3) | 2 |
| #859 | 2 |
| #419 | 3 |
| #217 | 4 |