Finite Group DatabasebetaFinite Group DatabasebetaOrder 192.
72 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | 2−1+4⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | GU(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #454 | — | 3 | 1 | no | maximal, Sylow |
| 5 | #41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 7 | GL(2,𝔽3) | — | 2 | 1 | no | — |
| 8 | #213 | — | 4 | 1 | no | maximal |
| 9 | 2−1+4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 11 | AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 12 | #124 | — | 3 | 1 | no | — |
| 13 | #104 | — | 3 | 1 | no | — |
| 14 | #104 | — | 3 | 1 | no | — |
| 15 | #101 | — | 3 | 1 | no | — |
| 16 | C3×D4 | — | 4 | 1 | no | — |
| 17 | D12 | — | 4 | 1 | no | — |
| 18 | SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 19 | C3⋊C8 | — | 4 | 1 | no | — |
| 20 | #41 | — | 3 | 1 | no | — |
| 21 | #41 | — | 6 | 1 | no | — |
| 22 | #41 | C2×D3 | 1 | 1 | yes | characteristic |
| 23 | C2×Q8 | — | 2 | 1 | no | — |
| 24 | C2×Q8 | — | 3 | 1 | no | — |
| 25 | C2×D4 | — | 3 | 1 | no | — |
| 26 | SD16 | — | 6 | 1 | no | — |
| 27 | SD16 | — | 6 | 1 | no | — |
| 28 | D8 | — | 6 | 1 | no | — |
| 29 | D8 | — | 6 | 1 | no | — |
| 30 | M16 | — | 3 | 1 | no | — |
| 31 | M16 | — | 3 | 1 | no | — |
| 32 | C22⋊C4 | — | 6 | 2 | no | — |
| 33 | C42 | — | 3 | 1 | no | — |
| 34 | C22×C3 | — | 8 | 1 | no | — |
| 35 | C2×D3 | — | 8 | 1 | no | — |
| 36 | C3×C4 | — | 4 | 1 | no | — |
| 37 | C23 | — | 6 | 1 | no | — |
| 38 | Q8 | — | 3 | 1 | no | — |
| 39 | Q8 | — | 6 | 1 | no | — |
| 40 | Q8 | #67 | 1 | 1 | yes | characteristic |
| 41 | D4 | — | 3 | 1 | no | — |
| 42 | D4 | — | 6 | 1 | no | — |
| 43 | D4 | — | 6 | 1 | no | — |
| 44 | D4 | — | 6 | 1 | no | — |
| 45 | D4 | S4 | 1 | 1 | yes | characteristic |
| 46 | C2×C4 | — | 3 | 1 | no | — |
| 47 | C2×C4 | — | 3 | 2 | no | — |
| 48 | C2×C4 | — | 6 | 1 | no | — |
| 49 | C2×C4 | — | 6 | 1 | no | — |
| 50 | C8 | — | 6 | 1 | no | — |
| 51 | C8 | — | 6 | 1 | no | — |
| 52 | C2×C3 | — | 4 | 1 | no | — |
| 53 | C2×C3 | — | 16 | 1 | no | — |
| 54 | D3 | — | 16 | 1 | no | — |
| 55 | C22 | — | 2 | 1 | no | — |
| 56 | C22 | — | 3 | 1 | no | — |
| 57 | C22 | — | 12 | 2 | no | — |
| 58 | C22 | — | 12 | 1 | no | — |
| 59 | C4 | — | 3 | 1 | no | — |
| 60 | C4 | — | 6 | 1 | no | — |
| 61 | C4 | — | 12 | 1 | no | — |
| 62 | C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 63 | C3 | — | 4 | 1 | no | Sylow |
| 64 | C2 | — | 4 | 1 | no | — |
| 65 | C2 | — | 6 | 1 | no | — |
| 66 | C2 | — | 24 | 1 | no | — |
| 67 | C2 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | GL(2,ℤ/4) |
| commutator, cosocle | #41⋊C3 | C22 |
| Frattini | C4 | C2×S4 |
| Fitting | 2−1+4 | D3 |
| hypercenter | D4 | S4 |
| hypocenter | SL(2,𝔽3) | D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. 2−1+4⋊C3 | C2 | semidirect |
| 2. #986 | C2 | semidirect |
| 3. GU(2,𝔽3) | C2 | semidirect |
| 9. 2−1+4 | D3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 2−1+4⋊C3 | 2 |
| #986 | 2 |
| GU(2,𝔽3) | 2 |
| #454 | 3 |
| #213 | 4 |