Finite Group DatabasebetaFinite Group DatabasebetaOrder 192.
73 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #994 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #131⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #576 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 4 | #41⋊C3 | — | 2 | 2 | no | — |
| 5 | #41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 6 | C3×Q16 | — | 4 | 1 | no | maximal |
| 7 | H(5,𝔽2) | C2×C3 | 1 | 2 | yes | — |
| 8 | AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 9 | #135 | — | 3 | 1 | no | — |
| 10 | C2×D8 | — | 3 | 1 | no | — |
| 11 | #131 | C2×C3 | 1 | 1 | yes | characteristic |
| 12 | C3×Q8 | — | 4 | 2 | no | — |
| 13 | SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 14 | C3×C8 | — | 4 | 1 | no | — |
| 15 | #41 | — | 2 | 2 | no | — |
| 16 | #41 | — | 3 | 2 | no | — |
| 17 | #41 | C22×C3 | 1 | 1 | yes | characteristic |
| 18 | C2×D4 | — | 3 | 2 | no | — |
| 19 | C2×D4 | — | 6 | 2 | no | — |
| 20 | Q16 | A4 | 1 | 1 | yes | characteristic |
| 21 | SD16 | — | 3 | 2 | no | — |
| 22 | D8 | — | 3 | 2 | no | — |
| 23 | D8 | — | 3 | 1 | no | — |
| 24 | M16 | — | 3 | 1 | no | — |
| 25 | C2×C8 | — | 3 | 1 | no | — |
| 26 | C3×C4 | — | 4 | 1 | no | — |
| 27 | C3×C4 | — | 8 | 2 | no | — |
| 28 | C23 | — | 6 | 2 | no | — |
| 29 | Q8 | C3×D4 | 1 | 1 | yes | characteristic |
| 30 | Q8 | C2×A4 | 1 | 2 | yes | — |
| 31 | D4 | — | 3 | 1 | no | — |
| 32 | D4 | — | 3 | 2 | no | — |
| 33 | D4 | — | 6 | 2 | no | — |
| 34 | D4 | — | 6 | 2 | no | — |
| 35 | C2×C4 | — | 3 | 1 | no | — |
| 36 | C2×C4 | — | 6 | 2 | no | — |
| 37 | C8 | — | 3 | 1 | no | — |
| 38 | C8 | C2×A4 | 1 | 1 | yes | characteristic |
| 39 | C2×C3 | — | 4 | 1 | no | — |
| 40 | C22 | — | 3 | 1 | no | — |
| 41 | C22 | — | 6 | 2 | no | — |
| 42 | C22 | — | 24 | 2 | no | — |
| 43 | C4 | — | 2 | 2 | no | — |
| 44 | C4 | — | 3 | 1 | no | — |
| 45 | C4 | C22×A4 | 1 | 1 | yes | characteristic |
| 46 | C3 | — | 4 | 1 | no | Sylow |
| 47 | C2 | — | 6 | 1 | no | — |
| 48 | C2 | — | 12 | 2 | no | — |
| 49 | C2 | D4×A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×A4 |
| commutator, cosocle | #41 | C22×C3 |
| Frattini | C4 | C22×A4 |
| Fitting | #576 | C3 |
| hypercenter | Q16 | A4 |
| hypocenter | Q8 | C3×D4 |
| LCS | G ▷ #41 ▷ Q8 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #994 | C2 | semidirect |
| 2. #131⋊C3 | C2 | semidirect |
| 3. #576 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #994 | 2 |
| #131⋊C3 | 2 |
| #576 | 3 |
| C3×Q16 | 4 |