Finite Group DatabasebetaFinite Group DatabasebetaOrder 192.
146 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #994 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #986 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #985 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #576 | — | 3 | 1 | no | maximal, Sylow |
| 5 | #239 | — | 4 | 1 | no | maximal |
| 6 | #41⋊C3 | C22 | 1 | 3 | yes | complemented |
| 7 | GL(2,𝔽3) | C22 | 1 | 3 | yes | complemented |
| 8 | #226 | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | H(5,𝔽2) | — | 3 | 1 | no | — |
| 10 | H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | AGL(1,ℤ/8) | — | 3 | 3 | no | — |
| 12 | AGL(1,ℤ/8) | — | 3 | 3 | no | — |
| 13 | #135 | — | 3 | 3 | no | — |
| 14 | C2×D8 | — | 3 | 3 | no | — |
| 15 | #131 | — | 3 | 1 | no | — |
| 16 | C3×Q8 | — | 4 | 1 | no | — |
| 17 | D12 | — | 4 | 3 | no | — |
| 18 | C4×D3 | — | 4 | 3 | no | — |
| 19 | SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 20 | #41 | — | 3 | 1 | no | — |
| 21 | #41 | — | 3 | 3 | no | — |
| 22 | #41 | — | 6 | 1 | no | — |
| 23 | #41 | C2×D3 | 1 | 3 | yes | — |
| 24 | C2×D4 | — | 3 | 3 | no | — |
| 25 | C2×D4 | — | 3 | 3 | no | — |
| 26 | C2×D4 | — | 6 | 3 | no | — |
| 27 | Q16 | — | 3 | 1 | no | — |
| 28 | SD16 | — | 3 | 3 | no | — |
| 29 | SD16 | — | 3 | 3 | no | — |
| 30 | D8 | — | 3 | 3 | no | — |
| 31 | D8 | — | 3 | 6 | no | — |
| 32 | M16 | — | 3 | 3 | no | — |
| 33 | C2×C8 | — | 3 | 3 | no | — |
| 34 | C2×D3 | — | 4 | 3 | no | — |
| 35 | C3×C4 | — | 4 | 3 | no | — |
| 36 | Dic3 | — | 4 | 1 | no | — |
| 37 | C23 | — | 6 | 1 | no | — |
| 38 | C23 | — | 6 | 3 | no | — |
| 39 | Q8 | — | 3 | 1 | no | — |
| 40 | Q8 | S4 | 1 | 1 | yes | characteristic |
| 41 | Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 42 | D4 | — | 3 | 3 | no | — |
| 43 | D4 | — | 3 | 3 | no | — |
| 44 | D4 | — | 3 | 3 | no | — |
| 45 | D4 | — | 6 | 3 | no | — |
| 46 | D4 | — | 6 | 3 | no | — |
| 47 | C2×C4 | — | 3 | 3 | no | — |
| 48 | C2×C4 | — | 6 | 3 | no | — |
| 49 | C8 | — | 3 | 1 | no | — |
| 50 | C8 | — | 3 | 3 | no | — |
| 51 | C2×C3 | — | 4 | 1 | no | — |
| 52 | D3 | — | 8 | 3 | no | — |
| 53 | C22 | — | 3 | 3 | no | — |
| 54 | C22 | — | 6 | 3 | no | — |
| 55 | C22 | — | 24 | 1 | no | — |
| 56 | C22 | — | 24 | 3 | no | — |
| 57 | C4 | — | 3 | 1 | no | — |
| 58 | C4 | — | 6 | 1 | no | — |
| 59 | C4 | C2×S4 | 1 | 3 | yes | — |
| 60 | C3 | — | 4 | 1 | no | Sylow |
| 61 | C2 | — | 6 | 3 | no | — |
| 62 | C2 | — | 12 | 3 | no | — |
| 63 | C2 | C22×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23 |
| Fitting | H(5,𝔽2) | D3 |
| hypercenter | Q8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #994 | C2 | semidirect |
| 2. #986 | C2 | semidirect |
| 3. #985 | C2 | semidirect |
| 6. #41⋊C3 | C22 | semidirect |
| 7. GL(2,𝔽3) | C22 | semidirect |
| 8. #226 | C22 | semidirect |
| 10. H(5,𝔽2) | D3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #994 | 2 |
| #986 | 2 |
| #985 | 2 |
| #576 | 3 |
| #239 | 4 |