Finite Group DatabasebetaFinite Group DatabasebetaOrder 192.
58 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #997 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | B(4,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 3 | C22×A4 | — | 4 | 1 | no | maximal |
| 4 | H(5,𝔽2) | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 5 | #120 | — | 3 | 1 | no | — |
| 6 | #99 | — | 3 | 1 | no | — |
| 7 | C2×A4 | — | 4 | 2 | no | — |
| 8 | C2×A4 | — | 4 | 1 | no | — |
| 9 | C2×A4 | — | 8 | 1 | no | — |
| 10 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 11 | C24 | A4 | 1 | 1 | yes | characteristic, complemented |
| 12 | #41 | — | 6 | 1 | no | — |
| 13 | C2×D4 | — | 3 | 1 | no | — |
| 14 | C2×D4 | — | 6 | 1 | no | — |
| 15 | C2×D4 | — | 6 | 1 | no | — |
| 16 | C22⋊C4 | — | 3 | 1 | no | — |
| 17 | C22⋊C4 | — | 6 | 1 | no | — |
| 18 | C22×C3 | — | 16 | 1 | no | — |
| 19 | A4 | — | 4 | 1 | no | — |
| 20 | A4 | — | 8 | 1 | no | — |
| 21 | C23 | — | 2 | 1 | no | — |
| 22 | C23 | — | 3 | 1 | no | — |
| 23 | C23 | — | 4 | 2 | no | — |
| 24 | C23 | — | 6 | 1 | no | — |
| 25 | C23 | C2×A4 | 1 | 1 | yes | characteristic |
| 26 | Q8 | — | 2 | 1 | no | — |
| 27 | D4 | — | 6 | 2 | no | — |
| 28 | D4 | — | 6 | 1 | no | — |
| 29 | D4 | — | 12 | 2 | no | — |
| 30 | C2×C4 | — | 3 | 1 | no | — |
| 31 | C2×C4 | — | 6 | 1 | no | — |
| 32 | C2×C4 | — | 6 | 1 | no | — |
| 33 | C2×C3 | — | 16 | 2 | no | — |
| 34 | C2×C3 | — | 16 | 1 | no | — |
| 35 | C22 | — | 3 | 1 | no | — |
| 36 | C22 | — | 4 | 1 | no | — |
| 37 | C22 | — | 4 | 1 | no | — |
| 38 | C22 | — | 6 | 2 | no | — |
| 39 | C22 | — | 6 | 1 | no | — |
| 40 | C22 | — | 8 | 1 | no | — |
| 41 | C22 | — | 12 | 1 | no | — |
| 42 | C22 | — | 12 | 1 | no | — |
| 43 | C4 | — | 6 | 1 | no | — |
| 44 | C4 | — | 12 | 1 | no | — |
| 45 | C3 | — | 16 | 1 | no | Sylow |
| 46 | C2 | — | 4 | 2 | no | — |
| 47 | C2 | — | 6 | 1 | no | — |
| 48 | C2 | — | 12 | 1 | no | — |
| 49 | C2 | #120⋊C3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2 | #120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C2×C3 |
| Frattini | C23 | C2×A4 |
| Fitting | B(4,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #997 | C2 | semidirect |
| 2. B(4,𝔽2) | C3 | semidirect |
| 4. H(5,𝔽2) | C2×C3 | semidirect |
| 11. C24 | A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #997 | 2 |
| B(4,𝔽2) | 3 |
| C22×A4 | 4 |