Finite Group DatabasebetaFinite Group DatabasebetaOrder 96.
39 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #226 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #137 | — | 3 | 1 | no | maximal, Sylow |
| 5 | #67 | — | 4 | 1 | no | maximal |
| 6 | SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 7 | #41 | — | 3 | 1 | no | — |
| 8 | C2×Q8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 9 | Q16 | — | 3 | 1 | no | — |
| 10 | Q16 | — | 3 | 1 | no | — |
| 11 | SD16 | — | 3 | 1 | no | — |
| 12 | SD16 | — | 3 | 1 | no | — |
| 13 | M16 | — | 3 | 1 | no | — |
| 14 | C22×C3 | — | 4 | 1 | no | — |
| 15 | C2×D3 | — | 4 | 1 | no | — |
| 16 | Dic3 | — | 4 | 1 | no | — |
| 17 | Q8 | — | 3 | 1 | no | — |
| 18 | Q8 | — | 3 | 1 | no | — |
| 19 | Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 20 | D4 | — | 3 | 1 | no | — |
| 21 | D4 | — | 6 | 1 | no | — |
| 22 | C2×C4 | — | 3 | 1 | no | — |
| 23 | C2×C4 | — | 6 | 1 | no | — |
| 24 | C8 | — | 3 | 1 | no | — |
| 25 | C8 | — | 3 | 1 | no | — |
| 26 | C2×C3 | — | 4 | 1 | no | — |
| 27 | C2×C3 | — | 8 | 1 | no | — |
| 28 | D3 | — | 8 | 1 | no | — |
| 29 | C22 | — | 6 | 1 | no | — |
| 30 | C22 | S4 | 1 | 1 | yes | characteristic |
| 31 | C4 | — | 3 | 1 | no | — |
| 32 | C4 | — | 3 | 1 | no | — |
| 33 | C4 | — | 6 | 1 | no | — |
| 34 | C3 | — | 4 | 1 | no | Sylow |
| 35 | C2 | — | 2 | 1 | no | — |
| 36 | C2 | — | 12 | 1 | no | — |
| 37 | C2 | C2×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C2×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C22 |
| Fitting | C2×Q8 | D3 |
| hypercenter | C22 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×SL(2,𝔽3) | C2 | semidirect |
| 2. GL(2,𝔽3) | C2 | semidirect |
| 3. #226 | C2 | semidirect |
| 8. C2×Q8 | D3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×SL(2,𝔽3) | 2 |
| GL(2,𝔽3) | 2 |
| #226 | 2 |
| #137 | 3 |
| #67 | 4 |