Finite Group DatabasebetaFinite Group DatabasebetaOrder 6048.
36 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #6249 | — | 28 | 1 | no | maximal |
| 2 | GL(3,𝔽2) | — | 36 | 1 | no | maximal |
| 3 | #1088 | — | 28 | 1 | no | — |
| 4 | GU(2,𝔽3) | — | 63 | 1 | no | maximal |
| 5 | #857 | — | 63 | 1 | no | maximal |
| 6 | #272 | — | 28 | 1 | no | — |
| 7 | #41⋊C3 | — | 63 | 1 | no | — |
| 8 | C42⋊C3 | — | 63 | 1 | no | — |
| 9 | #104 | — | 189 | 1 | no | Sylow |
| 10 | H(3,𝔽3) | — | 28 | 1 | no | Sylow |
| 11 | S4 | — | 252 | 1 | no | — |
| 12 | SL(2,𝔽3) | — | 63 | 1 | no | — |
| 13 | C3⋊C8 | — | 252 | 1 | no | — |
| 14 | ΓL(1,𝔽8) | — | 288 | 1 | no | — |
| 15 | C3×D3 | — | 336 | 1 | no | — |
| 16 | #41 | — | 63 | 1 | no | — |
| 17 | M16 | — | 189 | 1 | no | — |
| 18 | C42 | — | 63 | 1 | no | — |
| 19 | A4 | — | 252 | 1 | no | — |
| 20 | C3×C4 | — | 252 | 1 | no | — |
| 21 | C32 | — | 112 | 1 | no | — |
| 22 | Q8 | — | 63 | 1 | no | — |
| 23 | D4 | — | 189 | 1 | no | — |
| 24 | C2×C4 | — | 189 | 1 | no | — |
| 25 | C8 | — | 378 | 1 | no | — |
| 26 | C7 | — | 288 | 1 | no | Sylow |
| 27 | C2×C3 | — | 252 | 1 | no | — |
| 28 | D3 | — | 336 | 1 | no | — |
| 29 | C22 | — | 63 | 1 | no | — |
| 30 | C4 | — | 63 | 1 | no | — |
| 31 | C4 | — | 189 | 1 | no | — |
| 32 | C3 | — | 28 | 1 | no | — |
| 33 | C3 | — | 336 | 1 | no | — |
| 34 | C2 | — | 63 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6249 | 28 |
| GL(3,𝔽2) | 36 |
| GU(2,𝔽3) | 63 |
| #857 | 63 |