Finite Group DatabasebetaFinite Group DatabasebetaOrder 352.
62 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C11×#41 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C11×M16 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×C8×C11 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | Q8×C11 | C4 | 1 | 1 | yes | characteristic |
| 5 | D4×C11 | C4 | 1 | 3 | yes | — |
| 6 | C2×C4×C11 | C22 | 1 | 3 | yes | — |
| 7 | C8×C11 | C22 | 1 | 3 | yes | — |
| 8 | C8×C11 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×C11 | C2×C4 | 1 | 3 | yes | — |
| 10 | C4×C11 | C2×C4 | 1 | 3 | yes | — |
| 11 | C4×C11 | C23 | 1 | 1 | yes | characteristic |
| 12 | #131 | C11 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 13 | C2×C11 | — | 2 | 3 | no | — |
| 14 | C2×C11 | C22×C4 | 1 | 1 | yes | characteristic |
| 15 | #41 | C2×C11 | 1 | 1 | yes | characteristic |
| 16 | M16 | C2×C11 | 1 | 3 | yes | complemented |
| 17 | C2×C8 | C2×C11 | 1 | 3 | yes | complemented |
| 18 | C11 | #131 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 19 | Q8 | C4×C11 | 1 | 1 | yes | characteristic |
| 20 | D4 | C4×C11 | 1 | 3 | yes | — |
| 21 | C2×C4 | C22×C11 | 1 | 3 | yes | — |
| 22 | C8 | C22×C11 | 1 | 1 | yes | characteristic |
| 23 | C8 | C22×C11 | 1 | 3 | yes | — |
| 24 | C22 | C2×C4×C11 | 1 | 3 | yes | — |
| 25 | C4 | C2×C4×C11 | 1 | 3 | yes | — |
| 26 | C4 | C23×C11 | 1 | 1 | yes | characteristic |
| 27 | C2 | — | 2 | 3 | no | — |
| 28 | C2 | C22×C4×C11 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C8×C11 | C22 |
| commutator | C2 | C22×C4×C11 |
| Frattini, cosocle | C4 | C23×C11 |
| socle | C2×C11 | C22×C4 |
| prime radical | C11×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C11 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C11×M16 | C2 | semidirect |
| 3. C2×C8×C11 | C2 | semidirect |
| 12. #131 | C11 | direct (N × H) |
| 16. M16 | C2×C11 | semidirect |
| 17. C2×C8 | C2×C11 | semidirect |
| 18. C11 | #131 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C11×#41 | 2 |
| C11×M16 | 2 |
| C2×C8×C11 | 2 |
| #131 | 11 |