Finite Group DatabasebetaFinite Group DatabasebetaOrder 408.
32 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C3×C17 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×D51 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | Dic51 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C22⋊D17 | — | 3 | 1 | no | maximal, Hall |
| 5 | C2×C3×C17 | — | 2 | 1 | no | — |
| 6 | C2×C3×C17 | C22 | 1 | 1 | yes | characteristic |
| 7 | D51 | — | 2 | 1 | no | — |
| 8 | C22×C17 | D3 | 1 | 1 | yes | characteristic, complemented |
| 9 | C2×D17 | — | 3 | 1 | no | — |
| 10 | Dic17 | — | 3 | 1 | no | — |
| 11 | C3×C17 | D4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 12 | C2×C17 | — | 2 | 1 | no | — |
| 13 | C2×C17 | C2×D3 | 1 | 1 | yes | characteristic |
| 14 | D17 | — | 6 | 1 | no | — |
| 15 | #67 | — | 17 | 1 | no | maximal, Hall |
| 16 | C17 | #67 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 17 | C22×C3 | D17 | 1 | 1 | yes | characteristic, complemented |
| 18 | C2×D3 | — | 17 | 1 | no | — |
| 19 | Dic3 | — | 17 | 1 | no | — |
| 20 | D4 | — | 51 | 1 | no | Sylow |
| 21 | C2×C3 | — | 2 | 1 | no | — |
| 22 | C2×C3 | C2×D17 | 1 | 1 | yes | characteristic |
| 23 | D3 | — | 34 | 1 | no | — |
| 24 | C22 | — | 51 | 1 | no | — |
| 25 | C22 | D51 | 1 | 1 | yes | characteristic, complemented |
| 26 | C4 | — | 51 | 1 | no | — |
| 27 | C3 | C22⋊D17 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 28 | C2 | — | 2 | 1 | no | — |
| 29 | C2 | — | 102 | 1 | no | — |
| 30 | C2 | C2×D51 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D51 |
| commutator, socle, cosocle | C2×C3×C17 | C22 |
| Fitting | C22×C3×C17 | C2 |
| hypercenter | C22 | D51 |
| hypocenter | C3×C17 | D4 |
| LCS | G ▷ C2×C3×C17 ▷ C3×C17 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×C17 | C2 | semidirect |
| 2. C2×D51 | C2 | semidirect |
| 3. Dic51 | C2 | semidirect |
| 8. C22×C17 | D3 | semidirect |
| 11. C3×C17 | D4 | semidirect |
| 16. C17 | #67 | semidirect |
| 17. C22×C3 | D17 | semidirect |
| 25. C22 | D51 | semidirect |
| 27. C3 | C22⋊D17 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C17 | 2 |
| C2×D51 | 2 |
| Dic51 | 2 |
| C22⋊D17 | 3 |
| #67 | 17 |