Finite Group DatabasebetaFinite Group DatabasebetaOrder 224.
30 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×M16 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #1136 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C7⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×C4×C7 | C22 | 1 | 1 | yes | characteristic |
| 5 | C7×C8 | — | 2 | 1 | no | — |
| 6 | C7⋊C8 | — | 2 | 1 | no | — |
| 7 | C7⋊C8 | C4 | 1 | 2 | yes | — |
| 8 | #108 | — | 7 | 1 | no | maximal, Sylow |
| 9 | C22×C7 | Q8 | 1 | 1 | yes | characteristic |
| 10 | C4×C7 | C2×C4 | 1 | 1 | yes | characteristic |
| 11 | C4×C7 | D4 | 1 | 1 | yes | characteristic |
| 12 | M16 | — | 7 | 1 | no | — |
| 13 | M16 | D7 | 1 | 1 | yes | characteristic |
| 14 | C2×C8 | — | 7 | 1 | no | — |
| 15 | C2×C7 | — | 2 | 1 | no | — |
| 16 | C2×C7 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 17 | C2×C4 | C2×D7 | 1 | 1 | yes | characteristic |
| 18 | C8 | — | 2 | 1 | no | — |
| 19 | C8 | — | 7 | 2 | no | — |
| 20 | C8 | — | 14 | 1 | no | — |
| 21 | C7 | #108 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 22 | C22 | Dic14 | 1 | 1 | yes | characteristic |
| 23 | C4 | C4×D7 | 1 | 1 | yes | characteristic |
| 24 | C4 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 25 | C2 | — | 2 | 1 | no | — |
| 26 | C2 | Dic7⋊C4 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 21. C7 | #108 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×M16 | 2 |
| #1136 | 2 |
| C2×C7⋊C8 | 2 |
| #108 | 7 |