Finite Group DatabasebetaFinite Group DatabasebetaOrder 32.
118 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C24 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C2×D4 | C2 | 1 | 12 | yes | maximal, direct product factor |
| 3 | C22×C4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C23 | — | 2 | 8 | no | — |
| 5 | C23 | C22 | 1 | 1 | yes | characteristic |
| 6 | C23 | C22 | 1 | 12 | yes | complemented |
| 7 | D4 | C22 | 1 | 16 | yes | direct product factor |
| 8 | C2×C4 | C22 | 1 | 6 | yes | complemented |
| 9 | C22 | — | 2 | 24 | no | — |
| 10 | C22 | D4 | 1 | 4 | yes | direct product factor |
| 11 | C22 | C23 | 1 | 3 | yes | — |
| 12 | C22 | C23 | 1 | 8 | yes | complemented |
| 13 | C4 | C23 | 1 | 4 | yes | complemented |
| 14 | C2 | — | 2 | 8 | no | — |
| 15 | C2 | C2×D4 | 1 | 6 | yes | direct product factor |
| 16 | C2 | C24 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22 |
| commutator, Frattini, cosocle | C2 | C24 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24 | C2 | semidirect |
| 2. C2×D4 | C2 | direct (N × H) |
| 3. C22×C4 | C2 | semidirect |
| 6. C23 | C22 | semidirect |
| 7. D4 | C22 | direct (N × H) |
| 8. C2×C4 | C22 | semidirect |
| 10. C22 | D4 | direct (N × H) |
| 12. C22 | C23 | semidirect |
| 13. C4 | C23 | semidirect |
| 15. C2 | C2×D4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24 | 2 |
| C2×D4 | 2 |
| C22×C4 | 2 |