Finite Group DatabasebetaFinite Group DatabasebetaOrder 64.
425 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×Q8 | C2 | 1 | 28 | yes | maximal, direct product factor |
| 2 | C23×C4 | C2 | 1 | 3 | yes | maximal |
| 3 | C24 | C22 | 1 | 1 | yes | characteristic |
| 4 | C2×Q8 | C22 | 1 | 112 | yes | direct product factor |
| 5 | C22×C4 | C22 | 1 | 42 | yes | — |
| 6 | C23 | Q8 | 1 | 8 | yes | direct product factor |
| 7 | C23 | C23 | 1 | 7 | yes | — |
| 8 | Q8 | C23 | 1 | 64 | yes | direct product factor |
| 9 | C2×C4 | C23 | 1 | 84 | yes | — |
| 10 | C22 | C2×Q8 | 1 | 28 | yes | direct product factor |
| 11 | C22 | C24 | 1 | 7 | yes | — |
| 12 | C4 | C24 | 1 | 24 | yes | — |
| 13 | C2 | C22×Q8 | 1 | 14 | yes | direct product factor |
| 14 | C2 | C25 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22 |
| commutator, Frattini, cosocle | C2 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q8 | C2 | direct (N × H) |
| 4. C2×Q8 | C22 | direct (N × H) |
| 6. C23 | Q8 | direct (N × H) |
| 8. Q8 | C23 | direct (N × H) |
| 10. C22 | C2×Q8 | direct (N × H) |
| 13. C2 | C22×Q8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8 | 2 |
| C23×C4 | 2 |