Finite Group DatabasebetaFinite Group DatabasebetaOrder 64.
106 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C4⋊Q8 | C2 | 1 | 9 | yes | maximal |
| 2 | C4×Q8 | C2 | 1 | 6 | yes | maximal |
| 3 | C2×Q8 | C22 | 1 | 6 | yes | — |
| 4 | C2×Q8 | C22 | 1 | 2 | yes | — |
| 5 | C4⋊C4 | C22 | 1 | 18 | yes | — |
| 6 | C42 | C22 | 1 | 9 | yes | — |
| 7 | Q8 | — | 4 | 6 | no | — |
| 8 | Q8 | Q8 | 1 | 8 | yes | direct product factor |
| 9 | C2×C4 | C23 | 1 | 9 | yes | — |
| 10 | C2×C4 | C23 | 1 | 6 | yes | — |
| 11 | C22 | C24 | 1 | 1 | yes | characteristic |
| 12 | C4 | — | 2 | 9 | no | — |
| 13 | C4 | C2×Q8 | 1 | 12 | yes | — |
| 14 | C2 | C22×Q8 | 1 | 2 | yes | — |
| 15 | C2 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C22 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. Q8 | Q8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4⋊Q8 | 2 |
| C4×Q8 | 2 |