Finite Group DatabasebetaFinite Group DatabasebetaOrder 64.
405 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | 2−1+4 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | H(5,𝔽2) | C2 | 1 | 10 | yes | maximal, complemented |
| 3 | C2×#41 | C2 | 1 | 15 | yes | maximal, complemented |
| 4 | #41 | C22 | 1 | 20 | yes | complemented |
| 5 | #41 | C22 | 1 | 60 | yes | complemented |
| 6 | C2×Q8 | C22 | 1 | 15 | yes | complemented |
| 7 | C2×D4 | C22 | 1 | 45 | yes | complemented |
| 8 | C22×C4 | C22 | 1 | 15 | yes | complemented |
| 9 | C23 | C23 | 1 | 15 | yes | — |
| 10 | Q8 | C23 | 1 | 20 | yes | — |
| 11 | D4 | C23 | 1 | 60 | yes | — |
| 12 | C2×C4 | C23 | 1 | 45 | yes | — |
| 13 | C2×C4 | C23 | 1 | 15 | yes | — |
| 14 | C22 | — | 4 | 15 | no | — |
| 15 | C22 | C24 | 1 | 15 | yes | — |
| 16 | C4 | C24 | 1 | 15 | yes | — |
| 17 | C4 | C24 | 1 | 1 | yes | characteristic |
| 18 | C2 | — | 2 | 15 | no | — |
| 19 | C2 | C25 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C24 |
| commutator, Frattini, socle, cosocle | C2 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. 2−1+4 | C2 | semidirect |
| 2. H(5,𝔽2) | C2 | semidirect |
| 3. C2×#41 | C2 | semidirect |
| 4. #41 | C22 | semidirect |
| 5. #41 | C22 | semidirect |
| 6. C2×Q8 | C22 | semidirect |
| 7. C2×D4 | C22 | semidirect |
| 8. C22×C4 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 2−1+4 | 2 |
| H(5,𝔽2) | 2 |
| C2×#41 | 2 |