Finite Group DatabasebetaFinite Group DatabasebetaOrder 64.
134 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | H(5,𝔽2) | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | AGL(1,ℤ/8) | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #135 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C2×D8 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #131 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #41 | — | 2 | 2 | no | — |
| 7 | #41 | C22 | 1 | 1 | yes | characteristic |
| 8 | #41 | C22 | 1 | 6 | yes | complemented |
| 9 | C2×D4 | — | 2 | 6 | no | — |
| 10 | C2×D4 | C22 | 1 | 6 | yes | complemented |
| 11 | Q16 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | SD16 | C22 | 1 | 6 | yes | complemented |
| 13 | D8 | C22 | 1 | 3 | yes | complemented |
| 14 | D8 | C22 | 1 | 6 | yes | complemented |
| 15 | M16 | C22 | 1 | 3 | yes | complemented |
| 16 | C2×C8 | C22 | 1 | 3 | yes | complemented |
| 17 | C23 | — | 2 | 6 | no | — |
| 18 | Q8 | D4 | 1 | 1 | yes | characteristic |
| 19 | Q8 | C23 | 1 | 2 | yes | — |
| 20 | D4 | — | 2 | 6 | no | — |
| 21 | D4 | — | 2 | 6 | no | — |
| 22 | D4 | D4 | 1 | 3 | yes | — |
| 23 | D4 | C23 | 1 | 6 | yes | — |
| 24 | C2×C4 | — | 2 | 6 | no | — |
| 25 | C2×C4 | C23 | 1 | 3 | yes | — |
| 26 | C8 | C23 | 1 | 1 | yes | characteristic |
| 27 | C8 | C23 | 1 | 3 | yes | — |
| 28 | C22 | — | 2 | 6 | no | — |
| 29 | C22 | — | 8 | 6 | no | — |
| 30 | C22 | C2×D4 | 1 | 3 | yes | — |
| 31 | C4 | — | 2 | 2 | no | — |
| 32 | C4 | C2×D4 | 1 | 3 | yes | — |
| 33 | C4 | C24 | 1 | 1 | yes | characteristic |
| 34 | C2 | — | 2 | 3 | no | — |
| 35 | C2 | — | 4 | 6 | no | — |
| 36 | C2 | C22×D4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C24 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(5,𝔽2) | C2 | semidirect |
| 2. AGL(1,ℤ/8) | C2 | semidirect |
| 3. #135 | C2 | semidirect |
| 4. C2×D8 | C2 | semidirect |
| 5. #131 | C2 | semidirect |
| 8. #41 | C22 | semidirect |
| 10. C2×D4 | C22 | semidirect |
| 11. Q16 | C22 | semidirect |
| 12. SD16 | C22 | semidirect |
| 13. D8 | C22 | semidirect |
| 14. D8 | C22 | semidirect |
| 15. M16 | C22 | semidirect |
| 16. C2×C8 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(5,𝔽2) | 2 |
| AGL(1,ℤ/8) | 2 |
| #135 | 2 |
| C2×D8 | 2 |
| #131 | 2 |