Finite Group DatabasebetaFinite Group DatabasebetaOrder 64.
29 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #135 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | M32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #106 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #41 | — | 2 | 1 | no | — |
| 5 | Q16 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | SD16 | — | 2 | 1 | no | — |
| 7 | D8 | C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×C8 | C22 | 1 | 1 | yes | characteristic |
| 9 | C4⋊C4 | — | 2 | 1 | no | — |
| 10 | C16 | — | 2 | 1 | no | — |
| 11 | Q8 | — | 2 | 1 | no | — |
| 12 | D4 | — | 2 | 1 | no | — |
| 13 | D4 | — | 4 | 1 | no | — |
| 14 | C2×C4 | — | 4 | 1 | no | — |
| 15 | C2×C4 | — | 4 | 1 | no | — |
| 16 | C2×C4 | D4 | 1 | 1 | yes | characteristic |
| 17 | C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 18 | C8 | D4 | 1 | 1 | yes | characteristic |
| 19 | C22 | — | 4 | 1 | no | — |
| 20 | C22 | D8 | 1 | 1 | yes | characteristic |
| 21 | C4 | — | 4 | 1 | no | — |
| 22 | C4 | — | 8 | 1 | no | — |
| 23 | C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 24 | C4 | SD16 | 1 | 1 | yes | characteristic |
| 25 | C2 | — | 2 | 1 | no | — |
| 26 | C2 | — | 8 | 1 | no | — |
| 27 | C2 | #102 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #102 |
| commutator | C8 | C2×C4 |
| Frattini, cosocle | C2×C8 | C22 |
| prime radical | #135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. M32 | C2 | semidirect |
| 3. #106 | C2 | semidirect |
| 5. Q16 | C4 | semidirect |
| 7. D8 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #135 | 2 |
| M32 | 2 |
| #106 | 2 |