Finite Group DatabasebetaFinite Group DatabasebetaOrder 64.
34 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #123 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #100 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #99 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×D4 | C22 | 1 | 1 | yes | characteristic |
| 5 | C22×C4 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | M16 | — | 2 | 1 | no | — |
| 7 | C4⋊C4 | — | 2 | 1 | no | — |
| 8 | C22⋊C4 | — | 2 | 1 | no | — |
| 9 | C22⋊C4 | — | 2 | 1 | no | — |
| 10 | C22⋊C4 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | C23 | C2×C4 | 1 | 1 | yes | characteristic |
| 12 | C23 | D4 | 1 | 1 | yes | characteristic |
| 13 | D4 | — | 4 | 1 | no | — |
| 14 | C2×C4 | — | 2 | 1 | no | — |
| 15 | C2×C4 | — | 2 | 1 | no | — |
| 16 | C2×C4 | — | 4 | 1 | no | — |
| 17 | C2×C4 | — | 4 | 1 | no | — |
| 18 | C2×C4 | D4 | 1 | 1 | yes | characteristic |
| 19 | C8 | — | 4 | 1 | no | — |
| 20 | C22 | — | 2 | 1 | no | — |
| 21 | C22 | — | 2 | 1 | no | — |
| 22 | C22 | — | 4 | 1 | no | — |
| 23 | C22 | — | 4 | 1 | no | — |
| 24 | C22 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 25 | C4 | — | 2 | 1 | no | — |
| 26 | C4 | — | 4 | 1 | no | — |
| 27 | C4 | — | 4 | 1 | no | — |
| 28 | C4 | — | 8 | 1 | no | — |
| 29 | C2 | — | 2 | 1 | no | — |
| 30 | C2 | — | 4 | 1 | no | — |
| 31 | C2 | — | 4 | 1 | no | — |
| 32 | C2 | #99 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #99 |
| commutator | C23 | C2×C4 |
| Frattini, prime radical, cosocle | C2×D4 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C22×C4 | C4 | semidirect |
| 10. C22⋊C4 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #123 | 2 |
| #100 | 2 |
| #99 | 2 |