Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
89 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #1822 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #1320 | C2 | 1 | 2 | yes | maximal |
| 3 | C2×#117 | C4 | 1 | 1 | yes | characteristic |
| 4 | #407 | C4 | 1 | 1 | yes | characteristic |
| 5 | #407 | C22 | 1 | 1 | yes | characteristic |
| 6 | #349 | — | 2 | 2 | no | — |
| 7 | #343 | — | 2 | 2 | no | — |
| 8 | C23×C4 | C2×C4 | 1 | 1 | yes | characteristic |
| 9 | C2×M16 | — | 2 | 1 | no | — |
| 10 | C2×M16 | D4 | 1 | 2 | yes | — |
| 11 | #117 | — | 4 | 1 | no | — |
| 12 | #117 | — | 4 | 1 | no | — |
| 13 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 14 | C2×C22⋊C4 | C8 | 1 | 2 | yes | — |
| 15 | C2×C42 | — | 2 | 1 | no | — |
| 16 | M32 | — | 4 | 2 | no | — |
| 17 | C22⋊C8 | — | 4 | 1 | no | — |
| 18 | C22⋊C8 | — | 4 | 1 | no | — |
| 19 | C24 | C2×C8 | 1 | 1 | yes | characteristic |
| 20 | C22×C4 | — | 2 | 1 | no | — |
| 21 | C22×C4 | — | 2 | 1 | no | — |
| 22 | C22×C4 | — | 2 | 2 | no | — |
| 23 | C22×C4 | — | 4 | 2 | no | — |
| 24 | C22×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 25 | C22×C4 | M16 | 1 | 1 | yes | characteristic |
| 26 | M16 | — | 4 | 2 | no | — |
| 27 | M16 | — | 4 | 2 | no | — |
| 28 | C2×C8 | — | 2 | 2 | no | — |
| 29 | C2×C8 | — | 4 | 1 | no | — |
| 30 | C4⋊C4 | — | 4 | 1 | no | — |
| 31 | C4⋊C4 | — | 4 | 1 | no | — |
| 32 | C22⋊C4 | — | 4 | 2 | no | — |
| 33 | C42 | — | 4 | 2 | no | — |
| 34 | C16 | — | 8 | 2 | no | — |
| 35 | C23 | — | 2 | 1 | no | — |
| 36 | C23 | — | 4 | 1 | no | — |
| 37 | C23 | — | 8 | 1 | no | — |
| 38 | C23 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 39 | C2×C4 | — | 2 | 2 | no | — |
| 40 | C2×C4 | — | 2 | 1 | no | — |
| 41 | C2×C4 | — | 4 | 2 | no | — |
| 42 | C2×C4 | — | 4 | 2 | no | — |
| 43 | C2×C4 | — | 4 | 1 | no | — |
| 44 | C2×C4 | — | 4 | 2 | no | — |
| 45 | C2×C4 | — | 8 | 2 | no | — |
| 46 | C2×C4 | #99 | 1 | 1 | yes | characteristic |
| 47 | C2×C4 | #100 | 1 | 1 | yes | characteristic |
| 48 | C8 | — | 4 | 2 | no | — |
| 49 | C8 | — | 8 | 1 | no | — |
| 50 | C22 | — | 2 | 1 | no | — |
| 51 | C22 | — | 4 | 1 | no | — |
| 52 | C22 | — | 4 | 2 | no | — |
| 53 | C22 | — | 4 | 1 | no | — |
| 54 | C22 | — | 8 | 2 | no | — |
| 55 | C22 | #323 | 1 | 1 | yes | characteristic |
| 56 | C4 | — | 2 | 1 | no | — |
| 57 | C4 | — | 4 | 1 | no | — |
| 58 | C4 | — | 8 | 2 | no | — |
| 59 | C4 | #352 | 1 | 2 | yes | — |
| 60 | C2 | — | 2 | 1 | no | — |
| 61 | C2 | — | 4 | 1 | no | — |
| 62 | C2 | — | 8 | 1 | no | — |
| 63 | C2 | #1317 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1317 |
| commutator, prime radical | C24 | C2×C8 |
| Frattini, cosocle | #407 | C22 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #1822 | 2 |
| #1320 | 2 |