Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
239 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C4×#97 | C2 | 1 | 3 | yes | maximal |
| 2 | #1273 | C2 | 1 | 4 | yes | maximal |
| 3 | C2×#97 | C4 | 1 | 12 | yes | — |
| 4 | C2×C4×C8 | C22 | 1 | 6 | yes | — |
| 5 | C43 | C22 | 1 | 1 | yes | characteristic |
| 6 | C22×C8 | C2×C4 | 1 | 12 | yes | — |
| 7 | C2×C42 | C2×C4 | 1 | 6 | yes | — |
| 8 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 9 | #97 | — | 2 | 24 | no | — |
| 10 | C4×C8 | — | 2 | 12 | no | — |
| 11 | C22×C4 | C42 | 1 | 4 | yes | — |
| 12 | C22×C4 | C22×C4 | 1 | 3 | yes | — |
| 13 | C2×C8 | — | 2 | 12 | no | — |
| 14 | C2×C8 | — | 2 | 12 | no | — |
| 15 | C2×C8 | M16 | 1 | 24 | yes | — |
| 16 | C42 | — | 2 | 12 | no | — |
| 17 | C42 | #41 | 1 | 4 | yes | — |
| 18 | C23 | C2×C42 | 1 | 1 | yes | characteristic |
| 19 | C2×C4 | — | 2 | 12 | no | — |
| 20 | C2×C4 | #117 | 1 | 6 | yes | — |
| 21 | C2×C4 | C2×M16 | 1 | 12 | yes | — |
| 22 | C8 | — | 2 | 24 | no | — |
| 23 | C22 | #376 | 1 | 1 | yes | characteristic |
| 24 | C22 | C4×M16 | 1 | 6 | yes | — |
| 25 | C4 | — | 4 | 4 | no | — |
| 26 | C4 | #432 | 1 | 12 | yes | — |
| 27 | C2 | #1455 | 1 | 4 | yes | — |
| 28 | C2 | #1745 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C23 |
| commutator, socle, prime radical | C23 | C2×C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#97 | 2 |
| #1273 | 2 |