Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
639 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C43 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #1434 | C2 | 1 | 14 | yes | maximal |
| 3 | C22×C42 | C22 | 1 | 7 | yes | — |
| 4 | C2×#95 | C22 | 1 | 28 | yes | — |
| 5 | C43 | C4 | 1 | 8 | yes | complemented |
| 6 | C23×C4 | C23 | 1 | 8 | yes | — |
| 7 | C23×C4 | C23 | 1 | 7 | yes | — |
| 8 | C2×C42 | — | 2 | 28 | no | — |
| 9 | C2×C42 | C2×C4 | 1 | 28 | yes | — |
| 10 | #95 | — | 2 | 56 | no | — |
| 11 | C24 | C24 | 1 | 1 | yes | characteristic |
| 12 | C22×C4 | — | 2 | 28 | no | — |
| 13 | C22×C4 | — | 2 | 56 | no | — |
| 14 | C22×C4 | C22×C4 | 1 | 14 | yes | — |
| 15 | C22×C4 | #41 | 1 | 28 | yes | — |
| 16 | C42 | — | 2 | 56 | no | — |
| 17 | C23 | C23×C4 | 1 | 1 | yes | characteristic |
| 18 | C23 | C2×#41 | 1 | 14 | yes | — |
| 19 | C2×C4 | — | 2 | 28 | no | — |
| 20 | C2×C4 | — | 2 | 56 | no | — |
| 21 | C2×C4 | — | 4 | 56 | no | — |
| 22 | C2×C4 | #117 | 1 | 28 | yes | — |
| 23 | C22 | C2×#117 | 1 | 7 | yes | — |
| 24 | C22 | #529 | 1 | 28 | yes | — |
| 25 | C4 | — | 2 | 28 | no | — |
| 26 | C4 | — | 8 | 8 | no | — |
| 27 | C2 | #2283 | 1 | 7 | yes | — |
| 28 | C2 | #2865 | 1 | 8 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C43 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C43 | 2 |
| #1434 | 2 |