Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
218 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #2082 | C2 | 1 | 6 | yes | maximal |
| 2 | #2070 | C2 | 1 | 2 | yes | maximal |
| 3 | #1979 | C2 | 1 | 3 | yes | maximal |
| 4 | #1953 | C2 | 1 | 3 | yes | maximal |
| 5 | #1747 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C22×Q16 | C22 | 1 | 1 | yes | characteristic |
| 7 | #430 | C22 | 1 | 6 | yes | — |
| 8 | C2×#108 | C22 | 1 | 3 | yes | — |
| 9 | #419 | C22 | 1 | 6 | yes | — |
| 10 | C2×#101 | C22 | 1 | 6 | yes | — |
| 11 | #405 | C22 | 1 | 3 | yes | — |
| 12 | #344 | C22 | 1 | 6 | yes | — |
| 13 | #338 | C22 | 1 | 1 | yes | characteristic |
| 14 | H(3,ℤ/4) | C22 | 1 | 3 | yes | — |
| 15 | C22×Q8 | C23 | 1 | 2 | yes | — |
| 16 | C2×Q16 | — | 2 | 3 | no | — |
| 17 | C2×Q16 | — | 2 | 3 | no | — |
| 18 | C2×M16 | C23 | 1 | 3 | yes | — |
| 19 | C2×M16 | C23 | 1 | 6 | yes | — |
| 20 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 21 | #117 | C23 | 1 | 3 | yes | — |
| 22 | #108 | — | 2 | 3 | no | — |
| 23 | #108 | — | 2 | 3 | no | — |
| 24 | #108 | — | 4 | 6 | no | — |
| 25 | #103 | — | 4 | 6 | no | — |
| 26 | #101 | — | 4 | 6 | no | — |
| 27 | #97 | — | 2 | 3 | no | — |
| 28 | C4×C8 | — | 2 | 3 | no | — |
| 29 | C2×Q8 | — | 2 | 6 | no | — |
| 30 | C2×Q8 | — | 2 | 6 | no | — |
| 31 | C22×C4 | — | 2 | 2 | no | — |
| 32 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 33 | Q16 | — | 4 | 1 | no | — |
| 34 | Q16 | — | 4 | 3 | no | — |
| 35 | M16 | — | 2 | 6 | no | — |
| 36 | M16 | — | 2 | 3 | no | — |
| 37 | M16 | — | 2 | 6 | no | — |
| 38 | M16 | — | 2 | 3 | no | — |
| 39 | C2×C8 | — | 2 | 3 | no | — |
| 40 | C2×C8 | — | 2 | 6 | no | — |
| 41 | C2×C8 | #41 | 1 | 3 | yes | — |
| 42 | C2×C8 | #41 | 1 | 3 | yes | — |
| 43 | C4⋊C4 | — | 2 | 3 | no | — |
| 44 | C22⋊C4 | — | 2 | 3 | no | — |
| 45 | C42 | — | 2 | 3 | no | — |
| 46 | C23 | 2−1+4 | 1 | 1 | yes | characteristic |
| 47 | Q8 | — | 4 | 2 | no | — |
| 48 | Q8 | — | 4 | 6 | no | — |
| 49 | C2×C4 | — | 4 | 3 | no | — |
| 50 | C2×C4 | — | 4 | 6 | no | — |
| 51 | C2×C4 | C2×#41 | 1 | 3 | yes | — |
| 52 | C2×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 53 | C8 | — | 4 | 3 | no | — |
| 54 | C8 | — | 4 | 6 | no | — |
| 55 | C8 | #126 | 1 | 1 | yes | characteristic |
| 56 | C8 | #126 | 1 | 3 | yes | — |
| 57 | C22 | — | 4 | 1 | no | — |
| 58 | C22 | #542 | 1 | 3 | yes | — |
| 59 | C4 | — | 8 | 3 | no | — |
| 60 | C4 | — | 8 | 2 | no | — |
| 61 | C4 | C2×#126 | 1 | 1 | yes | characteristic |
| 62 | C4 | #540 | 1 | 3 | yes | — |
| 63 | C2 | — | 2 | 3 | no | — |
| 64 | C2 | #2644 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2644 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2082 | 2 |
| #2070 | 2 |
| #1979 | 2 |
| #1953 | 2 |
| #1747 | 2 |