Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
271 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #3374 | C2 | 1 | 4 | yes | maximal |
| 2 | #1710 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #1700 | C2 | 1 | 4 | yes | maximal |
| 4 | #1648 | C2 | 1 | 4 | yes | maximal |
| 5 | #1591 | C2 | 1 | 2 | yes | maximal |
| 6 | #557 | C22 | 1 | 4 | yes | — |
| 7 | #500 | C22 | 1 | 2 | yes | — |
| 8 | #491 | — | 2 | 4 | no | — |
| 9 | #487 | C22 | 1 | 4 | yes | — |
| 10 | #479 | — | 2 | 4 | no | — |
| 11 | #475 | C22 | 1 | 4 | yes | — |
| 12 | #462 | — | 2 | 4 | no | — |
| 13 | #462 | C22 | 1 | 4 | yes | — |
| 14 | #446 | C22 | 1 | 4 | yes | — |
| 15 | #441 | — | 2 | 4 | no | — |
| 16 | C4×Q16 | C22 | 1 | 2 | yes | — |
| 17 | #335 | C22 | 1 | 2 | yes | — |
| 18 | #332 | C22 | 1 | 4 | yes | — |
| 19 | Q8⋊C8 | C22 | 1 | 4 | yes | — |
| 20 | C82 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×Q16 | — | 2 | 4 | no | — |
| 22 | C2×Q16 | D4 | 1 | 2 | yes | — |
| 23 | C4⋊Q8 | — | 2 | 4 | no | — |
| 24 | C4⋊Q8 | C23 | 1 | 4 | yes | — |
| 25 | #125 | — | 2 | 4 | no | — |
| 26 | #125 | — | 4 | 4 | no | — |
| 27 | C4×Q8 | — | 2 | 4 | no | — |
| 28 | C4×Q8 | — | 2 | 4 | no | — |
| 29 | C4×Q8 | C23 | 1 | 4 | yes | — |
| 30 | #107 | — | 4 | 2 | no | — |
| 31 | #107 | D4 | 1 | 2 | yes | — |
| 32 | #106 | — | 2 | 4 | no | — |
| 33 | C4⋊C8 | — | 2 | 4 | no | — |
| 34 | C4⋊C8 | C23 | 1 | 4 | yes | — |
| 35 | #103 | — | 2 | 4 | no | — |
| 36 | #103 | — | 2 | 4 | no | — |
| 37 | #103 | — | 2 | 4 | no | — |
| 38 | #103 | — | 2 | 4 | no | — |
| 39 | #103 | D4 | 1 | 4 | yes | — |
| 40 | #97 | — | 2 | 4 | no | — |
| 41 | C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 42 | C4×C8 | C23 | 1 | 2 | yes | — |
| 43 | C2×Q8 | — | 2 | 4 | no | — |
| 44 | C2×Q8 | C2×D4 | 1 | 4 | yes | — |
| 45 | Q16 | — | 4 | 2 | no | — |
| 46 | Q16 | — | 4 | 8 | no | — |
| 47 | C2×C8 | — | 2 | 1 | no | — |
| 48 | C2×C8 | — | 2 | 4 | no | — |
| 49 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 50 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 51 | C4⋊C4 | — | 2 | 4 | no | — |
| 52 | C4⋊C4 | — | 2 | 4 | no | — |
| 53 | C4⋊C4 | — | 2 | 4 | no | — |
| 54 | C4⋊C4 | — | 4 | 4 | no | — |
| 55 | C4⋊C4 | — | 4 | 4 | no | — |
| 56 | C4⋊C4 | — | 4 | 4 | no | — |
| 57 | C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 58 | C42 | — | 2 | 4 | no | — |
| 59 | C42 | — | 4 | 4 | no | — |
| 60 | C42 | C24 | 1 | 1 | yes | characteristic |
| 61 | Q8 | — | 2 | 4 | no | — |
| 62 | Q8 | — | 2 | 4 | no | — |
| 63 | Q8 | — | 4 | 4 | no | — |
| 64 | Q8 | — | 4 | 4 | no | — |
| 65 | C2×C4 | — | 2 | 4 | no | — |
| 66 | C2×C4 | — | 2 | 4 | no | — |
| 67 | C2×C4 | — | 4 | 4 | no | — |
| 68 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 69 | C2×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 70 | C8 | — | 2 | 2 | no | — |
| 71 | C8 | — | 2 | 2 | no | — |
| 72 | C8 | — | 2 | 2 | no | — |
| 73 | C8 | — | 4 | 4 | no | — |
| 74 | C22 | D42 | 1 | 1 | yes | characteristic |
| 75 | C4 | — | 2 | 1 | no | — |
| 76 | C4 | — | 4 | 4 | no | — |
| 77 | C4 | — | 4 | 4 | no | — |
| 78 | C4 | — | 8 | 4 | no | — |
| 79 | C4 | #575 | 1 | 2 | yes | — |
| 80 | C4 | #578 | 1 | 2 | yes | — |
| 81 | C2 | #3277 | 1 | 2 | yes | — |
| 82 | C2 | #3293 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3374 | 2 |
| #1710 | 2 |
| #1700 | 2 |
| #1648 | 2 |
| #1591 | 2 |