Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
319 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #3396 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #1712 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #1668 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #1620 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C8×Q16 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #559 | C22 | 1 | 4 | yes | — |
| 7 | #495 | — | 2 | 4 | no | — |
| 8 | #493 | C22 | 1 | 4 | yes | — |
| 9 | #493 | C22 | 1 | 2 | yes | — |
| 10 | #463 | — | 2 | 8 | no | — |
| 11 | #460 | C22 | 1 | 8 | yes | — |
| 12 | C8×Q8 | C22 | 1 | 4 | yes | — |
| 13 | C4×Q16 | C22 | 1 | 2 | yes | — |
| 14 | C4×D8 | — | 2 | 4 | no | — |
| 15 | #335 | C22 | 1 | 2 | yes | — |
| 16 | #331 | C22 | 1 | 4 | yes | — |
| 17 | Q8⋊C8 | C22 | 1 | 4 | yes | — |
| 18 | C82 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×Q16 | D4 | 1 | 2 | yes | — |
| 20 | C2×SD16 | — | 2 | 8 | no | — |
| 21 | C2×D8 | — | 2 | 4 | no | — |
| 22 | C2×D8 | — | 2 | 4 | no | — |
| 23 | C2×D8 | — | 4 | 2 | no | — |
| 24 | #127 | C23 | 1 | 4 | yes | — |
| 25 | #124 | — | 2 | 8 | no | — |
| 26 | #123 | — | 4 | 4 | no | — |
| 27 | C4×Q8 | C23 | 1 | 4 | yes | — |
| 28 | C4×D4 | — | 2 | 8 | no | — |
| 29 | #107 | D4 | 1 | 2 | yes | — |
| 30 | C4⋊C8 | — | 2 | 4 | no | — |
| 31 | C4⋊C8 | C23 | 1 | 4 | yes | — |
| 32 | #103 | D4 | 1 | 4 | yes | — |
| 33 | #102 | — | 2 | 8 | no | — |
| 34 | C4×C8 | — | 2 | 4 | no | — |
| 35 | C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 36 | C4×C8 | C23 | 1 | 2 | yes | — |
| 37 | C2×Q8 | C2×D4 | 1 | 4 | yes | — |
| 38 | C2×D4 | — | 2 | 8 | no | — |
| 39 | C2×D4 | — | 2 | 4 | no | — |
| 40 | C22×C4 | — | 4 | 4 | no | — |
| 41 | Q16 | — | 2 | 4 | no | — |
| 42 | SD16 | — | 8 | 8 | no | — |
| 43 | D8 | — | 4 | 8 | no | — |
| 44 | D8 | — | 8 | 4 | no | — |
| 45 | D8 | — | 8 | 4 | no | — |
| 46 | C2×C8 | — | 2 | 1 | no | — |
| 47 | C2×C8 | — | 2 | 4 | no | — |
| 48 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 49 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 50 | C4⋊C4 | — | 2 | 4 | no | — |
| 51 | C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 52 | C22⋊C4 | — | 4 | 8 | no | — |
| 53 | C22⋊C4 | — | 4 | 4 | no | — |
| 54 | C42 | — | 2 | 4 | no | — |
| 55 | C42 | C24 | 1 | 1 | yes | characteristic |
| 56 | C23 | — | 4 | 4 | no | — |
| 57 | Q8 | — | 2 | 4 | no | — |
| 58 | Q8 | — | 2 | 4 | no | — |
| 59 | D4 | — | 4 | 8 | no | — |
| 60 | D4 | — | 4 | 8 | no | — |
| 61 | D4 | — | 8 | 4 | no | — |
| 62 | C2×C4 | — | 2 | 4 | no | — |
| 63 | C2×C4 | — | 2 | 4 | no | — |
| 64 | C2×C4 | — | 16 | 4 | no | — |
| 65 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 66 | C2×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 67 | C8 | — | 2 | 2 | no | — |
| 68 | C8 | — | 2 | 2 | no | — |
| 69 | C8 | — | 4 | 4 | no | — |
| 70 | C8 | #135 | 1 | 4 | yes | — |
| 71 | C22 | — | 8 | 8 | no | — |
| 72 | C22 | — | 8 | 4 | no | — |
| 73 | C22 | D42 | 1 | 1 | yes | characteristic |
| 74 | C4 | — | 2 | 1 | no | — |
| 75 | C4 | — | 4 | 4 | no | — |
| 76 | C4 | — | 4 | 4 | no | — |
| 77 | C4 | C2×#135 | 1 | 2 | yes | — |
| 78 | C4 | #576 | 1 | 2 | yes | — |
| 79 | C2 | — | 16 | 4 | no | — |
| 80 | C2 | #3287 | 1 | 2 | yes | — |
| 81 | C2 | #3292 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #1712 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3396 | C2 | semidirect |
| 3. #1668 | C2 | semidirect |
| 4. #1620 | C2 | semidirect |
| 5. C8×Q16 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3396 | 2 |
| #1712 | 2 |
| #1668 | 2 |
| #1620 | 2 |
| C8×Q16 | 2 |