Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
335 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#494 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#486 | C2 | 1 | 2 | yes | maximal |
| 3 | #1990 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #1957 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #1273 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×Q16 | — | 2 | 2 | no | — |
| 7 | C2×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 8 | C2×#127 | C22 | 1 | 1 | yes | characteristic |
| 9 | #494 | — | 2 | 4 | no | — |
| 10 | #486 | — | 2 | 4 | no | — |
| 11 | #486 | — | 2 | 4 | no | — |
| 12 | C2×#102 | — | 2 | 4 | no | — |
| 13 | C2×#102 | — | 2 | 2 | no | — |
| 14 | C2×C4×C8 | C22 | 1 | 2 | yes | — |
| 15 | C2×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 16 | #384 | C22 | 1 | 1 | yes | characteristic |
| 17 | #340 | — | 2 | 4 | no | — |
| 18 | C22×Q8 | — | 2 | 2 | no | — |
| 19 | C22×D4 | — | 2 | 1 | no | — |
| 20 | C22×D4 | — | 2 | 2 | no | — |
| 21 | C2×Q16 | — | 2 | 8 | no | — |
| 22 | C2×Q16 | — | 4 | 4 | no | — |
| 23 | C2×Q16 | — | 4 | 4 | no | — |
| 24 | C22×C8 | D4 | 1 | 4 | yes | — |
| 25 | C22×C8 | D4 | 1 | 2 | yes | — |
| 26 | C4⋊Q8 | — | 2 | 2 | no | — |
| 27 | C4⋊Q8 | — | 2 | 2 | no | — |
| 28 | C4⋊Q8 | — | 2 | 2 | no | — |
| 29 | C4⋊Q8 | D4 | 1 | 4 | yes | — |
| 30 | #127 | — | 2 | 2 | no | — |
| 31 | #127 | — | 2 | 1 | no | — |
| 32 | #127 | D4 | 1 | 2 | yes | — |
| 33 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 34 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 35 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 36 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 37 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 38 | #102 | — | 2 | 16 | no | — |
| 39 | #102 | — | 4 | 4 | no | — |
| 40 | C4×C8 | — | 2 | 2 | no | — |
| 41 | C4×C8 | — | 2 | 2 | no | — |
| 42 | C4×C8 | — | 2 | 2 | no | — |
| 43 | C24 | — | 4 | 1 | no | — |
| 44 | C2×Q8 | — | 2 | 4 | no | — |
| 45 | C2×Q8 | — | 4 | 2 | no | — |
| 46 | C2×Q8 | — | 4 | 4 | no | — |
| 47 | C2×Q8 | — | 4 | 4 | no | — |
| 48 | C2×D4 | — | 2 | 4 | no | — |
| 49 | C2×D4 | — | 2 | 2 | no | — |
| 50 | C2×D4 | — | 4 | 1 | no | — |
| 51 | C2×D4 | — | 4 | 2 | no | — |
| 52 | C2×D4 | — | 4 | 2 | no | — |
| 53 | C2×D4 | — | 4 | 4 | no | — |
| 54 | C2×D4 | — | 4 | 2 | no | — |
| 55 | C2×D4 | — | 4 | 4 | no | — |
| 56 | C22×C4 | — | 4 | 2 | no | — |
| 57 | C22×C4 | — | 4 | 1 | no | — |
| 58 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 59 | C22×C4 | C2×D4 | 1 | 2 | yes | — |
| 60 | Q16 | — | 4 | 16 | no | — |
| 61 | C2×C8 | — | 2 | 2 | no | — |
| 62 | C2×C8 | — | 2 | 2 | no | — |
| 63 | C2×C8 | — | 2 | 4 | no | — |
| 64 | C2×C8 | — | 2 | 4 | no | — |
| 65 | C2×C8 | — | 2 | 4 | no | — |
| 66 | C2×C8 | D8 | 1 | 4 | yes | — |
| 67 | C4⋊C4 | — | 2 | 4 | no | — |
| 68 | C4⋊C4 | — | 2 | 4 | no | — |
| 69 | C4⋊C4 | — | 4 | 2 | no | — |
| 70 | C4⋊C4 | — | 4 | 2 | no | — |
| 71 | C4⋊C4 | — | 4 | 2 | no | — |
| 72 | C4⋊C4 | — | 4 | 4 | no | — |
| 73 | C42 | C2×D4 | 1 | 2 | yes | — |
| 74 | C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C42 | #41 | 1 | 1 | yes | characteristic |
| 76 | C23 | — | 4 | 2 | no | — |
| 77 | C23 | — | 8 | 1 | no | — |
| 78 | C23 | — | 8 | 2 | no | — |
| 79 | C23 | — | 8 | 1 | no | — |
| 80 | C23 | — | 8 | 2 | no | — |
| 81 | C23 | #127 | 1 | 1 | yes | characteristic |
| 82 | Q8 | — | 4 | 8 | no | — |
| 83 | Q8 | — | 8 | 4 | no | — |
| 84 | D4 | — | 4 | 8 | no | — |
| 85 | D4 | — | 4 | 4 | no | — |
| 86 | D4 | — | 8 | 2 | no | — |
| 87 | D4 | — | 8 | 4 | no | — |
| 88 | C2×C4 | — | 4 | 4 | no | — |
| 89 | C2×C4 | — | 8 | 2 | no | — |
| 90 | C2×C4 | — | 8 | 2 | no | — |
| 91 | C2×C4 | — | 8 | 1 | no | — |
| 92 | C2×C4 | — | 8 | 2 | no | — |
| 93 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 94 | C2×C4 | #120 | 1 | 2 | yes | — |
| 95 | C2×C4 | #121 | 1 | 1 | yes | characteristic |
| 96 | C2×C4 | #121 | 1 | 2 | yes | — |
| 97 | C2×C4 | C2×D8 | 1 | 2 | yes | — |
| 98 | C2×C4 | #135 | 1 | 4 | yes | — |
| 99 | C2×C4 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 100 | C2×C4 | #137 | 1 | 4 | yes | — |
| 101 | C8 | — | 2 | 4 | no | — |
| 102 | C8 | — | 4 | 4 | no | — |
| 103 | C22 | — | 8 | 2 | no | — |
| 104 | C22 | — | 8 | 4 | no | — |
| 105 | C22 | — | 16 | 1 | no | — |
| 106 | C22 | — | 16 | 2 | no | — |
| 107 | C22 | — | 16 | 1 | no | — |
| 108 | C22 | #392 | 1 | 1 | yes | characteristic |
| 109 | C22 | #493 | 1 | 1 | yes | characteristic |
| 110 | C22 | #495 | 1 | 2 | yes | — |
| 111 | C22 | #496 | 1 | 1 | yes | characteristic |
| 112 | C22 | #497 | 1 | 2 | yes | — |
| 113 | C4 | — | 8 | 4 | no | — |
| 114 | C4 | — | 16 | 1 | no | — |
| 115 | C4 | #447 | 1 | 2 | yes | — |
| 116 | C4 | #452 | 1 | 4 | yes | — |
| 117 | C4 | #459 | 1 | 2 | yes | — |
| 118 | C4 | #464 | 1 | 4 | yes | — |
| 119 | C2 | — | 16 | 2 | no | — |
| 120 | C2 | #1709 | 1 | 1 | yes | characteristic |
| 121 | C2 | #1720 | 1 | 2 | yes | — |
| 122 | C2 | #1722 | 1 | 1 | yes | characteristic |
| 123 | C2 | #2011 | 1 | 1 | yes | characteristic |
| 124 | C2 | #2015 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#494 | C2 | semidirect |
| 3. #1990 | C2 | semidirect |
| 5. #1273 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#494 | 2 |
| C2×#486 | 2 |
| #1990 | 2 |
| #1957 | 2 |
| #1273 | 2 |