Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
149 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 23 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 | #2006 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #1393 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #1270 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 5 | #537 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×#104 | — | 2 | 1 | no | — |
| 7 | C2×#99 | — | 2 | 1 | no | — |
| 8 | #390 | C22 | 1 | 1 | yes | characteristic |
| 9 | #343 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | #336 | — | 2 | 1 | no | — |
| 11 | C2×#41 | — | 2 | 1 | no | — |
| 12 | C22×D4 | D4 | 1 | 1 | yes | characteristic |
| 13 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 14 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 15 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 16 | C2×SD16 | — | 4 | 1 | no | — |
| 17 | C2×D8 | — | 4 | 1 | no | — |
| 18 | C2×M16 | — | 2 | 1 | no | — |
| 19 | C22×C8 | — | 4 | 1 | no | — |
| 20 | C22⋊Q8 | — | 4 | 1 | no | — |
| 21 | #121 | — | 4 | 1 | no | — |
| 22 | #121 | — | 4 | 1 | no | — |
| 23 | C2×C4⋊C4 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 25 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 26 | C2×C42 | D4 | 1 | 1 | yes | characteristic |
| 27 | #104 | — | 4 | 2 | no | — |
| 28 | #99 | — | 2 | 2 | no | — |
| 29 | #99 | — | 4 | 1 | no | — |
| 30 | C24 | — | 2 | 1 | no | — |
| 31 | #41 | — | 4 | 1 | no | — |
| 32 | #41 | — | 4 | 1 | no | — |
| 33 | #41 | — | 4 | 1 | no | — |
| 34 | #41 | — | 4 | 1 | no | — |
| 35 | C2×Q8 | — | 2 | 1 | no | — |
| 36 | C2×D4 | — | 2 | 1 | no | — |
| 37 | C2×D4 | — | 2 | 1 | no | — |
| 38 | C2×D4 | — | 4 | 1 | no | — |
| 39 | C2×D4 | — | 4 | 1 | no | — |
| 40 | C2×D4 | — | 4 | 1 | no | — |
| 41 | C2×D4 | — | 4 | 1 | no | — |
| 42 | C2×D4 | D8 | 1 | 1 | yes | characteristic |
| 43 | C2×D4 | SD16 | 1 | 1 | yes | characteristic |
| 44 | C22×C4 | — | 2 | 1 | no | — |
| 45 | C22×C4 | — | 2 | 1 | no | — |
| 46 | C22×C4 | — | 4 | 1 | no | — |
| 47 | C22×C4 | — | 8 | 1 | no | — |
| 48 | C22×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 49 | SD16 | — | 4 | 2 | no | — |
| 50 | SD16 | — | 4 | 2 | no | — |
| 51 | D8 | — | 4 | 2 | no | — |
| 52 | D8 | — | 4 | 2 | no | — |
| 53 | M16 | — | 2 | 2 | no | — |
| 54 | M16 | — | 4 | 1 | no | — |
| 55 | C2×C8 | — | 4 | 1 | no | — |
| 56 | C2×C8 | — | 4 | 2 | no | — |
| 57 | C2×C8 | — | 8 | 1 | no | — |
| 58 | C2×C8 | — | 8 | 1 | no | — |
| 59 | C4⋊C4 | — | 4 | 1 | no | — |
| 60 | C4⋊C4 | — | 4 | 1 | no | — |
| 61 | C22⋊C4 | — | 4 | 2 | no | — |
| 62 | C22⋊C4 | — | 4 | 2 | no | — |
| 63 | C22⋊C4 | — | 8 | 1 | no | — |
| 64 | C22⋊C4 | — | 8 | 1 | no | — |
| 65 | C22⋊C4 | — | 8 | 1 | no | — |
| 66 | C42 | — | 4 | 1 | no | — |
| 67 | C23 | — | 2 | 1 | no | — |
| 68 | C23 | — | 2 | 1 | no | — |
| 69 | C23 | — | 4 | 1 | no | — |
| 70 | C23 | — | 4 | 1 | no | — |
| 71 | C23 | — | 4 | 1 | no | — |
| 72 | C23 | — | 8 | 1 | no | — |
| 73 | C23 | — | 8 | 1 | no | — |
| 74 | C23 | #99 | 1 | 1 | yes | characteristic |
| 75 | Q8 | — | 4 | 1 | no | — |
| 76 | Q8 | — | 4 | 1 | no | — |
| 77 | D4 | — | 4 | 1 | no | — |
| 78 | D4 | — | 4 | 1 | no | — |
| 79 | D4 | — | 4 | 1 | no | — |
| 80 | D4 | — | 4 | 1 | no | — |
| 81 | D4 | — | 8 | 1 | no | — |
| 82 | D4 | — | 8 | 1 | no | — |
| 83 | D4 | — | 8 | 1 | no | — |
| 84 | D4 | — | 16 | 1 | no | — |
| 85 | C2×C4 | — | 2 | 1 | no | — |
| 86 | C2×C4 | — | 2 | 1 | no | — |
| 87 | C2×C4 | — | 2 | 1 | no | — |
| 88 | C2×C4 | — | 2 | 1 | no | — |
| 89 | C2×C4 | — | 4 | 1 | no | — |
| 90 | C2×C4 | — | 4 | 1 | no | — |
| 91 | C2×C4 | — | 4 | 1 | no | — |
| 92 | C2×C4 | — | 4 | 1 | no | — |
| 93 | C2×C4 | — | 8 | 1 | no | — |
| 94 | C2×C4 | — | 8 | 1 | no | — |
| 95 | C2×C4 | — | 8 | 1 | no | — |
| 96 | C2×C4 | — | 8 | 2 | no | — |
| 97 | C2×C4 | — | 16 | 1 | no | — |
| 98 | C2×C4 | — | 16 | 1 | no | — |
| 99 | C2×C4 | #102 | 1 | 1 | yes | characteristic |
| 100 | C2×C4 | #104 | 1 | 1 | yes | characteristic |
| 101 | C8 | — | 4 | 2 | no | — |
| 102 | C8 | — | 8 | 2 | no | — |
| 103 | C22 | — | 2 | 1 | no | — |
| 104 | C22 | — | 2 | 1 | no | — |
| 105 | C22 | — | 4 | 1 | no | — |
| 106 | C22 | — | 4 | 1 | no | — |
| 107 | C22 | — | 8 | 1 | no | — |
| 108 | C22 | — | 8 | 1 | no | — |
| 109 | C22 | — | 8 | 1 | no | — |
| 110 | C22 | — | 8 | 1 | no | — |
| 111 | C22 | — | 8 | 1 | no | — |
| 112 | C22 | — | 8 | 1 | no | — |
| 113 | C22 | — | 8 | 1 | no | — |
| 114 | C22 | — | 16 | 1 | no | — |
| 115 | C22 | #327 | 1 | 1 | yes | characteristic |
| 116 | C22 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 117 | C22 | #353 | 1 | 1 | yes | characteristic |
| 118 | C4 | — | 2 | 1 | no | — |
| 119 | C4 | — | 2 | 1 | no | — |
| 120 | C4 | — | 4 | 1 | no | — |
| 121 | C4 | — | 4 | 1 | no | — |
| 122 | C4 | — | 8 | 1 | no | — |
| 123 | C4 | — | 8 | 1 | no | — |
| 124 | C4 | — | 16 | 2 | no | — |
| 125 | C2 | — | 2 | 1 | no | — |
| 126 | C2 | — | 2 | 1 | no | — |
| 127 | C2 | — | 8 | 1 | no | — |
| 128 | C2 | — | 8 | 1 | no | — |
| 129 | C2 | — | 16 | 1 | no | — |
| 130 | C2 | #1343 | 1 | 1 | yes | characteristic |
| 131 | C2 | #1402 | 1 | 1 | yes | characteristic |
| 132 | C2 | #1404 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C2×C4 |
| Frattini, cosocle | #390 | C22 |
| prime radical | #2006 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #1393 | C2 | semidirect |
| 3. #1270 | C2 | semidirect |
| 5. #537 | C4 | semidirect |
| 9. #343 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2006 | 2 |
| #1393 | 2 |
| #1270 | 2 |