Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
171 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 35 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 | #3199 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #2238 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #2098 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #1339 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #1321 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #535 | — | 2 | 1 | no | — |
| 7 | #482 | — | 2 | 2 | no | — |
| 8 | #469 | C22 | 1 | 2 | yes | — |
| 9 | #466 | — | 2 | 1 | no | — |
| 10 | #447 | — | 2 | 2 | no | — |
| 11 | #430 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | #424 | — | 2 | 1 | no | — |
| 13 | #421 | — | 2 | 1 | no | — |
| 14 | #408 | C22 | 1 | 1 | yes | characteristic |
| 15 | #365 | — | 2 | 2 | no | — |
| 16 | #361 | — | 2 | 2 | no | — |
| 17 | #349 | C22 | 1 | 2 | yes | complemented |
| 18 | H(3,ℤ/4) | C22 | 1 | 1 | yes | characteristic, complemented |
| 19 | C2×#41 | D4 | 1 | 1 | yes | characteristic |
| 20 | C22×D4 | — | 2 | 1 | no | — |
| 21 | C2×D8 | — | 2 | 1 | no | — |
| 22 | C2×D8 | D4 | 1 | 2 | yes | — |
| 23 | C2×M16 | — | 2 | 1 | no | — |
| 24 | C2×M16 | C23 | 1 | 1 | yes | characteristic |
| 25 | C22×C8 | D4 | 1 | 1 | yes | characteristic |
| 26 | #127 | — | 4 | 1 | no | — |
| 27 | #124 | — | 4 | 1 | no | — |
| 28 | #121 | — | 2 | 2 | no | — |
| 29 | #121 | — | 2 | 2 | no | — |
| 30 | #120 | — | 4 | 1 | no | — |
| 31 | #120 | — | 4 | 1 | no | — |
| 32 | #117 | — | 2 | 1 | no | — |
| 33 | #117 | — | 2 | 1 | no | — |
| 34 | M32 | — | 2 | 2 | no | — |
| 35 | #108 | — | 2 | 2 | no | — |
| 36 | #107 | — | 2 | 1 | no | — |
| 37 | #106 | D4 | 1 | 2 | yes | — |
| 38 | C4⋊C8 | — | 4 | 2 | no | — |
| 39 | #104 | — | 4 | 2 | no | — |
| 40 | #102 | — | 2 | 2 | no | — |
| 41 | #102 | — | 2 | 2 | no | — |
| 42 | C22⋊C8 | — | 2 | 2 | no | — |
| 43 | C24 | — | 4 | 1 | no | — |
| 44 | #41 | — | 4 | 1 | no | — |
| 45 | #41 | — | 4 | 1 | no | — |
| 46 | C2×Q8 | SD16 | 1 | 1 | yes | characteristic |
| 47 | C2×D4 | — | 2 | 1 | no | — |
| 48 | C2×D4 | — | 2 | 2 | no | — |
| 49 | C2×D4 | — | 4 | 1 | no | — |
| 50 | C2×D4 | — | 4 | 2 | no | — |
| 51 | C2×D4 | — | 4 | 2 | no | — |
| 52 | C2×D4 | — | 8 | 1 | no | — |
| 53 | C2×D4 | — | 8 | 1 | no | — |
| 54 | C2×D4 | SD16 | 1 | 1 | yes | characteristic |
| 55 | C22×C4 | — | 2 | 1 | no | — |
| 56 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 57 | D8 | — | 4 | 2 | no | — |
| 58 | D8 | — | 8 | 1 | no | — |
| 59 | M16 | — | 4 | 1 | no | — |
| 60 | M16 | — | 4 | 1 | no | — |
| 61 | M16 | — | 4 | 1 | no | — |
| 62 | C2×C8 | — | 2 | 1 | no | — |
| 63 | C2×C8 | — | 4 | 1 | no | — |
| 64 | C2×C8 | — | 4 | 1 | no | — |
| 65 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 66 | C4⋊C4 | — | 2 | 2 | no | — |
| 67 | C4⋊C4 | — | 4 | 1 | no | — |
| 68 | C22⋊C4 | — | 4 | 2 | no | — |
| 69 | C22⋊C4 | — | 4 | 1 | no | — |
| 70 | C22⋊C4 | — | 4 | 1 | no | — |
| 71 | C22⋊C4 | — | 4 | 2 | no | — |
| 72 | C42 | — | 2 | 2 | no | — |
| 73 | C42 | — | 4 | 1 | no | — |
| 74 | C16 | — | 4 | 2 | no | — |
| 75 | C23 | — | 2 | 1 | no | — |
| 76 | C23 | — | 4 | 2 | no | — |
| 77 | C23 | — | 8 | 2 | no | — |
| 78 | C23 | — | 16 | 2 | no | — |
| 79 | C23 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 80 | Q8 | — | 4 | 1 | no | — |
| 81 | D4 | — | 4 | 1 | no | — |
| 82 | D4 | — | 4 | 2 | no | — |
| 83 | D4 | — | 4 | 2 | no | — |
| 84 | D4 | — | 8 | 1 | no | — |
| 85 | D4 | — | 8 | 2 | no | — |
| 86 | D4 | — | 16 | 2 | no | — |
| 87 | D4 | — | 16 | 2 | no | — |
| 88 | C2×C4 | — | 2 | 1 | no | — |
| 89 | C2×C4 | — | 2 | 1 | no | — |
| 90 | C2×C4 | — | 2 | 1 | no | — |
| 91 | C2×C4 | — | 4 | 1 | no | — |
| 92 | C2×C4 | — | 4 | 1 | no | — |
| 93 | C2×C4 | — | 4 | 2 | no | — |
| 94 | C2×C4 | — | 4 | 2 | no | — |
| 95 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 96 | C2×C4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 97 | C8 | — | 2 | 2 | no | — |
| 98 | C8 | — | 4 | 1 | no | — |
| 99 | C8 | — | 8 | 1 | no | — |
| 100 | C22 | — | 2 | 1 | no | — |
| 101 | C22 | — | 4 | 1 | no | — |
| 102 | C22 | — | 4 | 2 | no | — |
| 103 | C22 | — | 4 | 1 | no | — |
| 104 | C22 | — | 8 | 2 | no | — |
| 105 | C22 | — | 8 | 2 | no | — |
| 106 | C22 | — | 8 | 2 | no | — |
| 107 | C22 | — | 16 | 4 | no | — |
| 108 | C22 | #450 | 1 | 1 | yes | characteristic |
| 109 | C4 | — | 2 | 1 | no | — |
| 110 | C4 | — | 4 | 1 | no | — |
| 111 | C4 | — | 8 | 2 | no | — |
| 112 | C4 | — | 8 | 2 | no | — |
| 113 | C4 | #453 | 1 | 1 | yes | characteristic |
| 114 | C4 | #458 | 1 | 1 | yes | characteristic |
| 115 | C2 | — | 2 | 1 | no | — |
| 116 | C2 | — | 4 | 1 | no | — |
| 117 | C2 | — | 8 | 1 | no | — |
| 118 | C2 | — | 16 | 2 | no | — |
| 119 | C2 | #1613 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1613 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| prime radical | #3199 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2238 | C2 | semidirect |
| 3. #2098 | C2 | semidirect |
| 4. #1339 | C2 | semidirect |
| 5. #1321 | C2 | semidirect |
| 11. #430 | C22 | semidirect |
| 17. #349 | C22 | semidirect |
| 18. H(3,ℤ/4) | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3199 | 2 |
| #2238 | 2 |
| #2098 | 2 |
| #1339 | 2 |
| #1321 | 2 |