Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
412 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 115 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 | #3556 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #3165 | C2 | 1 | 2 | yes | maximal |
| 3 | #3139 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #1663 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | #1526 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #1526 | C2 | 1 | 2 | yes | maximal |
| 7 | #564 | C22 | 1 | 4 | yes | — |
| 8 | #557 | — | 2 | 2 | no | — |
| 9 | #557 | — | 2 | 2 | no | — |
| 10 | #557 | — | 2 | 2 | no | — |
| 11 | #541 | — | 2 | 2 | no | — |
| 12 | #541 | — | 2 | 4 | no | — |
| 13 | #533 | C22 | 1 | 1 | yes | characteristic |
| 14 | #533 | C22 | 1 | 2 | yes | — |
| 15 | C2×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 16 | #501 | C22 | 1 | 8 | yes | — |
| 17 | #500 | — | 2 | 2 | no | — |
| 18 | #498 | — | 2 | 2 | no | — |
| 19 | #491 | C22 | 1 | 4 | yes | — |
| 20 | #487 | — | 2 | 2 | no | — |
| 21 | #428 | — | 2 | 4 | no | — |
| 22 | #419 | — | 2 | 2 | no | — |
| 23 | C4×M16 | C22 | 1 | 1 | yes | characteristic |
| 24 | C4×M16 | C22 | 1 | 2 | yes | — |
| 25 | #330 | C22 | 1 | 8 | yes | — |
| 26 | #330 | C22 | 1 | 4 | yes | — |
| 27 | C22×Q8 | — | 2 | 1 | no | — |
| 28 | C2×M16 | — | 2 | 2 | no | — |
| 29 | C2×M16 | — | 2 | 1 | no | — |
| 30 | C4⋊Q8 | — | 2 | 2 | no | — |
| 31 | C4⋊Q8 | — | 2 | 4 | no | — |
| 32 | C4⋊Q8 | — | 2 | 2 | no | — |
| 33 | C4⋊Q8 | — | 2 | 1 | no | — |
| 34 | C4⋊Q8 | D4 | 1 | 2 | yes | — |
| 35 | C4⋊Q8 | D4 | 1 | 2 | yes | — |
| 36 | C4⋊Q8 | D4 | 1 | 2 | yes | — |
| 37 | C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 38 | #126 | — | 2 | 8 | no | — |
| 39 | #125 | — | 2 | 2 | no | — |
| 40 | #125 | — | 2 | 4 | no | — |
| 41 | #125 | — | 2 | 8 | no | — |
| 42 | #125 | — | 2 | 8 | no | — |
| 43 | #125 | — | 2 | 8 | no | — |
| 44 | #125 | C23 | 1 | 2 | yes | — |
| 45 | #125 | C23 | 1 | 4 | yes | — |
| 46 | C22⋊Q8 | — | 2 | 2 | no | — |
| 47 | C22⋊Q8 | — | 2 | 2 | no | — |
| 48 | C22⋊Q8 | — | 2 | 2 | no | — |
| 49 | C4×Q8 | — | 2 | 4 | no | — |
| 50 | C4×Q8 | — | 2 | 4 | no | — |
| 51 | C4×Q8 | — | 2 | 4 | no | — |
| 52 | C4×Q8 | — | 2 | 2 | no | — |
| 53 | C4×Q8 | — | 2 | 2 | no | — |
| 54 | C4×Q8 | — | 2 | 2 | no | — |
| 55 | #117 | D4 | 1 | 2 | yes | — |
| 56 | #117 | D4 | 1 | 4 | yes | — |
| 57 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 58 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 59 | #107 | — | 2 | 8 | no | — |
| 60 | #106 | — | 2 | 8 | no | — |
| 61 | #103 | — | 2 | 8 | no | — |
| 62 | #97 | C23 | 1 | 2 | yes | — |
| 63 | #97 | C23 | 1 | 4 | yes | — |
| 64 | C4×C8 | — | 2 | 1 | no | — |
| 65 | C4×C8 | — | 2 | 2 | no | — |
| 66 | C2×Q8 | — | 2 | 2 | no | — |
| 67 | C2×Q8 | — | 2 | 2 | no | — |
| 68 | C2×Q8 | — | 2 | 2 | no | — |
| 69 | C2×Q8 | — | 2 | 2 | no | — |
| 70 | C2×Q8 | — | 4 | 4 | no | — |
| 71 | C2×Q8 | — | 4 | 1 | no | — |
| 72 | C22×C4 | — | 4 | 1 | no | — |
| 73 | C22×C4 | C2×D4 | 1 | 2 | yes | — |
| 74 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | M16 | — | 2 | 8 | no | — |
| 76 | M16 | — | 4 | 2 | no | — |
| 77 | C2×C8 | — | 2 | 4 | no | — |
| 78 | C2×C8 | — | 2 | 2 | no | — |
| 79 | C4⋊C4 | — | 2 | 8 | no | — |
| 80 | C4⋊C4 | — | 2 | 8 | no | — |
| 81 | C4⋊C4 | — | 2 | 8 | no | — |
| 82 | C4⋊C4 | — | 2 | 2 | no | — |
| 83 | C4⋊C4 | — | 2 | 2 | no | — |
| 84 | C4⋊C4 | — | 2 | 2 | no | — |
| 85 | C4⋊C4 | — | 2 | 4 | no | — |
| 86 | C4⋊C4 | — | 2 | 4 | no | — |
| 87 | C4⋊C4 | — | 2 | 4 | no | — |
| 88 | C4⋊C4 | — | 2 | 4 | no | — |
| 89 | C4⋊C4 | — | 4 | 2 | no | — |
| 90 | C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 91 | C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 92 | C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 93 | C22⋊C4 | — | 2 | 2 | no | — |
| 94 | C22⋊C4 | — | 2 | 4 | no | — |
| 95 | C42 | — | 2 | 8 | no | — |
| 96 | C42 | — | 2 | 2 | no | — |
| 97 | C42 | — | 2 | 4 | no | — |
| 98 | C42 | C2×D4 | 1 | 2 | yes | — |
| 99 | C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C42 | C24 | 1 | 1 | yes | characteristic |
| 101 | C23 | #120 | 1 | 1 | yes | characteristic |
| 102 | Q8 | — | 4 | 4 | no | — |
| 103 | Q8 | — | 4 | 4 | no | — |
| 104 | Q8 | — | 4 | 4 | no | — |
| 105 | Q8 | — | 4 | 4 | no | — |
| 106 | Q8 | — | 8 | 2 | no | — |
| 107 | C2×C4 | — | 2 | 2 | no | — |
| 108 | C2×C4 | — | 2 | 4 | no | — |
| 109 | C2×C4 | — | 2 | 4 | no | — |
| 110 | C2×C4 | — | 2 | 4 | no | — |
| 111 | C2×C4 | — | 2 | 4 | no | — |
| 112 | C2×C4 | — | 4 | 2 | no | — |
| 113 | C2×C4 | — | 8 | 2 | no | — |
| 114 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 115 | C2×C4 | #120 | 1 | 2 | yes | — |
| 116 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 117 | C2×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 118 | C8 | — | 2 | 8 | no | — |
| 119 | C8 | — | 4 | 2 | no | — |
| 120 | C22 | — | 2 | 1 | no | — |
| 121 | C22 | — | 2 | 2 | no | — |
| 122 | C22 | C2×#120 | 1 | 1 | yes | characteristic |
| 123 | C4 | — | 2 | 1 | no | — |
| 124 | C4 | — | 2 | 2 | no | — |
| 125 | C4 | — | 4 | 4 | no | — |
| 126 | C4 | — | 4 | 4 | no | — |
| 127 | C4 | — | 4 | 4 | no | — |
| 128 | C4 | — | 8 | 2 | no | — |
| 129 | C4 | #454 | 1 | 2 | yes | — |
| 130 | C4 | #456 | 1 | 4 | yes | — |
| 131 | C2 | — | 4 | 1 | no | — |
| 132 | C2 | C2×#454 | 1 | 1 | yes | characteristic |
| 133 | C2 | C2×#456 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C23 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #1663 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3556 | 2 |
| #3165 | 2 |
| #3139 | 2 |
| #1663 | 2 |
| #1526 | 2 |