Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
222 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 49 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 | #3237 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #3113 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #3104 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #1537 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #1497 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #1467 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #1287 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #533 | C22 | 1 | 1 | yes | characteristic |
| 9 | #533 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C4×Q8 | C22 | 1 | 1 | yes | characteristic |
| 11 | #501 | — | 2 | 1 | no | — |
| 12 | #491 | — | 2 | 1 | no | — |
| 13 | #485 | — | 2 | 2 | no | — |
| 14 | #484 | — | 2 | 1 | no | — |
| 15 | #483 | — | 2 | 1 | no | — |
| 16 | #477 | — | 2 | 1 | no | — |
| 17 | #477 | — | 2 | 1 | no | — |
| 18 | #475 | — | 2 | 1 | no | — |
| 19 | #475 | — | 2 | 1 | no | — |
| 20 | #462 | — | 2 | 1 | no | — |
| 21 | #462 | — | 2 | 1 | no | — |
| 22 | #451 | — | 2 | 1 | no | — |
| 23 | #451 | — | 2 | 1 | no | — |
| 24 | #441 | — | 2 | 2 | no | — |
| 25 | C4×Q16 | — | 2 | 2 | no | — |
| 26 | #432 | C22 | 1 | 1 | yes | characteristic |
| 27 | #432 | C22 | 1 | 1 | yes | characteristic |
| 28 | #432 | C22 | 1 | 1 | yes | characteristic |
| 29 | #431 | C22 | 1 | 1 | yes | characteristic |
| 30 | #330 | — | 2 | 2 | no | — |
| 31 | Q8⋊C8 | — | 2 | 2 | no | — |
| 32 | #324 | — | 2 | 2 | no | — |
| 33 | C22×Q8 | D4 | 1 | 1 | yes | characteristic |
| 34 | C2×Q16 | — | 2 | 2 | no | — |
| 35 | C2×Q16 | — | 2 | 2 | no | — |
| 36 | C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 37 | C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 38 | C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 39 | C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 40 | #125 | — | 2 | 1 | no | — |
| 41 | #125 | — | 2 | 1 | no | — |
| 42 | C22⋊Q8 | — | 2 | 1 | no | — |
| 43 | C22⋊Q8 | — | 2 | 1 | no | — |
| 44 | C22⋊Q8 | — | 2 | 1 | no | — |
| 45 | C22⋊Q8 | — | 2 | 1 | no | — |
| 46 | C4×Q8 | — | 2 | 1 | no | — |
| 47 | C4×Q8 | — | 2 | 1 | no | — |
| 48 | C4×Q8 | — | 2 | 1 | no | — |
| 49 | C4×Q8 | — | 2 | 1 | no | — |
| 50 | C4×Q8 | — | 2 | 1 | no | — |
| 51 | C4×Q8 | — | 2 | 1 | no | — |
| 52 | C4×Q8 | — | 2 | 1 | no | — |
| 53 | C4×Q8 | — | 2 | 1 | no | — |
| 54 | #117 | — | 2 | 1 | no | — |
| 55 | #117 | — | 2 | 1 | no | — |
| 56 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 57 | C2×C4⋊C4 | D4 | 1 | 1 | yes | characteristic |
| 58 | C2×C42 | — | 2 | 1 | no | — |
| 59 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 60 | #107 | — | 2 | 2 | no | — |
| 61 | #107 | — | 2 | 2 | no | — |
| 62 | #106 | — | 2 | 2 | no | — |
| 63 | #106 | — | 2 | 2 | no | — |
| 64 | C4⋊C8 | — | 2 | 1 | no | — |
| 65 | C4⋊C8 | — | 2 | 1 | no | — |
| 66 | C4⋊C8 | — | 2 | 1 | no | — |
| 67 | C4⋊C8 | — | 2 | 1 | no | — |
| 68 | #103 | — | 2 | 2 | no | — |
| 69 | #103 | — | 2 | 2 | no | — |
| 70 | #103 | — | 2 | 2 | no | — |
| 71 | #103 | — | 2 | 2 | no | — |
| 72 | #103 | — | 2 | 2 | no | — |
| 73 | #103 | — | 2 | 2 | no | — |
| 74 | C22⋊C8 | — | 2 | 1 | no | — |
| 75 | C22⋊C8 | — | 2 | 1 | no | — |
| 76 | C22⋊C8 | — | 2 | 1 | no | — |
| 77 | C22⋊C8 | D4 | 1 | 2 | yes | — |
| 78 | #97 | — | 2 | 1 | no | — |
| 79 | #97 | — | 2 | 1 | no | — |
| 80 | #97 | D4 | 1 | 2 | yes | — |
| 81 | C4×C8 | D4 | 1 | 2 | yes | — |
| 82 | C2×Q8 | — | 2 | 1 | no | — |
| 83 | C2×Q8 | — | 2 | 1 | no | — |
| 84 | C2×Q8 | — | 2 | 1 | no | — |
| 85 | C2×Q8 | — | 2 | 1 | no | — |
| 86 | C2×Q8 | — | 2 | 1 | no | — |
| 87 | C2×Q8 | — | 2 | 1 | no | — |
| 88 | C2×Q8 | — | 4 | 2 | no | — |
| 89 | C22×C4 | — | 2 | 1 | no | — |
| 90 | C22×C4 | — | 2 | 1 | no | — |
| 91 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C22×C4 | #41 | 1 | 1 | yes | characteristic |
| 94 | Q16 | — | 4 | 4 | no | — |
| 95 | Q16 | — | 4 | 4 | no | — |
| 96 | C2×C8 | — | 2 | 2 | no | — |
| 97 | C2×C8 | — | 2 | 2 | no | — |
| 98 | C2×C8 | — | 2 | 2 | no | — |
| 99 | C2×C8 | — | 4 | 1 | no | — |
| 100 | C4⋊C4 | — | 2 | 1 | no | — |
| 101 | C4⋊C4 | — | 2 | 1 | no | — |
| 102 | C4⋊C4 | — | 2 | 1 | no | — |
| 103 | C4⋊C4 | — | 2 | 1 | no | — |
| 104 | C4⋊C4 | — | 2 | 1 | no | — |
| 105 | C4⋊C4 | — | 2 | 1 | no | — |
| 106 | C4⋊C4 | — | 2 | 1 | no | — |
| 107 | C4⋊C4 | — | 2 | 1 | no | — |
| 108 | C4⋊C4 | — | 2 | 1 | no | — |
| 109 | C4⋊C4 | — | 2 | 1 | no | — |
| 110 | C4⋊C4 | — | 4 | 1 | no | — |
| 111 | C4⋊C4 | — | 4 | 1 | no | — |
| 112 | C4⋊C4 | — | 4 | 1 | no | — |
| 113 | C4⋊C4 | — | 4 | 1 | no | — |
| 114 | C4⋊C4 | — | 4 | 1 | no | — |
| 115 | C4⋊C4 | — | 4 | 1 | no | — |
| 116 | C22⋊C4 | — | 4 | 1 | no | — |
| 117 | C22⋊C4 | — | 4 | 1 | no | — |
| 118 | C42 | — | 4 | 1 | no | — |
| 119 | C42 | — | 4 | 1 | no | — |
| 120 | C42 | — | 4 | 1 | no | — |
| 121 | C42 | — | 4 | 1 | no | — |
| 122 | C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 123 | C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 124 | C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 125 | C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 126 | C23 | #121 | 1 | 1 | yes | characteristic |
| 127 | Q8 | — | 2 | 2 | no | — |
| 128 | Q8 | — | 2 | 2 | no | — |
| 129 | Q8 | — | 4 | 1 | no | — |
| 130 | Q8 | — | 4 | 1 | no | — |
| 131 | Q8 | — | 4 | 2 | no | — |
| 132 | Q8 | — | 4 | 2 | no | — |
| 133 | Q8 | — | 4 | 2 | no | — |
| 134 | Q8 | — | 4 | 2 | no | — |
| 135 | C2×C4 | — | 2 | 2 | no | — |
| 136 | C2×C4 | — | 2 | 2 | no | — |
| 137 | C2×C4 | — | 4 | 1 | no | — |
| 138 | C2×C4 | — | 4 | 1 | no | — |
| 139 | C2×C4 | — | 4 | 1 | no | — |
| 140 | C2×C4 | — | 4 | 1 | no | — |
| 141 | C2×C4 | — | 4 | 1 | no | — |
| 142 | C2×C4 | — | 4 | 1 | no | — |
| 143 | C2×C4 | — | 4 | 1 | no | — |
| 144 | C2×C4 | — | 4 | 1 | no | — |
| 145 | C2×C4 | — | 4 | 1 | no | — |
| 146 | C2×C4 | — | 8 | 1 | no | — |
| 147 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 148 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 149 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 150 | C2×C4 | #121 | 1 | 1 | yes | characteristic |
| 151 | C2×C4 | #121 | 1 | 1 | yes | characteristic |
| 152 | C2×C4 | #127 | 1 | 1 | yes | characteristic |
| 153 | C8 | — | 4 | 2 | no | — |
| 154 | C8 | — | 4 | 2 | no | — |
| 155 | C8 | — | 4 | 2 | no | — |
| 156 | C8 | — | 8 | 1 | no | — |
| 157 | C22 | — | 2 | 1 | no | — |
| 158 | C22 | — | 2 | 1 | no | — |
| 159 | C22 | — | 2 | 1 | no | — |
| 160 | C22 | #392 | 1 | 1 | yes | characteristic |
| 161 | C4 | — | 2 | 1 | no | — |
| 162 | C4 | — | 2 | 1 | no | — |
| 163 | C4 | — | 2 | 1 | no | — |
| 164 | C4 | — | 2 | 1 | no | — |
| 165 | C4 | — | 2 | 1 | no | — |
| 166 | C4 | — | 4 | 2 | no | — |
| 167 | C4 | — | 8 | 1 | no | — |
| 168 | C4 | — | 8 | 1 | no | — |
| 169 | C4 | — | 8 | 1 | no | — |
| 170 | C4 | — | 8 | 1 | no | — |
| 171 | C4 | — | 8 | 1 | no | — |
| 172 | C4 | #473 | 1 | 2 | yes | — |
| 173 | C2 | — | 4 | 1 | no | — |
| 174 | C2 | #2009 | 1 | 1 | yes | characteristic |
| 175 | C2 | #2010 | 1 | 1 | yes | characteristic |
| 176 | C2 | #2020 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3237 | 2 |
| #3113 | 2 |
| #3104 | 2 |
| #1537 | 2 |
| #1497 | 2 |
| #1467 | 2 |
| #1287 | 2 |