Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
306 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 68 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 | #23250 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11478 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #8962 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #8865 | C2 | 1 | 2 | yes | maximal |
| 5 | #7286 | C2 | 1 | 2 | yes | maximal |
| 6 | #2983 | C22 | 1 | 1 | yes | characteristic |
| 7 | C4×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 8 | #1985 | — | 2 | 1 | no | — |
| 9 | #1985 | C22 | 1 | 2 | yes | — |
| 10 | #1939 | — | 2 | 2 | no | — |
| 11 | #1770 | — | 2 | 2 | no | — |
| 12 | C4×C4⋊C8 | — | 2 | 1 | no | — |
| 13 | #1573 | — | 2 | 2 | no | — |
| 14 | #1479 | — | 2 | 1 | no | — |
| 15 | #1461 | C22 | 1 | 1 | yes | characteristic |
| 16 | #1461 | C22 | 1 | 2 | yes | — |
| 17 | #1325 | C4 | 1 | 2 | yes | — |
| 18 | #1319 | C4 | 1 | 2 | yes | — |
| 19 | #1302 | — | 2 | 2 | no | — |
| 20 | #1276 | — | 2 | 1 | no | — |
| 21 | C2×C4⋊Q8 | Q8 | 1 | 1 | yes | characteristic |
| 22 | C2×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 23 | C2×C4×Q8 | C23 | 1 | 1 | yes | characteristic |
| 24 | #446 | — | 4 | 1 | no | — |
| 25 | #446 | — | 4 | 1 | no | — |
| 26 | #432 | — | 2 | 2 | no | — |
| 27 | #431 | — | 2 | 2 | no | — |
| 28 | #431 | — | 2 | 1 | no | — |
| 29 | #431 | — | 2 | 1 | no | — |
| 30 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 31 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 32 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 33 | C2×C4⋊C8 | D4 | 1 | 2 | yes | — |
| 34 | C2×#97 | — | 2 | 1 | no | — |
| 35 | C2×#97 | D4 | 1 | 2 | yes | — |
| 36 | C2×C4×C8 | — | 2 | 2 | no | — |
| 37 | #391 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | #391 | Q8 | 1 | 1 | yes | characteristic |
| 39 | #389 | — | 2 | 2 | no | — |
| 40 | #384 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 42 | C43 | D4 | 1 | 1 | yes | characteristic |
| 43 | #336 | — | 2 | 2 | no | — |
| 44 | #336 | — | 2 | 2 | no | — |
| 45 | #336 | — | 2 | 2 | no | — |
| 46 | #335 | — | 4 | 2 | no | — |
| 47 | #334 | — | 4 | 2 | no | — |
| 48 | Q8⋊C8 | — | 4 | 1 | no | — |
| 49 | Q8⋊C8 | — | 4 | 1 | no | — |
| 50 | #324 | — | 2 | 2 | no | — |
| 51 | #324 | — | 2 | 2 | no | — |
| 52 | #324 | C2×C4 | 1 | 4 | yes | — |
| 53 | C22×Q8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 54 | C22×Q8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 55 | C22×C8 | — | 2 | 2 | no | — |
| 56 | C22×C8 | — | 2 | 2 | no | — |
| 57 | C4⋊Q8 | — | 2 | 2 | no | — |
| 58 | C4⋊Q8 | — | 4 | 1 | no | — |
| 59 | C4×Q8 | — | 4 | 1 | no | — |
| 60 | C4×Q8 | — | 4 | 1 | no | — |
| 61 | C4×Q8 | — | 4 | 1 | no | — |
| 62 | C4×Q8 | — | 4 | 1 | no | — |
| 63 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 64 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 65 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 66 | C2×C4⋊C4 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 67 | C2×C4⋊C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | C2×C4⋊C4 | #41 | 1 | 2 | yes | — |
| 69 | C2×C42 | — | 2 | 1 | no | — |
| 70 | C2×C42 | — | 2 | 1 | no | — |
| 71 | C2×C42 | — | 2 | 1 | no | — |
| 72 | C2×C42 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 73 | C2×C42 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 74 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 76 | C4⋊C8 | — | 2 | 4 | no | — |
| 77 | C4⋊C8 | — | 2 | 1 | no | — |
| 78 | C4⋊C8 | — | 2 | 1 | no | — |
| 79 | C4⋊C8 | — | 2 | 2 | no | — |
| 80 | C4⋊C8 | — | 2 | 2 | no | — |
| 81 | C4⋊C8 | — | 4 | 2 | no | — |
| 82 | C4⋊C8 | — | 4 | 2 | no | — |
| 83 | C4⋊C8 | — | 4 | 2 | no | — |
| 84 | C4⋊C8 | — | 4 | 1 | no | — |
| 85 | C4⋊C8 | — | 8 | 1 | no | — |
| 86 | C22⋊C8 | — | 2 | 4 | no | — |
| 87 | C22⋊C8 | — | 4 | 1 | no | — |
| 88 | C22⋊C8 | — | 4 | 1 | no | — |
| 89 | #97 | — | 4 | 2 | no | — |
| 90 | #97 | — | 4 | 2 | no | — |
| 91 | #97 | — | 8 | 1 | no | — |
| 92 | C4×C8 | — | 4 | 2 | no | — |
| 93 | C4×C8 | — | 4 | 4 | no | — |
| 94 | C4×C8 | — | 4 | 1 | no | — |
| 95 | C4×C8 | — | 4 | 1 | no | — |
| 96 | #95 | — | 2 | 2 | no | — |
| 97 | #95 | — | 2 | 2 | no | — |
| 98 | C2×Q8 | — | 2 | 1 | no | — |
| 99 | C2×Q8 | — | 2 | 1 | no | — |
| 100 | C2×Q8 | — | 2 | 1 | no | — |
| 101 | C2×Q8 | — | 2 | 1 | no | — |
| 102 | C2×Q8 | — | 4 | 1 | no | — |
| 103 | C2×Q8 | — | 4 | 1 | no | — |
| 104 | C2×Q8 | — | 4 | 1 | no | — |
| 105 | C2×Q8 | — | 4 | 1 | no | — |
| 106 | C22×C4 | — | 2 | 1 | no | — |
| 107 | C22×C4 | — | 2 | 1 | no | — |
| 108 | C22×C4 | — | 2 | 1 | no | — |
| 109 | C22×C4 | — | 2 | 1 | no | — |
| 110 | C22×C4 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 111 | C22×C4 | C4×D4 | 1 | 2 | yes | — |
| 112 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 113 | C22×C4 | C22⋊Q8 | 1 | 2 | yes | — |
| 114 | C22×C4 | #123 | 1 | 1 | yes | characteristic |
| 115 | C2×C8 | — | 2 | 4 | no | — |
| 116 | C2×C8 | — | 4 | 2 | no | — |
| 117 | C2×C8 | — | 4 | 2 | no | — |
| 118 | C2×C8 | — | 4 | 2 | no | — |
| 119 | C2×C8 | — | 4 | 4 | no | — |
| 120 | C2×C8 | — | 4 | 2 | no | — |
| 121 | C2×C8 | — | 4 | 1 | no | — |
| 122 | C2×C8 | — | 4 | 1 | no | — |
| 123 | C2×C8 | — | 4 | 2 | no | — |
| 124 | C2×C8 | — | 8 | 1 | no | — |
| 125 | C4⋊C4 | — | 4 | 1 | no | — |
| 126 | C4⋊C4 | — | 4 | 2 | no | — |
| 127 | C4⋊C4 | — | 4 | 4 | no | — |
| 128 | C4⋊C4 | — | 4 | 1 | no | — |
| 129 | C4⋊C4 | — | 4 | 4 | no | — |
| 130 | C4⋊C4 | — | 8 | 1 | no | — |
| 131 | C42 | — | 2 | 2 | no | — |
| 132 | C42 | — | 2 | 2 | no | — |
| 133 | C42 | — | 2 | 1 | no | — |
| 134 | C42 | — | 2 | 1 | no | — |
| 135 | C42 | — | 2 | 2 | no | — |
| 136 | C42 | — | 2 | 2 | no | — |
| 137 | C42 | — | 2 | 1 | no | — |
| 138 | C42 | — | 4 | 1 | no | — |
| 139 | C42 | — | 4 | 2 | no | — |
| 140 | C42 | — | 4 | 1 | no | — |
| 141 | C42 | — | 4 | 1 | no | — |
| 142 | C42 | #108 | 1 | 2 | yes | — |
| 143 | C23 | #385 | 1 | 1 | yes | characteristic |
| 144 | Q8 | — | 4 | 2 | no | — |
| 145 | Q8 | — | 8 | 1 | no | — |
| 146 | Q8 | — | 8 | 1 | no | — |
| 147 | Q8 | — | 8 | 1 | no | — |
| 148 | C2×C4 | — | 2 | 2 | no | — |
| 149 | C2×C4 | — | 2 | 2 | no | — |
| 150 | C2×C4 | — | 2 | 2 | no | — |
| 151 | C2×C4 | — | 2 | 1 | no | — |
| 152 | C2×C4 | — | 2 | 1 | no | — |
| 153 | C2×C4 | — | 2 | 1 | no | — |
| 154 | C2×C4 | — | 2 | 1 | no | — |
| 155 | C2×C4 | — | 2 | 2 | no | — |
| 156 | C2×C4 | — | 2 | 2 | no | — |
| 157 | C2×C4 | — | 2 | 2 | no | — |
| 158 | C2×C4 | — | 2 | 1 | no | — |
| 159 | C2×C4 | — | 2 | 2 | no | — |
| 160 | C2×C4 | — | 2 | 2 | no | — |
| 161 | C2×C4 | — | 4 | 1 | no | — |
| 162 | C2×C4 | — | 4 | 1 | no | — |
| 163 | C2×C4 | — | 4 | 1 | no | — |
| 164 | C2×C4 | — | 4 | 2 | no | — |
| 165 | C2×C4 | — | 4 | 2 | no | — |
| 166 | C2×C4 | — | 4 | 1 | no | — |
| 167 | C2×C4 | — | 4 | 1 | no | — |
| 168 | C2×C4 | — | 4 | 1 | no | — |
| 169 | C2×C4 | — | 4 | 1 | no | — |
| 170 | C2×C4 | C2×#108 | 1 | 1 | yes | characteristic |
| 171 | C2×C4 | #430 | 1 | 1 | yes | characteristic |
| 172 | C2×C4 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 173 | C2×C4 | #458 | 1 | 1 | yes | characteristic |
| 174 | C8 | — | 4 | 4 | no | — |
| 175 | C8 | — | 8 | 4 | no | — |
| 176 | C8 | — | 8 | 1 | no | — |
| 177 | C8 | — | 8 | 2 | no | — |
| 178 | C8 | — | 8 | 1 | no | — |
| 179 | C8 | — | 16 | 1 | no | — |
| 180 | C22 | — | 2 | 1 | no | — |
| 181 | C22 | — | 2 | 1 | no | — |
| 182 | C22 | #1855 | 1 | 1 | yes | characteristic |
| 183 | C22 | #1858 | 1 | 1 | yes | characteristic |
| 184 | C22 | #1867 | 1 | 1 | yes | characteristic |
| 185 | C4 | — | 2 | 4 | no | — |
| 186 | C4 | — | 2 | 1 | no | — |
| 187 | C4 | — | 2 | 1 | no | — |
| 188 | C4 | — | 2 | 1 | no | — |
| 189 | C4 | — | 4 | 2 | no | — |
| 190 | C4 | — | 4 | 2 | no | — |
| 191 | C4 | — | 4 | 2 | no | — |
| 192 | C4 | — | 4 | 1 | no | — |
| 193 | C4 | — | 8 | 1 | no | — |
| 194 | C4 | — | 8 | 1 | no | — |
| 195 | C4 | #2204 | 1 | 1 | yes | characteristic |
| 196 | C4 | #2205 | 1 | 1 | yes | characteristic |
| 197 | C2 | — | 2 | 1 | no | — |
| 198 | C2 | — | 2 | 1 | no | — |
| 199 | C2 | #8860 | 1 | 1 | yes | characteristic |
| 200 | C2 | #12968 | 1 | 1 | yes | characteristic |
| 201 | C2 | #12973 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | #1867 |
| prime radical | C23 | #385 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23250 | 2 |
| #11478 | 2 |
| #8962 | 2 |
| #8865 | 2 |
| #7286 | 2 |