← #496723
Subgroups — #496723
Order 512.
439 conjugacy classes of subgroups.
The subgroup lattice has 223 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #33421 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33401 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20787 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #13164 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #13162 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #12916 | C2 | 1 | 2 | yes | maximal |
| 7 | #12915 | C2 | 1 | 2 | yes | maximal |
| 8 | #12902 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #12885 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #3407 | C22 | 1 | 2 | yes | — |
| 11 | C4×#131 | C22 | 1 | 1 | yes | characteristic |
| 12 | #2164 | — | 2 | 2 | no | — |
| 13 | #2163 | — | 2 | 2 | no | — |
| 14 | #2153 | — | 2 | 2 | no | — |
| 15 | #2153 | C22 | 1 | 2 | yes | — |
| 16 | #2153 | C22 | 1 | 4 | yes | — |
| 17 | C2×#364 | — | 2 | 2 | no | — |
| 18 | C2×#364 | C22 | 1 | 2 | yes | — |
| 19 | #2151 | C22 | 1 | 1 | yes | characteristic |
| 20 | #2151 | C22 | 1 | 1 | yes | characteristic |
| 21 | #2151 | C22 | 1 | 4 | yes | — |
| 22 | #2113 | C22 | 1 | 1 | yes | characteristic |
| 23 | #2113 | C22 | 1 | 1 | yes | characteristic |
| 24 | C4×M32 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#346 | C22 | 1 | 1 | yes | characteristic |
| 26 | #1841 | C4 | 1 | 2 | yes | — |
| 27 | #1833 | C4 | 1 | 2 | yes | — |
| 28 | #1827 | C4 | 1 | 2 | yes | — |
| 29 | #1813 | C4 | 1 | 1 | yes | characteristic |
| 30 | #1812 | C4 | 1 | 1 | yes | characteristic |
| 31 | #1772 | C22 | 1 | 2 | yes | — |
| 32 | #1757 | C22 | 1 | 2 | yes | — |
| 33 | #1747 | C22 | 1 | 2 | yes | — |
| 34 | #1379 | C22 | 1 | 4 | yes | — |
| 35 | #1378 | C22 | 1 | 4 | yes | — |
| 36 | C2×#131 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 38 | C4×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | #504 | — | 4 | 4 | no | — |
| 40 | C2×M32 | — | 2 | 4 | no | — |
| 41 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 42 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 43 | C2×M32 | C23 | 1 | 4 | yes | — |
| 44 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 45 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×#108 | C2×C4 | 1 | 2 | yes | — |
| 47 | #427 | C2×C4 | 1 | 2 | yes | — |
| 48 | #413 | C2×C4 | 1 | 2 | yes | — |
| 49 | #410 | C2×C4 | 1 | 2 | yes | — |
| 50 | #405 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | #405 | D4 | 1 | 2 | yes | — |
| 52 | #405 | Q8 | 1 | 2 | yes | — |
| 53 | #405 | C23 | 1 | 4 | yes | — |
| 54 | #405 | C23 | 1 | 1 | yes | characteristic |
| 55 | C4×M16 | C2×C4 | 1 | 1 | yes | characteristic |
| 56 | C4×M16 | Q8 | 1 | 2 | yes | — |
| 57 | C2×C4×C8 | D4 | 1 | 2 | yes | — |
| 58 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 59 | #364 | — | 2 | 2 | no | — |
| 60 | #364 | — | 2 | 4 | no | — |
| 61 | #364 | — | 2 | 2 | no | — |
| 62 | #364 | — | 4 | 4 | no | — |
| 63 | C4⋊C16 | — | 2 | 2 | no | — |
| 64 | C4⋊C16 | — | 2 | 2 | no | — |
| 65 | C4⋊C16 | — | 4 | 4 | no | — |
| 66 | C22⋊C16 | — | 2 | 2 | no | — |
| 67 | C22⋊C16 | — | 2 | 2 | no | — |
| 68 | #346 | — | 2 | 1 | no | — |
| 69 | #346 | — | 2 | 1 | no | — |
| 70 | #346 | — | 2 | 1 | no | — |
| 71 | C4×C16 | — | 2 | 1 | no | — |
| 72 | #344 | C2×C4 | 1 | 4 | yes | — |
| 73 | #339 | C2×C4 | 1 | 4 | yes | — |
| 74 | #338 | C2×C4 | 1 | 2 | yes | — |
| 75 | H(3,ℤ/4) | C2×C4 | 1 | 2 | yes | — |
| 76 | H(3,ℤ/4) | C2×C4 | 1 | 4 | yes | — |
| 77 | C2×#41 | C22×C4 | 1 | 1 | yes | characteristic |
| 78 | #131 | — | 2 | 2 | no | — |
| 79 | #131 | — | 2 | 2 | no | — |
| 80 | #131 | — | 2 | 2 | no | — |
| 81 | #131 | — | 2 | 2 | no | — |
| 82 | C2×M16 | C4⋊C4 | 1 | 2 | yes | — |
| 83 | C2×M16 | C22×C4 | 1 | 4 | yes | — |
| 84 | C2×M16 | C22×C4 | 1 | 1 | yes | characteristic |
| 85 | C2×M16 | C22×C4 | 1 | 1 | yes | characteristic |
| 86 | C2×M16 | C2×Q8 | 1 | 2 | yes | — |
| 87 | C22×C8 | C4⋊C4 | 1 | 2 | yes | — |
| 88 | C22×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 89 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 90 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 91 | C4×Q8 | — | 2 | 1 | no | — |
| 92 | C4×D4 | — | 2 | 1 | no | — |
| 93 | C4×D4 | — | 2 | 2 | no | — |
| 94 | #117 | C4⋊C4 | 1 | 2 | yes | — |
| 95 | #117 | C22×C4 | 1 | 4 | yes | — |
| 96 | #117 | C22×C4 | 1 | 1 | yes | characteristic |
| 97 | C2×C42 | C4⋊C4 | 1 | 2 | yes | — |
| 98 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 99 | M32 | — | 2 | 4 | no | — |
| 100 | M32 | — | 2 | 2 | no | — |
| 101 | M32 | — | 2 | 4 | no | — |
| 102 | M32 | — | 2 | 2 | no | — |
| 103 | M32 | — | 4 | 4 | no | — |
| 104 | C2×C16 | — | 2 | 1 | no | — |
| 105 | C2×C16 | — | 2 | 1 | no | — |
| 106 | C2×C16 | — | 2 | 1 | no | — |
| 107 | C2×C16 | — | 2 | 1 | no | — |
| 108 | C2×C16 | — | 2 | 2 | no | — |
| 109 | C2×C16 | — | 2 | 4 | no | — |
| 110 | C2×C16 | — | 2 | 2 | no | — |
| 111 | C2×C16 | — | 4 | 4 | no | — |
| 112 | #108 | — | 2 | 2 | no | — |
| 113 | #108 | — | 2 | 2 | no | — |
| 114 | #107 | — | 2 | 2 | no | — |
| 115 | #106 | — | 2 | 2 | no | — |
| 116 | #101 | — | 2 | 2 | no | — |
| 117 | #100 | — | 2 | 2 | no | — |
| 118 | #99 | — | 2 | 4 | no | — |
| 119 | #97 | — | 2 | 4 | no | — |
| 120 | #97 | C4⋊C4 | 1 | 2 | yes | — |
| 121 | #97 | C4⋊C4 | 1 | 2 | yes | — |
| 122 | #97 | C2×Q8 | 1 | 2 | yes | — |
| 123 | C4×C8 | — | 2 | 4 | no | — |
| 124 | C4×C8 | C4⋊C4 | 1 | 2 | yes | — |
| 125 | C4×C8 | C4⋊C4 | 1 | 2 | yes | — |
| 126 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 127 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 128 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 129 | C4×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 130 | C4×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 131 | #41 | — | 2 | 2 | no | — |
| 132 | #41 | — | 2 | 2 | no | — |
| 133 | C2×Q8 | #131 | 1 | 1 | yes | characteristic |
| 134 | C2×D4 | #131 | 1 | 1 | yes | characteristic |
| 135 | C2×D4 | #131 | 1 | 2 | yes | — |
| 136 | C22×C4 | — | 2 | 1 | no | — |
| 137 | C22×C4 | — | 2 | 2 | no | — |
| 138 | C22×C4 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 139 | C22×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 140 | M16 | — | 2 | 2 | no | — |
| 141 | M16 | — | 2 | 2 | no | — |
| 142 | M16 | — | 2 | 4 | no | — |
| 143 | M16 | — | 2 | 4 | no | — |
| 144 | M16 | — | 2 | 4 | no | — |
| 145 | M16 | — | 2 | 4 | no | — |
| 146 | C2×C8 | — | 2 | 2 | no | — |
| 147 | C2×C8 | — | 2 | 2 | no | — |
| 148 | C2×C8 | — | 2 | 2 | no | — |
| 149 | C2×C8 | — | 2 | 2 | no | — |
| 150 | C2×C8 | — | 2 | 4 | no | — |
| 151 | C2×C8 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 152 | C2×C8 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 153 | C2×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 154 | C2×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 155 | C2×C8 | #131 | 1 | 2 | yes | — |
| 156 | C2×C8 | #131 | 1 | 2 | yes | — |
| 157 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 158 | C2×C8 | C22×Q8 | 1 | 1 | yes | characteristic |
| 159 | C2×C8 | H(5,𝔽2) | 1 | 2 | yes | — |
| 160 | C2×C8 | 2−1+4 | 1 | 2 | yes | — |
| 161 | C4⋊C4 | — | 2 | 1 | no | — |
| 162 | C4⋊C4 | — | 2 | 2 | no | — |
| 163 | C4⋊C4 | — | 2 | 4 | no | — |
| 164 | C22⋊C4 | — | 2 | 1 | no | — |
| 165 | C22⋊C4 | — | 2 | 2 | no | — |
| 166 | C22⋊C4 | — | 2 | 4 | no | — |
| 167 | C42 | — | 2 | 1 | no | — |
| 168 | C42 | — | 2 | 2 | no | — |
| 169 | C42 | — | 2 | 4 | no | — |
| 170 | C42 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 171 | C42 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 172 | C42 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 173 | C16 | — | 4 | 1 | no | — |
| 174 | C16 | — | 4 | 4 | no | — |
| 175 | C16 | — | 4 | 1 | no | — |
| 176 | C16 | — | 4 | 1 | no | — |
| 177 | C16 | — | 4 | 1 | no | — |
| 178 | C23 | #424 | 1 | 2 | yes | — |
| 179 | C23 | C2×#131 | 1 | 1 | yes | characteristic |
| 180 | Q8 | — | 2 | 2 | no | — |
| 181 | D4 | — | 2 | 4 | no | — |
| 182 | D4 | — | 2 | 2 | no | — |
| 183 | C2×C4 | — | 2 | 2 | no | — |
| 184 | C2×C4 | — | 2 | 2 | no | — |
| 185 | C2×C4 | — | 2 | 1 | no | — |
| 186 | C2×C4 | — | 2 | 1 | no | — |
| 187 | C2×C4 | — | 2 | 2 | no | — |
| 188 | C2×C4 | — | 4 | 4 | no | — |
| 189 | C2×C4 | — | 8 | 1 | no | — |
| 190 | C2×C4 | — | 8 | 2 | no | — |
| 191 | C2×C4 | #424 | 1 | 2 | yes | — |
| 192 | C2×C4 | C22×C4⋊C4 | 1 | 1 | yes | characteristic |
| 193 | C2×C4 | #520 | 1 | 2 | yes | — |
| 194 | C2×C4 | C2×#131 | 1 | 2 | yes | — |
| 195 | C2×C4 | C2×#131 | 1 | 1 | yes | characteristic |
| 196 | C8 | — | 2 | 2 | no | — |
| 197 | C8 | — | 2 | 2 | no | — |
| 198 | C8 | — | 4 | 4 | no | — |
| 199 | C8 | #424 | 1 | 1 | yes | characteristic |
| 200 | C8 | #424 | 1 | 1 | yes | characteristic |
| 201 | C8 | #424 | 1 | 2 | yes | — |
| 202 | C8 | #535 | 1 | 1 | yes | characteristic |
| 203 | C8 | #536 | 1 | 1 | yes | characteristic |
| 204 | C8 | #544 | 1 | 2 | yes | — |
| 205 | C22 | — | 2 | 2 | no | — |
| 206 | C22 | — | 4 | 1 | no | — |
| 207 | C22 | — | 4 | 2 | no | — |
| 208 | C22 | C2×#424 | 1 | 1 | yes | characteristic |
| 209 | C22 | #2994 | 1 | 2 | yes | — |
| 210 | C4 | — | 2 | 2 | no | — |
| 211 | C4 | — | 4 | 1 | no | — |
| 212 | C4 | — | 4 | 2 | no | — |
| 213 | C4 | — | 4 | 1 | no | — |
| 214 | C4 | — | 8 | 4 | no | — |
| 215 | C4 | #2317 | 1 | 1 | yes | characteristic |
| 216 | C4 | C2×#424 | 1 | 1 | yes | characteristic |
| 217 | C4 | #2994 | 1 | 2 | yes | — |
| 218 | C2 | — | 2 | 1 | no | — |
| 219 | C2 | — | 2 | 2 | no | — |
| 220 | C2 | — | 4 | 2 | no | — |
| 221 | C2 | #21973 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta