← #128694
Subgroups — #128694
Order 512.
316 conjugacy classes of subgroups.
The subgroup lattice has 228 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#2171 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #12665 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #12642 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#1336 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×#1313 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #7182 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7161 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#504 | — | 2 | 1 | no | — |
| 9 | C2×#443 | C22 | 1 | 1 | yes | characteristic |
| 10 | #2171 | — | 2 | 2 | no | — |
| 11 | #2171 | C4 | 1 | 4 | yes | complemented |
| 12 | C2×#364 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#364 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#359 | C4 | 1 | 1 | yes | characteristic |
| 15 | #2138 | C4 | 1 | 1 | yes | characteristic |
| 16 | C2×#350 | — | 2 | 1 | no | — |
| 17 | #2108 | — | 2 | 1 | no | — |
| 18 | C2×#346 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#346 | C22 | 1 | 1 | yes | characteristic |
| 20 | #1772 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#322 | C22 | 1 | 1 | yes | characteristic |
| 22 | #1382 | C4 | 1 | 1 | yes | characteristic |
| 23 | #1381 | C4 | 1 | 1 | yes | characteristic |
| 24 | #1379 | — | 2 | 1 | no | — |
| 25 | #1336 | C4 | 1 | 4 | yes | complemented |
| 26 | #1313 | — | 2 | 2 | no | — |
| 27 | C2×#135 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C2×#131 | — | 2 | 1 | no | — |
| 29 | #504 | — | 2 | 4 | no | — |
| 30 | #504 | — | 4 | 2 | no | — |
| 31 | C2×M32 | — | 2 | 1 | no | — |
| 32 | C2×M32 | — | 2 | 1 | no | — |
| 33 | C2×M32 | — | 4 | 1 | no | — |
| 34 | C22×C16 | — | 4 | 1 | no | — |
| 35 | C22×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C22×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C22×C16 | D4 | 1 | 1 | yes | characteristic |
| 38 | C22×C16 | D4 | 1 | 1 | yes | characteristic |
| 39 | #443 | — | 2 | 1 | no | — |
| 40 | #443 | — | 2 | 1 | no | — |
| 41 | #443 | C2×C4 | 1 | 2 | yes | — |
| 42 | #443 | C2×C4 | 1 | 2 | yes | — |
| 43 | C2×#108 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | C2×#108 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | #427 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | C2×#104 | — | 2 | 1 | no | — |
| 47 | #405 | — | 2 | 1 | no | — |
| 48 | C2×C4×C8 | — | 2 | 1 | no | — |
| 49 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 50 | #364 | — | 2 | 1 | no | — |
| 51 | #364 | — | 2 | 1 | no | — |
| 52 | #364 | C2×C4 | 1 | 2 | yes | — |
| 53 | #364 | C2×C4 | 1 | 2 | yes | — |
| 54 | #359 | C8 | 1 | 4 | yes | complemented |
| 55 | #358 | C8 | 1 | 2 | yes | complemented |
| 56 | #357 | C8 | 1 | 2 | yes | complemented |
| 57 | #350 | — | 2 | 4 | no | — |
| 58 | #346 | — | 2 | 1 | no | — |
| 59 | #346 | — | 2 | 1 | no | — |
| 60 | #346 | — | 4 | 1 | no | — |
| 61 | #346 | C2×C4 | 1 | 2 | yes | — |
| 62 | #346 | D4 | 1 | 2 | yes | — |
| 63 | C4×C16 | — | 4 | 1 | no | — |
| 64 | #339 | — | 2 | 1 | no | — |
| 65 | #322 | — | 2 | 1 | no | — |
| 66 | #322 | C2×C4 | 1 | 2 | yes | — |
| 67 | C2×#41 | — | 2 | 1 | no | — |
| 68 | #135 | — | 2 | 1 | no | — |
| 69 | #135 | — | 2 | 1 | no | — |
| 70 | #135 | C42 | 1 | 2 | yes | — |
| 71 | #135 | C2×C8 | 1 | 2 | yes | — |
| 72 | C2×Q16 | C2×C8 | 1 | 1 | yes | characteristic |
| 73 | C2×SD16 | — | 2 | 1 | no | — |
| 74 | C2×D8 | C2×C8 | 1 | 1 | yes | characteristic |
| 75 | #131 | — | 2 | 2 | no | — |
| 76 | #131 | — | 2 | 2 | no | — |
| 77 | #131 | — | 4 | 1 | no | — |
| 78 | #131 | — | 4 | 1 | no | — |
| 79 | C2×M16 | — | 2 | 1 | no | — |
| 80 | C2×M16 | — | 2 | 1 | no | — |
| 81 | C2×M16 | — | 4 | 1 | no | — |
| 82 | C22×C8 | — | 4 | 1 | no | — |
| 83 | C22×C8 | — | 4 | 1 | no | — |
| 84 | C22×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 85 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | #117 | — | 2 | 1 | no | — |
| 87 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 88 | M32 | — | 2 | 2 | no | — |
| 89 | M32 | — | 2 | 2 | no | — |
| 90 | M32 | — | 4 | 4 | no | — |
| 91 | M32 | — | 4 | 1 | no | — |
| 92 | M32 | — | 4 | 1 | no | — |
| 93 | C2×C16 | — | 2 | 1 | no | — |
| 94 | C2×C16 | — | 2 | 1 | no | — |
| 95 | C2×C16 | — | 2 | 1 | no | — |
| 96 | C2×C16 | — | 2 | 1 | no | — |
| 97 | C2×C16 | — | 2 | 1 | no | — |
| 98 | C2×C16 | — | 2 | 1 | no | — |
| 99 | C2×C16 | — | 2 | 1 | no | — |
| 100 | C2×C16 | — | 2 | 1 | no | — |
| 101 | C2×C16 | — | 4 | 4 | no | — |
| 102 | C2×C16 | — | 4 | 1 | no | — |
| 103 | C2×C16 | — | 4 | 1 | no | — |
| 104 | C2×C16 | C22⋊C4 | 1 | 2 | yes | — |
| 105 | C2×C16 | C22⋊C4 | 1 | 2 | yes | — |
| 106 | C2×C16 | C2×C8 | 1 | 2 | yes | — |
| 107 | C2×C16 | C2×C8 | 1 | 2 | yes | — |
| 108 | #108 | — | 2 | 1 | no | — |
| 109 | #108 | — | 2 | 1 | no | — |
| 110 | #108 | C42 | 1 | 2 | yes | — |
| 111 | #108 | C2×C8 | 1 | 2 | yes | — |
| 112 | #107 | C2×C8 | 1 | 2 | yes | — |
| 113 | #106 | — | 2 | 1 | no | — |
| 114 | #104 | — | 2 | 2 | no | — |
| 115 | #104 | — | 2 | 2 | no | — |
| 116 | #97 | — | 4 | 1 | no | — |
| 117 | C4×C8 | — | 2 | 2 | no | — |
| 118 | C4×C8 | — | 4 | 1 | no | — |
| 119 | C4×C8 | — | 4 | 1 | no | — |
| 120 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 121 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 122 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 123 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 124 | #41 | — | 2 | 2 | no | — |
| 125 | #41 | — | 2 | 2 | no | — |
| 126 | #41 | — | 4 | 1 | no | — |
| 127 | #41 | — | 4 | 1 | no | — |
| 128 | C2×Q8 | — | 2 | 1 | no | — |
| 129 | C2×D4 | — | 2 | 1 | no | — |
| 130 | C2×D4 | — | 4 | 1 | no | — |
| 131 | C22×C4 | — | 2 | 1 | no | — |
| 132 | C22×C4 | — | 4 | 1 | no | — |
| 133 | C22×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 134 | Q16 | — | 2 | 1 | no | — |
| 135 | Q16 | C4×C8 | 1 | 2 | yes | — |
| 136 | SD16 | — | 2 | 2 | no | — |
| 137 | SD16 | — | 2 | 2 | no | — |
| 138 | D8 | — | 2 | 1 | no | — |
| 139 | D8 | C4×C8 | 1 | 2 | yes | — |
| 140 | M16 | — | 2 | 2 | no | — |
| 141 | M16 | — | 2 | 2 | no | — |
| 142 | M16 | — | 4 | 1 | no | — |
| 143 | M16 | — | 4 | 1 | no | — |
| 144 | M16 | — | 4 | 2 | no | — |
| 145 | M16 | — | 4 | 2 | no | — |
| 146 | C2×C8 | — | 4 | 1 | no | — |
| 147 | C2×C8 | — | 4 | 1 | no | — |
| 148 | C2×C8 | — | 4 | 2 | no | — |
| 149 | C2×C8 | — | 4 | 2 | no | — |
| 150 | C2×C8 | — | 4 | 2 | no | — |
| 151 | C2×C8 | — | 8 | 2 | no | — |
| 152 | C2×C8 | C2×C42 | 1 | 1 | yes | characteristic |
| 153 | C2×C8 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 154 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 155 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 156 | C2×C8 | C22×C8 | 1 | 1 | yes | characteristic |
| 157 | C2×C8 | C22×C8 | 1 | 1 | yes | characteristic |
| 158 | C2×C8 | #131 | 1 | 1 | yes | characteristic |
| 159 | C2×C8 | #131 | 1 | 1 | yes | characteristic |
| 160 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 161 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 162 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 163 | C2×C8 | #137 | 1 | 1 | yes | characteristic |
| 164 | C4⋊C4 | — | 2 | 2 | no | — |
| 165 | C22⋊C4 | — | 4 | 1 | no | — |
| 166 | C42 | — | 2 | 1 | no | — |
| 167 | C42 | — | 4 | 1 | no | — |
| 168 | C42 | #117 | 1 | 1 | yes | characteristic |
| 169 | C42 | C4×D4 | 1 | 1 | yes | characteristic |
| 170 | C16 | — | 2 | 2 | no | — |
| 171 | C16 | — | 2 | 2 | no | — |
| 172 | C16 | — | 2 | 2 | no | — |
| 173 | C16 | — | 2 | 2 | no | — |
| 174 | C16 | — | 4 | 2 | no | — |
| 175 | C16 | — | 4 | 2 | no | — |
| 176 | C23 | — | 4 | 1 | no | — |
| 177 | C23 | #444 | 1 | 1 | yes | characteristic |
| 178 | Q8 | — | 2 | 2 | no | — |
| 179 | Q8 | — | 4 | 1 | no | — |
| 180 | D4 | — | 2 | 2 | no | — |
| 181 | D4 | — | 4 | 2 | no | — |
| 182 | D4 | — | 4 | 2 | no | — |
| 183 | D4 | — | 4 | 1 | no | — |
| 184 | C2×C4 | — | 2 | 1 | no | — |
| 185 | C2×C4 | — | 2 | 1 | no | — |
| 186 | C2×C4 | — | 4 | 2 | no | — |
| 187 | C2×C4 | — | 4 | 1 | no | — |
| 188 | C2×C4 | — | 4 | 2 | no | — |
| 189 | C2×C4 | — | 4 | 2 | no | — |
| 190 | C2×C4 | — | 4 | 2 | no | — |
| 191 | C2×C4 | C4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 192 | C2×C4 | C8×D4 | 1 | 1 | yes | characteristic |
| 193 | C2×C4 | C8×D4 | 1 | 1 | yes | characteristic |
| 194 | C2×C4 | #440 | 1 | 1 | yes | characteristic |
| 195 | C2×C4 | #442 | 1 | 1 | yes | characteristic |
| 196 | C2×C4 | #444 | 1 | 1 | yes | characteristic |
| 197 | C8 | — | 4 | 2 | no | — |
| 198 | C8 | — | 4 | 2 | no | — |
| 199 | C8 | — | 8 | 2 | no | — |
| 200 | C8 | C2×C4×C8 | 1 | 1 | yes | characteristic |
| 201 | C8 | #405 | 1 | 1 | yes | characteristic |
| 202 | C8 | #417 | 1 | 1 | yes | characteristic |
| 203 | C8 | #418 | 1 | 1 | yes | characteristic |
| 204 | C8 | C8×D4 | 1 | 1 | yes | characteristic |
| 205 | C8 | C8×D4 | 1 | 1 | yes | characteristic |
| 206 | C8 | #440 | 1 | 1 | yes | characteristic |
| 207 | C8 | #442 | 1 | 1 | yes | characteristic |
| 208 | C22 | — | 2 | 2 | no | — |
| 209 | C22 | — | 4 | 2 | no | — |
| 210 | C22 | — | 8 | 2 | no | — |
| 211 | C22 | SD16⋊C8 | 1 | 1 | yes | characteristic |
| 212 | C22 | #1580 | 1 | 1 | yes | characteristic |
| 213 | C22 | #1765 | 1 | 1 | yes | characteristic |
| 214 | C4 | — | 4 | 1 | no | — |
| 215 | C4 | — | 4 | 1 | no | — |
| 216 | C4 | — | 4 | 2 | no | — |
| 217 | C4 | — | 8 | 2 | no | — |
| 218 | C4 | SD16⋊C8 | 1 | 1 | yes | characteristic |
| 219 | C4 | #1580 | 1 | 1 | yes | characteristic |
| 220 | C4 | C8×C22⋊C4 | 1 | 1 | yes | characteristic |
| 221 | C4 | #1762 | 1 | 1 | yes | characteristic |
| 222 | C2 | — | 2 | 1 | no | — |
| 223 | C2 | — | 2 | 1 | no | — |
| 224 | C2 | — | 8 | 2 | no | — |
| 225 | C2 | #8200 | 1 | 1 | yes | characteristic |
| 226 | C2 | #12068 | 1 | 2 | yes | — |
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