← #483513
Subgroups — #483513
Order 512.
475 conjugacy classes of subgroups.
The subgroup lattice has 222 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#2178 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33443 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #12047 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #12046 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C2×#1373 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11602 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #11399 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#508 | C4 | 1 | 1 | yes | characteristic |
| 9 | #3407 | — | 2 | 2 | no | — |
| 10 | C2×#443 | C22 | 1 | 2 | yes | — |
| 11 | #2178 | C4 | 1 | 2 | yes | — |
| 12 | #2178 | C4 | 1 | 2 | yes | — |
| 13 | #2178 | C22 | 1 | 2 | yes | — |
| 14 | #2178 | C22 | 1 | 2 | yes | — |
| 15 | #2171 | — | 2 | 4 | no | — |
| 16 | #2170 | C22 | 1 | 8 | yes | complemented |
| 17 | C2×#368 | C4 | 1 | 1 | yes | characteristic |
| 18 | C2×#364 | C22 | 1 | 2 | yes | — |
| 19 | C2×#359 | C4 | 1 | 2 | yes | — |
| 20 | #2117 | — | 2 | 2 | no | — |
| 21 | C4×M32 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 23 | #1450 | C22 | 1 | 1 | yes | characteristic |
| 24 | #1373 | C22 | 1 | 2 | yes | complemented |
| 25 | #1373 | C22 | 1 | 2 | yes | complemented |
| 26 | #1335 | C22 | 1 | 8 | yes | complemented |
| 27 | #1311 | C22 | 1 | 2 | yes | complemented |
| 28 | C8×C16 | C22 | 1 | 2 | yes | complemented |
| 29 | C2×#135 | C2×C4 | 1 | 2 | yes | — |
| 30 | C2×#131 | — | 2 | 2 | no | — |
| 31 | #508 | C8 | 1 | 2 | yes | complemented |
| 32 | #508 | C8 | 1 | 2 | yes | complemented |
| 33 | #508 | C2×C4 | 1 | 2 | yes | — |
| 34 | #508 | C2×C4 | 1 | 2 | yes | — |
| 35 | C2×Q32 | C8 | 1 | 1 | yes | characteristic, complemented |
| 36 | C2×SD32 | C8 | 1 | 2 | yes | complemented |
| 37 | C2×D16 | C8 | 1 | 1 | yes | characteristic, complemented |
| 38 | #504 | — | 2 | 8 | no | — |
| 39 | #504 | — | 4 | 4 | no | — |
| 40 | C2×M32 | — | 2 | 2 | no | — |
| 41 | C2×M32 | — | 4 | 4 | no | — |
| 42 | C2×M32 | D4 | 1 | 1 | yes | characteristic |
| 43 | C2×M32 | D4 | 1 | 1 | yes | characteristic |
| 44 | C22×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | C22×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | #443 | — | 2 | 2 | no | — |
| 47 | #443 | — | 2 | 2 | no | — |
| 48 | #443 | C2×C4 | 1 | 4 | yes | — |
| 49 | #443 | C23 | 1 | 4 | yes | — |
| 50 | #431 | — | 2 | 1 | no | — |
| 51 | C2×#108 | C2×C4 | 1 | 2 | yes | — |
| 52 | C2×#104 | — | 2 | 2 | no | — |
| 53 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 54 | #368 | C2×C4 | 1 | 2 | yes | — |
| 55 | #368 | C2×C4 | 1 | 2 | yes | — |
| 56 | #364 | — | 2 | 2 | no | — |
| 57 | #364 | C23 | 1 | 4 | yes | — |
| 58 | #359 | C2×C4 | 1 | 4 | yes | — |
| 59 | #359 | C2×C4 | 1 | 4 | yes | — |
| 60 | #350 | — | 2 | 8 | no | — |
| 61 | #346 | D4 | 1 | 2 | yes | — |
| 62 | C4×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 63 | C4×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 64 | C4×C16 | C23 | 1 | 1 | yes | characteristic |
| 65 | C4×C16 | C23 | 1 | 1 | yes | characteristic |
| 66 | C4×C16 | C23 | 1 | 2 | yes | — |
| 67 | #322 | — | 2 | 1 | no | — |
| 68 | C82 | C23 | 1 | 2 | yes | — |
| 69 | C2×#41 | — | 2 | 2 | no | — |
| 70 | #135 | — | 2 | 2 | no | — |
| 71 | #135 | — | 2 | 2 | no | — |
| 72 | #135 | C2×C8 | 1 | 4 | yes | — |
| 73 | #135 | C22×C4 | 1 | 4 | yes | — |
| 74 | C2×Q16 | C2×C8 | 1 | 2 | yes | — |
| 75 | C2×SD16 | — | 2 | 2 | no | — |
| 76 | C2×D8 | C2×C8 | 1 | 2 | yes | — |
| 77 | #131 | — | 2 | 4 | no | — |
| 78 | #131 | — | 2 | 4 | no | — |
| 79 | #131 | — | 4 | 2 | no | — |
| 80 | #131 | — | 4 | 2 | no | — |
| 81 | C2×M16 | — | 2 | 2 | no | — |
| 82 | C2×M16 | — | 4 | 2 | no | — |
| 83 | C22×C8 | — | 4 | 2 | no | — |
| 84 | C22×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 85 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 87 | Q32 | C2×C8 | 1 | 2 | yes | — |
| 88 | Q32 | C2×C8 | 1 | 2 | yes | — |
| 89 | SD32 | C2×C8 | 1 | 4 | yes | — |
| 90 | SD32 | C2×C8 | 1 | 4 | yes | — |
| 91 | D16 | C2×C8 | 1 | 2 | yes | — |
| 92 | D16 | C2×C8 | 1 | 2 | yes | — |
| 93 | M32 | — | 2 | 1 | no | — |
| 94 | M32 | — | 2 | 1 | no | — |
| 95 | M32 | — | 2 | 4 | no | — |
| 96 | M32 | — | 4 | 2 | no | — |
| 97 | M32 | — | 4 | 8 | no | — |
| 98 | M32 | D8 | 1 | 2 | yes | — |
| 99 | M32 | D8 | 1 | 2 | yes | — |
| 100 | C2×C16 | — | 2 | 1 | no | — |
| 101 | C2×C16 | — | 2 | 1 | no | — |
| 102 | C2×C16 | — | 4 | 2 | no | — |
| 103 | C2×C16 | — | 4 | 8 | no | — |
| 104 | C2×C16 | C2×C8 | 1 | 1 | yes | characteristic |
| 105 | C2×C16 | C2×C8 | 1 | 1 | yes | characteristic |
| 106 | C2×C16 | C2×C8 | 1 | 1 | yes | characteristic |
| 107 | C2×C16 | C2×C8 | 1 | 1 | yes | characteristic |
| 108 | C2×C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 109 | C2×C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 110 | C2×C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 111 | C2×C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 112 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 113 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 114 | #108 | — | 2 | 2 | no | — |
| 115 | #108 | C22×C4 | 1 | 4 | yes | — |
| 116 | C4⋊C8 | — | 2 | 2 | no | — |
| 117 | #104 | — | 2 | 4 | no | — |
| 118 | #104 | — | 2 | 4 | no | — |
| 119 | C22⋊C8 | — | 2 | 2 | no | — |
| 120 | C4×C8 | — | 2 | 2 | no | — |
| 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 | C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 125 | #41 | — | 2 | 4 | no | — |
| 126 | #41 | — | 2 | 4 | no | — |
| 127 | #41 | — | 4 | 2 | no | — |
| 128 | #41 | — | 4 | 2 | no | — |
| 129 | C2×Q8 | — | 2 | 2 | no | — |
| 130 | C2×D4 | — | 2 | 2 | no | — |
| 131 | C2×D4 | — | 4 | 2 | no | — |
| 132 | C22×C4 | — | 2 | 1 | no | — |
| 133 | C22×C4 | — | 4 | 2 | no | — |
| 134 | C22×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 135 | Q16 | — | 2 | 2 | no | — |
| 136 | Q16 | C22×C8 | 1 | 4 | yes | — |
| 137 | SD16 | — | 2 | 4 | no | — |
| 138 | SD16 | — | 2 | 4 | no | — |
| 139 | D8 | — | 2 | 2 | no | — |
| 140 | D8 | C22×C8 | 1 | 4 | yes | — |
| 141 | M16 | — | 2 | 4 | no | — |
| 142 | M16 | — | 4 | 2 | no | — |
| 143 | M16 | — | 4 | 4 | no | — |
| 144 | M16 | — | 4 | 4 | no | — |
| 145 | C2×C8 | — | 2 | 4 | no | — |
| 146 | C2×C8 | — | 4 | 2 | no | — |
| 147 | C2×C8 | — | 4 | 4 | no | — |
| 148 | C2×C8 | — | 4 | 4 | no | — |
| 149 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 150 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 151 | C2×C8 | C22×C8 | 1 | 1 | yes | characteristic |
| 152 | C2×C8 | C22×C8 | 1 | 1 | yes | characteristic |
| 153 | C2×C8 | #131 | 1 | 1 | yes | characteristic |
| 154 | C2×C8 | #131 | 1 | 1 | yes | characteristic |
| 155 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 156 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 157 | C2×C8 | #135 | 1 | 1 | yes | characteristic |
| 158 | C2×C8 | #135 | 1 | 1 | yes | characteristic |
| 159 | C2×C8 | C23×C4 | 1 | 1 | yes | characteristic |
| 160 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 161 | C42 | — | 2 | 1 | no | — |
| 162 | C42 | C4×D4 | 1 | 1 | yes | characteristic |
| 163 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 164 | C16 | — | 2 | 1 | no | — |
| 165 | C16 | — | 2 | 1 | no | — |
| 166 | C16 | — | 2 | 2 | no | — |
| 167 | C16 | — | 4 | 4 | no | — |
| 168 | C16 | C22×C8 | 1 | 1 | yes | characteristic |
| 169 | C16 | C22×C8 | 1 | 1 | yes | characteristic |
| 170 | C16 | C22×C8 | 1 | 1 | yes | characteristic |
| 171 | C16 | C22×C8 | 1 | 1 | yes | characteristic |
| 172 | C16 | C2×D8 | 1 | 4 | yes | — |
| 173 | C23 | — | 4 | 2 | no | — |
| 174 | C23 | #443 | 1 | 1 | yes | characteristic |
| 175 | Q8 | — | 2 | 4 | no | — |
| 176 | Q8 | — | 4 | 2 | no | — |
| 177 | D4 | — | 2 | 4 | no | — |
| 178 | D4 | — | 4 | 4 | no | — |
| 179 | D4 | — | 4 | 4 | no | — |
| 180 | D4 | — | 4 | 2 | no | — |
| 181 | C2×C4 | — | 2 | 1 | no | — |
| 182 | C2×C4 | — | 2 | 1 | no | — |
| 183 | C2×C4 | — | 4 | 2 | no | — |
| 184 | C2×C4 | — | 4 | 2 | no | — |
| 185 | C2×C4 | — | 4 | 4 | no | — |
| 186 | C2×C4 | — | 4 | 4 | no | — |
| 187 | C2×C4 | C8×D4 | 1 | 1 | yes | characteristic |
| 188 | C2×C4 | C8×D4 | 1 | 1 | yes | characteristic |
| 189 | C2×C4 | C4×D8 | 1 | 1 | yes | characteristic |
| 190 | C2×C4 | C4×D8 | 1 | 1 | yes | characteristic |
| 191 | C2×C4 | #443 | 1 | 1 | yes | characteristic |
| 192 | C2×C4 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 193 | C8 | — | 4 | 4 | no | — |
| 194 | C8 | — | 4 | 4 | no | — |
| 195 | C8 | C8×D4 | 1 | 1 | yes | characteristic |
| 196 | C8 | C8×D4 | 1 | 1 | yes | characteristic |
| 197 | C8 | C4×D8 | 1 | 1 | yes | characteristic |
| 198 | C8 | C4×D8 | 1 | 1 | yes | characteristic |
| 199 | C8 | C23×C8 | 1 | 1 | yes | characteristic |
| 200 | C8 | C2×#131 | 1 | 1 | yes | characteristic |
| 201 | C8 | C22×D8 | 1 | 1 | yes | characteristic |
| 202 | C8 | C2×#135 | 1 | 1 | yes | characteristic |
| 203 | C22 | — | 4 | 1 | no | — |
| 204 | C22 | — | 4 | 4 | no | — |
| 205 | C22 | — | 16 | 2 | no | — |
| 206 | C22 | C8×D8 | 1 | 1 | yes | characteristic |
| 207 | C22 | C8×D8 | 1 | 1 | yes | characteristic |
| 208 | C22 | C2×#443 | 1 | 1 | yes | characteristic |
| 209 | C4 | — | 2 | 2 | no | — |
| 210 | C4 | — | 4 | 1 | no | — |
| 211 | C4 | — | 4 | 4 | no | — |
| 212 | C4 | C8×D8 | 1 | 1 | yes | characteristic |
| 213 | C4 | C8×D8 | 1 | 1 | yes | characteristic |
| 214 | C4 | C2×C8×D4 | 1 | 1 | yes | characteristic |
| 215 | C4 | C2×C4×D8 | 1 | 1 | yes | characteristic |
| 216 | C2 | — | 2 | 1 | no | — |
| 217 | C2 | — | 2 | 1 | no | — |
| 218 | C2 | — | 2 | 1 | no | — |
| 219 | C2 | — | 8 | 4 | no | — |
| 220 | C2 | C2×C8×D8 | 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