← #12068596
Subgroups — #12068596
Order 1280.
612 conjugacy classes of subgroups.
The subgroup lattice has 240 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C5×#2991 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10614254 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #41×#168 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10601541 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #10601538 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #10601288 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #10601283 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C4×C5×#41 | C22 | 1 | 1 | yes | characteristic |
| 9 | C5×C8×Q8 | — | 2 | 1 | no | — |
| 10 | C5×#436 | — | 2 | 3 | no | — |
| 11 | C5×#431 | C22 | 1 | 3 | yes | — |
| 12 | C5×#423 | C22 | 1 | 3 | yes | — |
| 13 | Q8×#168 | — | 2 | 1 | no | — |
| 14 | Q8×#168 | C4 | 1 | 1 | yes | characteristic |
| 15 | Q8×#168 | C4 | 1 | 1 | yes | characteristic |
| 16 | #65301 | — | 2 | 3 | no | — |
| 17 | D4×#168 | C4 | 1 | 3 | yes | — |
| 18 | D4×#168 | C4 | 1 | 3 | yes | — |
| 19 | #65214 | C22 | 1 | 3 | yes | — |
| 20 | #65214 | C22 | 1 | 3 | yes | — |
| 21 | #65213 | C22 | 1 | 3 | yes | — |
| 22 | C2×C4×#168 | C22 | 1 | 3 | yes | — |
| 23 | #64700 | C22 | 1 | 1 | yes | characteristic |
| 24 | #64700 | C22 | 1 | 3 | yes | — |
| 25 | #64700 | C22 | 1 | 6 | yes | — |
| 26 | #64699 | C22 | 1 | 3 | yes | — |
| 27 | #64699 | C22 | 1 | 3 | yes | — |
| 28 | #17835 | — | 5 | 1 | no | maximal, Sylow |
| 29 | C2×C5×#41 | Q8 | 1 | 1 | yes | characteristic |
| 30 | C2×C5×M16 | — | 2 | 3 | no | — |
| 31 | C4×C5×Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C4×C5×Q8 | D4 | 1 | 1 | yes | characteristic |
| 33 | C4×C5×D4 | C2×C4 | 1 | 3 | yes | — |
| 34 | C4×C5×D4 | D4 | 1 | 3 | yes | — |
| 35 | C5×#117 | Q8 | 1 | 3 | yes | — |
| 36 | C2×C42×C5 | C23 | 1 | 3 | yes | — |
| 37 | C2×#681 | — | 2 | 3 | no | — |
| 38 | C22×#168 | C2×C4 | 1 | 3 | yes | — |
| 39 | C22×#168 | C2×C4 | 1 | 3 | yes | — |
| 40 | C5×C4⋊C8 | — | 2 | 3 | no | — |
| 41 | C5×C4⋊C8 | C23 | 1 | 3 | yes | — |
| 42 | C5×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 43 | C5×C22⋊C8 | — | 2 | 3 | no | — |
| 44 | C4×C5×C8 | — | 2 | 3 | no | — |
| 45 | C22⋊#168 | — | 2 | 3 | no | — |
| 46 | C22⋊#168 | C2×C4 | 1 | 6 | yes | — |
| 47 | C4⋊#168 | — | 2 | 3 | no | — |
| 48 | C4⋊#168 | C2×C4 | 1 | 3 | yes | — |
| 49 | C4⋊#168 | C2×C4 | 1 | 3 | yes | — |
| 50 | C4⋊#168 | C23 | 1 | 1 | yes | characteristic |
| 51 | C4⋊#168 | C23 | 1 | 3 | yes | — |
| 52 | C4×#168 | — | 2 | 3 | no | — |
| 53 | C4×#168 | C2×C4 | 1 | 3 | yes | — |
| 54 | C4×#168 | C2×C4 | 1 | 3 | yes | — |
| 55 | C4×#168 | C23 | 1 | 1 | yes | characteristic |
| 56 | C4×#168 | C23 | 1 | 3 | yes | — |
| 57 | #2991 | — | 5 | 1 | no | — |
| 58 | #2991 | D5 | 1 | 1 | yes | characteristic |
| 59 | C8×#41 | — | 5 | 1 | no | — |
| 60 | #1574 | — | 5 | 3 | no | — |
| 61 | #1574 | — | 5 | 3 | no | — |
| 62 | #1566 | — | 5 | 3 | no | — |
| 63 | #1564 | — | 5 | 3 | no | — |
| 64 | C5×#41 | — | 2 | 4 | no | — |
| 65 | C2×C5×Q8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 66 | C2×C5×D4 | C4⋊C4 | 1 | 3 | yes | — |
| 67 | C22×C4×C5 | C22×C4 | 1 | 3 | yes | — |
| 68 | C22×C4×C5 | C2×D4 | 1 | 3 | yes | — |
| 69 | C22×C4×C5 | C2×Q8 | 1 | 3 | yes | — |
| 70 | C5×M16 | — | 8 | 3 | no | — |
| 71 | C2×C5×C8 | — | 2 | 3 | no | — |
| 72 | C2×C5×C8 | — | 2 | 1 | no | — |
| 73 | C5×C4⋊C4 | C4⋊C4 | 1 | 3 | yes | — |
| 74 | C5×C4⋊C4 | C4⋊C4 | 1 | 3 | yes | — |
| 75 | C5×C22⋊C4 | C4⋊C4 | 1 | 6 | yes | — |
| 76 | C42×C5 | C22×C4 | 1 | 3 | yes | — |
| 77 | C42×C5 | C2×D4 | 1 | 3 | yes | — |
| 78 | C42×C5 | C2×Q8 | 1 | 3 | yes | — |
| 79 | C42×C5 | C24 | 1 | 1 | yes | characteristic |
| 80 | #681 | — | 8 | 3 | no | — |
| 81 | C2×#168 | — | 2 | 6 | no | — |
| 82 | C2×#168 | — | 2 | 1 | no | — |
| 83 | C2×#168 | — | 2 | 3 | no | — |
| 84 | C2×#168 | — | 4 | 3 | no | — |
| 85 | C2×#168 | C22×C4 | 1 | 1 | yes | characteristic |
| 86 | C2×#168 | C22×C4 | 1 | 3 | yes | — |
| 87 | C2×#168 | C22×C4 | 1 | 3 | yes | — |
| 88 | C2×#168 | C22×C4 | 1 | 1 | yes | characteristic |
| 89 | C4×#41 | C2×D5 | 1 | 1 | yes | characteristic |
| 90 | C8×Q8 | — | 2 | 1 | no | — |
| 91 | C8×Q8 | — | 5 | 1 | no | — |
| 92 | C8×Q8 | — | 5 | 1 | no | — |
| 93 | C8×Q8 | — | 10 | 1 | no | — |
| 94 | #436 | — | 2 | 3 | no | — |
| 95 | #436 | — | 10 | 3 | no | — |
| 96 | C8×D4 | — | 5 | 3 | no | — |
| 97 | C8×D4 | — | 5 | 3 | no | — |
| 98 | #431 | — | 5 | 3 | no | — |
| 99 | #431 | — | 5 | 3 | no | — |
| 100 | #431 | C2×D5 | 1 | 3 | yes | — |
| 101 | #423 | — | 5 | 3 | no | — |
| 102 | #423 | C2×D5 | 1 | 3 | yes | — |
| 103 | C2×C4×C8 | — | 5 | 3 | no | — |
| 104 | #335 | — | 5 | 3 | no | — |
| 105 | #335 | — | 5 | 3 | no | — |
| 106 | #334 | — | 5 | 1 | no | — |
| 107 | #334 | — | 5 | 3 | no | — |
| 108 | #334 | — | 5 | 6 | no | — |
| 109 | C23×C5 | C2×C4⋊C4 | 1 | 3 | yes | — |
| 110 | C5×Q8 | #108 | 1 | 4 | yes | — |
| 111 | C5×D4 | — | 2 | 6 | no | — |
| 112 | C2×C4×C5 | — | 2 | 3 | no | — |
| 113 | C2×C4×C5 | — | 2 | 3 | no | — |
| 114 | C2×C4×C5 | — | 2 | 3 | no | — |
| 115 | C2×C4×C5 | — | 2 | 3 | no | — |
| 116 | C2×C4×C5 | — | 2 | 6 | no | — |
| 117 | C2×C4×C5 | C2×C4⋊C4 | 1 | 3 | yes | — |
| 118 | C2×C4×C5 | C2×C4⋊C4 | 1 | 3 | yes | — |
| 119 | C2×C4×C5 | C2×C4⋊C4 | 1 | 3 | yes | — |
| 120 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C5 | C22×Q8 | 1 | 1 | yes | characteristic |
| 123 | C5×C8 | — | 4 | 1 | no | — |
| 124 | C5×C8 | — | 4 | 3 | no | — |
| 125 | #168 | — | 2 | 3 | no | — |
| 126 | #168 | — | 2 | 1 | no | — |
| 127 | #168 | — | 2 | 3 | no | — |
| 128 | #168 | — | 4 | 1 | no | — |
| 129 | #168 | — | 4 | 3 | no | — |
| 130 | #168 | #131 | 1 | 2 | yes | — |
| 131 | C2×#41 | Dic10 | 1 | 1 | yes | characteristic |
| 132 | C2×M16 | — | 2 | 3 | no | — |
| 133 | C2×M16 | — | 10 | 3 | no | — |
| 134 | C22×C8 | — | 5 | 3 | no | — |
| 135 | C22×C8 | — | 5 | 3 | no | — |
| 136 | C4×Q8 | C4×D5 | 1 | 1 | yes | characteristic |
| 137 | C4×Q8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 138 | C4×D4 | C4×D5 | 1 | 3 | yes | — |
| 139 | C4×D4 | C22⋊D5 | 1 | 3 | yes | — |
| 140 | #117 | Dic10 | 1 | 3 | yes | — |
| 141 | C2×C42 | C22×D5 | 1 | 3 | yes | — |
| 142 | C4⋊C8 | — | 2 | 3 | no | — |
| 143 | C4⋊C8 | — | 5 | 3 | no | — |
| 144 | C4⋊C8 | — | 5 | 3 | no | — |
| 145 | C4⋊C8 | — | 5 | 3 | no | — |
| 146 | C4⋊C8 | — | 5 | 1 | no | — |
| 147 | C4⋊C8 | — | 10 | 3 | no | — |
| 148 | C4⋊C8 | C22×D5 | 1 | 3 | yes | — |
| 149 | C4⋊C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 150 | C22⋊C8 | — | 2 | 3 | no | — |
| 151 | C22⋊C8 | — | 5 | 6 | no | — |
| 152 | C22⋊C8 | — | 10 | 3 | no | — |
| 153 | C4×C8 | — | 2 | 3 | no | — |
| 154 | C4×C8 | — | 5 | 3 | no | — |
| 155 | C4×C8 | — | 5 | 3 | no | — |
| 156 | C4×C8 | — | 5 | 1 | no | — |
| 157 | C4×C8 | — | 5 | 3 | no | — |
| 158 | C4×C8 | — | 10 | 3 | no | — |
| 159 | C22×C5 | — | 2 | 3 | no | — |
| 160 | C22×C5 | — | 2 | 3 | no | — |
| 161 | C22×C5 | — | 2 | 3 | no | — |
| 162 | C22×C5 | C22×C4⋊C4 | 1 | 1 | yes | characteristic |
| 163 | C4×C5 | — | 2 | 3 | no | — |
| 164 | C4×C5 | — | 2 | 3 | no | — |
| 165 | C4×C5 | — | 2 | 1 | no | — |
| 166 | C4×C5 | C2×#108 | 1 | 6 | yes | — |
| 167 | C4×C5 | C2×#131 | 1 | 1 | yes | characteristic |
| 168 | C4×C5 | #568 | 1 | 1 | yes | characteristic |
| 169 | C4×C5 | #577 | 1 | 1 | yes | characteristic |
| 170 | C4×C5 | #577 | 1 | 1 | yes | characteristic |
| 171 | #41 | — | 2 | 4 | no | — |
| 172 | C2×Q8 | #683 | 1 | 1 | yes | characteristic |
| 173 | C2×D4 | #683 | 1 | 3 | yes | — |
| 174 | C22×C4 | C2×Dic10 | 1 | 3 | yes | — |
| 175 | C22×C4 | C2×C4×D5 | 1 | 3 | yes | — |
| 176 | C22×C4 | C2×C22⋊D5 | 1 | 3 | yes | — |
| 177 | M16 | — | 8 | 3 | no | — |
| 178 | M16 | — | 40 | 3 | no | — |
| 179 | C2×C8 | — | 2 | 1 | no | — |
| 180 | C2×C8 | — | 2 | 3 | no | — |
| 181 | C2×C8 | — | 5 | 1 | no | — |
| 182 | C2×C8 | — | 5 | 1 | no | — |
| 183 | C2×C8 | — | 5 | 3 | no | — |
| 184 | C2×C8 | — | 5 | 3 | no | — |
| 185 | C2×C8 | — | 10 | 6 | no | — |
| 186 | C2×C8 | — | 10 | 1 | no | — |
| 187 | C2×C8 | — | 10 | 3 | no | — |
| 188 | C2×C8 | — | 20 | 3 | no | — |
| 189 | C4⋊C4 | #683 | 1 | 3 | yes | — |
| 190 | C4⋊C4 | #683 | 1 | 3 | yes | — |
| 191 | C22⋊C4 | #683 | 1 | 6 | yes | — |
| 192 | C42 | C2×Dic10 | 1 | 3 | yes | — |
| 193 | C42 | C2×C4×D5 | 1 | 3 | yes | — |
| 194 | C42 | C2×C22⋊D5 | 1 | 3 | yes | — |
| 195 | C42 | C23×D5 | 1 | 1 | yes | characteristic |
| 196 | C2×C5 | — | 4 | 3 | no | — |
| 197 | C2×C5 | #2906 | 1 | 1 | yes | characteristic |
| 198 | C2×C5 | #2912 | 1 | 1 | yes | characteristic |
| 199 | C2×C5 | C22×#108 | 1 | 1 | yes | characteristic |
| 200 | C23 | C2×#683 | 1 | 3 | yes | — |
| 201 | Q8 | #3954 | 1 | 4 | yes | — |
| 202 | D4 | — | 2 | 6 | no | — |
| 203 | C2×C4 | — | 2 | 3 | no | — |
| 204 | C2×C4 | — | 2 | 3 | no | — |
| 205 | C2×C4 | — | 2 | 3 | no | — |
| 206 | C2×C4 | — | 2 | 3 | no | — |
| 207 | C2×C4 | — | 2 | 6 | no | — |
| 208 | C2×C4 | C2×#683 | 1 | 3 | yes | — |
| 209 | C2×C4 | C2×#683 | 1 | 3 | yes | — |
| 210 | C2×C4 | C2×#683 | 1 | 3 | yes | — |
| 211 | C2×C4 | C22×Dic10 | 1 | 1 | yes | characteristic |
| 212 | C2×C4 | C22×C4×D5 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | C22×C22⋊D5 | 1 | 1 | yes | characteristic |
| 214 | C8 | — | 4 | 3 | no | — |
| 215 | C8 | — | 4 | 1 | no | — |
| 216 | C8 | — | 5 | 2 | no | — |
| 217 | C8 | — | 10 | 3 | no | — |
| 218 | C8 | — | 10 | 1 | no | — |
| 219 | C8 | — | 10 | 3 | no | — |
| 220 | C8 | — | 20 | 1 | no | — |
| 221 | C8 | — | 20 | 3 | no | — |
| 222 | C5 | #17835 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 223 | C22 | — | 2 | 3 | no | — |
| 224 | C22 | — | 2 | 3 | no | — |
| 225 | C22 | — | 2 | 3 | no | — |
| 226 | C22 | C22×#683 | 1 | 1 | yes | characteristic |
| 227 | C4 | — | 2 | 3 | no | — |
| 228 | C4 | — | 2 | 1 | no | — |
| 229 | C4 | — | 2 | 3 | no | — |
| 230 | C4 | C2×#3954 | 1 | 6 | yes | — |
| 231 | C4 | D5×#131 | 1 | 1 | yes | characteristic |
| 232 | C4 | #66084 | 1 | 1 | yes | characteristic |
| 233 | C4 | #66170 | 1 | 1 | yes | characteristic |
| 234 | C4 | #66171 | 1 | 1 | yes | characteristic |
| 235 | C2 | — | 4 | 3 | no | — |
| 236 | C2 | #10614292 | 1 | 1 | yes | characteristic |
| 237 | C2 | C22×#3954 | 1 | 1 | yes | characteristic |
| 238 | C2 | #10615514 | 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