Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
1258 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 231 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 | #10617775 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10610437 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C5×#1830 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10602452 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10600907 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10599833 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #10599829 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | C2×#4147 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×#713 | C22 | 1 | 1 | yes | characteristic |
| 10 | C5×#519 | C22 | 1 | 1 | yes | characteristic |
| 11 | #66079 | C22 | 1 | 4 | yes | — |
| 12 | #65911 | — | 2 | 4 | no | — |
| 13 | #65910 | — | 2 | 4 | no | — |
| 14 | #65859 | C4 | 1 | 4 | yes | — |
| 15 | #65859 | C22 | 1 | 2 | yes | — |
| 16 | #65858 | C4 | 1 | 4 | yes | — |
| 17 | #65858 | C22 | 1 | 2 | yes | — |
| 18 | C2×C5×#101 | C22 | 1 | 2 | yes | — |
| 19 | C2×#3968 | C22 | 1 | 2 | yes | — |
| 20 | #65421 | C22 | 1 | 4 | yes | — |
| 21 | C2×#3955 | C22 | 1 | 4 | yes | — |
| 22 | #65415 | C22 | 1 | 4 | yes | — |
| 23 | #65040 | — | 2 | 4 | no | — |
| 24 | D5×#101 | — | 2 | 4 | no | — |
| 25 | C5×#344 | C22 | 1 | 4 | yes | — |
| 26 | #64753 | C22 | 1 | 4 | yes | — |
| 27 | #21781 | — | 5 | 1 | no | maximal, Sylow |
| 28 | C22×C5×Q8 | C23 | 1 | 1 | yes | characteristic |
| 29 | #4147 | — | 4 | 4 | no | — |
| 30 | C2×#713 | — | 2 | 4 | no | — |
| 31 | C2×#713 | — | 4 | 4 | no | — |
| 32 | C2×#713 | D4 | 1 | 4 | yes | — |
| 33 | C2×Q8×D5 | D4 | 1 | 4 | yes | — |
| 34 | C2×#709 | C2×C4 | 1 | 4 | yes | — |
| 35 | C2×#709 | C23 | 1 | 2 | yes | — |
| 36 | C22×D20 | C2×C4 | 1 | 2 | yes | — |
| 37 | C22×D20 | C23 | 1 | 1 | yes | characteristic |
| 38 | C22×C4×D5 | C2×C4 | 1 | 2 | yes | — |
| 39 | C22×C4×D5 | C23 | 1 | 1 | yes | characteristic |
| 40 | C2×C5×M16 | C23 | 1 | 4 | yes | — |
| 41 | C4×C5×Q8 | — | 2 | 4 | no | — |
| 42 | C5×#117 | C2×C4 | 1 | 4 | yes | — |
| 43 | C5×#117 | C23 | 1 | 2 | yes | — |
| 44 | C2×#681 | C23 | 1 | 4 | yes | — |
| 45 | #4053 | — | 4 | 4 | no | — |
| 46 | D5×M16 | — | 2 | 8 | no | — |
| 47 | #4047 | — | 4 | 4 | no | — |
| 48 | #4043 | — | 2 | 4 | no | — |
| 49 | #4043 | — | 2 | 2 | no | — |
| 50 | #4041 | — | 2 | 4 | no | — |
| 51 | #4041 | — | 2 | 2 | no | — |
| 52 | #4041 | — | 2 | 8 | no | — |
| 53 | #4041 | — | 2 | 4 | no | — |
| 54 | #4032 | C2×C4 | 1 | 4 | yes | — |
| 55 | #4028 | C2×C4 | 1 | 8 | yes | — |
| 56 | #4028 | C2×C4 | 1 | 4 | yes | — |
| 57 | #4021 | — | 2 | 4 | no | — |
| 58 | #4021 | — | 2 | 2 | no | — |
| 59 | #4020 | — | 2 | 4 | no | — |
| 60 | #4020 | — | 2 | 2 | no | — |
| 61 | C4×D20 | — | 2 | 8 | no | — |
| 62 | C4×D20 | — | 2 | 4 | no | — |
| 63 | C5×#101 | — | 2 | 4 | no | — |
| 64 | #3968 | — | 2 | 4 | no | — |
| 65 | #3956 | — | 2 | 4 | no | — |
| 66 | #3955 | — | 2 | 4 | no | — |
| 67 | #3955 | — | 4 | 4 | no | — |
| 68 | #3952 | — | 4 | 4 | no | — |
| 69 | #3487 | — | 5 | 1 | no | — |
| 70 | #2889 | — | 5 | 2 | no | — |
| 71 | #1914 | — | 5 | 4 | no | — |
| 72 | #1891 | — | 5 | 4 | no | — |
| 73 | #1830 | — | 5 | 1 | no | — |
| 74 | #1830 | D5 | 1 | 1 | yes | characteristic, complemented |
| 75 | #1785 | — | 5 | 2 | no | — |
| 76 | C23×D5 | C22×C4 | 1 | 2 | yes | — |
| 77 | C23×D5 | C22×C4 | 1 | 1 | yes | characteristic |
| 78 | C2×C5×Q8 | — | 2 | 4 | no | — |
| 79 | C2×C5×Q8 | C2×D4 | 1 | 4 | yes | — |
| 80 | C22×C4×C5 | C22×C4 | 1 | 2 | yes | — |
| 81 | C22×C4×C5 | C24 | 1 | 1 | yes | characteristic |
| 82 | C2×C22⋊D5 | C22×C4 | 1 | 4 | yes | — |
| 83 | C22×Dic5 | C22×C4 | 1 | 1 | yes | characteristic |
| 84 | #713 | — | 4 | 8 | no | — |
| 85 | #713 | — | 4 | 16 | no | — |
| 86 | Q8×D5 | — | 4 | 8 | no | — |
| 87 | #709 | — | 2 | 4 | no | — |
| 88 | #709 | — | 2 | 4 | no | — |
| 89 | #709 | — | 4 | 8 | no | — |
| 90 | C2×D20 | — | 2 | 4 | no | — |
| 91 | C2×D20 | — | 2 | 2 | no | — |
| 92 | C2×D20 | — | 2 | 4 | no | — |
| 93 | C2×D20 | — | 2 | 8 | no | — |
| 94 | C2×D20 | — | 2 | 4 | no | — |
| 95 | C2×D20 | — | 4 | 4 | no | — |
| 96 | C2×D20 | C22⋊C4 | 1 | 4 | yes | — |
| 97 | C2×D20 | C2×D4 | 1 | 2 | yes | — |
| 98 | C2×C4×D5 | — | 2 | 4 | no | — |
| 99 | C2×C4×D5 | — | 2 | 2 | no | — |
| 100 | C2×C4×D5 | — | 2 | 4 | no | — |
| 101 | C2×C4×D5 | — | 4 | 4 | no | — |
| 102 | C2×C4×D5 | C22⋊C4 | 1 | 8 | yes | — |
| 103 | C2×C4×D5 | C2×D4 | 1 | 4 | yes | — |
| 104 | C2×Dic10 | C22⋊C4 | 1 | 4 | yes | — |
| 105 | C2×Dic10 | C2×D4 | 1 | 2 | yes | — |
| 106 | C5×M16 | — | 2 | 4 | no | — |
| 107 | C5×M16 | — | 2 | 4 | no | — |
| 108 | C2×C5×C8 | — | 2 | 4 | no | — |
| 109 | C5×C4⋊C4 | — | 2 | 4 | no | — |
| 110 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 111 | C5×C22⋊C4 | C22×C4 | 1 | 4 | yes | — |
| 112 | C42×C5 | — | 2 | 4 | no | — |
| 113 | C42×C5 | — | 2 | 2 | no | — |
| 114 | #685 | — | 2 | 4 | no | — |
| 115 | #685 | — | 2 | 8 | no | — |
| 116 | #685 | — | 2 | 4 | no | — |
| 117 | C4⋊Dic5 | — | 2 | 4 | no | — |
| 118 | #681 | — | 2 | 4 | no | — |
| 119 | #681 | — | 2 | 4 | no | — |
| 120 | C2×#168 | — | 2 | 4 | no | — |
| 121 | #676 | — | 4 | 8 | no | — |
| 122 | C8×D5 | — | 4 | 8 | no | — |
| 123 | C2×2−1+4 | — | 5 | 1 | no | — |
| 124 | C22×#41 | — | 5 | 1 | no | — |
| 125 | #568 | — | 5 | 4 | no | — |
| 126 | #550 | — | 10 | 4 | no | — |
| 127 | #548 | — | 10 | 4 | no | — |
| 128 | #535 | — | 5 | 4 | no | — |
| 129 | #535 | — | 5 | 2 | no | — |
| 130 | #525 | — | 5 | 4 | no | — |
| 131 | #525 | — | 5 | 2 | no | — |
| 132 | #519 | C2×D5 | 1 | 1 | yes | characteristic |
| 133 | #413 | — | 5 | 4 | no | — |
| 134 | #413 | — | 10 | 4 | no | — |
| 135 | C2×#101 | — | 5 | 2 | no | — |
| 136 | C2×#101 | — | 10 | 4 | no | — |
| 137 | C2×#101 | C2×D5 | 1 | 2 | yes | — |
| 138 | C2×#100 | — | 5 | 4 | no | — |
| 139 | #407 | — | 5 | 4 | no | — |
| 140 | #344 | — | 5 | 4 | no | — |
| 141 | #344 | C2×D5 | 1 | 4 | yes | — |
| 142 | C23×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 143 | C22×D5 | — | 2 | 2 | no | — |
| 144 | C22×D5 | — | 2 | 1 | no | — |
| 145 | C22×D5 | — | 2 | 2 | no | — |
| 146 | C22×D5 | — | 2 | 4 | no | — |
| 147 | C22×D5 | — | 8 | 1 | no | — |
| 148 | C22×D5 | — | 8 | 2 | no | — |
| 149 | C22×D5 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 150 | C5×Q8 | — | 2 | 8 | no | — |
| 151 | C2×C4×C5 | — | 2 | 4 | no | — |
| 152 | C2×C4×C5 | — | 2 | 4 | no | — |
| 153 | C2×C4×C5 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 154 | C2×C4×C5 | C22×D4 | 1 | 2 | yes | — |
| 155 | C2×C4×C5 | H(5,𝔽2) | 1 | 4 | yes | — |
| 156 | C22⋊D5 | — | 2 | 4 | no | — |
| 157 | C22⋊D5 | — | 2 | 4 | no | — |
| 158 | C2×Dic5 | — | 2 | 1 | no | — |
| 159 | C2×Dic5 | — | 2 | 2 | no | — |
| 160 | C2×Dic5 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 161 | D20 | — | 2 | 4 | no | — |
| 162 | D20 | — | 4 | 4 | no | — |
| 163 | D20 | — | 4 | 8 | no | — |
| 164 | D20 | — | 4 | 4 | no | — |
| 165 | D20 | — | 8 | 4 | no | — |
| 166 | D20 | — | 8 | 8 | no | — |
| 167 | C4×D5 | — | 2 | 8 | no | — |
| 168 | C4×D5 | — | 4 | 8 | no | — |
| 169 | C4×D5 | — | 4 | 8 | no | — |
| 170 | C4×D5 | — | 4 | 4 | no | — |
| 171 | Dic10 | — | 2 | 4 | no | — |
| 172 | Dic10 | — | 4 | 4 | no | — |
| 173 | C5×C8 | — | 4 | 4 | no | — |
| 174 | #168 | — | 4 | 4 | no | — |
| 175 | 2−1+4 | — | 20 | 4 | no | — |
| 176 | C2×#41 | — | 5 | 4 | no | — |
| 177 | C2×#41 | — | 5 | 4 | no | — |
| 178 | C2×#41 | — | 5 | 2 | no | — |
| 179 | C2×#41 | — | 10 | 4 | no | — |
| 180 | C2×#41 | — | 20 | 4 | no | — |
| 181 | C22×Q8 | — | 5 | 4 | no | — |
| 182 | C22×Q8 | C22×D5 | 1 | 1 | yes | characteristic |
| 183 | C22×D4 | — | 5 | 2 | no | — |
| 184 | C22×D4 | — | 5 | 1 | no | — |
| 185 | C23×C4 | — | 5 | 1 | no | — |
| 186 | C23×C4 | — | 5 | 2 | no | — |
| 187 | #131 | — | 20 | 4 | no | — |
| 188 | #131 | — | 20 | 4 | no | — |
| 189 | C2×M16 | — | 5 | 4 | no | — |
| 190 | C2×M16 | — | 10 | 8 | no | — |
| 191 | C2×M16 | C22×D5 | 1 | 4 | yes | — |
| 192 | #127 | — | 10 | 2 | no | — |
| 193 | #127 | — | 10 | 4 | no | — |
| 194 | #124 | — | 10 | 2 | no | — |
| 195 | #124 | — | 10 | 4 | no | — |
| 196 | #123 | — | 5 | 4 | no | — |
| 197 | C22⋊Q8 | — | 10 | 2 | no | — |
| 198 | C22⋊Q8 | — | 10 | 4 | no | — |
| 199 | #121 | — | 10 | 4 | no | — |
| 200 | #121 | — | 10 | 8 | no | — |
| 201 | #121 | — | 10 | 2 | no | — |
| 202 | #121 | — | 10 | 4 | no | — |
| 203 | #120 | — | 5 | 8 | no | — |
| 204 | #120 | — | 5 | 4 | no | — |
| 205 | C4×Q8 | — | 2 | 4 | no | — |
| 206 | C4×D4 | — | 10 | 4 | no | — |
| 207 | C4×D4 | — | 10 | 8 | no | — |
| 208 | #117 | C4×D5 | 1 | 4 | yes | — |
| 209 | #117 | C22×D5 | 1 | 2 | yes | — |
| 210 | #101 | — | 2 | 4 | no | — |
| 211 | #101 | — | 10 | 4 | no | — |
| 212 | #101 | — | 10 | 4 | no | — |
| 213 | #100 | — | 10 | 4 | no | — |
| 214 | #100 | — | 20 | 4 | no | — |
| 215 | C22⋊C8 | — | 20 | 4 | no | — |
| 216 | C22×C5 | — | 4 | 1 | no | — |
| 217 | C22×C5 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 218 | C22×C5 | #518 | 1 | 2 | yes | — |
| 219 | C2×D5 | — | 2 | 4 | no | — |
| 220 | C2×D5 | — | 4 | 4 | no | — |
| 221 | C2×D5 | — | 4 | 2 | no | — |
| 222 | C2×D5 | — | 4 | 4 | no | — |
| 223 | C2×D5 | — | 4 | 4 | no | — |
| 224 | C2×D5 | — | 4 | 1 | no | — |
| 225 | C2×D5 | — | 4 | 2 | no | — |
| 226 | C2×D5 | — | 8 | 4 | no | — |
| 227 | C4×C5 | — | 2 | 4 | no | — |
| 228 | C4×C5 | — | 4 | 4 | no | — |
| 229 | C4×C5 | #534 | 1 | 4 | yes | — |
| 230 | Dic5 | — | 2 | 4 | no | — |
| 231 | Dic5 | — | 4 | 1 | no | — |
| 232 | C24 | — | 5 | 1 | no | — |
| 233 | C24 | — | 5 | 2 | no | — |
| 234 | #41 | — | 10 | 4 | no | — |
| 235 | #41 | — | 10 | 4 | no | — |
| 236 | #41 | — | 20 | 8 | no | — |
| 237 | #41 | — | 20 | 8 | no | — |
| 238 | #41 | — | 20 | 16 | no | — |
| 239 | C2×Q8 | — | 2 | 4 | no | — |
| 240 | C2×Q8 | — | 5 | 2 | no | — |
| 241 | C2×Q8 | — | 5 | 4 | no | — |
| 242 | C2×Q8 | — | 20 | 8 | no | — |
| 243 | C2×Q8 | D4×D5 | 1 | 4 | yes | — |
| 244 | C2×D4 | — | 5 | 4 | no | — |
| 245 | C2×D4 | — | 5 | 2 | no | — |
| 246 | C2×D4 | — | 5 | 4 | no | — |
| 247 | C2×D4 | — | 10 | 4 | no | — |
| 248 | C2×D4 | — | 10 | 8 | no | — |
| 249 | C2×D4 | — | 10 | 2 | no | — |
| 250 | C2×D4 | — | 10 | 4 | no | — |
| 251 | C2×D4 | — | 10 | 4 | no | — |
| 252 | C2×D4 | — | 20 | 4 | no | — |
| 253 | C22×C4 | — | 5 | 4 | no | — |
| 254 | C22×C4 | — | 5 | 8 | no | — |
| 255 | C22×C4 | — | 5 | 1 | no | — |
| 256 | C22×C4 | — | 10 | 2 | no | — |
| 257 | C22×C4 | — | 10 | 4 | no | — |
| 258 | C22×C4 | — | 10 | 4 | no | — |
| 259 | C22×C4 | — | 20 | 4 | no | — |
| 260 | C22×C4 | C2×C4×D5 | 1 | 2 | yes | — |
| 261 | C22×C4 | C23×D5 | 1 | 1 | yes | characteristic |
| 262 | M16 | — | 2 | 4 | no | — |
| 263 | M16 | — | 2 | 4 | no | — |
| 264 | M16 | — | 10 | 4 | no | — |
| 265 | M16 | — | 10 | 4 | no | — |
| 266 | M16 | — | 20 | 8 | no | — |
| 267 | C2×C8 | — | 2 | 4 | no | — |
| 268 | C2×C8 | — | 10 | 4 | no | — |
| 269 | C2×C8 | — | 20 | 8 | no | — |
| 270 | C4⋊C4 | — | 2 | 2 | no | — |
| 271 | C4⋊C4 | — | 2 | 4 | no | — |
| 272 | C4⋊C4 | — | 10 | 4 | no | — |
| 273 | C22⋊C4 | — | 10 | 4 | no | — |
| 274 | C22⋊C4 | — | 10 | 8 | no | — |
| 275 | C22⋊C4 | — | 10 | 4 | no | — |
| 276 | C22⋊C4 | C2×C4×D5 | 1 | 4 | yes | — |
| 277 | C42 | — | 2 | 2 | no | — |
| 278 | C42 | — | 2 | 4 | no | — |
| 279 | C2×C5 | — | 2 | 1 | no | — |
| 280 | C2×C5 | — | 2 | 2 | no | — |
| 281 | C2×C5 | #2308 | 1 | 1 | yes | characteristic |
| 282 | D5 | — | 4 | 4 | no | — |
| 283 | D5 | — | 8 | 1 | no | — |
| 284 | D5 | — | 8 | 2 | no | — |
| 285 | C23 | — | 5 | 4 | no | — |
| 286 | C23 | — | 10 | 1 | no | — |
| 287 | C23 | — | 10 | 2 | no | — |
| 288 | C23 | — | 10 | 2 | no | — |
| 289 | C23 | — | 10 | 4 | no | — |
| 290 | C23 | — | 40 | 1 | no | — |
| 291 | C23 | — | 40 | 2 | no | — |
| 292 | C23 | C22×C4×D5 | 1 | 1 | yes | characteristic |
| 293 | Q8 | — | 2 | 8 | no | — |
| 294 | Q8 | — | 10 | 4 | no | — |
| 295 | Q8 | — | 20 | 4 | no | — |
| 296 | D4 | — | 10 | 4 | no | — |
| 297 | D4 | — | 10 | 4 | no | — |
| 298 | D4 | — | 10 | 4 | no | — |
| 299 | D4 | — | 20 | 4 | no | — |
| 300 | D4 | — | 20 | 8 | no | — |
| 301 | D4 | — | 20 | 4 | no | — |
| 302 | D4 | — | 40 | 4 | no | — |
| 303 | D4 | — | 40 | 8 | no | — |
| 304 | C2×C4 | — | 2 | 4 | no | — |
| 305 | C2×C4 | — | 2 | 4 | no | — |
| 306 | C2×C4 | — | 5 | 4 | no | — |
| 307 | C2×C4 | — | 10 | 8 | no | — |
| 308 | C2×C4 | — | 10 | 1 | no | — |
| 309 | C2×C4 | — | 10 | 2 | no | — |
| 310 | C2×C4 | — | 20 | 8 | no | — |
| 311 | C2×C4 | — | 20 | 8 | no | — |
| 312 | C2×C4 | — | 20 | 4 | no | — |
| 313 | C2×C4 | D5×C22⋊C4 | 1 | 4 | yes | — |
| 314 | C2×C4 | C2×D4×D5 | 1 | 2 | yes | — |
| 315 | C2×C4 | #4149 | 1 | 4 | yes | — |
| 316 | C8 | — | 4 | 4 | no | — |
| 317 | C8 | — | 20 | 4 | no | — |
| 318 | C5 | #21781 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 319 | C22 | — | 4 | 1 | no | — |
| 320 | C22 | — | 10 | 4 | no | — |
| 321 | C22 | — | 20 | 4 | no | — |
| 322 | C22 | — | 20 | 2 | no | — |
| 323 | C22 | — | 20 | 4 | no | — |
| 324 | C22 | — | 20 | 4 | no | — |
| 325 | C22 | — | 20 | 1 | no | — |
| 326 | C22 | — | 20 | 2 | no | — |
| 327 | C22 | — | 40 | 4 | no | — |
| 328 | C22 | C2×D5×C22⋊C4 | 1 | 1 | yes | characteristic |
| 329 | C22 | #65855 | 1 | 2 | yes | — |
| 330 | C4 | — | 2 | 4 | no | — |
| 331 | C4 | — | 4 | 4 | no | — |
| 332 | C4 | — | 10 | 4 | no | — |
| 333 | C4 | — | 20 | 1 | no | — |
| 334 | C4 | #65987 | 1 | 4 | yes | — |
| 335 | C2 | — | 2 | 1 | no | — |
| 336 | C2 | — | 2 | 2 | no | — |
| 337 | C2 | — | 20 | 4 | no | — |
| 338 | C2 | — | 40 | 1 | no | — |
| 339 | C2 | — | 40 | 2 | no | — |
| 340 | C2 | #10605850 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10610437 | C2 | semidirect |
| 3. C5×#1830 | C2 | semidirect |
| 4. #10602452 | C2 | semidirect |
| 5. #10600907 | C2 | semidirect |
| 6. #10599833 | C2 | semidirect |
| 7. #10599829 | C2 | semidirect |
| 74. #1830 | D5 | semidirect |
| 318. C5 | #21781 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10617775 | 2 |
| #10610437 | 2 |
| C5×#1830 | 2 |
| #10602452 | 2 |
| #10600907 | 2 |
| #10599833 | 2 |
| #10599829 | 2 |
| #21781 | 5 |