Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
960 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 960 conjugacy classes of subgroups — too many to draw. Only its 224 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 | #10618943 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10617198 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10617184 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10604637 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #10604634 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #10604432 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #10604431 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C2×#4133 | — | 2 | 1 | no | — |
| 9 | C5×Q82 | C4 | 1 | 1 | yes | characteristic |
| 10 | #66068 | C4 | 1 | 3 | yes | — |
| 11 | #66067 | — | 2 | 3 | no | — |
| 12 | #66066 | — | 2 | 3 | no | — |
| 13 | #66062 | C22 | 1 | 3 | yes | — |
| 14 | #65918 | C4 | 1 | 3 | yes | — |
| 15 | #65914 | C22 | 1 | 3 | yes | — |
| 16 | #65910 | — | 2 | 1 | no | — |
| 17 | Q8×D20 | C4 | 1 | 1 | yes | characteristic |
| 18 | C4×#713 | C22 | 1 | 1 | yes | characteristic |
| 19 | #65906 | — | 2 | 1 | no | — |
| 20 | Q8×#170 | — | 2 | 1 | no | — |
| 21 | #65705 | — | 2 | 3 | no | — |
| 22 | #65701 | — | 2 | 3 | no | — |
| 23 | #65683 | C22 | 1 | 3 | yes | — |
| 24 | C4×#700 | C22 | 1 | 3 | yes | — |
| 25 | C4×#700 | C22 | 1 | 3 | yes | — |
| 26 | #65676 | C22 | 1 | 3 | yes | — |
| 27 | #64927 | C22 | 1 | 1 | yes | characteristic |
| 28 | #64927 | C22 | 1 | 3 | yes | — |
| 29 | #64870 | C22 | 1 | 3 | yes | — |
| 30 | #64870 | C22 | 1 | 6 | yes | — |
| 31 | #64870 | C22 | 1 | 3 | yes | — |
| 32 | #17654 | — | 5 | 1 | no | maximal, Sylow |
| 33 | C2×#713 | D4 | 1 | 1 | yes | characteristic |
| 34 | C2×Q8×D5 | — | 2 | 1 | no | — |
| 35 | #4133 | — | 2 | 8 | no | — |
| 36 | C2×#700 | — | 2 | 3 | no | — |
| 37 | C2×#700 | — | 2 | 3 | no | — |
| 38 | C2×D5⋊C8 | — | 2 | 3 | no | — |
| 39 | C5×C4⋊Q8 | — | 2 | 3 | no | — |
| 40 | C5×C4⋊Q8 | C2×C4 | 1 | 3 | yes | — |
| 41 | C4×C5×Q8 | — | 2 | 1 | no | — |
| 42 | C4×C5×Q8 | C2×C4 | 1 | 3 | yes | — |
| 43 | C4×C5×Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | #4093 | — | 2 | 6 | no | — |
| 45 | #4093 | C23 | 1 | 1 | yes | characteristic |
| 46 | #4092 | — | 2 | 3 | no | — |
| 47 | #4092 | C2×C4 | 1 | 3 | yes | — |
| 48 | Q8×Dic5 | — | 2 | 3 | no | — |
| 49 | Q8×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 50 | #4044 | — | 2 | 6 | no | — |
| 51 | #4044 | — | 2 | 3 | no | — |
| 52 | #4043 | — | 2 | 3 | no | — |
| 53 | #4043 | — | 2 | 3 | no | — |
| 54 | #4041 | — | 2 | 3 | no | — |
| 55 | #4041 | C2×C4 | 1 | 3 | yes | — |
| 56 | #4040 | — | 2 | 3 | no | — |
| 57 | #4039 | D4 | 1 | 3 | yes | — |
| 58 | #4038 | — | 2 | 3 | no | — |
| 59 | #4038 | C2×C4 | 1 | 6 | yes | — |
| 60 | #4038 | D4 | 1 | 3 | yes | — |
| 61 | #4036 | C23 | 1 | 3 | yes | — |
| 62 | #4033 | — | 2 | 3 | no | — |
| 63 | #4021 | — | 2 | 3 | no | — |
| 64 | C4×D20 | — | 2 | 3 | no | — |
| 65 | C4×D20 | C2×C4 | 1 | 3 | yes | — |
| 66 | C4×D20 | C2×C4 | 1 | 3 | yes | — |
| 67 | #4018 | — | 2 | 3 | no | — |
| 68 | C42×D5 | C23 | 1 | 3 | yes | — |
| 69 | #4015 | C2×C4 | 1 | 3 | yes | — |
| 70 | C4×Dic10 | — | 2 | 3 | no | — |
| 71 | #4003 | — | 2 | 3 | no | — |
| 72 | #4002 | — | 2 | 3 | no | — |
| 73 | #4002 | C23 | 1 | 1 | yes | characteristic |
| 74 | #4002 | C23 | 1 | 3 | yes | — |
| 75 | #4002 | C23 | 1 | 1 | yes | characteristic |
| 76 | #4002 | C23 | 1 | 3 | yes | — |
| 77 | C4⋊#170 | — | 2 | 3 | no | — |
| 78 | C4×#170 | — | 2 | 3 | no | — |
| 79 | C4×#170 | — | 2 | 3 | no | — |
| 80 | #3518 | — | 5 | 1 | no | — |
| 81 | #2971 | — | 5 | 1 | no | — |
| 82 | #2875 | — | 5 | 1 | no | — |
| 83 | #1531 | — | 5 | 3 | no | — |
| 84 | #1529 | — | 5 | 3 | no | — |
| 85 | #1526 | — | 5 | 3 | no | — |
| 86 | #1524 | — | 5 | 3 | no | — |
| 87 | C2×C5×Q8 | — | 2 | 3 | no | — |
| 88 | C2×C5×Q8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 89 | C2×C5×Q8 | C22×C4 | 1 | 1 | yes | characteristic |
| 90 | #713 | — | 2 | 4 | no | — |
| 91 | Q8×D5 | — | 4 | 4 | no | — |
| 92 | C2×D20 | — | 2 | 3 | no | — |
| 93 | C2×D20 | C22⋊C4 | 1 | 3 | yes | — |
| 94 | C2×D20 | C22×C4 | 1 | 1 | yes | characteristic |
| 95 | C2×C4×D5 | — | 2 | 3 | no | — |
| 96 | C2×C4×D5 | C22×C4 | 1 | 3 | yes | — |
| 97 | C2×C4×D5 | C2×D4 | 1 | 3 | yes | — |
| 98 | C2×C4×D5 | C2×D4 | 1 | 3 | yes | — |
| 99 | C2×Dic10 | — | 2 | 3 | no | — |
| 100 | C2×#170 | — | 2 | 1 | no | — |
| 101 | C2×#170 | — | 2 | 3 | no | — |
| 102 | C2×#170 | — | 2 | 1 | no | — |
| 103 | C2×#170 | — | 2 | 3 | no | — |
| 104 | #700 | — | 2 | 12 | no | — |
| 105 | #700 | — | 4 | 6 | no | — |
| 106 | D5⋊C8 | — | 2 | 12 | no | — |
| 107 | C5×C4⋊C4 | — | 2 | 3 | no | — |
| 108 | C5×C4⋊C4 | — | 2 | 3 | no | — |
| 109 | C5×C4⋊C4 | C22⋊C4 | 1 | 3 | yes | — |
| 110 | C5×C4⋊C4 | C22×C4 | 1 | 3 | yes | — |
| 111 | C42×C5 | — | 2 | 3 | no | — |
| 112 | C42×C5 | C22×C4 | 1 | 3 | yes | — |
| 113 | #685 | — | 2 | 3 | no | — |
| 114 | #685 | — | 2 | 6 | no | — |
| 115 | #685 | — | 2 | 3 | no | — |
| 116 | #685 | — | 2 | 3 | no | — |
| 117 | #685 | — | 2 | 1 | no | — |
| 118 | #685 | — | 2 | 1 | no | — |
| 119 | #685 | C22⋊C4 | 1 | 6 | yes | — |
| 120 | C4⋊Dic5 | — | 2 | 3 | no | — |
| 121 | C4⋊Dic5 | — | 2 | 3 | no | — |
| 122 | C4⋊Dic5 | C22⋊C4 | 1 | 3 | yes | — |
| 123 | C4⋊Dic5 | C22×C4 | 1 | 3 | yes | — |
| 124 | #683 | — | 2 | 3 | no | — |
| 125 | #683 | — | 2 | 3 | no | — |
| 126 | C4×Dic5 | — | 2 | 6 | no | — |
| 127 | C4×Dic5 | C2×D4 | 1 | 3 | yes | — |
| 128 | C4×Dic5 | C2×D4 | 1 | 3 | yes | — |
| 129 | C4×Dic5 | C24 | 1 | 1 | yes | characteristic |
| 130 | C2×#131 | — | 10 | 1 | no | — |
| 131 | Q82 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 132 | #556 | — | 5 | 3 | no | — |
| 133 | #556 | — | 10 | 3 | no | — |
| 134 | #555 | — | 5 | 3 | no | — |
| 135 | D4×Q8 | — | 5 | 1 | no | — |
| 136 | #548 | — | 10 | 1 | no | — |
| 137 | #542 | — | 10 | 3 | no | — |
| 138 | #533 | — | 5 | 3 | no | — |
| 139 | #529 | — | 5 | 3 | no | — |
| 140 | #519 | — | 10 | 1 | no | — |
| 141 | C4×#41 | — | 5 | 1 | no | — |
| 142 | C8×Q8 | — | 10 | 1 | no | — |
| 143 | #436 | — | 10 | 3 | no | — |
| 144 | #432 | — | 5 | 3 | no | — |
| 145 | #405 | — | 10 | 3 | no | — |
| 146 | C4×M16 | — | 5 | 3 | no | — |
| 147 | C4×M16 | — | 5 | 3 | no | — |
| 148 | C2×#97 | — | 5 | 3 | no | — |
| 149 | #330 | — | 5 | 3 | no | — |
| 150 | #330 | — | 5 | 3 | no | — |
| 151 | #330 | — | 5 | 6 | no | — |
| 152 | #329 | — | 5 | 1 | no | — |
| 153 | #329 | — | 5 | 3 | no | — |
| 154 | C22×D5 | — | 2 | 1 | no | — |
| 155 | C22×D5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 156 | C5×Q8 | — | 4 | 1 | no | — |
| 157 | C5×Q8 | — | 8 | 3 | no | — |
| 158 | C5×Q8 | #104 | 1 | 4 | yes | — |
| 159 | C2×C4×C5 | — | 2 | 3 | no | — |
| 160 | C2×C4×C5 | — | 2 | 1 | no | — |
| 161 | C2×C4×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 162 | C2×C4×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 163 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 164 | C2×Dic5 | — | 2 | 3 | no | — |
| 165 | C2×Dic5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 166 | C2×Dic5 | C22×D4 | 1 | 1 | yes | characteristic |
| 167 | C2×Dic5 | C22×D4 | 1 | 1 | yes | characteristic |
| 168 | D20 | — | 2 | 6 | no | — |
| 169 | D20 | — | 4 | 1 | no | — |
| 170 | D20 | — | 8 | 3 | no | — |
| 171 | C4×D5 | — | 2 | 6 | no | — |
| 172 | C4×D5 | — | 4 | 3 | no | — |
| 173 | C4×D5 | — | 4 | 3 | no | — |
| 174 | C4×D5 | — | 4 | 6 | no | — |
| 175 | Dic10 | — | 4 | 6 | no | — |
| 176 | #170 | — | 2 | 2 | no | — |
| 177 | #170 | — | 2 | 6 | no | — |
| 178 | #170 | — | 2 | 2 | no | — |
| 179 | #170 | — | 4 | 3 | no | — |
| 180 | C2×#41 | — | 5 | 1 | no | — |
| 181 | C22×Q8 | — | 10 | 1 | no | — |
| 182 | #131 | — | 10 | 8 | no | — |
| 183 | C2×M16 | — | 10 | 3 | no | — |
| 184 | C2×M16 | — | 10 | 3 | no | — |
| 185 | C22×C8 | — | 10 | 3 | no | — |
| 186 | C4⋊Q8 | — | 2 | 3 | no | — |
| 187 | C4⋊Q8 | — | 5 | 3 | no | — |
| 188 | C4⋊Q8 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 189 | #126 | — | 10 | 3 | no | — |
| 190 | #126 | — | 10 | 6 | no | — |
| 191 | #125 | — | 5 | 3 | no | — |
| 192 | #124 | — | 5 | 1 | no | — |
| 193 | #124 | — | 10 | 3 | no | — |
| 194 | #124 | — | 10 | 6 | no | — |
| 195 | #123 | — | 10 | 3 | no | — |
| 196 | C22⋊Q8 | — | 5 | 3 | no | — |
| 197 | C22⋊Q8 | — | 10 | 3 | no | — |
| 198 | C22⋊Q8 | — | 10 | 3 | no | — |
| 199 | C22⋊Q8 | — | 10 | 3 | no | — |
| 200 | #121 | — | 5 | 3 | no | — |
| 201 | #121 | — | 10 | 3 | no | — |
| 202 | C4×Q8 | — | 2 | 1 | no | — |
| 203 | C4×Q8 | — | 5 | 1 | no | — |
| 204 | C4×Q8 | — | 10 | 3 | no | — |
| 205 | C4×Q8 | — | 10 | 3 | no | — |
| 206 | C4×Q8 | — | 10 | 3 | no | — |
| 207 | C4×Q8 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 208 | C4×Q8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 209 | C4×D4 | — | 5 | 3 | no | — |
| 210 | C4×D4 | — | 5 | 3 | no | — |
| 211 | C4×D4 | — | 5 | 3 | no | — |
| 212 | C4×D4 | — | 10 | 3 | no | — |
| 213 | #117 | — | 5 | 3 | no | — |
| 214 | #117 | — | 5 | 6 | no | — |
| 215 | #117 | — | 10 | 3 | no | — |
| 216 | #117 | — | 10 | 3 | no | — |
| 217 | C2×C42 | — | 5 | 3 | no | — |
| 218 | C4⋊C8 | — | 10 | 3 | no | — |
| 219 | C22⋊C8 | — | 10 | 3 | no | — |
| 220 | #97 | — | 5 | 1 | no | — |
| 221 | #97 | — | 5 | 3 | no | — |
| 222 | #97 | — | 5 | 1 | no | — |
| 223 | #97 | — | 5 | 3 | no | — |
| 224 | #97 | — | 10 | 3 | no | — |
| 225 | C4×C8 | — | 10 | 3 | no | — |
| 226 | C4×C8 | — | 10 | 3 | no | — |
| 227 | C22×C5 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 228 | C2×D5 | — | 2 | 3 | no | — |
| 229 | C2×D5 | — | 2 | 3 | no | — |
| 230 | C2×D5 | — | 2 | 3 | no | — |
| 231 | C2×D5 | — | 4 | 1 | no | — |
| 232 | C2×D5 | — | 4 | 1 | no | — |
| 233 | C2×D5 | — | 4 | 1 | no | — |
| 234 | C4×C5 | — | 2 | 1 | no | — |
| 235 | C4×C5 | — | 2 | 3 | no | — |
| 236 | C4×C5 | — | 4 | 1 | no | — |
| 237 | C4×C5 | — | 4 | 3 | no | — |
| 238 | C4×C5 | C2×#104 | 1 | 6 | yes | — |
| 239 | Dic5 | — | 2 | 3 | no | — |
| 240 | Dic5 | — | 2 | 1 | no | — |
| 241 | Dic5 | — | 2 | 1 | no | — |
| 242 | Dic5 | — | 4 | 3 | no | — |
| 243 | #41 | — | 10 | 4 | no | — |
| 244 | C2×Q8 | — | 2 | 3 | no | — |
| 245 | C2×Q8 | — | 10 | 3 | no | — |
| 246 | C2×Q8 | — | 20 | 4 | no | — |
| 247 | C2×Q8 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 248 | C2×Q8 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 249 | C2×D4 | — | 5 | 3 | no | — |
| 250 | C2×D4 | — | 5 | 1 | no | — |
| 251 | C2×D4 | — | 10 | 3 | no | — |
| 252 | C22×C4 | — | 5 | 3 | no | — |
| 253 | C22×C4 | — | 5 | 3 | no | — |
| 254 | C22×C4 | — | 5 | 3 | no | — |
| 255 | C22×C4 | — | 10 | 3 | no | — |
| 256 | M16 | — | 10 | 12 | no | — |
| 257 | M16 | — | 20 | 6 | no | — |
| 258 | C2×C8 | — | 10 | 1 | no | — |
| 259 | C2×C8 | — | 10 | 3 | no | — |
| 260 | C2×C8 | — | 10 | 1 | no | — |
| 261 | C2×C8 | — | 10 | 12 | no | — |
| 262 | C2×C8 | — | 10 | 3 | no | — |
| 263 | C4⋊C4 | — | 2 | 3 | no | — |
| 264 | C4⋊C4 | — | 2 | 3 | no | — |
| 265 | C4⋊C4 | — | 5 | 3 | no | — |
| 266 | C4⋊C4 | — | 5 | 3 | no | — |
| 267 | C4⋊C4 | — | 10 | 3 | no | — |
| 268 | C4⋊C4 | — | 10 | 3 | no | — |
| 269 | C4⋊C4 | — | 10 | 3 | no | — |
| 270 | C4⋊C4 | — | 10 | 3 | no | — |
| 271 | C4⋊C4 | C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 272 | C4⋊C4 | C22×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 273 | C22⋊C4 | — | 5 | 6 | no | — |
| 274 | C22⋊C4 | — | 10 | 3 | no | — |
| 275 | C22⋊C4 | — | 10 | 1 | no | — |
| 276 | C22⋊C4 | — | 10 | 1 | no | — |
| 277 | C22⋊C4 | — | 10 | 6 | no | — |
| 278 | C22⋊C4 | — | 10 | 3 | no | — |
| 279 | C22⋊C4 | — | 10 | 3 | no | — |
| 280 | C42 | — | 2 | 3 | no | — |
| 281 | C42 | — | 5 | 3 | no | — |
| 282 | C42 | — | 5 | 3 | no | — |
| 283 | C42 | — | 5 | 1 | no | — |
| 284 | C42 | — | 10 | 6 | no | — |
| 285 | C42 | C22×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 286 | C2×C5 | #2889 | 1 | 1 | yes | characteristic |
| 287 | C2×C5 | C22×#104 | 1 | 1 | yes | characteristic |
| 288 | C2×C5 | #2901 | 1 | 1 | yes | characteristic |
| 289 | D5 | — | 4 | 3 | no | — |
| 290 | D5 | — | 8 | 1 | no | — |
| 291 | C23 | — | 5 | 3 | no | — |
| 292 | C23 | — | 10 | 1 | no | — |
| 293 | Q8 | — | 4 | 1 | no | — |
| 294 | Q8 | — | 8 | 3 | no | — |
| 295 | Q8 | — | 20 | 6 | no | — |
| 296 | Q8 | #4010 | 1 | 4 | yes | — |
| 297 | D4 | — | 10 | 6 | no | — |
| 298 | D4 | — | 20 | 1 | no | — |
| 299 | D4 | — | 40 | 3 | no | — |
| 300 | C2×C4 | — | 2 | 1 | no | — |
| 301 | C2×C4 | — | 2 | 3 | no | — |
| 302 | C2×C4 | — | 5 | 1 | no | — |
| 303 | C2×C4 | — | 5 | 1 | no | — |
| 304 | C2×C4 | — | 5 | 3 | no | — |
| 305 | C2×C4 | — | 10 | 6 | no | — |
| 306 | C2×C4 | — | 10 | 3 | no | — |
| 307 | C2×C4 | — | 20 | 3 | no | — |
| 308 | C2×C4 | — | 20 | 3 | no | — |
| 309 | C2×C4 | — | 20 | 6 | no | — |
| 310 | C2×C4 | C2×C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 311 | C2×C4 | C2×C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 312 | C2×C4 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 313 | C8 | — | 10 | 2 | no | — |
| 314 | C8 | — | 10 | 6 | no | — |
| 315 | C8 | — | 10 | 2 | no | — |
| 316 | C8 | — | 20 | 3 | no | — |
| 317 | C5 | #17654 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 318 | C22 | — | 10 | 3 | no | — |
| 319 | C22 | — | 10 | 3 | no | — |
| 320 | C22 | — | 10 | 3 | no | — |
| 321 | C22 | — | 20 | 1 | no | — |
| 322 | C22 | — | 20 | 1 | no | — |
| 323 | C22 | — | 20 | 1 | no | — |
| 324 | C22 | C22×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 325 | C4 | — | 2 | 1 | no | — |
| 326 | C4 | — | 2 | 3 | no | — |
| 327 | C4 | — | 4 | 1 | no | — |
| 328 | C4 | — | 4 | 3 | no | — |
| 329 | C4 | — | 10 | 1 | no | — |
| 330 | C4 | — | 10 | 3 | no | — |
| 331 | C4 | — | 10 | 1 | no | — |
| 332 | C4 | — | 20 | 3 | no | — |
| 333 | C4 | C2×#4010 | 1 | 6 | yes | — |
| 334 | C2 | — | 20 | 3 | no | — |
| 335 | C2 | — | 40 | 1 | no | — |
| 336 | C2 | C22×#4010 | 1 | 1 | yes | characteristic |
| 337 | C2 | #10617525 | 1 | 1 | yes | characteristic |
| 338 | C2 | #10617560 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10617198 | C2 | semidirect |
| 3. #10617184 | C2 | semidirect |
| 4. #10604637 | C2 | semidirect |
| 5. #10604634 | C2 | semidirect |
| 6. #10604432 | C2 | semidirect |
| 7. #10604431 | C2 | semidirect |
| 317. C5 | #17654 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10618943 | 2 |
| #10617198 | 2 |
| #10617184 | 2 |
| #10604637 | 2 |
| #10604634 | 2 |
| #10604432 | 2 |
| #10604431 | 2 |
| #17654 | 5 |