Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
760 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 86 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 | #10618986 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10604569 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #10604545 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #66260 | C22 | 1 | 1 | yes | characteristic, complemented |
| 5 | C5×H(3,𝔽4) | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | #66018 | — | 2 | 3 | no | — |
| 7 | #66014 | — | 2 | 3 | no | — |
| 8 | #66006 | C4 | 1 | 3 | yes | complemented |
| 9 | #66002 | C22 | 1 | 3 | yes | — |
| 10 | #65929 | — | 2 | 3 | no | — |
| 11 | #65920 | — | 2 | 3 | no | — |
| 12 | #65785 | C22 | 1 | 3 | yes | complemented |
| 13 | #65769 | — | 2 | 3 | no | — |
| 14 | #64922 | — | 2 | 3 | no | — |
| 15 | #64862 | — | 2 | 3 | no | — |
| 16 | #64854 | — | 2 | 3 | no | — |
| 17 | #12772 | — | 5 | 1 | no | maximal, Sylow |
| 18 | C23×Dic5 | — | 2 | 1 | no | — |
| 19 | C2×#713 | C23 | 1 | 1 | yes | characteristic |
| 20 | C2×#711 | D4 | 1 | 3 | yes | — |
| 21 | #4133 | — | 4 | 2 | no | — |
| 22 | #4130 | — | 2 | 3 | no | — |
| 23 | D5⋊M16 | — | 2 | 3 | no | — |
| 24 | C5×#124 | — | 2 | 3 | no | — |
| 25 | C5×#124 | C2×C4 | 1 | 3 | yes | — |
| 26 | C5×#120 | — | 2 | 3 | no | — |
| 27 | #4099 | — | 2 | 3 | no | — |
| 28 | #4092 | — | 2 | 3 | no | — |
| 29 | Q8×Dic5 | — | 2 | 1 | no | — |
| 30 | #4090 | C2×C4 | 1 | 3 | yes | — |
| 31 | #4086 | D4 | 1 | 3 | yes | — |
| 32 | #4085 | — | 2 | 6 | no | — |
| 33 | #4084 | — | 2 | 3 | no | — |
| 34 | #4081 | — | 2 | 3 | no | — |
| 35 | D4×Dic5 | — | 2 | 3 | no | — |
| 36 | #4032 | — | 2 | 3 | no | — |
| 37 | #4030 | — | 2 | 6 | no | — |
| 38 | #4029 | — | 2 | 6 | no | — |
| 39 | #4027 | — | 2 | 6 | no | — |
| 40 | #4025 | — | 2 | 6 | no | — |
| 41 | #4024 | — | 2 | 6 | no | — |
| 42 | #4023 | — | 2 | 3 | no | — |
| 43 | #4021 | D4 | 1 | 3 | yes | — |
| 44 | C42×D5 | D4 | 1 | 3 | yes | — |
| 45 | #4016 | — | 2 | 3 | no | — |
| 46 | #4008 | — | 4 | 6 | no | — |
| 47 | #3999 | — | 2 | 6 | no | — |
| 48 | #3529 | — | 5 | 1 | no | — |
| 49 | #2126 | — | 5 | 3 | no | — |
| 50 | #1884 | — | 5 | 3 | no | — |
| 51 | C24×C5 | — | 2 | 1 | no | — |
| 52 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 53 | C2×C5×Q8 | C22×C4 | 1 | 1 | yes | characteristic |
| 54 | C2×C5×D4 | — | 2 | 3 | no | — |
| 55 | C2×C5×D4 | C22⋊C4 | 1 | 3 | yes | — |
| 56 | C2×C22⋊D5 | — | 2 | 3 | no | — |
| 57 | C2×C22⋊D5 | — | 2 | 6 | no | — |
| 58 | C22×Dic5 | — | 2 | 3 | no | — |
| 59 | C22×Dic5 | — | 2 | 3 | no | — |
| 60 | C22×Dic5 | — | 8 | 1 | no | — |
| 61 | C22×Dic5 | — | 8 | 1 | no | — |
| 62 | #713 | — | 4 | 1 | no | — |
| 63 | #713 | — | 4 | 1 | no | — |
| 64 | #711 | — | 4 | 3 | no | — |
| 65 | #711 | — | 4 | 3 | no | — |
| 66 | C2×D20 | C2×D4 | 1 | 3 | yes | — |
| 67 | C2×C4×D5 | — | 2 | 3 | no | — |
| 68 | C2×C4×D5 | C2×D4 | 1 | 3 | yes | — |
| 69 | C2×Dic10 | C22⋊C4 | 1 | 3 | yes | — |
| 70 | C22⋊AGL(1,𝔽5) | — | 4 | 3 | no | — |
| 71 | #704 | — | 4 | 1 | no | — |
| 72 | C4⋊AGL(1,𝔽5) | — | 4 | 3 | no | — |
| 73 | C4×AGL(1,𝔽5) | — | 4 | 3 | no | — |
| 74 | #700 | — | 2 | 6 | no | — |
| 75 | D5⋊C8 | — | 4 | 3 | no | — |
| 76 | C5×C22⋊C4 | — | 2 | 3 | no | — |
| 77 | C5×C22⋊C4 | — | 2 | 6 | no | — |
| 78 | C42×C5 | C22⋊C4 | 1 | 3 | yes | — |
| 79 | C22⋊Dic5 | — | 2 | 3 | no | — |
| 80 | C22⋊Dic5 | — | 2 | 6 | no | — |
| 81 | #685 | — | 2 | 6 | no | — |
| 82 | C4⋊Dic5 | — | 2 | 3 | no | — |
| 83 | C4⋊Dic5 | — | 2 | 3 | no | — |
| 84 | #683 | — | 2 | 6 | no | — |
| 85 | #683 | — | 2 | 6 | no | — |
| 86 | C4×Dic5 | — | 2 | 3 | no | — |
| 87 | C4×Dic5 | C22⋊C4 | 1 | 3 | yes | — |
| 88 | #568 | — | 5 | 1 | no | — |
| 89 | #563 | — | 5 | 3 | no | — |
| 90 | #562 | — | 10 | 3 | no | — |
| 91 | H(3,𝔽4) | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 92 | #560 | — | 10 | 3 | no | — |
| 93 | #532 | — | 5 | 3 | no | — |
| 94 | #529 | — | 10 | 3 | no | — |
| 95 | #525 | — | 10 | 3 | no | — |
| 96 | #421 | — | 10 | 3 | no | — |
| 97 | #410 | — | 5 | 3 | no | — |
| 98 | #354 | — | 10 | 3 | no | — |
| 99 | #353 | — | 10 | 3 | no | — |
| 100 | #339 | — | 10 | 3 | no | — |
| 101 | C23×C5 | — | 2 | 3 | no | — |
| 102 | C23×C5 | — | 8 | 1 | no | — |
| 103 | C23×C5 | — | 8 | 2 | no | — |
| 104 | C22×D5 | #120 | 1 | 3 | yes | — |
| 105 | C2×AGL(1,𝔽5) | — | 8 | 3 | no | — |
| 106 | C5×Q8 | — | 4 | 1 | no | — |
| 107 | C5×Q8 | — | 8 | 1 | no | — |
| 108 | C5×D4 | — | 4 | 3 | no | — |
| 109 | C5×D4 | — | 8 | 3 | no | — |
| 110 | C2×C4×C5 | — | 2 | 3 | no | — |
| 111 | C2×C4×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 112 | C22⋊D5 | — | 8 | 3 | no | — |
| 113 | C22⋊D5 | — | 8 | 6 | no | — |
| 114 | C2×Dic5 | — | 2 | 3 | no | — |
| 115 | C2×Dic5 | — | 2 | 3 | no | — |
| 116 | C2×Dic5 | — | 4 | 3 | no | — |
| 117 | C2×Dic5 | — | 4 | 3 | no | — |
| 118 | C2×Dic5 | — | 8 | 3 | no | — |
| 119 | C2×Dic5 | #120 | 1 | 1 | yes | characteristic |
| 120 | D20 | — | 4 | 3 | no | — |
| 121 | C4×D5 | — | 2 | 3 | no | — |
| 122 | C4×D5 | — | 2 | 3 | no | — |
| 123 | C4×D5 | — | 8 | 3 | no | — |
| 124 | Dic10 | — | 4 | 3 | no | — |
| 125 | #170 | — | 4 | 2 | no | — |
| 126 | C2×#41 | — | 5 | 1 | no | — |
| 127 | C2×#41 | — | 5 | 3 | no | — |
| 128 | C23×C4 | — | 10 | 1 | no | — |
| 129 | #131 | — | 20 | 2 | no | — |
| 130 | C2×M16 | — | 10 | 3 | no | — |
| 131 | C4⋊Q8 | — | 5 | 3 | no | — |
| 132 | #127 | — | 5 | 3 | no | — |
| 133 | #126 | — | 10 | 6 | no | — |
| 134 | #125 | — | 10 | 3 | no | — |
| 135 | #124 | — | 2 | 3 | no | — |
| 136 | #124 | — | 5 | 3 | no | — |
| 137 | #124 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 138 | #123 | — | 10 | 3 | no | — |
| 139 | #123 | — | 10 | 6 | no | — |
| 140 | #123 | — | 10 | 3 | no | — |
| 141 | C22⋊Q8 | — | 10 | 6 | no | — |
| 142 | C22⋊Q8 | — | 10 | 3 | no | — |
| 143 | #121 | — | 10 | 6 | no | — |
| 144 | #121 | — | 10 | 6 | no | — |
| 145 | #121 | — | 10 | 3 | no | — |
| 146 | #120 | — | 2 | 3 | no | — |
| 147 | #120 | — | 10 | 3 | no | — |
| 148 | C4×Q8 | — | 10 | 1 | no | — |
| 149 | C4×D4 | — | 10 | 6 | no | — |
| 150 | C4×D4 | — | 10 | 3 | no | — |
| 151 | #117 | — | 10 | 3 | no | — |
| 152 | #117 | — | 10 | 3 | no | — |
| 153 | C2×C42 | — | 5 | 3 | no | — |
| 154 | #104 | — | 20 | 6 | no | — |
| 155 | #99 | — | 10 | 6 | no | — |
| 156 | C22×C5 | — | 4 | 6 | no | — |
| 157 | C22×C5 | — | 4 | 3 | no | — |
| 158 | C22×C5 | — | 8 | 4 | no | — |
| 159 | C22×C5 | #379 | 1 | 1 | yes | characteristic |
| 160 | C2×D5 | — | 2 | 3 | no | — |
| 161 | C2×D5 | — | 4 | 3 | no | — |
| 162 | AGL(1,𝔽5) | — | 16 | 3 | no | — |
| 163 | C4×C5 | — | 2 | 3 | no | — |
| 164 | C4×C5 | — | 4 | 3 | no | — |
| 165 | Dic5 | — | 4 | 3 | no | — |
| 166 | Dic5 | — | 4 | 3 | no | — |
| 167 | Dic5 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 168 | Dic5 | #458 | 1 | 1 | yes | characteristic |
| 169 | C24 | — | 2 | 1 | no | — |
| 170 | #41 | — | 20 | 1 | no | — |
| 171 | #41 | — | 20 | 1 | no | — |
| 172 | #41 | — | 20 | 3 | no | — |
| 173 | #41 | — | 20 | 3 | no | — |
| 174 | C2×Q8 | — | 2 | 1 | no | — |
| 175 | C2×Q8 | — | 5 | 3 | no | — |
| 176 | C2×Q8 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 177 | C2×D4 | — | 2 | 3 | no | — |
| 178 | C2×D4 | — | 5 | 3 | no | — |
| 179 | C2×D4 | — | 10 | 6 | no | — |
| 180 | C2×D4 | — | 10 | 3 | no | — |
| 181 | C2×D4 | C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 182 | C22×C4 | — | 5 | 3 | no | — |
| 183 | C22×C4 | — | 10 | 3 | no | — |
| 184 | C22×C4 | — | 10 | 3 | no | — |
| 185 | C22×C4 | — | 10 | 3 | no | — |
| 186 | C22×C4 | — | 40 | 1 | no | — |
| 187 | C22×C4 | — | 40 | 1 | no | — |
| 188 | M16 | — | 10 | 6 | no | — |
| 189 | M16 | — | 20 | 1 | no | — |
| 190 | C2×C8 | — | 20 | 3 | no | — |
| 191 | C4⋊C4 | — | 10 | 6 | no | — |
| 192 | C4⋊C4 | — | 10 | 3 | no | — |
| 193 | C4⋊C4 | — | 10 | 3 | no | — |
| 194 | C4⋊C4 | — | 10 | 6 | no | — |
| 195 | C4⋊C4 | — | 20 | 3 | no | — |
| 196 | C22⋊C4 | — | 2 | 3 | no | — |
| 197 | C22⋊C4 | — | 2 | 6 | no | — |
| 198 | C22⋊C4 | — | 10 | 6 | no | — |
| 199 | C22⋊C4 | — | 10 | 3 | no | — |
| 200 | C22⋊C4 | — | 10 | 6 | no | — |
| 201 | C22⋊C4 | — | 20 | 3 | no | — |
| 202 | C42 | — | 5 | 3 | no | — |
| 203 | C42 | — | 10 | 3 | no | — |
| 204 | C42 | — | 20 | 3 | no | — |
| 205 | C42 | C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 206 | C2×C5 | — | 2 | 1 | no | — |
| 207 | C2×C5 | — | 8 | 3 | no | — |
| 208 | C2×C5 | #1790 | 1 | 1 | yes | characteristic |
| 209 | D5 | — | 4 | 3 | no | — |
| 210 | C23 | — | 2 | 3 | no | — |
| 211 | C23 | — | 5 | 3 | no | — |
| 212 | C23 | — | 8 | 1 | no | — |
| 213 | C23 | — | 8 | 2 | no | — |
| 214 | Q8 | — | 4 | 1 | no | — |
| 215 | Q8 | — | 8 | 1 | no | — |
| 216 | Q8 | — | 20 | 3 | no | — |
| 217 | D4 | — | 4 | 3 | no | — |
| 218 | D4 | — | 8 | 3 | no | — |
| 219 | D4 | — | 20 | 3 | no | — |
| 220 | D4 | — | 40 | 3 | no | — |
| 221 | D4 | — | 40 | 6 | no | — |
| 222 | C2×C4 | — | 2 | 3 | no | — |
| 223 | C2×C4 | — | 5 | 1 | no | — |
| 224 | C2×C4 | — | 10 | 3 | no | — |
| 225 | C2×C4 | — | 10 | 3 | no | — |
| 226 | C2×C4 | — | 10 | 3 | no | — |
| 227 | C2×C4 | — | 10 | 3 | no | — |
| 228 | C2×C4 | — | 20 | 3 | no | — |
| 229 | C2×C4 | — | 20 | 3 | no | — |
| 230 | C2×C4 | — | 40 | 3 | no | — |
| 231 | C2×C4 | — | 40 | 3 | no | — |
| 232 | C2×C4 | — | 40 | 3 | no | — |
| 233 | C2×C4 | C2×C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 234 | C8 | — | 20 | 2 | no | — |
| 235 | C5 | #12772 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 236 | C22 | — | 4 | 6 | no | — |
| 237 | C22 | — | 4 | 3 | no | — |
| 238 | C22 | — | 8 | 4 | no | — |
| 239 | C22 | — | 10 | 3 | no | — |
| 240 | C22 | — | 20 | 3 | no | — |
| 241 | C22 | #65800 | 1 | 1 | yes | characteristic |
| 242 | C4 | — | 2 | 3 | no | — |
| 243 | C4 | — | 4 | 3 | no | — |
| 244 | C4 | — | 5 | 1 | no | — |
| 245 | C4 | — | 5 | 1 | no | — |
| 246 | C4 | — | 20 | 3 | no | — |
| 247 | C4 | — | 20 | 3 | no | — |
| 248 | C4 | — | 80 | 3 | no | — |
| 249 | C2 | — | 2 | 1 | no | — |
| 250 | C2 | — | 8 | 3 | no | — |
| 251 | C2 | — | 20 | 3 | no | — |
| 252 | C2 | #10605017 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10604569 | C2 | semidirect |
| 3. #10604545 | C2 | semidirect |
| 4. #66260 | C22 | semidirect |
| 5. C5×H(3,𝔽4) | C4 | semidirect |
| 8. #66006 | C4 | semidirect |
| 12. #65785 | C22 | semidirect |
| 91. H(3,𝔽4) | AGL(1,𝔽5) | semidirect |
| 235. C5 | #12772 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10618986 | 2 |
| #10604569 | 2 |
| #10604545 | 2 |
| #12772 | 5 |