Finite Group DatabasebetaFinite Group DatabasebetaOrder 6400.
3976 conjugacy classes of subgroups; only the 659 normal subgroups are shown.
This group has 3976 conjugacy classes of subgroups — too many to draw. Only its 659 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 | #72223086 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #72223053 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #72223030 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #72221162 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #72221128 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C52×#1455 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #72220990 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C2×#11717228 | — | 2 | 6 | no | — |
| 9 | #13107005 | C22 | 1 | 3 | yes | — |
| 10 | C2×#11717194 | C22 | 1 | 1 | yes | characteristic |
| 11 | C4×C52×M16 | C22 | 1 | 3 | yes | — |
| 12 | #13102997 | — | 2 | 6 | no | — |
| 13 | #13102576 | — | 2 | 6 | no | — |
| 14 | #13102575 | — | 2 | 6 | no | — |
| 15 | #13102558 | C22 | 1 | 3 | yes | — |
| 16 | #13102558 | C22 | 1 | 3 | yes | — |
| 17 | #13102558 | C22 | 1 | 3 | yes | complemented |
| 18 | #13102557 | C22 | 1 | 6 | yes | complemented |
| 19 | #13102556 | C22 | 1 | 3 | yes | — |
| 20 | #13102556 | C22 | 1 | 3 | yes | — |
| 21 | #13102543 | C22 | 1 | 3 | yes | complemented |
| 22 | C52×#322 | C22 | 1 | 3 | yes | complemented |
| 23 | C52×C82 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | #12088541 | — | 5 | 6 | no | maximal |
| 25 | C2×#87900 | — | 2 | 3 | no | — |
| 26 | C22×C4⋊(C5⋊D5) | — | 2 | 3 | no | — |
| 27 | C2×C52×M16 | D4 | 1 | 6 | yes | — |
| 28 | C2×C42×C52 | C23 | 1 | 1 | yes | characteristic |
| 29 | #11717250 | — | 2 | 6 | no | — |
| 30 | #11717228 | — | 4 | 24 | no | — |
| 31 | C2×#87801 | — | 2 | 6 | no | — |
| 32 | C2×#87801 | — | 2 | 6 | no | — |
| 33 | C2×#87801 | — | 2 | 6 | no | — |
| 34 | C2×#87800 | — | 2 | 6 | no | — |
| 35 | C2×#87800 | — | 2 | 6 | no | — |
| 36 | #11717195 | — | 2 | 3 | no | — |
| 37 | #11717194 | — | 2 | 3 | no | — |
| 38 | #11717194 | C23 | 1 | 1 | yes | characteristic |
| 39 | #11717194 | C23 | 1 | 1 | yes | characteristic |
| 40 | #11717194 | C23 | 1 | 3 | yes | — |
| 41 | C4×C4⋊(C5⋊D5) | — | 2 | 6 | no | — |
| 42 | #11717189 | C23 | 1 | 3 | yes | — |
| 43 | C52×#97 | D4 | 1 | 6 | yes | — |
| 44 | C4×C52×C8 | C23 | 1 | 3 | yes | — |
| 45 | C4×C52×C8 | C23 | 1 | 3 | yes | — |
| 46 | #11716688 | — | 2 | 12 | no | — |
| 47 | #11716684 | — | 2 | 6 | no | — |
| 48 | #10610266 | — | 5 | 18 | no | — |
| 49 | #10610261 | — | 5 | 18 | no | — |
| 50 | C5×#1455 | D5 | 1 | 6 | yes | complemented |
| 51 | #10598887 | — | 5 | 18 | no | — |
| 52 | #10598884 | — | 5 | 18 | no | — |
| 53 | #10598855 | — | 5 | 6 | no | — |
| 54 | #10598850 | — | 5 | 6 | no | — |
| 55 | C23×C5⋊D5 | — | 4 | 1 | no | — |
| 56 | C22×C4×C52 | C2×D4 | 1 | 3 | yes | — |
| 57 | C2×C22⋊(C5⋊D5) | — | 4 | 3 | no | — |
| 58 | #87900 | — | 4 | 12 | no | — |
| 59 | C2×C4⋊(C5⋊D5) | — | 2 | 6 | no | — |
| 60 | C2×C4⋊(C5⋊D5) | — | 2 | 3 | no | — |
| 61 | C2×C4⋊(C5⋊D5) | — | 2 | 3 | no | — |
| 62 | C2×C4⋊(C5⋊D5) | — | 2 | 3 | no | — |
| 63 | C2×C4⋊(C5⋊D5) | — | 4 | 12 | no | — |
| 64 | C2×C4⋊(C5⋊D5) | — | 4 | 3 | no | — |
| 65 | C2×C4×C5⋊D5 | — | 4 | 3 | no | — |
| 66 | C2×#6045 | — | 2 | 3 | no | — |
| 67 | C52×M16 | — | 2 | 12 | no | — |
| 68 | C2×C52×C8 | C2×D4 | 1 | 6 | yes | — |
| 69 | C2×C52×C8 | C2×D4 | 1 | 6 | yes | — |
| 70 | C42×C52 | C2×D4 | 1 | 3 | yes | — |
| 71 | C42×C52 | C24 | 1 | 1 | yes | characteristic |
| 72 | #87808 | — | 4 | 6 | no | — |
| 73 | C4⋊#1039 | — | 2 | 6 | no | — |
| 74 | #87801 | — | 4 | 12 | no | — |
| 75 | #87801 | — | 4 | 12 | no | — |
| 76 | #87801 | — | 4 | 12 | no | — |
| 77 | #87800 | — | 4 | 12 | no | — |
| 78 | #87800 | — | 4 | 12 | no | — |
| 79 | C2×#4054 | — | 10 | 36 | no | — |
| 80 | #65811 | — | 5 | 18 | no | — |
| 81 | C2×#4020 | — | 5 | 6 | no | — |
| 82 | C4×C5×M16 | C2×D5 | 1 | 18 | yes | — |
| 83 | #65424 | — | 10 | 36 | no | — |
| 84 | #65001 | — | 10 | 36 | no | — |
| 85 | #65000 | — | 10 | 36 | no | — |
| 86 | #64984 | — | 5 | 18 | no | — |
| 87 | #64984 | — | 5 | 18 | no | — |
| 88 | #64984 | — | 5 | 18 | no | — |
| 89 | #64983 | — | 5 | 36 | no | — |
| 90 | #64982 | — | 5 | 18 | no | — |
| 91 | #64982 | — | 5 | 18 | no | — |
| 92 | #64969 | — | 5 | 18 | no | — |
| 93 | C5×#322 | C2×D5 | 1 | 18 | yes | complemented |
| 94 | C5×C82 | C2×D5 | 1 | 6 | yes | complemented |
| 95 | #19745 | — | 25 | 1 | no | Sylow |
| 96 | C23×C52 | #127 | 1 | 1 | yes | characteristic |
| 97 | C22×C5⋊D5 | — | 4 | 3 | no | — |
| 98 | C22×C5⋊D5 | — | 4 | 1 | no | — |
| 99 | C22×C5⋊D5 | — | 4 | 1 | no | — |
| 100 | C22×C5⋊D5 | — | 8 | 6 | no | — |
| 101 | C2×C4×C52 | — | 2 | 6 | no | — |
| 102 | C2×C4×C52 | #127 | 1 | 3 | yes | — |
| 103 | C2×C4×C52 | C22×D4 | 1 | 3 | yes | — |
| 104 | C22⋊(C5⋊D5) | — | 8 | 6 | no | — |
| 105 | C2×#1039 | — | 4 | 3 | no | — |
| 106 | C4⋊(C5⋊D5) | — | 4 | 6 | no | — |
| 107 | C4⋊(C5⋊D5) | — | 4 | 6 | no | — |
| 108 | C4⋊(C5⋊D5) | — | 4 | 6 | no | — |
| 109 | C4⋊(C5⋊D5) | — | 4 | 12 | no | — |
| 110 | C4⋊(C5⋊D5) | — | 8 | 6 | no | — |
| 111 | C4×C5⋊D5 | — | 8 | 6 | no | — |
| 112 | #6045 | — | 4 | 6 | no | — |
| 113 | C52×C8 | — | 2 | 6 | no | — |
| 114 | C52×C8 | AGL(1,ℤ/8) | 1 | 12 | yes | complemented |
| 115 | C2×#709 | — | 10 | 18 | no | — |
| 116 | C22×D20 | — | 10 | 18 | no | — |
| 117 | C2×C5×M16 | D20 | 1 | 36 | yes | — |
| 118 | C2×C42×C5 | C22×D5 | 1 | 6 | yes | — |
| 119 | #4076 | — | 10 | 36 | no | — |
| 120 | #4054 | — | 20 | 144 | no | — |
| 121 | C2×D40 | — | 10 | 36 | no | — |
| 122 | C2×D40 | — | 10 | 36 | no | — |
| 123 | C2×D40 | — | 10 | 36 | no | — |
| 124 | C2×#677 | — | 10 | 36 | no | — |
| 125 | C2×#677 | — | 10 | 36 | no | — |
| 126 | #4021 | — | 10 | 18 | no | — |
| 127 | #4020 | — | 5 | 6 | no | — |
| 128 | #4020 | — | 5 | 6 | no | — |
| 129 | #4020 | — | 5 | 18 | no | — |
| 130 | #4020 | — | 10 | 18 | no | — |
| 131 | C4×D20 | — | 10 | 36 | no | — |
| 132 | #4015 | — | 5 | 18 | no | — |
| 133 | C5×#97 | D20 | 1 | 36 | yes | — |
| 134 | C4×C5×C8 | C22×D5 | 1 | 18 | yes | — |
| 135 | C4×C5×C8 | C22×D5 | 1 | 18 | yes | — |
| 136 | #3953 | — | 10 | 72 | no | — |
| 137 | #3949 | — | 10 | 36 | no | — |
| 138 | #3184 | — | 25 | 3 | no | — |
| 139 | #3151 | — | 25 | 3 | no | — |
| 140 | #1721 | — | 25 | 3 | no | — |
| 141 | #1715 | — | 25 | 3 | no | — |
| 142 | #1712 | — | 25 | 1 | no | — |
| 143 | #1705 | — | 25 | 1 | no | — |
| 144 | #1455 | C5⋊D5 | 1 | 1 | yes | characteristic, complemented |
| 145 | C22×C52 | — | 2 | 3 | no | — |
| 146 | C22×C52 | C2×#127 | 1 | 1 | yes | characteristic |
| 147 | C2×C5⋊D5 | — | 8 | 3 | no | — |
| 148 | C2×C5⋊D5 | — | 8 | 3 | no | — |
| 149 | C2×C5⋊D5 | — | 8 | 3 | no | — |
| 150 | C2×C5⋊D5 | — | 8 | 6 | no | — |
| 151 | C2×C5⋊D5 | — | 16 | 4 | no | — |
| 152 | C4×C52 | — | 2 | 3 | no | — |
| 153 | C4×C52 | C2×AGL(1,ℤ/8) | 1 | 6 | yes | — |
| 154 | #1039 | — | 8 | 3 | no | — |
| 155 | C23×D5 | — | 20 | 6 | no | — |
| 156 | C22×C4×C5 | C2×D20 | 1 | 18 | yes | — |
| 157 | C2×C22⋊D5 | — | 20 | 18 | no | — |
| 158 | #709 | — | 20 | 72 | no | — |
| 159 | C2×D20 | — | 10 | 18 | no | — |
| 160 | C2×D20 | — | 10 | 18 | no | — |
| 161 | C2×D20 | — | 10 | 18 | no | — |
| 162 | C2×D20 | — | 10 | 36 | no | — |
| 163 | C2×D20 | — | 20 | 72 | no | — |
| 164 | C2×D20 | — | 20 | 18 | no | — |
| 165 | C2×C4×D5 | — | 20 | 18 | no | — |
| 166 | C2×Dic10 | — | 10 | 18 | no | — |
| 167 | C5×M16 | — | 2 | 72 | no | — |
| 168 | C2×C5×C8 | C2×D20 | 1 | 36 | yes | — |
| 169 | C2×C5×C8 | C2×D20 | 1 | 36 | yes | — |
| 170 | C42×C5 | C2×D20 | 1 | 18 | yes | — |
| 171 | C42×C5 | C23×D5 | 1 | 6 | yes | — |
| 172 | #685 | — | 20 | 36 | no | — |
| 173 | C4⋊Dic5 | — | 10 | 36 | no | — |
| 174 | D40 | — | 20 | 72 | no | — |
| 175 | D40 | — | 20 | 72 | no | — |
| 176 | D40 | — | 20 | 72 | no | — |
| 177 | #677 | — | 20 | 72 | no | — |
| 178 | #677 | — | 20 | 72 | no | — |
| 179 | C2×AGL(1,ℤ/8) | — | 50 | 6 | no | — |
| 180 | #532 | — | 25 | 3 | no | — |
| 181 | C2×#127 | — | 25 | 1 | no | — |
| 182 | #498 | — | 25 | 3 | no | — |
| 183 | #496 | — | 50 | 6 | no | — |
| 184 | #493 | — | 25 | 3 | no | — |
| 185 | #493 | — | 25 | 3 | no | — |
| 186 | #493 | — | 25 | 3 | no | — |
| 187 | #492 | — | 25 | 3 | no | — |
| 188 | #492 | — | 25 | 3 | no | — |
| 189 | #486 | — | 25 | 6 | no | — |
| 190 | #469 | — | 50 | 6 | no | — |
| 191 | #442 | — | 50 | 6 | no | — |
| 192 | C4×M16 | C2×C5⋊D5 | 1 | 3 | yes | — |
| 193 | #322 | C2×C5⋊D5 | 1 | 3 | yes | complemented |
| 194 | C82 | C2×C5⋊D5 | 1 | 1 | yes | characteristic, complemented |
| 195 | C2×C52 | — | 4 | 1 | no | — |
| 196 | C2×C52 | #3151 | 1 | 3 | yes | — |
| 197 | C5⋊D5 | — | 16 | 3 | no | — |
| 198 | C5⋊D5 | — | 16 | 1 | no | — |
| 199 | C5⋊D5 | — | 16 | 1 | no | — |
| 200 | C23×C5 | #4020 | 1 | 6 | yes | — |
| 201 | C22×D5 | — | 20 | 18 | no | — |
| 202 | C22×D5 | — | 20 | 6 | no | — |
| 203 | C22×D5 | — | 20 | 6 | no | — |
| 204 | C22×D5 | — | 40 | 36 | no | — |
| 205 | C2×C4×C5 | — | 2 | 36 | no | — |
| 206 | C2×C4×C5 | #4020 | 1 | 18 | yes | — |
| 207 | C2×C4×C5 | C22×D20 | 1 | 18 | yes | — |
| 208 | C22⋊D5 | — | 40 | 36 | no | — |
| 209 | C2×Dic5 | — | 20 | 18 | no | — |
| 210 | D20 | — | 20 | 36 | no | — |
| 211 | D20 | — | 20 | 36 | no | — |
| 212 | D20 | — | 20 | 36 | no | — |
| 213 | D20 | — | 20 | 72 | no | — |
| 214 | D20 | — | 40 | 36 | no | — |
| 215 | C4×D5 | — | 40 | 36 | no | — |
| 216 | Dic10 | — | 20 | 36 | no | — |
| 217 | C5×C8 | — | 2 | 36 | no | — |
| 218 | C5×C8 | #4054 | 1 | 72 | yes | complemented |
| 219 | C2×#41 | — | 50 | 3 | no | — |
| 220 | C22×D4 | — | 50 | 3 | no | — |
| 221 | AGL(1,ℤ/8) | — | 100 | 24 | no | — |
| 222 | C2×SD16 | — | 50 | 6 | no | — |
| 223 | C2×SD16 | — | 50 | 6 | no | — |
| 224 | C2×D8 | — | 50 | 6 | no | — |
| 225 | C2×D8 | — | 50 | 6 | no | — |
| 226 | C2×D8 | — | 50 | 6 | no | — |
| 227 | C2×M16 | C4⋊(C5⋊D5) | 1 | 6 | yes | — |
| 228 | C4⋊Q8 | — | 25 | 3 | no | — |
| 229 | #127 | — | 25 | 1 | no | — |
| 230 | #127 | — | 25 | 1 | no | — |
| 231 | #127 | — | 25 | 3 | no | — |
| 232 | #127 | — | 50 | 3 | no | — |
| 233 | #124 | — | 50 | 3 | no | — |
| 234 | #121 | — | 50 | 6 | no | — |
| 235 | C4×D4 | — | 50 | 6 | no | — |
| 236 | C2×C42 | C22×C5⋊D5 | 1 | 1 | yes | characteristic |
| 237 | #106 | — | 50 | 6 | no | — |
| 238 | #102 | — | 50 | 12 | no | — |
| 239 | #97 | C4⋊(C5⋊D5) | 1 | 6 | yes | — |
| 240 | C4×C8 | C22×C5⋊D5 | 1 | 3 | yes | — |
| 241 | C4×C8 | C22×C5⋊D5 | 1 | 3 | yes | — |
| 242 | C52 | #19745 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 243 | C22×C5 | — | 2 | 18 | no | — |
| 244 | C22×C5 | C2×#4020 | 1 | 6 | yes | — |
| 245 | C2×D5 | — | 40 | 18 | no | — |
| 246 | C2×D5 | — | 40 | 18 | no | — |
| 247 | C2×D5 | — | 40 | 18 | no | — |
| 248 | C2×D5 | — | 40 | 36 | no | — |
| 249 | C2×D5 | — | 80 | 24 | no | — |
| 250 | C4×C5 | — | 2 | 18 | no | — |
| 251 | C4×C5 | C2×#4054 | 1 | 36 | yes | — |
| 252 | Dic5 | — | 40 | 18 | no | — |
| 253 | C24 | — | 100 | 1 | no | — |
| 254 | #41 | — | 100 | 12 | no | — |
| 255 | C2×Q8 | — | 50 | 3 | no | — |
| 256 | C2×D4 | — | 50 | 6 | no | — |
| 257 | C2×D4 | — | 50 | 3 | no | — |
| 258 | C2×D4 | — | 50 | 3 | no | — |
| 259 | C2×D4 | — | 50 | 3 | no | — |
| 260 | C2×D4 | — | 100 | 12 | no | — |
| 261 | C2×D4 | — | 100 | 3 | no | — |
| 262 | C2×D4 | — | 100 | 3 | no | — |
| 263 | C22×C4 | — | 100 | 3 | no | — |
| 264 | C22×C4 | C2×C4⋊(C5⋊D5) | 1 | 3 | yes | — |
| 265 | SD16 | — | 100 | 12 | no | — |
| 266 | SD16 | — | 100 | 12 | no | — |
| 267 | D8 | — | 100 | 12 | no | — |
| 268 | D8 | — | 100 | 12 | no | — |
| 269 | D8 | — | 100 | 12 | no | — |
| 270 | M16 | — | 2 | 12 | no | — |
| 271 | C2×C8 | C2×C4⋊(C5⋊D5) | 1 | 6 | yes | — |
| 272 | C2×C8 | C2×C4⋊(C5⋊D5) | 1 | 6 | yes | — |
| 273 | C4⋊C4 | — | 50 | 6 | no | — |
| 274 | C22⋊C4 | — | 100 | 6 | no | — |
| 275 | C42 | C2×C4⋊(C5⋊D5) | 1 | 3 | yes | — |
| 276 | C42 | C23×C5⋊D5 | 1 | 1 | yes | characteristic |
| 277 | C2×C5 | — | 4 | 6 | no | — |
| 278 | C2×C5 | #10610261 | 1 | 18 | yes | — |
| 279 | D5 | — | 80 | 18 | no | — |
| 280 | D5 | — | 80 | 6 | no | — |
| 281 | D5 | — | 80 | 6 | no | — |
| 282 | C23 | — | 100 | 3 | no | — |
| 283 | C23 | — | 100 | 1 | no | — |
| 284 | C23 | — | 100 | 1 | no | — |
| 285 | C23 | — | 200 | 6 | no | — |
| 286 | C23 | #11717194 | 1 | 1 | yes | characteristic |
| 287 | Q8 | — | 100 | 6 | no | — |
| 288 | D4 | — | 100 | 6 | no | — |
| 289 | D4 | — | 100 | 6 | no | — |
| 290 | D4 | — | 100 | 6 | no | — |
| 291 | D4 | — | 100 | 12 | no | — |
| 292 | D4 | — | 200 | 6 | no | — |
| 293 | D4 | — | 200 | 6 | no | — |
| 294 | C2×C4 | — | 2 | 6 | no | — |
| 295 | C2×C4 | — | 100 | 3 | no | — |
| 296 | C2×C4 | — | 200 | 6 | no | — |
| 297 | C2×C4 | #11717194 | 1 | 3 | yes | — |
| 298 | C2×C4 | C22×C4⋊(C5⋊D5) | 1 | 3 | yes | — |
| 299 | C8 | — | 2 | 6 | no | — |
| 300 | C8 | #11717228 | 1 | 12 | yes | complemented |
| 301 | C5 | #12088541 | 1 | 6 | yes | complemented |
| 302 | C22 | — | 2 | 3 | no | — |
| 303 | C22 | — | 400 | 4 | no | — |
| 304 | C22 | — | 200 | 3 | no | — |
| 305 | C22 | — | 200 | 3 | no | — |
| 306 | C22 | — | 200 | 3 | no | — |
| 307 | C22 | — | 200 | 6 | no | — |
| 308 | C22 | C2×#11717194 | 1 | 1 | yes | characteristic |
| 309 | C4 | — | 2 | 3 | no | — |
| 310 | C4 | — | 200 | 3 | no | — |
| 311 | C4 | C2×#11717228 | 1 | 6 | yes | — |
| 312 | C2 | — | 4 | 1 | no | — |
| 313 | C2 | — | 400 | 1 | no | — |
| 314 | C2 | — | 400 | 1 | no | — |
| 315 | C2 | — | 400 | 3 | no | — |
| 316 | C2 | #72220990 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72223086 | C2 | semidirect |
| 2. #72223053 | C2 | semidirect |
| 3. #72223030 | C2 | semidirect |
| 4. #72221162 | C2 | semidirect |
| 5. #72221128 | C2 | semidirect |
| 6. C52×#1455 | C2 | semidirect |
| 7. #72220990 | C2 | semidirect |
| 17. #13102558 | C22 | semidirect |
| 18. #13102557 | C22 | semidirect |
| 21. #13102543 | C22 | semidirect |
| 22. C52×#322 | C22 | semidirect |
| 23. C52×C82 | C22 | semidirect |
| 50. C5×#1455 | D5 | semidirect |
| 93. C5×#322 | C2×D5 | semidirect |
| 94. C5×C82 | C2×D5 | semidirect |
| 114. C52×C8 | AGL(1,ℤ/8) | semidirect |
| 144. #1455 | C5⋊D5 | semidirect |
| 193. #322 | C2×C5⋊D5 | semidirect |
| 194. C82 | C2×C5⋊D5 | semidirect |
| 218. C5×C8 | #4054 | semidirect |
| 242. C52 | #19745 | semidirect |
| 300. C8 | #11717228 | semidirect |
| 301. C5 | #12088541 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72223086 | 2 |
| #72223053 | 2 |
| #72223030 | 2 |
| #72221162 | 2 |
| #72221128 | 2 |
| C52×#1455 | 2 |
| #72220990 | 2 |
| #12088541 | 5 |