Finite Group DatabasebetaFinite Group DatabasebetaOrder 4352.
454 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 60 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 | #14545077 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #14545057 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #14543912 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #14539236 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #11763795 | — | 2 | 1 | no | — |
| 6 | #11763788 | — | 2 | 1 | no | — |
| 7 | #11763720 | — | 2 | 1 | no | — |
| 8 | #11763718 | C22 | 1 | 2 | yes | complemented |
| 9 | #11763716 | C22 | 1 | 2 | yes | complemented |
| 10 | #11763714 | — | 2 | 1 | no | — |
| 11 | C2×#10587594 | — | 2 | 1 | no | — |
| 12 | C2×#10587594 | — | 2 | 1 | no | — |
| 13 | #11763359 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#10587560 | C4 | 1 | 1 | yes | characteristic, complemented |
| 15 | #11763198 | C4 | 1 | 2 | yes | complemented |
| 16 | #11763198 | C22 | 1 | 2 | yes | — |
| 17 | #11763186 | — | 2 | 1 | no | — |
| 18 | #11763182 | — | 2 | 1 | no | — |
| 19 | C4×M16×C17 | C4 | 1 | 1 | yes | characteristic, complemented |
| 20 | C2×#63252 | D4 | 1 | 1 | yes | characteristic |
| 21 | C2×#63252 | D4 | 1 | 1 | yes | characteristic |
| 22 | C22×D68 | — | 2 | 1 | no | — |
| 23 | C22×D68 | — | 4 | 1 | no | — |
| 24 | #10587677 | — | 4 | 1 | no | — |
| 25 | D17⋊M16 | — | 4 | 1 | no | — |
| 26 | C2×M16×C17 | — | 2 | 1 | no | — |
| 27 | C2×C42×C17 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | #10587616 | — | 4 | 1 | no | — |
| 29 | #10587594 | — | 4 | 2 | no | — |
| 30 | #10587594 | — | 4 | 2 | no | — |
| 31 | #10587594 | — | 4 | 2 | no | — |
| 32 | #10587594 | — | 4 | 2 | no | — |
| 33 | C2×D136 | — | 2 | 2 | no | — |
| 34 | C2×D136 | — | 2 | 2 | no | — |
| 35 | C2×#63219 | — | 2 | 2 | no | — |
| 36 | C2×#63219 | — | 2 | 2 | no | — |
| 37 | #10587561 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | #10587561 | C23 | 1 | 1 | yes | characteristic |
| 39 | #10587560 | — | 2 | 1 | no | — |
| 40 | #10587560 | — | 4 | 1 | no | — |
| 41 | #10587560 | C2×C4 | 1 | 2 | yes | — |
| 42 | #10587560 | D4 | 1 | 1 | yes | characteristic |
| 43 | C4×D68 | — | 4 | 1 | no | — |
| 44 | #10587555 | D4 | 1 | 1 | yes | characteristic |
| 45 | #10587553 | — | 4 | 1 | no | — |
| 46 | #10587551 | — | 4 | 1 | no | — |
| 47 | #10587545 | — | 2 | 2 | no | — |
| 48 | #10587539 | — | 2 | 2 | no | — |
| 49 | C17×#97 | C2×C4 | 1 | 2 | yes | — |
| 50 | C4×C8×C17 | — | 2 | 1 | no | — |
| 51 | C23×D17 | — | 4 | 1 | no | — |
| 52 | C22×C4×C17 | — | 2 | 1 | no | — |
| 53 | C22×C4×C17 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 54 | C2×C22⋊D17 | — | 2 | 1 | no | — |
| 55 | C2×C22⋊D17 | — | 2 | 1 | no | — |
| 56 | #63252 | — | 4 | 1 | no | — |
| 57 | #63252 | — | 4 | 1 | no | — |
| 58 | #63252 | — | 4 | 1 | no | — |
| 59 | #63252 | — | 4 | 1 | no | — |
| 60 | C2×D68 | — | 2 | 2 | no | — |
| 61 | C2×D68 | — | 4 | 1 | no | — |
| 62 | C2×D68 | — | 4 | 1 | no | — |
| 63 | C2×D68 | — | 4 | 1 | no | — |
| 64 | C2×D68 | — | 4 | 1 | no | — |
| 65 | C2×D68 | — | 4 | 1 | no | — |
| 66 | C2×D68 | — | 4 | 2 | no | — |
| 67 | C2×D68 | — | 4 | 2 | no | — |
| 68 | C2×D68 | — | 4 | 1 | no | — |
| 69 | C2×D68 | — | 8 | 4 | no | — |
| 70 | C2×D68 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 71 | C2×D68 | C2×D4 | 1 | 1 | yes | characteristic |
| 72 | C2×C4×D17 | — | 2 | 1 | no | — |
| 73 | C2×C4×D17 | — | 2 | 1 | no | — |
| 74 | C2×Dic34 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 75 | C2×Dic34 | C2×D4 | 1 | 1 | yes | characteristic |
| 76 | C22⋊#595 | — | 4 | 2 | no | — |
| 77 | #63247 | — | 4 | 2 | no | — |
| 78 | C4⋊#595 | — | 8 | 1 | no | — |
| 79 | C4×#595 | — | 8 | 1 | no | — |
| 80 | #63243 | — | 8 | 1 | no | — |
| 81 | #63242 | — | 8 | 1 | no | — |
| 82 | M16×C17 | — | 2 | 2 | no | — |
| 83 | M16×C17 | — | 2 | 2 | no | — |
| 84 | C2×C8×C17 | — | 2 | 2 | no | — |
| 85 | C42×C17 | — | 2 | 1 | no | — |
| 86 | C42×C17 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 87 | C42×C17 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | #63227 | — | 4 | 1 | no | — |
| 89 | #63227 | — | 4 | 1 | no | — |
| 90 | C4⋊Dic17 | — | 4 | 1 | no | — |
| 91 | D136 | — | 4 | 4 | no | — |
| 92 | D136 | — | 4 | 4 | no | — |
| 93 | #63219 | — | 4 | 4 | no | — |
| 94 | #63219 | — | 4 | 4 | no | — |
| 95 | #13579 | — | 17 | 1 | no | maximal, Sylow |
| 96 | C23×C17 | #99 | 1 | 1 | yes | characteristic |
| 97 | C22×D17 | — | 2 | 1 | no | — |
| 98 | C22×D17 | — | 2 | 1 | no | — |
| 99 | C22×D17 | — | 4 | 2 | no | — |
| 100 | C22×D17 | — | 8 | 1 | no | — |
| 101 | C22×D17 | — | 8 | 1 | no | — |
| 102 | C22×D17 | — | 8 | 4 | no | — |
| 103 | C2×#595 | — | 8 | 2 | no | — |
| 104 | C2×C4×C17 | — | 2 | 1 | no | — |
| 105 | C2×C4×C17 | — | 2 | 1 | no | — |
| 106 | C2×C4×C17 | — | 2 | 1 | no | — |
| 107 | C2×C4×C17 | — | 2 | 1 | no | — |
| 108 | C2×C4×C17 | — | 4 | 2 | no | — |
| 109 | C2×C4×C17 | #99 | 1 | 1 | yes | characteristic |
| 110 | C2×C4×C17 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 111 | C22⋊D17 | — | 8 | 1 | no | — |
| 112 | C22⋊D17 | — | 8 | 1 | no | — |
| 113 | C2×Dic17 | — | 2 | 1 | no | — |
| 114 | C2×Dic17 | — | 2 | 1 | no | — |
| 115 | D68 | — | 4 | 1 | no | — |
| 116 | D68 | — | 4 | 1 | no | — |
| 117 | D68 | — | 4 | 2 | no | — |
| 118 | D68 | — | 4 | 2 | no | — |
| 119 | D68 | — | 8 | 2 | no | — |
| 120 | D68 | — | 8 | 4 | no | — |
| 121 | D68 | — | 8 | 4 | no | — |
| 122 | D68 | — | 8 | 1 | no | — |
| 123 | D68 | — | 8 | 1 | no | — |
| 124 | C4×D17 | — | 4 | 1 | no | — |
| 125 | C4×D17 | — | 4 | 1 | no | — |
| 126 | C4×D17 | — | 4 | 1 | no | — |
| 127 | C4×D17 | — | 4 | 1 | no | — |
| 128 | Dic34 | — | 4 | 1 | no | — |
| 129 | Dic34 | — | 4 | 1 | no | — |
| 130 | #3624 | — | 8 | 2 | no | — |
| 131 | C8×C17 | — | 2 | 4 | no | — |
| 132 | #3151 | — | 17 | 1 | no | — |
| 133 | #2132 | — | 17 | 1 | no | — |
| 134 | #2126 | — | 17 | 1 | no | — |
| 135 | #1406 | — | 17 | 4 | no | — |
| 136 | C22×C17 | — | 2 | 1 | no | — |
| 137 | C22×C17 | — | 4 | 1 | no | — |
| 138 | C22×C17 | C2×#99 | 1 | 1 | yes | characteristic |
| 139 | C2×D17 | — | 4 | 1 | no | — |
| 140 | C2×D17 | — | 4 | 1 | no | — |
| 141 | C2×D17 | — | 8 | 1 | no | — |
| 142 | C2×D17 | — | 8 | 1 | no | — |
| 143 | C2×D17 | — | 8 | 4 | no | — |
| 144 | C2×D17 | — | 8 | 2 | no | — |
| 145 | C2×D17 | — | 16 | 2 | no | — |
| 146 | C2×D17 | — | 16 | 2 | no | — |
| 147 | #595 | — | 16 | 2 | no | — |
| 148 | C4×C17 | — | 2 | 2 | no | — |
| 149 | C4×C17 | — | 2 | 1 | no | — |
| 150 | C4×C17 | — | 4 | 1 | no | — |
| 151 | C4×C17 | #353 | 1 | 1 | yes | characteristic |
| 152 | C4×C17 | #353 | 1 | 1 | yes | characteristic |
| 153 | Dic17 | — | 4 | 1 | no | — |
| 154 | Dic17 | — | 4 | 1 | no | — |
| 155 | C2×AGL(1,ℤ/8) | — | 34 | 1 | no | — |
| 156 | C2×AGL(1,ℤ/8) | — | 34 | 1 | no | — |
| 157 | #532 | — | 17 | 1 | no | — |
| 158 | C2×#127 | — | 17 | 1 | no | — |
| 159 | #496 | — | 17 | 2 | no | — |
| 160 | #496 | — | 17 | 2 | no | — |
| 161 | #493 | — | 34 | 1 | no | — |
| 162 | #492 | — | 34 | 1 | no | — |
| 163 | #413 | — | 34 | 1 | no | — |
| 164 | #410 | — | 34 | 1 | no | — |
| 165 | C4×M16 | #595 | 1 | 1 | yes | characteristic, complemented |
| 166 | #356 | — | 34 | 1 | no | — |
| 167 | #355 | — | 17 | 2 | no | — |
| 168 | #354 | — | 17 | 2 | no | — |
| 169 | #353 | — | 34 | 1 | no | — |
| 170 | C2×C17 | — | 2 | 1 | no | — |
| 171 | C2×C17 | — | 4 | 1 | no | — |
| 172 | C2×C17 | C2×#353 | 1 | 1 | yes | characteristic |
| 173 | D17 | — | 8 | 1 | no | — |
| 174 | D17 | — | 8 | 1 | no | — |
| 175 | D17 | — | 16 | 2 | no | — |
| 176 | C2×#41 | — | 17 | 1 | no | — |
| 177 | C2×#41 | — | 17 | 1 | no | — |
| 178 | C22×D4 | — | 34 | 1 | no | — |
| 179 | C22×D4 | — | 68 | 1 | no | — |
| 180 | AGL(1,ℤ/8) | — | 68 | 2 | no | — |
| 181 | AGL(1,ℤ/8) | — | 68 | 2 | no | — |
| 182 | AGL(1,ℤ/8) | — | 68 | 2 | no | — |
| 183 | AGL(1,ℤ/8) | — | 68 | 2 | no | — |
| 184 | C2×SD16 | — | 34 | 2 | no | — |
| 185 | C2×SD16 | — | 34 | 2 | no | — |
| 186 | C2×D8 | — | 34 | 2 | no | — |
| 187 | C2×D8 | — | 34 | 2 | no | — |
| 188 | C2×M16 | — | 2 | 1 | no | — |
| 189 | C2×M16 | — | 68 | 1 | no | — |
| 190 | C4⋊Q8 | — | 17 | 1 | no | — |
| 191 | #127 | — | 17 | 1 | no | — |
| 192 | #127 | — | 17 | 2 | no | — |
| 193 | #127 | — | 34 | 1 | no | — |
| 194 | #127 | — | 68 | 1 | no | — |
| 195 | #124 | — | 17 | 1 | no | — |
| 196 | #124 | — | 17 | 1 | no | — |
| 197 | #121 | — | 68 | 1 | no | — |
| 198 | C4×D4 | — | 68 | 1 | no | — |
| 199 | #117 | — | 68 | 1 | no | — |
| 200 | C2×C42 | C2×#595 | 1 | 1 | yes | characteristic |
| 201 | #101 | — | 34 | 2 | no | — |
| 202 | #100 | — | 68 | 1 | no | — |
| 203 | #99 | — | 34 | 2 | no | — |
| 204 | #99 | — | 68 | 1 | no | — |
| 205 | #97 | C2×#595 | 1 | 2 | yes | — |
| 206 | C4×C8 | — | 2 | 1 | no | — |
| 207 | C17 | #13579 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 208 | C24 | — | 68 | 1 | no | — |
| 209 | #41 | — | 68 | 1 | no | — |
| 210 | #41 | — | 68 | 1 | no | — |
| 211 | #41 | — | 68 | 1 | no | — |
| 212 | #41 | — | 68 | 1 | no | — |
| 213 | C2×Q8 | — | 17 | 1 | no | — |
| 214 | C2×Q8 | — | 17 | 1 | no | — |
| 215 | C2×D4 | — | 17 | 1 | no | — |
| 216 | C2×D4 | — | 17 | 1 | no | — |
| 217 | C2×D4 | — | 34 | 1 | no | — |
| 218 | C2×D4 | — | 34 | 1 | no | — |
| 219 | C2×D4 | — | 34 | 2 | no | — |
| 220 | C2×D4 | — | 68 | 1 | no | — |
| 221 | C2×D4 | — | 68 | 1 | no | — |
| 222 | C2×D4 | — | 68 | 1 | no | — |
| 223 | C2×D4 | — | 68 | 1 | no | — |
| 224 | C2×D4 | — | 68 | 1 | no | — |
| 225 | C2×D4 | — | 68 | 2 | no | — |
| 226 | C2×D4 | — | 68 | 2 | no | — |
| 227 | C2×D4 | — | 68 | 1 | no | — |
| 228 | C2×D4 | — | 136 | 4 | no | — |
| 229 | C22×C4 | — | 2 | 1 | no | — |
| 230 | C22×C4 | — | 34 | 1 | no | — |
| 231 | C22×C4 | — | 34 | 1 | no | — |
| 232 | C22×C4 | C22⋊#595 | 1 | 1 | yes | characteristic |
| 233 | SD16 | — | 68 | 4 | no | — |
| 234 | SD16 | — | 68 | 4 | no | — |
| 235 | D8 | — | 68 | 4 | no | — |
| 236 | D8 | — | 68 | 4 | no | — |
| 237 | M16 | — | 2 | 2 | no | — |
| 238 | M16 | — | 2 | 2 | no | — |
| 239 | M16 | — | 68 | 2 | no | — |
| 240 | M16 | — | 136 | 1 | no | — |
| 241 | C2×C8 | — | 2 | 2 | no | — |
| 242 | C2×C8 | — | 136 | 1 | no | — |
| 243 | C4⋊C4 | — | 68 | 1 | no | — |
| 244 | C4⋊C4 | — | 136 | 1 | no | — |
| 245 | C22⋊C4 | — | 68 | 1 | no | — |
| 246 | C22⋊C4 | — | 68 | 1 | no | — |
| 247 | C22⋊C4 | — | 68 | 2 | no | — |
| 248 | C42 | — | 2 | 1 | no | — |
| 249 | C42 | — | 136 | 1 | no | — |
| 250 | C42 | C22⋊#595 | 1 | 1 | yes | characteristic |
| 251 | C42 | C22×#595 | 1 | 1 | yes | characteristic |
| 252 | C23 | — | 34 | 1 | no | — |
| 253 | C23 | — | 34 | 1 | no | — |
| 254 | C23 | — | 68 | 2 | no | — |
| 255 | C23 | — | 136 | 1 | no | — |
| 256 | C23 | — | 136 | 1 | no | — |
| 257 | C23 | — | 136 | 4 | no | — |
| 258 | C23 | #10587539 | 1 | 1 | yes | characteristic |
| 259 | Q8 | — | 68 | 1 | no | — |
| 260 | Q8 | — | 68 | 1 | no | — |
| 261 | D4 | — | 68 | 1 | no | — |
| 262 | D4 | — | 68 | 1 | no | — |
| 263 | D4 | — | 68 | 2 | no | — |
| 264 | D4 | — | 68 | 2 | no | — |
| 265 | D4 | — | 136 | 4 | no | — |
| 266 | D4 | — | 136 | 2 | no | — |
| 267 | D4 | — | 136 | 4 | no | — |
| 268 | D4 | — | 136 | 1 | no | — |
| 269 | D4 | — | 136 | 1 | no | — |
| 270 | D4 | — | 136 | 1 | no | — |
| 271 | D4 | — | 136 | 1 | no | — |
| 272 | C2×C4 | — | 2 | 1 | no | — |
| 273 | C2×C4 | — | 2 | 1 | no | — |
| 274 | C2×C4 | — | 2 | 1 | no | — |
| 275 | C2×C4 | — | 2 | 1 | no | — |
| 276 | C2×C4 | — | 4 | 2 | no | — |
| 277 | C2×C4 | — | 34 | 1 | no | — |
| 278 | C2×C4 | — | 34 | 1 | no | — |
| 279 | C2×C4 | — | 68 | 1 | no | — |
| 280 | C2×C4 | — | 68 | 1 | no | — |
| 281 | C2×C4 | — | 68 | 1 | no | — |
| 282 | C2×C4 | — | 68 | 1 | no | — |
| 283 | C2×C4 | — | 136 | 2 | no | — |
| 284 | C2×C4 | #10587539 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | C2×C22⋊#595 | 1 | 1 | yes | characteristic |
| 286 | C8 | — | 2 | 4 | no | — |
| 287 | C8 | — | 136 | 2 | no | — |
| 288 | C22 | — | 2 | 1 | no | — |
| 289 | C22 | — | 4 | 1 | no | — |
| 290 | C22 | — | 272 | 2 | no | — |
| 291 | C22 | — | 272 | 2 | no | — |
| 292 | C22 | — | 68 | 1 | no | — |
| 293 | C22 | — | 68 | 1 | no | — |
| 294 | C22 | — | 136 | 4 | no | — |
| 295 | C22 | — | 136 | 1 | no | — |
| 296 | C22 | — | 136 | 1 | no | — |
| 297 | C22 | — | 136 | 2 | no | — |
| 298 | C22 | C2×#10587539 | 1 | 1 | yes | characteristic |
| 299 | C4 | — | 2 | 2 | no | — |
| 300 | C4 | — | 2 | 1 | no | — |
| 301 | C4 | — | 4 | 1 | no | — |
| 302 | C4 | — | 272 | 2 | no | — |
| 303 | C4 | — | 68 | 1 | no | — |
| 304 | C4 | — | 68 | 1 | no | — |
| 305 | C4 | #11763714 | 1 | 1 | yes | characteristic |
| 306 | C4 | #11763714 | 1 | 1 | yes | characteristic |
| 307 | C2 | — | 2 | 1 | no | — |
| 308 | C2 | — | 4 | 1 | no | — |
| 309 | C2 | — | 272 | 2 | no | — |
| 310 | C2 | — | 136 | 1 | no | — |
| 311 | C2 | — | 136 | 1 | no | — |
| 312 | C2 | C2×#11763714 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #14545077 | C2 | semidirect |
| 2. #14545057 | C2 | semidirect |
| 3. #14543912 | C2 | semidirect |
| 8. #11763718 | C22 | semidirect |
| 9. #11763716 | C22 | semidirect |
| 14. C2×#10587560 | C4 | semidirect |
| 15. #11763198 | C4 | semidirect |
| 19. C4×M16×C17 | C4 | semidirect |
| 165. C4×M16 | #595 | semidirect |
| 207. C17 | #13579 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #14545077 | 2 |
| #14545057 | 2 |
| #14543912 | 2 |
| #14539236 | 2 |
| #13579 | 17 |