← #16769465
Subgroups — #16769465
Order 3328.
2026 conjugacy classes of subgroups.
The subgroup lattice has 336 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #13129610 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #13127987 | C2 | 1 | 8 | yes | maximal, complemented |
| 3 | #13127980 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | #13127970 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C2×#11718840 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #13127939 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C2×C13×#578 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×C13×2−1+4 | — | 2 | 1 | no | — |
| 9 | #11719648 | — | 2 | 1 | no | — |
| 10 | C13×#578 | — | 2 | 4 | no | — |
| 11 | C13×#578 | C22 | 1 | 8 | yes | complemented |
| 12 | C2×C13×#137 | — | 2 | 3 | no | — |
| 13 | C2×C13×#135 | C22 | 1 | 3 | yes | complemented |
| 14 | C22×C13×Q16 | C22 | 1 | 3 | yes | complemented |
| 15 | C2×C13×#131 | C22 | 1 | 1 | yes | characteristic |
| 16 | #11719458 | C22 | 1 | 4 | yes | complemented |
| 17 | #11719457 | C22 | 1 | 4 | yes | complemented |
| 18 | #11719456 | — | 2 | 4 | no | — |
| 19 | #11719453 | — | 2 | 3 | no | — |
| 20 | C2×#88158 | C22 | 1 | 6 | yes | — |
| 21 | #11718856 | C22 | 1 | 12 | yes | complemented |
| 22 | #11718855 | C22 | 1 | 24 | yes | complemented |
| 23 | #11718854 | C22 | 1 | 12 | yes | complemented |
| 24 | C2×#88069 | C22 | 1 | 12 | yes | complemented |
| 25 | #11718840 | C22 | 1 | 12 | yes | complemented |
| 26 | #11718840 | C22 | 1 | 24 | yes | complemented |
| 27 | C2×#88068 | C22 | 1 | 12 | yes | complemented |
| 28 | #11718808 | C22 | 1 | 4 | yes | complemented |
| 29 | #11718801 | C22 | 1 | 8 | yes | complemented |
| 30 | C2×#88052 | C22 | 1 | 6 | yes | complemented |
| 31 | C13×2−1+4 | — | 2 | 8 | no | — |
| 32 | C13×2−1+4 | — | 4 | 4 | no | — |
| 33 | C2×C13×#41 | — | 2 | 3 | no | — |
| 34 | C2×C13×#41 | — | 4 | 3 | no | — |
| 35 | C2×C13×#41 | D4 | 1 | 1 | yes | characteristic |
| 36 | C22×Q8×C13 | — | 2 | 3 | no | — |
| 37 | C22×Q8×C13 | — | 4 | 1 | no | — |
| 38 | #88257 | — | 2 | 4 | no | — |
| 39 | #88256 | — | 2 | 4 | no | — |
| 40 | #41×D13 | — | 4 | 4 | no | — |
| 41 | #88254 | — | 4 | 1 | no | — |
| 42 | #88251 | — | 4 | 3 | no | — |
| 43 | C2×#6127 | — | 2 | 6 | no | — |
| 44 | C13×#137 | — | 2 | 24 | no | — |
| 45 | C13×#135 | — | 2 | 6 | no | — |
| 46 | C13×#135 | C23 | 1 | 12 | yes | — |
| 47 | C2×C13×Q16 | — | 2 | 3 | no | — |
| 48 | C2×C13×Q16 | — | 2 | 6 | no | — |
| 49 | C2×C13×Q16 | C23 | 1 | 3 | yes | — |
| 50 | C2×C13×Q16 | C23 | 1 | 12 | yes | — |
| 51 | C2×C13×SD16 | — | 2 | 3 | no | — |
| 52 | C2×C13×D8 | C23 | 1 | 1 | yes | characteristic |
| 53 | C13×#131 | D4 | 1 | 4 | yes | — |
| 54 | C13×#131 | C23 | 1 | 4 | yes | — |
| 55 | C2×C13×M16 | D4 | 1 | 3 | yes | — |
| 56 | C22×C8×C13 | C23 | 1 | 3 | yes | — |
| 57 | #88163 | — | 2 | 12 | no | — |
| 58 | #88162 | — | 2 | 12 | no | — |
| 59 | C2×Dic52 | C23 | 1 | 6 | yes | — |
| 60 | #88158 | — | 2 | 6 | no | — |
| 61 | #88158 | — | 2 | 6 | no | — |
| 62 | #88158 | C23 | 1 | 4 | yes | — |
| 63 | #88158 | C23 | 1 | 12 | yes | — |
| 64 | C2×D104 | C23 | 1 | 6 | yes | — |
| 65 | C2×#6095 | — | 2 | 6 | no | — |
| 66 | #88069 | C23 | 1 | 24 | yes | — |
| 67 | #88068 | C23 | 1 | 24 | yes | — |
| 68 | #88067 | C23 | 1 | 8 | yes | — |
| 69 | #88066 | C23 | 1 | 8 | yes | — |
| 70 | #88052 | C23 | 1 | 4 | yes | — |
| 71 | #88052 | C23 | 1 | 12 | yes | — |
| 72 | C2×#6090 | C23 | 1 | 12 | yes | — |
| 73 | #62637 | — | 13 | 1 | no | maximal, Sylow |
| 74 | C13×#41 | — | 2 | 12 | no | — |
| 75 | C13×#41 | — | 4 | 24 | no | — |
| 76 | C13×#41 | — | 4 | 6 | no | — |
| 77 | C13×#41 | D8 | 1 | 4 | yes | — |
| 78 | C13×#41 | C2×D4 | 1 | 4 | yes | — |
| 79 | C2×Q8×C13 | — | 2 | 3 | no | — |
| 80 | C2×Q8×C13 | — | 2 | 12 | no | — |
| 81 | C2×Q8×C13 | — | 4 | 8 | no | — |
| 82 | C2×Q8×C13 | — | 4 | 3 | no | — |
| 83 | C2×Q8×C13 | — | 4 | 6 | no | — |
| 84 | C2×Q8×C13 | D8 | 1 | 1 | yes | characteristic |
| 85 | C2×D4×C13 | — | 2 | 1 | no | — |
| 86 | C2×D4×C13 | — | 4 | 3 | no | — |
| 87 | C2×D4×C13 | D8 | 1 | 3 | yes | — |
| 88 | C22×C4×C13 | — | 4 | 3 | no | — |
| 89 | C22×C4×C13 | — | 4 | 3 | no | — |
| 90 | C22×C4×C13 | C2×D4 | 1 | 3 | yes | — |
| 91 | C2×C22⋊D13 | — | 4 | 6 | no | — |
| 92 | #6131 | — | 4 | 4 | no | — |
| 93 | Q8×D13 | — | 4 | 4 | no | — |
| 94 | #6129 | — | 4 | 12 | no | — |
| 95 | D4×D13 | — | 4 | 12 | no | — |
| 96 | #6127 | — | 2 | 4 | no | — |
| 97 | #6127 | — | 2 | 12 | no | — |
| 98 | #6127 | — | 4 | 6 | no | — |
| 99 | #6127 | — | 4 | 6 | no | — |
| 100 | C2×D52 | — | 2 | 6 | no | — |
| 101 | C2×C4×D13 | — | 4 | 6 | no | — |
| 102 | C2×Dic26 | — | 2 | 6 | no | — |
| 103 | C13×Q16 | — | 2 | 3 | no | — |
| 104 | C13×Q16 | — | 2 | 12 | no | — |
| 105 | C13×Q16 | C24 | 1 | 6 | yes | — |
| 106 | C13×SD16 | — | 2 | 12 | no | — |
| 107 | C13×D8 | — | 2 | 1 | no | — |
| 108 | C13×D8 | C24 | 1 | 2 | yes | — |
| 109 | C13×M16 | C2×D4 | 1 | 12 | yes | — |
| 110 | C2×C8×C13 | C2×D4 | 1 | 6 | yes | — |
| 111 | C2×C8×C13 | C2×D4 | 1 | 3 | yes | — |
| 112 | C2×C8×C13 | C24 | 1 | 6 | yes | — |
| 113 | C2×C8×C13 | C24 | 1 | 1 | yes | characteristic |
| 114 | Dic52 | — | 2 | 6 | no | — |
| 115 | Dic52 | C24 | 1 | 4 | yes | — |
| 116 | D104 | — | 2 | 6 | no | — |
| 117 | D104 | C24 | 1 | 4 | yes | — |
| 118 | #6095 | — | 2 | 4 | no | — |
| 119 | #6095 | — | 2 | 12 | no | — |
| 120 | #6090 | C24 | 1 | 8 | yes | — |
| 121 | #3584 | — | 13 | 1 | no | — |
| 122 | C2×#578 | D13 | 1 | 1 | yes | characteristic, complemented |
| 123 | #3417 | — | 13 | 8 | no | — |
| 124 | #3416 | — | 13 | 8 | no | — |
| 125 | #3414 | — | 13 | 6 | no | — |
| 126 | C2×#510 | — | 13 | 6 | no | — |
| 127 | #3407 | — | 13 | 1 | no | — |
| 128 | C23×C13 | — | 4 | 1 | no | — |
| 129 | C23×C13 | C2×D8 | 1 | 3 | yes | — |
| 130 | C22×D13 | — | 4 | 6 | no | — |
| 131 | Q8×C13 | — | 2 | 6 | no | — |
| 132 | Q8×C13 | — | 4 | 12 | no | — |
| 133 | Q8×C13 | — | 4 | 3 | no | — |
| 134 | Q8×C13 | C2×D8 | 1 | 4 | yes | — |
| 135 | D4×C13 | — | 2 | 2 | no | — |
| 136 | D4×C13 | — | 4 | 12 | no | — |
| 137 | D4×C13 | — | 4 | 1 | no | — |
| 138 | D4×C13 | C2×D8 | 1 | 12 | yes | — |
| 139 | C2×C4×C13 | — | 4 | 3 | no | — |
| 140 | C2×C4×C13 | — | 4 | 12 | no | — |
| 141 | C2×C4×C13 | — | 4 | 12 | no | — |
| 142 | C2×C4×C13 | C2×D8 | 1 | 6 | yes | — |
| 143 | C2×C4×C13 | C2×D8 | 1 | 3 | yes | — |
| 144 | C2×C4×C13 | C22×D4 | 1 | 1 | yes | characteristic |
| 145 | C2×C4×C13 | C22×D4 | 1 | 6 | yes | — |
| 146 | C22⋊D13 | — | 4 | 4 | no | — |
| 147 | C22⋊D13 | — | 4 | 12 | no | — |
| 148 | C2×Dic13 | — | 4 | 6 | no | — |
| 149 | D52 | — | 2 | 4 | no | — |
| 150 | D52 | — | 4 | 6 | no | — |
| 151 | C4×D13 | — | 4 | 4 | no | — |
| 152 | C4×D13 | — | 4 | 12 | no | — |
| 153 | Dic26 | — | 2 | 4 | no | — |
| 154 | Dic26 | — | 4 | 6 | no | — |
| 155 | C8×C13 | C22×D4 | 1 | 1 | yes | characteristic |
| 156 | C8×C13 | C22×D4 | 1 | 6 | yes | — |
| 157 | C8×C13 | C25 | 1 | 1 | yes | characteristic |
| 158 | #585 | — | 26 | 1 | no | — |
| 159 | C2×2−1+4 | — | 2 | 1 | no | — |
| 160 | #578 | — | 2 | 4 | no | — |
| 161 | #578 | — | 13 | 4 | no | — |
| 162 | #578 | C2×D13 | 1 | 8 | yes | complemented |
| 163 | #577 | — | 26 | 4 | no | — |
| 164 | #576 | — | 13 | 4 | no | — |
| 165 | #575 | — | 26 | 3 | no | — |
| 166 | C2×#137 | — | 2 | 3 | no | — |
| 167 | C2×#135 | — | 13 | 6 | no | — |
| 168 | C2×#135 | C2×D13 | 1 | 3 | yes | complemented |
| 169 | C22×Q16 | C2×D13 | 1 | 3 | yes | complemented |
| 170 | C2×#131 | C2×D13 | 1 | 1 | yes | characteristic |
| 171 | #510 | — | 13 | 12 | no | — |
| 172 | #510 | — | 13 | 12 | no | — |
| 173 | #510 | — | 13 | 24 | no | — |
| 174 | #509 | — | 13 | 12 | no | — |
| 175 | #509 | — | 13 | 4 | no | — |
| 176 | #508 | — | 13 | 24 | no | — |
| 177 | C2×Q32 | — | 13 | 12 | no | — |
| 178 | C2×SD32 | — | 13 | 12 | no | — |
| 179 | #504 | — | 13 | 8 | no | — |
| 180 | C2×M32 | — | 13 | 6 | no | — |
| 181 | C22×C13 | — | 4 | 3 | no | — |
| 182 | C22×C13 | — | 4 | 2 | no | — |
| 183 | C22×C13 | — | 8 | 2 | no | — |
| 184 | C22×C13 | C22×D8 | 1 | 6 | yes | — |
| 185 | C22×C13 | C22×D8 | 1 | 1 | yes | characteristic |
| 186 | C2×D13 | — | 4 | 4 | no | — |
| 187 | C2×D13 | — | 16 | 6 | no | — |
| 188 | C4×C13 | — | 4 | 6 | no | — |
| 189 | C4×C13 | C22×D8 | 1 | 6 | yes | — |
| 190 | C4×C13 | C22×D8 | 1 | 1 | yes | characteristic |
| 191 | C4×C13 | C23×D4 | 1 | 1 | yes | characteristic |
| 192 | Dic13 | — | 4 | 4 | no | — |
| 193 | 2−1+4 | — | 2 | 8 | no | — |
| 194 | 2−1+4 | — | 4 | 4 | no | — |
| 195 | 2−1+4 | — | 26 | 4 | no | — |
| 196 | 2−1+4 | — | 52 | 1 | no | — |
| 197 | H(5,𝔽2) | — | 26 | 4 | no | — |
| 198 | H(5,𝔽2) | — | 52 | 3 | no | — |
| 199 | C2×#41 | — | 2 | 3 | no | — |
| 200 | C2×#41 | — | 4 | 3 | no | — |
| 201 | C2×#41 | — | 26 | 6 | no | — |
| 202 | C2×#41 | — | 52 | 4 | no | — |
| 203 | C2×#41 | C22⋊D13 | 1 | 1 | yes | characteristic |
| 204 | C22×Q8 | — | 2 | 3 | no | — |
| 205 | C22×Q8 | — | 4 | 1 | no | — |
| 206 | #137 | — | 2 | 24 | no | — |
| 207 | #137 | — | 26 | 12 | no | — |
| 208 | AGL(1,ℤ/8) | — | 26 | 12 | no | — |
| 209 | #135 | — | 2 | 6 | no | — |
| 210 | #135 | — | 13 | 4 | no | — |
| 211 | #135 | — | 13 | 12 | no | — |
| 212 | #135 | — | 26 | 6 | no | — |
| 213 | #135 | — | 26 | 6 | no | — |
| 214 | #135 | C22×D13 | 1 | 12 | yes | — |
| 215 | C2×Q16 | — | 2 | 3 | no | — |
| 216 | C2×Q16 | — | 2 | 6 | no | — |
| 217 | C2×Q16 | — | 13 | 6 | no | — |
| 218 | C2×Q16 | C22×D13 | 1 | 3 | yes | — |
| 219 | C2×Q16 | C22×D13 | 1 | 12 | yes | — |
| 220 | C2×SD16 | — | 2 | 3 | no | — |
| 221 | C2×SD16 | — | 26 | 6 | no | — |
| 222 | C2×D8 | — | 13 | 6 | no | — |
| 223 | C2×D8 | C22×D13 | 1 | 1 | yes | characteristic |
| 224 | #131 | C22⋊D13 | 1 | 4 | yes | — |
| 225 | #131 | C22×D13 | 1 | 4 | yes | — |
| 226 | C2×M16 | C22⋊D13 | 1 | 3 | yes | — |
| 227 | C22×C8 | C22×D13 | 1 | 3 | yes | — |
| 228 | Q32 | — | 13 | 24 | no | — |
| 229 | SD32 | — | 13 | 24 | no | — |
| 230 | SD32 | — | 13 | 8 | no | — |
| 231 | D16 | — | 13 | 8 | no | — |
| 232 | M32 | — | 13 | 4 | no | — |
| 233 | M32 | — | 13 | 12 | no | — |
| 234 | C2×C16 | — | 13 | 12 | no | — |
| 235 | C2×C13 | — | 2 | 6 | no | — |
| 236 | C2×C13 | — | 2 | 1 | no | — |
| 237 | C2×C13 | — | 8 | 2 | no | — |
| 238 | C2×C13 | C23×D8 | 1 | 1 | yes | characteristic |
| 239 | D13 | — | 8 | 4 | no | — |
| 240 | #41 | — | 2 | 12 | no | — |
| 241 | #41 | — | 4 | 24 | no | — |
| 242 | #41 | — | 4 | 6 | no | — |
| 243 | #41 | — | 26 | 4 | no | — |
| 244 | #41 | — | 26 | 12 | no | — |
| 245 | #41 | — | 52 | 4 | no | — |
| 246 | #41 | — | 52 | 12 | no | — |
| 247 | #41 | — | 52 | 6 | no | — |
| 248 | #41 | — | 52 | 6 | no | — |
| 249 | #41 | #6104 | 1 | 4 | yes | — |
| 250 | #41 | C2×C22⋊D13 | 1 | 4 | yes | — |
| 251 | C2×Q8 | — | 2 | 3 | no | — |
| 252 | C2×Q8 | — | 2 | 12 | no | — |
| 253 | C2×Q8 | — | 4 | 8 | no | — |
| 254 | C2×Q8 | — | 4 | 3 | no | — |
| 255 | C2×Q8 | — | 4 | 6 | no | — |
| 256 | C2×Q8 | — | 26 | 6 | no | — |
| 257 | C2×Q8 | — | 52 | 4 | no | — |
| 258 | C2×Q8 | #6104 | 1 | 1 | yes | characteristic |
| 259 | C2×D4 | — | 2 | 1 | no | — |
| 260 | C2×D4 | — | 4 | 3 | no | — |
| 261 | C2×D4 | — | 26 | 6 | no | — |
| 262 | C2×D4 | — | 52 | 6 | no | — |
| 263 | C2×D4 | — | 52 | 12 | no | — |
| 264 | C2×D4 | #6104 | 1 | 3 | yes | — |
| 265 | C22×C4 | — | 4 | 3 | no | — |
| 266 | C22×C4 | — | 4 | 3 | no | — |
| 267 | C22×C4 | — | 52 | 6 | no | — |
| 268 | C22×C4 | C2×C22⋊D13 | 1 | 3 | yes | — |
| 269 | Q16 | — | 2 | 3 | no | — |
| 270 | Q16 | — | 2 | 12 | no | — |
| 271 | Q16 | — | 13 | 4 | no | — |
| 272 | Q16 | — | 26 | 6 | no | — |
| 273 | Q16 | C23×D13 | 1 | 6 | yes | — |
| 274 | SD16 | — | 2 | 12 | no | — |
| 275 | SD16 | — | 26 | 4 | no | — |
| 276 | SD16 | — | 26 | 12 | no | — |
| 277 | D8 | — | 2 | 1 | no | — |
| 278 | D8 | — | 13 | 4 | no | — |
| 279 | D8 | — | 26 | 6 | no | — |
| 280 | D8 | C23×D13 | 1 | 2 | yes | — |
| 281 | M16 | C2×C22⋊D13 | 1 | 12 | yes | — |
| 282 | C2×C8 | C2×C22⋊D13 | 1 | 3 | yes | — |
| 283 | C2×C8 | C2×C22⋊D13 | 1 | 6 | yes | — |
| 284 | C2×C8 | C23×D13 | 1 | 6 | yes | — |
| 285 | C2×C8 | C23×D13 | 1 | 1 | yes | characteristic |
| 286 | C16 | — | 13 | 8 | no | — |
| 287 | C13 | #62637 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 288 | C23 | — | 4 | 1 | no | — |
| 289 | C23 | — | 52 | 6 | no | — |
| 290 | C23 | C2×#6104 | 1 | 3 | yes | — |
| 291 | Q8 | — | 2 | 6 | no | — |
| 292 | Q8 | — | 4 | 12 | no | — |
| 293 | Q8 | — | 4 | 3 | no | — |
| 294 | Q8 | — | 26 | 4 | no | — |
| 295 | Q8 | — | 52 | 6 | no | — |
| 296 | Q8 | C2×#6104 | 1 | 4 | yes | — |
| 297 | D4 | — | 2 | 2 | no | — |
| 298 | D4 | — | 4 | 12 | no | — |
| 299 | D4 | — | 4 | 1 | no | — |
| 300 | D4 | — | 26 | 4 | no | — |
| 301 | D4 | — | 52 | 4 | no | — |
| 302 | D4 | — | 52 | 12 | no | — |
| 303 | D4 | — | 52 | 6 | no | — |
| 304 | D4 | C2×#6104 | 1 | 12 | yes | — |
| 305 | C2×C4 | — | 4 | 3 | no | — |
| 306 | C2×C4 | — | 4 | 12 | no | — |
| 307 | C2×C4 | — | 4 | 12 | no | — |
| 308 | C2×C4 | — | 52 | 6 | no | — |
| 309 | C2×C4 | — | 52 | 4 | no | — |
| 310 | C2×C4 | — | 52 | 12 | no | — |
| 311 | C2×C4 | C2×#6104 | 1 | 3 | yes | — |
| 312 | C2×C4 | C2×#6104 | 1 | 6 | yes | — |
| 313 | C2×C4 | C22×C22⋊D13 | 1 | 1 | yes | characteristic |
| 314 | C2×C4 | C22×C22⋊D13 | 1 | 6 | yes | — |
| 315 | C8 | C22×C22⋊D13 | 1 | 1 | yes | characteristic |
| 316 | C8 | C22×C22⋊D13 | 1 | 6 | yes | — |
| 317 | C8 | C24×D13 | 1 | 1 | yes | characteristic |
| 318 | C22 | — | 4 | 3 | no | — |
| 319 | C22 | — | 4 | 2 | no | — |
| 320 | C22 | — | 8 | 2 | no | — |
| 321 | C22 | — | 52 | 4 | no | — |
| 322 | C22 | — | 208 | 6 | no | — |
| 323 | C22 | C22×#6104 | 1 | 1 | yes | characteristic |
| 324 | C22 | C22×#6104 | 1 | 6 | yes | — |
| 325 | C4 | — | 4 | 6 | no | — |
| 326 | C4 | — | 52 | 4 | no | — |
| 327 | C4 | C22×#6104 | 1 | 6 | yes | — |
| 328 | C4 | C22×#6104 | 1 | 1 | yes | characteristic |
| 329 | C4 | C23×C22⋊D13 | 1 | 1 | yes | characteristic |
| 330 | C2 | — | 2 | 1 | no | — |
| 331 | C2 | — | 2 | 6 | no | — |
| 332 | C2 | — | 8 | 2 | no | — |
| 333 | C2 | — | 104 | 4 | no | — |
| 334 | C2 | C23×#6104 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta