Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
4390 conjugacy classes of subgroups; only the 416 normal subgroups are shown.
This group has 4390 conjugacy classes of subgroups — too many to draw. Only its 416 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 | C2×#16811565 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | #20289905 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #20289897 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | C22×#41×D53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C22×C4⋊#6157 | C22 | 1 | 4 | yes | — |
| 6 | C22×C22⋊#6157 | C22 | 1 | 12 | yes | — |
| 7 | #16811565 | — | 2 | 24 | no | — |
| 8 | C2×(C2×#6157)⋊C4 | C22 | 1 | 12 | yes | — |
| 9 | C22×#11719949 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | C22×#11719945 | C4 | 1 | 3 | yes | complemented |
| 11 | C22×#11719947 | C4 | 1 | 3 | yes | complemented |
| 12 | C2×#41×D53 | — | 2 | 24 | no | — |
| 13 | C22×Q8×D53 | C22 | 1 | 1 | yes | characteristic |
| 14 | C22×D4×D53 | C22 | 1 | 3 | yes | — |
| 15 | C23×C4×D53 | C22 | 1 | 3 | yes | — |
| 16 | C22×#41×C53 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | C23×#6157 | C23 | 1 | 8 | yes | — |
| 18 | C2×C4⋊#6157 | — | 2 | 24 | no | — |
| 19 | C2×C22⋊#6157 | — | 2 | 24 | no | — |
| 20 | C2×C22⋊#6157 | — | 4 | 24 | no | — |
| 21 | (C2×#6157)⋊C4 | — | 2 | 24 | no | — |
| 22 | C24×D53 | C23 | 1 | 3 | yes | — |
| 23 | C2×#11719949 | — | 2 | 12 | no | — |
| 24 | C2×#11719945 | — | 2 | 12 | no | — |
| 25 | C2×#11719945 | — | 2 | 24 | no | — |
| 26 | C2×#11719947 | — | 2 | 24 | no | — |
| 27 | C2×#11719947 | — | 2 | 12 | no | — |
| 28 | #41×D53 | — | 2 | 128 | no | — |
| 29 | C22×Dic106 | C2×C4 | 1 | 3 | yes | — |
| 30 | C2×Q8×D53 | — | 2 | 12 | no | — |
| 31 | C22×D212 | C2×C4 | 1 | 3 | yes | — |
| 32 | C22×C22⋊D53 | C2×C4 | 1 | 6 | yes | — |
| 33 | C2×D4×D53 | — | 2 | 24 | no | — |
| 34 | C2×D4×D53 | — | 2 | 12 | no | — |
| 35 | C23×Dic53 | C2×C4 | 1 | 3 | yes | — |
| 36 | C22×C4×D53 | — | 2 | 24 | no | — |
| 37 | C22×C4×D53 | C2×C4 | 1 | 3 | yes | — |
| 38 | C22×C4×D53 | C2×C4 | 1 | 3 | yes | — |
| 39 | C22×C4×D53 | D4 | 1 | 24 | yes | — |
| 40 | C22×C4×D53 | C23 | 1 | 3 | yes | — |
| 41 | C22×C4×D53 | C23 | 1 | 1 | yes | characteristic |
| 42 | C2×#41×C53 | — | 2 | 12 | no | — |
| 43 | C22×Q8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | C22×D4×C53 | C2×C4 | 1 | 3 | yes | — |
| 45 | C23×C4×C53 | C2×C4 | 1 | 3 | yes | — |
| 46 | C24×C53 | C22×C4 | 1 | 3 | yes | — |
| 47 | C23×D53 | — | 2 | 12 | no | — |
| 48 | C23×D53 | — | 2 | 12 | no | — |
| 49 | C23×D53 | — | 2 | 12 | no | — |
| 50 | C23×D53 | C22×C4 | 1 | 3 | yes | — |
| 51 | C23×D53 | C2×D4 | 1 | 12 | yes | — |
| 52 | C23×D53 | C24 | 1 | 1 | yes | characteristic |
| 53 | C22×#6157 | — | 2 | 24 | no | — |
| 54 | C22×#6157 | — | 2 | 24 | no | — |
| 55 | C22×#6157 | — | 2 | 8 | no | — |
| 56 | #41×C53 | — | 2 | 32 | no | — |
| 57 | C2×Q8×C53 | — | 2 | 6 | no | — |
| 58 | C2×D4×C53 | — | 2 | 12 | no | — |
| 59 | C2×D4×C53 | — | 2 | 6 | no | — |
| 60 | C22×C4×C53 | — | 2 | 12 | no | — |
| 61 | C22×C4×C53 | C22⋊C4 | 1 | 12 | yes | complemented |
| 62 | C22×C4×C53 | C22×C4 | 1 | 3 | yes | — |
| 63 | C22×C4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 64 | C2×C22⋊D53 | — | 2 | 12 | no | — |
| 65 | C2×C22⋊D53 | — | 2 | 12 | no | — |
| 66 | C2×C22⋊D53 | — | 2 | 12 | no | — |
| 67 | C22×Dic53 | — | 2 | 12 | no | — |
| 68 | C22×Dic53 | C22⋊C4 | 1 | 12 | yes | complemented |
| 69 | C22×Dic53 | C22×C4 | 1 | 3 | yes | — |
| 70 | C22×Dic53 | C22×C4 | 1 | 1 | yes | characteristic |
| 71 | #11719949 | — | 2 | 32 | no | — |
| 72 | Q8×D53 | — | 2 | 32 | no | — |
| 73 | #11719947 | — | 2 | 96 | no | — |
| 74 | D4×D53 | — | 2 | 96 | no | — |
| 75 | #11719945 | — | 2 | 96 | no | — |
| 76 | C2×D212 | — | 2 | 12 | no | — |
| 77 | C2×D212 | — | 2 | 6 | no | — |
| 78 | C2×C4×D53 | — | 2 | 12 | no | — |
| 79 | C2×C4×D53 | — | 2 | 12 | no | — |
| 80 | C2×C4×D53 | — | 2 | 12 | no | — |
| 81 | C2×C4×D53 | — | 2 | 96 | no | — |
| 82 | C2×C4×D53 | C22⋊C4 | 1 | 12 | yes | complemented |
| 83 | C2×C4×D53 | C22⋊C4 | 1 | 12 | yes | complemented |
| 84 | C2×C4×D53 | C2×D4 | 1 | 12 | yes | — |
| 85 | C2×C4×D53 | C2×D4 | 1 | 12 | yes | — |
| 86 | C2×Dic106 | — | 2 | 6 | no | — |
| 87 | C2×Dic106 | — | 2 | 12 | no | — |
| 88 | C22⋊#6157 | — | 4 | 48 | no | — |
| 89 | C4⋊#6157 | — | 4 | 16 | no | — |
| 90 | C23×C53 | — | 2 | 6 | no | — |
| 91 | C23×C53 | — | 2 | 12 | no | — |
| 92 | C23×C53 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 93 | C23×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 94 | C22×D53 | — | 2 | 6 | no | — |
| 95 | C22×D53 | — | 2 | 12 | no | — |
| 96 | C22×D53 | — | 2 | 24 | no | — |
| 97 | C22×D53 | — | 2 | 12 | no | — |
| 98 | C22×D53 | — | 2 | 12 | no | — |
| 99 | C22×D53 | — | 2 | 24 | no | — |
| 100 | C22×D53 | — | 4 | 12 | no | — |
| 101 | C22×D53 | — | 4 | 12 | no | — |
| 102 | C22×D53 | — | 4 | 24 | no | — |
| 103 | C22×D53 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 104 | C22×D53 | C22×D4 | 1 | 6 | yes | — |
| 105 | C22×D53 | H(5,𝔽2) | 1 | 4 | yes | — |
| 106 | C22×D53 | 2−1+4 | 1 | 4 | yes | — |
| 107 | C2×#6157 | — | 4 | 8 | no | — |
| 108 | C2×#6157 | — | 4 | 24 | no | — |
| 109 | C2×#6157 | — | 4 | 24 | no | — |
| 110 | Q8×C53 | — | 2 | 8 | no | — |
| 111 | D4×C53 | — | 2 | 24 | no | — |
| 112 | C2×C4×C53 | — | 2 | 6 | no | — |
| 113 | C2×C4×C53 | — | 2 | 24 | no | — |
| 114 | C2×C4×C53 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 115 | C2×C4×C53 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 116 | C22⋊D53 | — | 2 | 48 | no | — |
| 117 | C2×Dic53 | — | 2 | 6 | no | — |
| 118 | C2×Dic53 | — | 2 | 24 | no | — |
| 119 | C2×Dic53 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 120 | C2×Dic53 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 121 | D212 | — | 2 | 24 | no | — |
| 122 | C4×D53 | — | 2 | 24 | no | — |
| 123 | C4×D53 | — | 2 | 24 | no | — |
| 124 | C4×D53 | — | 2 | 24 | no | — |
| 125 | C4×D53 | #120 | 1 | 16 | yes | — |
| 126 | Dic106 | — | 2 | 24 | no | — |
| 127 | #14337 | — | 53 | 1 | no | maximal, Sylow |
| 128 | C22×C53 | — | 2 | 12 | no | — |
| 129 | C22×C53 | — | 2 | 6 | no | — |
| 130 | C22×C53 | — | 4 | 12 | no | — |
| 131 | C22×C53 | C22×C22⋊C4 | 1 | 3 | yes | — |
| 132 | C22×C53 | #520 | 1 | 4 | yes | — |
| 133 | C2×D53 | — | 2 | 12 | no | — |
| 134 | C2×D53 | — | 2 | 6 | no | — |
| 135 | C2×D53 | — | 4 | 12 | no | — |
| 136 | C2×D53 | — | 4 | 24 | no | — |
| 137 | C2×D53 | — | 4 | 24 | no | — |
| 138 | C2×D53 | C2×#120 | 1 | 4 | yes | — |
| 139 | C2×D53 | #535 | 1 | 6 | yes | — |
| 140 | C2×D53 | #536 | 1 | 6 | yes | — |
| 141 | C2×D53 | #537 | 1 | 12 | yes | — |
| 142 | #6157 | — | 8 | 8 | no | — |
| 143 | C4×C53 | — | 2 | 6 | no | — |
| 144 | C4×C53 | #379 | 1 | 4 | yes | — |
| 145 | Dic53 | — | 2 | 6 | no | — |
| 146 | Dic53 | #379 | 1 | 4 | yes | — |
| 147 | C23×#41 | — | 53 | 1 | no | — |
| 148 | C2×#380 | — | 53 | 6 | no | — |
| 149 | #1446 | — | 53 | 2 | no | — |
| 150 | #1438 | — | 53 | 6 | no | — |
| 151 | C2×C53 | — | 4 | 6 | no | — |
| 152 | C2×C53 | C2×#379 | 1 | 1 | yes | characteristic |
| 153 | C2×C53 | #2311 | 1 | 6 | yes | — |
| 154 | D53 | — | 4 | 6 | no | — |
| 155 | D53 | #2615 | 1 | 4 | yes | — |
| 156 | D53 | #2616 | 1 | 4 | yes | — |
| 157 | C22×#41 | — | 53 | 1 | no | — |
| 158 | C22×#41 | — | 53 | 3 | no | — |
| 159 | C22×#41 | — | 53 | 3 | no | — |
| 160 | C22×#41 | — | 106 | 24 | no | — |
| 161 | C22×#41 | #6157 | 1 | 1 | yes | characteristic, complemented |
| 162 | C23×Q8 | — | 53 | 1 | no | — |
| 163 | C23×D4 | — | 53 | 3 | no | — |
| 164 | C24×C4 | — | 53 | 3 | no | — |
| 165 | C22×C4⋊C4 | — | 53 | 4 | no | — |
| 166 | C22×C22⋊C4 | — | 53 | 12 | no | — |
| 167 | #380 | — | 106 | 24 | no | — |
| 168 | C2×#95 | — | 53 | 12 | no | — |
| 169 | C53 | #14337 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 170 | C25 | — | 53 | 3 | no | — |
| 171 | C2×#41 | — | 2 | 12 | no | — |
| 172 | C2×#41 | — | 106 | 24 | no | — |
| 173 | C2×#41 | — | 106 | 24 | no | — |
| 174 | C2×#41 | — | 106 | 12 | no | — |
| 175 | C2×#41 | — | 106 | 12 | no | — |
| 176 | C2×#41 | — | 106 | 128 | no | — |
| 177 | C2×#41 | — | 106 | 12 | no | — |
| 178 | C22×Q8 | — | 53 | 3 | no | — |
| 179 | C22×Q8 | — | 106 | 12 | no | — |
| 180 | C22×Q8 | C2×#6157 | 1 | 1 | yes | characteristic |
| 181 | C22×D4 | — | 53 | 6 | no | — |
| 182 | C22×D4 | — | 53 | 3 | no | — |
| 183 | C22×D4 | — | 106 | 12 | no | — |
| 184 | C22×D4 | — | 106 | 24 | no | — |
| 185 | C22×D4 | C2×#6157 | 1 | 3 | yes | — |
| 186 | C23×C4 | — | 53 | 1 | no | — |
| 187 | C23×C4 | — | 53 | 3 | no | — |
| 188 | C23×C4 | — | 53 | 24 | no | — |
| 189 | C23×C4 | — | 53 | 3 | no | — |
| 190 | C23×C4 | — | 53 | 3 | no | — |
| 191 | C23×C4 | — | 53 | 3 | no | — |
| 192 | C23×C4 | — | 53 | 8 | no | — |
| 193 | C23×C4 | — | 106 | 24 | no | — |
| 194 | C23×C4 | C2×#6157 | 1 | 3 | yes | — |
| 195 | C2×C4⋊C4 | — | 106 | 24 | no | — |
| 196 | C2×C22⋊C4 | — | 106 | 24 | no | — |
| 197 | C2×C22⋊C4 | — | 212 | 24 | no | — |
| 198 | #95 | — | 106 | 24 | no | — |
| 199 | C24 | — | 53 | 1 | no | — |
| 200 | C24 | — | 53 | 12 | no | — |
| 201 | C24 | — | 53 | 3 | no | — |
| 202 | C24 | — | 106 | 12 | no | — |
| 203 | C24 | — | 106 | 12 | no | — |
| 204 | C24 | — | 106 | 12 | no | — |
| 205 | C24 | C22×#6157 | 1 | 3 | yes | — |
| 206 | #41 | — | 2 | 32 | no | — |
| 207 | #41 | — | 106 | 96 | no | — |
| 208 | #41 | — | 106 | 96 | no | — |
| 209 | #41 | — | 106 | 32 | no | — |
| 210 | C2×Q8 | — | 2 | 6 | no | — |
| 211 | C2×Q8 | — | 106 | 12 | no | — |
| 212 | C2×Q8 | — | 106 | 32 | no | — |
| 213 | C2×Q8 | — | 106 | 6 | no | — |
| 214 | C2×D4 | — | 2 | 12 | no | — |
| 215 | C2×D4 | — | 2 | 6 | no | — |
| 216 | C2×D4 | — | 106 | 12 | no | — |
| 217 | C2×D4 | — | 106 | 12 | no | — |
| 218 | C2×D4 | — | 106 | 96 | no | — |
| 219 | C2×D4 | — | 106 | 12 | no | — |
| 220 | C2×D4 | — | 106 | 12 | no | — |
| 221 | C2×D4 | — | 106 | 6 | no | — |
| 222 | C22×C4 | — | 2 | 12 | no | — |
| 223 | C22×C4 | — | 53 | 1 | no | — |
| 224 | C22×C4 | — | 53 | 12 | no | — |
| 225 | C22×C4 | — | 53 | 12 | no | — |
| 226 | C22×C4 | — | 53 | 3 | no | — |
| 227 | C22×C4 | — | 53 | 12 | no | — |
| 228 | C22×C4 | — | 53 | 12 | no | — |
| 229 | C22×C4 | — | 53 | 12 | no | — |
| 230 | C22×C4 | — | 106 | 12 | no | — |
| 231 | C22×C4 | — | 106 | 96 | no | — |
| 232 | C22×C4 | — | 106 | 12 | no | — |
| 233 | C22×C4 | — | 106 | 12 | no | — |
| 234 | C22×C4 | — | 106 | 12 | no | — |
| 235 | C22×C4 | — | 106 | 24 | no | — |
| 236 | C22×C4 | — | 106 | 24 | no | — |
| 237 | C22×C4 | — | 106 | 8 | no | — |
| 238 | C22×C4 | C22⋊#6157 | 1 | 12 | yes | complemented |
| 239 | C22×C4 | C22×#6157 | 1 | 3 | yes | — |
| 240 | C22×C4 | C22×#6157 | 1 | 1 | yes | characteristic |
| 241 | C4⋊C4 | — | 212 | 16 | no | — |
| 242 | C22⋊C4 | — | 212 | 48 | no | — |
| 243 | C23 | — | 2 | 6 | no | — |
| 244 | C23 | — | 2 | 12 | no | — |
| 245 | C23 | — | 53 | 6 | no | — |
| 246 | C23 | — | 53 | 4 | no | — |
| 247 | C23 | — | 53 | 4 | no | — |
| 248 | C23 | — | 53 | 6 | no | — |
| 249 | C23 | — | 106 | 12 | no | — |
| 250 | C23 | — | 106 | 24 | no | — |
| 251 | C23 | — | 106 | 24 | no | — |
| 252 | C23 | — | 106 | 12 | no | — |
| 253 | C23 | — | 106 | 6 | no | — |
| 254 | C23 | — | 106 | 12 | no | — |
| 255 | C23 | — | 212 | 12 | no | — |
| 256 | C23 | — | 212 | 24 | no | — |
| 257 | C23 | — | 212 | 12 | no | — |
| 258 | C23 | C2×C22⋊#6157 | 1 | 6 | yes | — |
| 259 | C23 | C23×#6157 | 1 | 1 | yes | characteristic |
| 260 | Q8 | — | 2 | 8 | no | — |
| 261 | Q8 | — | 106 | 24 | no | — |
| 262 | D4 | — | 2 | 24 | no | — |
| 263 | D4 | — | 106 | 24 | no | — |
| 264 | D4 | — | 106 | 48 | no | — |
| 265 | C2×C4 | — | 2 | 6 | no | — |
| 266 | C2×C4 | — | 2 | 24 | no | — |
| 267 | C2×C4 | — | 53 | 6 | no | — |
| 268 | C2×C4 | — | 53 | 16 | no | — |
| 269 | C2×C4 | — | 53 | 6 | no | — |
| 270 | C2×C4 | — | 106 | 24 | no | — |
| 271 | C2×C4 | — | 106 | 24 | no | — |
| 272 | C2×C4 | — | 106 | 24 | no | — |
| 273 | C2×C4 | — | 106 | 6 | no | — |
| 274 | C2×C4 | — | 106 | 24 | no | — |
| 275 | C2×C4 | — | 212 | 24 | no | — |
| 276 | C2×C4 | — | 212 | 8 | no | — |
| 277 | C2×C4 | — | 212 | 24 | no | — |
| 278 | C2×C4 | C2×C22⋊#6157 | 1 | 6 | yes | — |
| 279 | C2×C4 | C2×C22⋊#6157 | 1 | 6 | yes | — |
| 280 | C22 | — | 2 | 6 | no | — |
| 281 | C22 | — | 2 | 12 | no | — |
| 282 | C22 | — | 4 | 12 | no | — |
| 283 | C22 | — | 53 | 4 | no | — |
| 284 | C22 | — | 53 | 6 | no | — |
| 285 | C22 | — | 53 | 12 | no | — |
| 286 | C22 | — | 53 | 6 | no | — |
| 287 | C22 | — | 106 | 6 | no | — |
| 288 | C22 | — | 106 | 12 | no | — |
| 289 | C22 | — | 212 | 24 | no | — |
| 290 | C22 | — | 212 | 24 | no | — |
| 291 | C22 | — | 212 | 12 | no | — |
| 292 | C22 | C22×C22⋊#6157 | 1 | 3 | yes | — |
| 293 | C22 | #16811692 | 1 | 4 | yes | — |
| 294 | C4 | — | 2 | 6 | no | — |
| 295 | C4 | — | 53 | 4 | no | — |
| 296 | C4 | — | 106 | 6 | no | — |
| 297 | C4 | — | 424 | 8 | no | — |
| 298 | C4 | #16811564 | 1 | 4 | yes | — |
| 299 | C2 | — | 4 | 6 | no | — |
| 300 | C2 | — | 53 | 4 | no | — |
| 301 | C2 | — | 53 | 4 | no | — |
| 302 | C2 | — | 212 | 6 | no | — |
| 303 | C2 | C2×#16811564 | 1 | 1 | yes | characteristic |
| 304 | C2 | #20291190 | 1 | 6 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16811565 | C2 | semidirect |
| 2. #20289905 | C2 | semidirect |
| 3. #20289897 | C2 | semidirect |
| 9. C22×#11719949 | C4 | semidirect |
| 10. C22×#11719945 | C4 | semidirect |
| 11. C22×#11719947 | C4 | semidirect |
| 16. C22×#41×C53 | C4 | semidirect |
| 61. C22×C4×C53 | C22⋊C4 | semidirect |
| 68. C22×Dic53 | C22⋊C4 | semidirect |
| 82. C2×C4×D53 | C22⋊C4 | semidirect |
| 83. C2×C4×D53 | C22⋊C4 | semidirect |
| 161. C22×#41 | #6157 | semidirect |
| 169. C53 | #14337 | semidirect |
| 238. C22×C4 | C22⋊#6157 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16811565 | 2 |
| #20289905 | 2 |
| #20289897 | 2 |
| C22×#41×D53 | 2 |
| #14337 | 53 |