Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
1018 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 135 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 | H(5,𝔽2)×D131 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #120×D131 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #22553500 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #22553499 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #22553498 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #99×D131 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | B(4,𝔽2)×C131 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C24×D131 | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | #16862292 | — | 2 | 3 | no | — |
| 10 | #16862291 | — | 2 | 3 | no | — |
| 11 | #16862291 | C22 | 1 | 3 | yes | complemented |
| 12 | #41×D131 | — | 2 | 3 | no | — |
| 13 | C2×D4×D131 | — | 2 | 3 | no | — |
| 14 | C2×D4×D131 | — | 2 | 3 | no | — |
| 15 | C2×D4×D131 | C22 | 1 | 3 | yes | — |
| 16 | #16862216 | — | 2 | 3 | no | — |
| 17 | #16862216 | C22 | 1 | 3 | yes | complemented |
| 18 | #16862215 | C22 | 1 | 3 | yes | complemented |
| 19 | #16862214 | — | 2 | 3 | no | — |
| 20 | #16862214 | C22 | 1 | 3 | yes | complemented |
| 21 | C22⋊C4×D131 | — | 2 | 3 | no | — |
| 22 | C22⋊C4×D131 | C22 | 1 | 3 | yes | complemented |
| 23 | #16862161 | C22 | 1 | 3 | yes | complemented |
| 24 | #16862160 | C22 | 1 | 6 | yes | complemented |
| 25 | H(5,𝔽2)×C131 | C22 | 1 | 1 | yes | characteristic, complemented |
| 26 | #120×C131 | C22 | 1 | 3 | yes | complemented |
| 27 | #99×C131 | C22 | 1 | 3 | yes | complemented |
| 28 | C23×D131 | — | 2 | 1 | no | — |
| 29 | C23×D131 | — | 2 | 3 | no | — |
| 30 | C23×D131 | — | 2 | 3 | no | — |
| 31 | C23×D131 | — | 4 | 4 | no | — |
| 32 | C23×D131 | D4 | 1 | 3 | yes | complemented |
| 33 | C23×D131 | C23 | 1 | 1 | yes | characteristic |
| 34 | C23×D131 | C23 | 1 | 1 | yes | characteristic, complemented |
| 35 | #14523514 | — | 2 | 3 | no | — |
| 36 | #14523513 | — | 2 | 6 | no | — |
| 37 | #14523513 | — | 2 | 3 | no | — |
| 38 | #14523512 | — | 2 | 3 | no | — |
| 39 | #14523512 | — | 2 | 6 | no | — |
| 40 | Q8×D131 | — | 2 | 1 | no | — |
| 41 | C2×D524 | — | 2 | 3 | no | — |
| 42 | C2×D524 | — | 2 | 3 | no | — |
| 43 | C2×D524 | D4 | 1 | 3 | yes | complemented |
| 44 | C2×C22⋊D131 | — | 2 | 3 | no | — |
| 45 | C2×C22⋊D131 | — | 2 | 3 | no | — |
| 46 | C2×C22⋊D131 | — | 2 | 3 | no | — |
| 47 | C2×C22⋊D131 | — | 2 | 3 | no | — |
| 48 | C2×C22⋊D131 | D4 | 1 | 3 | yes | complemented |
| 49 | C2×C22⋊D131 | C23 | 1 | 3 | yes | — |
| 50 | D4×D131 | — | 2 | 3 | no | — |
| 51 | D4×D131 | — | 2 | 6 | no | — |
| 52 | D4×D131 | — | 2 | 6 | no | — |
| 53 | D4×D131 | — | 2 | 6 | no | — |
| 54 | D4×D131 | — | 2 | 6 | no | — |
| 55 | D4×D131 | — | 4 | 12 | no | — |
| 56 | C2×C4×D131 | — | 2 | 3 | no | — |
| 57 | C2×C4×D131 | — | 2 | 3 | no | — |
| 58 | C2×C4×D131 | D4 | 1 | 3 | yes | complemented |
| 59 | C22⋊Dic131 | — | 2 | 3 | no | — |
| 60 | C22⋊Dic131 | C23 | 1 | 3 | yes | complemented |
| 61 | (C2×D131)⋊C4 | — | 2 | 3 | no | — |
| 62 | (C2×D131)⋊C4 | — | 2 | 3 | no | — |
| 63 | (C2×D131)⋊C4 | — | 2 | 3 | no | — |
| 64 | C24×C131 | C23 | 1 | 1 | yes | characteristic, complemented |
| 65 | #41×C131 | — | 2 | 3 | no | — |
| 66 | C2×D4×C131 | — | 2 | 3 | no | — |
| 67 | C2×D4×C131 | — | 2 | 3 | no | — |
| 68 | C2×D4×C131 | C23 | 1 | 3 | yes | — |
| 69 | C22⋊C4×C131 | — | 2 | 3 | no | — |
| 70 | C22⋊C4×C131 | C23 | 1 | 3 | yes | complemented |
| 71 | C22×D131 | — | 2 | 12 | no | — |
| 72 | C22×D131 | — | 2 | 3 | no | — |
| 73 | C22×D131 | — | 2 | 3 | no | — |
| 74 | C22×D131 | — | 2 | 3 | no | — |
| 75 | C22×D131 | — | 2 | 3 | no | — |
| 76 | C22×D131 | — | 4 | 2 | no | — |
| 77 | C22×D131 | — | 4 | 2 | no | — |
| 78 | C22×D131 | — | 4 | 6 | no | — |
| 79 | C22×D131 | — | 4 | 6 | no | — |
| 80 | C22×D131 | — | 4 | 6 | no | — |
| 81 | C22×D131 | — | 4 | 6 | no | — |
| 82 | C22×D131 | — | 4 | 3 | no | — |
| 83 | C22×D131 | — | 4 | 1 | no | — |
| 84 | C22×D131 | — | 8 | 2 | no | — |
| 85 | C22×D131 | C2×D4 | 1 | 3 | yes | — |
| 86 | C22×D131 | C2×D4 | 1 | 3 | yes | — |
| 87 | C22×D131 | C2×D4 | 1 | 3 | yes | complemented |
| 88 | Dic262 | — | 2 | 3 | no | — |
| 89 | D524 | — | 2 | 3 | no | — |
| 90 | D524 | — | 2 | 6 | no | — |
| 91 | D524 | — | 4 | 6 | no | — |
| 92 | C22⋊D131 | — | 2 | 6 | no | — |
| 93 | C22⋊D131 | — | 2 | 6 | no | — |
| 94 | C22⋊D131 | — | 2 | 6 | no | — |
| 95 | C22⋊D131 | — | 4 | 6 | no | — |
| 96 | C22⋊D131 | — | 4 | 6 | no | — |
| 97 | C2×Dic131 | — | 2 | 3 | no | — |
| 98 | C2×Dic131 | — | 2 | 3 | no | — |
| 99 | C2×Dic131 | C2×D4 | 1 | 3 | yes | complemented |
| 100 | C4×D131 | — | 2 | 6 | no | — |
| 101 | C4×D131 | — | 2 | 3 | no | — |
| 102 | C4×D131 | — | 2 | 3 | no | — |
| 103 | C4×D131 | — | 4 | 6 | no | — |
| 104 | C23×C131 | — | 2 | 1 | no | — |
| 105 | C23×C131 | — | 2 | 3 | no | — |
| 106 | C23×C131 | — | 4 | 2 | no | — |
| 107 | C23×C131 | C2×D4 | 1 | 3 | yes | complemented |
| 108 | C23×C131 | C24 | 1 | 1 | yes | characteristic |
| 109 | Q8×C131 | — | 2 | 1 | no | — |
| 110 | D4×C131 | — | 2 | 3 | no | — |
| 111 | D4×C131 | — | 2 | 6 | no | — |
| 112 | D4×C131 | — | 4 | 6 | no | — |
| 113 | C2×C4×C131 | — | 2 | 3 | no | — |
| 114 | C2×C4×C131 | — | 2 | 3 | no | — |
| 115 | C2×C4×C131 | C2×D4 | 1 | 3 | yes | complemented |
| 116 | C22×C131 | — | 2 | 6 | no | — |
| 117 | C22×C131 | — | 2 | 3 | no | — |
| 118 | C22×C131 | — | 4 | 1 | no | — |
| 119 | C22×C131 | — | 4 | 3 | no | — |
| 120 | C22×C131 | — | 4 | 3 | no | — |
| 121 | C22×C131 | — | 4 | 1 | no | — |
| 122 | C22×C131 | — | 8 | 1 | no | — |
| 123 | C22×C131 | C22×D4 | 1 | 3 | yes | — |
| 124 | C2×D131 | — | 2 | 6 | no | — |
| 125 | C2×D131 | — | 2 | 6 | no | — |
| 126 | C2×D131 | — | 2 | 3 | no | — |
| 127 | C2×D131 | — | 4 | 3 | no | — |
| 128 | C2×D131 | — | 4 | 12 | no | — |
| 129 | C2×D131 | — | 4 | 3 | no | — |
| 130 | C2×D131 | — | 4 | 1 | no | — |
| 131 | C2×D131 | — | 4 | 4 | no | — |
| 132 | C2×D131 | — | 4 | 6 | no | — |
| 133 | C2×D131 | — | 4 | 3 | no | — |
| 134 | C2×D131 | — | 8 | 3 | no | — |
| 135 | C2×D131 | — | 8 | 3 | no | — |
| 136 | C2×D131 | #120 | 1 | 3 | yes | — |
| 137 | C2×D131 | #120 | 1 | 1 | yes | characteristic |
| 138 | C4×C131 | — | 2 | 3 | no | — |
| 139 | C4×C131 | — | 4 | 3 | no | — |
| 140 | Dic131 | — | 2 | 3 | no | — |
| 141 | Dic131 | — | 4 | 3 | no | — |
| 142 | C2×C131 | — | 2 | 3 | no | — |
| 143 | C2×C131 | — | 4 | 2 | no | — |
| 144 | C2×C131 | — | 4 | 3 | no | — |
| 145 | C2×C131 | C2×#120 | 1 | 1 | yes | characteristic |
| 146 | D131 | — | 2 | 3 | no | — |
| 147 | D131 | — | 4 | 2 | no | — |
| 148 | D131 | — | 4 | 3 | no | — |
| 149 | D131 | B(4,𝔽2) | 1 | 2 | yes | direct product factor |
| 150 | C131 | C2×B(4,𝔽2) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 151 | C2×B(4,𝔽2) | — | 131 | 1 | no | maximal, Sylow |
| 152 | C2×H(5,𝔽2) | — | 131 | 1 | no | — |
| 153 | C2×#120 | — | 131 | 3 | no | — |
| 154 | B(4,𝔽2) | — | 131 | 1 | no | — |
| 155 | B(4,𝔽2) | — | 131 | 3 | no | — |
| 156 | B(4,𝔽2) | — | 131 | 3 | no | — |
| 157 | B(4,𝔽2) | D131 | 1 | 1 | yes | characteristic, direct product factor |
| 158 | C2×#99 | — | 131 | 3 | no | — |
| 159 | C25 | — | 131 | 1 | no | — |
| 160 | H(5,𝔽2) | — | 262 | 3 | no | — |
| 161 | H(5,𝔽2) | — | 262 | 3 | no | — |
| 162 | H(5,𝔽2) | — | 131 | 3 | no | — |
| 163 | H(5,𝔽2) | C2×D131 | 1 | 1 | yes | characteristic, complemented |
| 164 | C2×#41 | — | 262 | 3 | no | — |
| 165 | C22×D4 | — | 262 | 3 | no | — |
| 166 | C22×D4 | — | 262 | 3 | no | — |
| 167 | C22×D4 | — | 131 | 3 | no | — |
| 168 | #120 | — | 262 | 3 | no | — |
| 169 | #120 | — | 262 | 3 | no | — |
| 170 | #120 | — | 131 | 3 | no | — |
| 171 | #120 | — | 131 | 3 | no | — |
| 172 | #120 | — | 131 | 3 | no | — |
| 173 | #120 | C2×D131 | 1 | 3 | yes | complemented |
| 174 | C2×C22⋊C4 | — | 262 | 3 | no | — |
| 175 | C2×C22⋊C4 | — | 131 | 3 | no | — |
| 176 | #99 | — | 131 | 3 | no | — |
| 177 | #99 | — | 131 | 6 | no | — |
| 178 | #99 | C2×D131 | 1 | 3 | yes | complemented |
| 179 | C24 | — | 262 | 1 | no | — |
| 180 | C24 | — | 262 | 3 | no | — |
| 181 | C24 | — | 262 | 3 | no | — |
| 182 | C24 | — | 524 | 4 | no | — |
| 183 | C24 | — | 131 | 1 | no | — |
| 184 | C24 | — | 131 | 3 | no | — |
| 185 | C24 | — | 131 | 1 | no | — |
| 186 | C24 | C22×D131 | 1 | 1 | yes | characteristic, complemented |
| 187 | #41 | — | 2 | 3 | no | — |
| 188 | #41 | — | 262 | 6 | no | — |
| 189 | #41 | — | 262 | 3 | no | — |
| 190 | #41 | — | 262 | 3 | no | — |
| 191 | #41 | — | 262 | 6 | no | — |
| 192 | #41 | — | 262 | 3 | no | — |
| 193 | C2×Q8 | — | 262 | 1 | no | — |
| 194 | C2×D4 | — | 2 | 3 | no | — |
| 195 | C2×D4 | — | 2 | 3 | no | — |
| 196 | C2×D4 | — | 262 | 3 | no | — |
| 197 | C2×D4 | — | 262 | 3 | no | — |
| 198 | C2×D4 | — | 262 | 3 | no | — |
| 199 | C2×D4 | — | 262 | 6 | no | — |
| 200 | C2×D4 | — | 262 | 6 | no | — |
| 201 | C2×D4 | — | 262 | 6 | no | — |
| 202 | C2×D4 | — | 262 | 6 | no | — |
| 203 | C2×D4 | — | 262 | 3 | no | — |
| 204 | C2×D4 | — | 262 | 3 | no | — |
| 205 | C2×D4 | — | 262 | 3 | no | — |
| 206 | C2×D4 | — | 262 | 3 | no | — |
| 207 | C2×D4 | — | 524 | 12 | no | — |
| 208 | C2×D4 | — | 131 | 3 | no | — |
| 209 | C2×D4 | — | 131 | 3 | no | — |
| 210 | C2×D4 | — | 131 | 3 | no | — |
| 211 | C2×D4 | C22×D131 | 1 | 3 | yes | — |
| 212 | C22×C4 | — | 262 | 3 | no | — |
| 213 | C22×C4 | — | 262 | 3 | no | — |
| 214 | C22×C4 | — | 131 | 3 | no | — |
| 215 | C22⋊C4 | — | 2 | 3 | no | — |
| 216 | C22⋊C4 | — | 262 | 3 | no | — |
| 217 | C22⋊C4 | — | 262 | 3 | no | — |
| 218 | C22⋊C4 | — | 262 | 3 | no | — |
| 219 | C22⋊C4 | — | 262 | 3 | no | — |
| 220 | C22⋊C4 | — | 131 | 3 | no | — |
| 221 | C22⋊C4 | C22×D131 | 1 | 3 | yes | complemented |
| 222 | C23 | — | 2 | 1 | no | — |
| 223 | C23 | — | 2 | 3 | no | — |
| 224 | C23 | — | 4 | 2 | no | — |
| 225 | C23 | — | 262 | 12 | no | — |
| 226 | C23 | — | 262 | 3 | no | — |
| 227 | C23 | — | 262 | 3 | no | — |
| 228 | C23 | — | 262 | 3 | no | — |
| 229 | C23 | — | 262 | 3 | no | — |
| 230 | C23 | — | 524 | 6 | no | — |
| 231 | C23 | — | 524 | 2 | no | — |
| 232 | C23 | — | 524 | 2 | no | — |
| 233 | C23 | — | 524 | 6 | no | — |
| 234 | C23 | — | 524 | 6 | no | — |
| 235 | C23 | — | 524 | 6 | no | — |
| 236 | C23 | — | 524 | 3 | no | — |
| 237 | C23 | — | 524 | 1 | no | — |
| 238 | C23 | — | 1048 | 2 | no | — |
| 239 | C23 | — | 131 | 3 | no | — |
| 240 | C23 | — | 131 | 3 | no | — |
| 241 | C23 | — | 131 | 3 | no | — |
| 242 | C23 | D4×D131 | 1 | 3 | yes | complemented |
| 243 | C23 | C23×D131 | 1 | 1 | yes | characteristic |
| 244 | Q8 | — | 2 | 1 | no | — |
| 245 | Q8 | — | 262 | 3 | no | — |
| 246 | D4 | — | 2 | 3 | no | — |
| 247 | D4 | — | 2 | 6 | no | — |
| 248 | D4 | — | 4 | 6 | no | — |
| 249 | D4 | — | 262 | 3 | no | — |
| 250 | D4 | — | 262 | 6 | no | — |
| 251 | D4 | — | 262 | 6 | no | — |
| 252 | D4 | — | 262 | 6 | no | — |
| 253 | D4 | — | 262 | 6 | no | — |
| 254 | D4 | — | 524 | 6 | no | — |
| 255 | D4 | — | 524 | 6 | no | — |
| 256 | D4 | — | 524 | 6 | no | — |
| 257 | C2×C4 | — | 2 | 3 | no | — |
| 258 | C2×C4 | — | 2 | 3 | no | — |
| 259 | C2×C4 | — | 262 | 6 | no | — |
| 260 | C2×C4 | — | 262 | 3 | no | — |
| 261 | C2×C4 | — | 262 | 3 | no | — |
| 262 | C2×C4 | — | 262 | 3 | no | — |
| 263 | C2×C4 | — | 262 | 3 | no | — |
| 264 | C2×C4 | — | 524 | 6 | no | — |
| 265 | C2×C4 | — | 131 | 3 | no | — |
| 266 | C2×C4 | D4×D131 | 1 | 3 | yes | complemented |
| 267 | C22 | — | 2 | 6 | no | — |
| 268 | C22 | — | 2 | 3 | no | — |
| 269 | C22 | — | 4 | 1 | no | — |
| 270 | C22 | — | 4 | 3 | no | — |
| 271 | C22 | — | 4 | 3 | no | — |
| 272 | C22 | — | 4 | 1 | no | — |
| 273 | C22 | — | 262 | 6 | no | — |
| 274 | C22 | — | 262 | 6 | no | — |
| 275 | C22 | — | 262 | 3 | no | — |
| 276 | C22 | — | 8 | 1 | no | — |
| 277 | C22 | — | 524 | 3 | no | — |
| 278 | C22 | — | 524 | 3 | no | — |
| 279 | C22 | — | 524 | 3 | no | — |
| 280 | C22 | — | 524 | 12 | no | — |
| 281 | C22 | — | 524 | 4 | no | — |
| 282 | C22 | — | 524 | 6 | no | — |
| 283 | C22 | — | 524 | 1 | no | — |
| 284 | C22 | — | 1048 | 3 | no | — |
| 285 | C22 | — | 1048 | 3 | no | — |
| 286 | C22 | — | 131 | 3 | no | — |
| 287 | C22 | — | 131 | 1 | no | — |
| 288 | C22 | C2×D4×D131 | 1 | 3 | yes | — |
| 289 | C4 | — | 2 | 3 | no | — |
| 290 | C4 | — | 4 | 3 | no | — |
| 291 | C4 | — | 262 | 3 | no | — |
| 292 | C4 | — | 524 | 3 | no | — |
| 293 | C2 | — | 2 | 3 | no | — |
| 294 | C2 | — | 4 | 2 | no | — |
| 295 | C2 | — | 4 | 3 | no | — |
| 296 | C2 | — | 262 | 3 | no | — |
| 297 | C2 | — | 524 | 2 | no | — |
| 298 | C2 | — | 524 | 3 | no | — |
| 299 | C2 | — | 131 | 2 | no | — |
| 300 | C2 | #120×D131 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(5,𝔽2)×D131 | C2 | semidirect |
| 2. #120×D131 | C2 | semidirect |
| 3. #22553500 | C2 | semidirect |
| 4. #22553499 | C2 | semidirect |
| 5. #22553498 | C2 | semidirect |
| 6. #99×D131 | C2 | semidirect |
| 7. B(4,𝔽2)×C131 | C2 | semidirect |
| 8. C24×D131 | C22 | semidirect |
| 11. #16862291 | C22 | semidirect |
| 17. #16862216 | C22 | semidirect |
| 18. #16862215 | C22 | semidirect |
| 20. #16862214 | C22 | semidirect |
| 22. C22⋊C4×D131 | C22 | semidirect |
| 23. #16862161 | C22 | semidirect |
| 24. #16862160 | C22 | semidirect |
| 25. H(5,𝔽2)×C131 | C22 | semidirect |
| 26. #120×C131 | C22 | semidirect |
| 27. #99×C131 | C22 | semidirect |
| 32. C23×D131 | D4 | semidirect |
| 34. C23×D131 | C23 | semidirect |
| 43. C2×D524 | D4 | semidirect |
| 48. C2×C22⋊D131 | D4 | semidirect |
| 58. C2×C4×D131 | D4 | semidirect |
| 60. C22⋊Dic131 | C23 | semidirect |
| 64. C24×C131 | C23 | semidirect |
| 70. C22⋊C4×C131 | C23 | semidirect |
| 87. C22×D131 | C2×D4 | semidirect |
| 99. C2×Dic131 | C2×D4 | semidirect |
| 107. C23×C131 | C2×D4 | semidirect |
| 115. C2×C4×C131 | C2×D4 | semidirect |
| 149. D131 | B(4,𝔽2) | direct (N × H) |
| 150. C131 | C2×B(4,𝔽2) | semidirect |
| 157. B(4,𝔽2) | D131 | direct (N × H) |
| 163. H(5,𝔽2) | C2×D131 | semidirect |
| 173. #120 | C2×D131 | semidirect |
| 178. #99 | C2×D131 | semidirect |
| 186. C24 | C22×D131 | semidirect |
| 221. C22⋊C4 | C22×D131 | semidirect |
| 242. C23 | D4×D131 | semidirect |
| 266. C2×C4 | D4×D131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(5,𝔽2)×D131 | 2 |
| #120×D131 | 2 |
| #22553500 | 2 |
| #22553499 | 2 |
| #22553498 | 2 |
| #99×D131 | 2 |
| B(4,𝔽2)×C131 | 2 |
| C2×B(4,𝔽2) | 131 |