Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
438 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 438 conjugacy classes of subgroups — too many to draw. Only its 54 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 | D97⋊#100 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D97⋊#99 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20261636 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #20261635 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #120×D97 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | D97⋊M16 | — | 2 | 1 | no | — |
| 7 | C2×C22⋊#87604 | — | 2 | 1 | no | — |
| 8 | #15690870 | — | 2 | 1 | no | — |
| 9 | #15690870 | C22 | 1 | 2 | yes | complemented |
| 10 | #15690869 | — | 2 | 1 | no | — |
| 11 | #15690868 | C22 | 1 | 2 | yes | complemented |
| 12 | C24×D97 | C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×D4×D97 | — | 2 | 1 | no | — |
| 14 | C2×D4×D97 | C22 | 1 | 1 | yes | characteristic |
| 15 | #15690780 | — | 2 | 1 | no | — |
| 16 | #15690780 | C22 | 1 | 1 | yes | characteristic |
| 17 | #15690779 | C22 | 1 | 1 | yes | characteristic |
| 18 | #15690778 | — | 2 | 1 | no | — |
| 19 | #15690778 | C4 | 1 | 1 | yes | characteristic, complemented |
| 20 | C22⋊C4×D97 | — | 2 | 1 | no | — |
| 21 | C22⋊C4×D97 | C4 | 1 | 1 | yes | characteristic, complemented |
| 22 | #120×C97 | C4 | 1 | 1 | yes | characteristic, complemented |
| 23 | C22×#87604 | — | 4 | 1 | no | — |
| 24 | #13095521 | — | 4 | 1 | no | — |
| 25 | #13095520 | — | 2 | 2 | no | — |
| 26 | #13095519 | — | 4 | 1 | no | — |
| 27 | C22⋊#87604 | — | 2 | 2 | no | — |
| 28 | C22⋊#87604 | — | 4 | 1 | no | — |
| 29 | C23×D97 | — | 2 | 1 | no | — |
| 30 | C23×D97 | — | 2 | 1 | no | — |
| 31 | C23×D97 | — | 4 | 4 | no | — |
| 32 | C23×D97 | — | 4 | 1 | no | — |
| 33 | C23×D97 | — | 4 | 1 | no | — |
| 34 | C23×D97 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | C23×D97 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 36 | C23×D97 | D4 | 1 | 1 | yes | characteristic |
| 37 | C2×D388 | — | 2 | 1 | no | — |
| 38 | C2×D388 | D4 | 1 | 1 | yes | characteristic |
| 39 | C2×C22⋊D97 | — | 2 | 1 | no | — |
| 40 | C2×C22⋊D97 | — | 2 | 1 | no | — |
| 41 | C2×C22⋊D97 | D4 | 1 | 1 | yes | characteristic |
| 42 | C2×C22⋊D97 | C23 | 1 | 1 | yes | characteristic |
| 43 | D4×D97 | — | 4 | 1 | no | — |
| 44 | D4×D97 | — | 4 | 1 | no | — |
| 45 | D4×D97 | — | 4 | 4 | no | — |
| 46 | C2×C4×D97 | — | 2 | 1 | no | — |
| 47 | C2×C4×D97 | D4 | 1 | 1 | yes | characteristic |
| 48 | C22⋊Dic97 | — | 2 | 1 | no | — |
| 49 | C22⋊Dic97 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 50 | #13095488 | — | 2 | 1 | no | — |
| 51 | #13095488 | — | 2 | 1 | no | — |
| 52 | #13095488 | — | 2 | 1 | no | — |
| 53 | C24×C97 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 54 | C2×D4×C97 | — | 2 | 1 | no | — |
| 55 | C2×D4×C97 | C2×C4 | 1 | 1 | yes | characteristic |
| 56 | C22⋊C4×C97 | — | 2 | 1 | no | — |
| 57 | C22⋊C4×C97 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 58 | C23×C97 | — | 2 | 1 | no | — |
| 59 | C23×C97 | — | 4 | 2 | no | — |
| 60 | C23×C97 | — | 4 | 1 | no | — |
| 61 | C23×C97 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 62 | C23×C97 | C22×C4 | 1 | 1 | yes | characteristic |
| 63 | C22×D97 | — | 2 | 4 | no | — |
| 64 | C22×D97 | — | 2 | 1 | no | — |
| 65 | C22×D97 | — | 2 | 1 | no | — |
| 66 | C22×D97 | — | 2 | 1 | no | — |
| 67 | C22×D97 | — | 2 | 1 | no | — |
| 68 | C22×D97 | — | 2 | 1 | no | — |
| 69 | C22×D97 | — | 4 | 2 | no | — |
| 70 | C22×D97 | — | 4 | 2 | no | — |
| 71 | C22×D97 | — | 4 | 2 | no | — |
| 72 | C22×D97 | — | 4 | 2 | no | — |
| 73 | C22×D97 | — | 4 | 2 | no | — |
| 74 | C22×D97 | — | 4 | 2 | no | — |
| 75 | C22×D97 | — | 4 | 2 | no | — |
| 76 | C22×D97 | — | 4 | 4 | no | — |
| 77 | C22×D97 | — | 4 | 4 | no | — |
| 78 | C22×D97 | — | 4 | 4 | no | — |
| 79 | C22×D97 | — | 4 | 1 | no | — |
| 80 | C22×D97 | — | 4 | 1 | no | — |
| 81 | C22×D97 | — | 4 | 1 | no | — |
| 82 | C22×D97 | — | 4 | 4 | no | — |
| 83 | C22×D97 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 84 | C22×D97 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 85 | C22×D97 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | C2×#87604 | — | 4 | 2 | no | — |
| 87 | C2×#87604 | — | 8 | 2 | no | — |
| 88 | D4×C97 | — | 4 | 1 | no | — |
| 89 | D4×C97 | — | 4 | 2 | no | — |
| 90 | C2×C4×C97 | — | 2 | 1 | no | — |
| 91 | C2×C4×C97 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 92 | C22⋊D97 | — | 4 | 1 | no | — |
| 93 | C22⋊D97 | — | 4 | 1 | no | — |
| 94 | C22⋊D97 | — | 4 | 2 | no | — |
| 95 | C22⋊D97 | — | 4 | 2 | no | — |
| 96 | C2×Dic97 | — | 2 | 1 | no | — |
| 97 | C2×Dic97 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | D388 | — | 4 | 1 | no | — |
| 99 | D388 | — | 4 | 2 | no | — |
| 100 | C4×D97 | — | 2 | 1 | no | — |
| 101 | C4×D97 | — | 2 | 1 | no | — |
| 102 | C4×D97 | — | 4 | 2 | no | — |
| 103 | #11715897 | — | 4 | 2 | no | — |
| 104 | C22×C97 | — | 2 | 2 | no | — |
| 105 | C22×C97 | — | 2 | 1 | no | — |
| 106 | C22×C97 | — | 2 | 1 | no | — |
| 107 | C22×C97 | — | 4 | 1 | no | — |
| 108 | C22×C97 | — | 4 | 1 | no | — |
| 109 | C22×C97 | — | 4 | 1 | no | — |
| 110 | C22×C97 | — | 4 | 2 | no | — |
| 111 | C22×C97 | — | 4 | 2 | no | — |
| 112 | C22×C97 | — | 4 | 1 | no | — |
| 113 | C22×C97 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 114 | C2×D97 | — | 2 | 2 | no | — |
| 115 | C2×D97 | — | 2 | 2 | no | — |
| 116 | C2×D97 | — | 2 | 1 | no | — |
| 117 | C2×D97 | — | 2 | 1 | no | — |
| 118 | C2×D97 | — | 4 | 4 | no | — |
| 119 | C2×D97 | — | 4 | 2 | no | — |
| 120 | C2×D97 | — | 4 | 1 | no | — |
| 121 | C2×D97 | — | 4 | 8 | no | — |
| 122 | C2×D97 | — | 4 | 2 | no | — |
| 123 | C2×D97 | — | 4 | 1 | no | — |
| 124 | C2×D97 | — | 4 | 2 | no | — |
| 125 | C2×D97 | — | 4 | 4 | no | — |
| 126 | C2×D97 | — | 4 | 2 | no | — |
| 127 | C2×D97 | — | 4 | 1 | no | — |
| 128 | C2×D97 | — | 4 | 1 | no | — |
| 129 | C2×D97 | — | 4 | 2 | no | — |
| 130 | C2×D97 | — | 8 | 1 | no | — |
| 131 | C2×D97 | #99 | 1 | 1 | yes | characteristic |
| 132 | C2×D97 | #99 | 1 | 1 | yes | characteristic |
| 133 | #87604 | — | 8 | 2 | no | — |
| 134 | C4×C97 | — | 2 | 1 | no | — |
| 135 | C4×C97 | — | 4 | 1 | no | — |
| 136 | Dic97 | — | 2 | 1 | no | — |
| 137 | Dic97 | — | 4 | 1 | no | — |
| 138 | C2×C97 | — | 2 | 1 | no | — |
| 139 | C2×C97 | — | 4 | 2 | no | — |
| 140 | C2×C97 | — | 4 | 1 | no | — |
| 141 | C2×C97 | — | 4 | 1 | no | — |
| 142 | C2×C97 | C2×#99 | 1 | 1 | yes | characteristic |
| 143 | D97 | — | 2 | 1 | no | — |
| 144 | D97 | — | 4 | 1 | no | — |
| 145 | D97 | — | 4 | 2 | no | — |
| 146 | D97 | — | 4 | 1 | no | — |
| 147 | D97 | AΣL(1,𝔽16) | 1 | 2 | yes | complemented |
| 148 | C2×AΣL(1,𝔽16) | — | 97 | 1 | no | maximal, Sylow |
| 149 | C97 | C2×AΣL(1,𝔽16) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 150 | C2×#120 | — | 97 | 1 | no | — |
| 151 | C2×#100 | — | 97 | 1 | no | — |
| 152 | C2×#99 | — | 97 | 1 | no | — |
| 153 | AΣL(1,𝔽16) | — | 97 | 2 | no | — |
| 154 | AΣL(1,𝔽16) | — | 97 | 2 | no | — |
| 155 | C25 | — | 97 | 1 | no | — |
| 156 | C22×D4 | — | 97 | 1 | no | — |
| 157 | C22×D4 | — | 194 | 1 | no | — |
| 158 | C2×M16 | — | 194 | 1 | no | — |
| 159 | #120 | — | 97 | 1 | no | — |
| 160 | #120 | — | 97 | 1 | no | — |
| 161 | #120 | — | 97 | 1 | no | — |
| 162 | #120 | — | 194 | 1 | no | — |
| 163 | #120 | — | 194 | 1 | no | — |
| 164 | #120 | #87604 | 1 | 1 | yes | characteristic, complemented |
| 165 | C2×C22⋊C4 | — | 97 | 1 | no | — |
| 166 | C2×C22⋊C4 | — | 194 | 1 | no | — |
| 167 | C2×C22⋊C4 | — | 194 | 1 | no | — |
| 168 | #100 | — | 97 | 2 | no | — |
| 169 | #100 | — | 194 | 1 | no | — |
| 170 | #99 | — | 97 | 2 | no | — |
| 171 | #99 | — | 194 | 1 | no | — |
| 172 | C24 | — | 97 | 1 | no | — |
| 173 | C24 | — | 97 | 1 | no | — |
| 174 | C24 | — | 97 | 1 | no | — |
| 175 | C24 | — | 388 | 1 | no | — |
| 176 | C24 | — | 388 | 1 | no | — |
| 177 | C24 | — | 388 | 4 | no | — |
| 178 | C24 | — | 194 | 1 | no | — |
| 179 | C24 | — | 194 | 1 | no | — |
| 180 | C24 | C2×#87604 | 1 | 1 | yes | characteristic, complemented |
| 181 | C2×D4 | — | 2 | 1 | no | — |
| 182 | C2×D4 | — | 97 | 1 | no | — |
| 183 | C2×D4 | — | 97 | 1 | no | — |
| 184 | C2×D4 | — | 97 | 1 | no | — |
| 185 | C2×D4 | — | 388 | 4 | no | — |
| 186 | C2×D4 | — | 388 | 1 | no | — |
| 187 | C2×D4 | — | 388 | 1 | no | — |
| 188 | C2×D4 | — | 194 | 1 | no | — |
| 189 | C2×D4 | — | 194 | 1 | no | — |
| 190 | C2×D4 | — | 194 | 1 | no | — |
| 191 | C2×D4 | C2×#87604 | 1 | 1 | yes | characteristic |
| 192 | C22×C4 | — | 97 | 1 | no | — |
| 193 | C22×C4 | — | 388 | 1 | no | — |
| 194 | C22×C4 | — | 194 | 1 | no | — |
| 195 | M16 | — | 388 | 1 | no | — |
| 196 | M16 | — | 194 | 2 | no | — |
| 197 | C2×C8 | — | 388 | 1 | no | — |
| 198 | C22⋊C4 | — | 2 | 1 | no | — |
| 199 | C22⋊C4 | — | 97 | 1 | no | — |
| 200 | C22⋊C4 | — | 388 | 1 | no | — |
| 201 | C22⋊C4 | — | 194 | 1 | no | — |
| 202 | C22⋊C4 | — | 194 | 1 | no | — |
| 203 | C22⋊C4 | — | 194 | 1 | no | — |
| 204 | C22⋊C4 | — | 194 | 2 | no | — |
| 205 | C22⋊C4 | — | 194 | 1 | no | — |
| 206 | C22⋊C4 | C2×#87604 | 1 | 1 | yes | characteristic, complemented |
| 207 | C23 | — | 2 | 1 | no | — |
| 208 | C23 | — | 4 | 2 | no | — |
| 209 | C23 | — | 4 | 1 | no | — |
| 210 | C23 | — | 97 | 1 | no | — |
| 211 | C23 | — | 97 | 1 | no | — |
| 212 | C23 | — | 97 | 1 | no | — |
| 213 | C23 | — | 388 | 1 | no | — |
| 214 | C23 | — | 388 | 2 | no | — |
| 215 | C23 | — | 388 | 2 | no | — |
| 216 | C23 | — | 388 | 1 | no | — |
| 217 | C23 | — | 388 | 1 | no | — |
| 218 | C23 | — | 388 | 4 | no | — |
| 219 | C23 | — | 388 | 4 | no | — |
| 220 | C23 | — | 388 | 4 | no | — |
| 221 | C23 | — | 388 | 4 | no | — |
| 222 | C23 | — | 388 | 2 | no | — |
| 223 | C23 | — | 388 | 2 | no | — |
| 224 | C23 | — | 388 | 2 | no | — |
| 225 | C23 | — | 388 | 2 | no | — |
| 226 | C23 | — | 388 | 2 | no | — |
| 227 | C23 | — | 194 | 1 | no | — |
| 228 | C23 | — | 194 | 1 | no | — |
| 229 | C23 | — | 194 | 1 | no | — |
| 230 | C23 | — | 194 | 1 | no | — |
| 231 | C23 | — | 194 | 4 | no | — |
| 232 | C23 | — | 194 | 1 | no | — |
| 233 | C23 | C22⋊#87604 | 1 | 1 | yes | characteristic |
| 234 | C23 | C22×#87604 | 1 | 1 | yes | characteristic |
| 235 | D4 | — | 4 | 1 | no | — |
| 236 | D4 | — | 4 | 2 | no | — |
| 237 | D4 | — | 388 | 1 | no | — |
| 238 | D4 | — | 388 | 2 | no | — |
| 239 | D4 | — | 388 | 2 | no | — |
| 240 | D4 | — | 388 | 2 | no | — |
| 241 | D4 | — | 388 | 1 | no | — |
| 242 | D4 | — | 388 | 1 | no | — |
| 243 | C2×C4 | — | 2 | 1 | no | — |
| 244 | C2×C4 | — | 776 | 2 | no | — |
| 245 | C2×C4 | — | 97 | 1 | no | — |
| 246 | C2×C4 | — | 388 | 2 | no | — |
| 247 | C2×C4 | — | 388 | 2 | no | — |
| 248 | C2×C4 | — | 194 | 1 | no | — |
| 249 | C2×C4 | — | 194 | 1 | no | — |
| 250 | C2×C4 | — | 194 | 1 | no | — |
| 251 | C2×C4 | C22⋊#87604 | 1 | 1 | yes | characteristic |
| 252 | C8 | — | 388 | 2 | no | — |
| 253 | C22 | — | 2 | 2 | no | — |
| 254 | C22 | — | 2 | 1 | no | — |
| 255 | C22 | — | 2 | 1 | no | — |
| 256 | C22 | — | 4 | 1 | no | — |
| 257 | C22 | — | 4 | 1 | no | — |
| 258 | C22 | — | 4 | 1 | no | — |
| 259 | C22 | — | 4 | 2 | no | — |
| 260 | C22 | — | 4 | 2 | no | — |
| 261 | C22 | — | 4 | 1 | no | — |
| 262 | C22 | — | 776 | 1 | no | — |
| 263 | C22 | — | 97 | 1 | no | — |
| 264 | C22 | — | 97 | 1 | no | — |
| 265 | C22 | — | 388 | 1 | no | — |
| 266 | C22 | — | 388 | 1 | no | — |
| 267 | C22 | — | 388 | 1 | no | — |
| 268 | C22 | — | 388 | 2 | no | — |
| 269 | C22 | — | 388 | 4 | no | — |
| 270 | C22 | — | 388 | 2 | no | — |
| 271 | C22 | — | 388 | 2 | no | — |
| 272 | C22 | — | 388 | 2 | no | — |
| 273 | C22 | — | 388 | 8 | no | — |
| 274 | C22 | — | 388 | 4 | no | — |
| 275 | C22 | — | 388 | 2 | no | — |
| 276 | C22 | — | 388 | 1 | no | — |
| 277 | C22 | — | 194 | 2 | no | — |
| 278 | C22 | — | 194 | 2 | no | — |
| 279 | C22 | — | 194 | 1 | no | — |
| 280 | C22 | — | 194 | 1 | no | — |
| 281 | C22 | C2×C22⋊#87604 | 1 | 1 | yes | characteristic |
| 282 | C4 | — | 2 | 1 | no | — |
| 283 | C4 | — | 4 | 1 | no | — |
| 284 | C4 | — | 776 | 2 | no | — |
| 285 | C4 | — | 388 | 1 | no | — |
| 286 | C4 | — | 194 | 1 | no | — |
| 287 | C2 | — | 2 | 1 | no | — |
| 288 | C2 | — | 4 | 2 | no | — |
| 289 | C2 | — | 4 | 1 | no | — |
| 290 | C2 | — | 4 | 1 | no | — |
| 291 | C2 | — | 97 | 2 | no | — |
| 292 | C2 | — | 388 | 1 | no | — |
| 293 | C2 | — | 388 | 1 | no | — |
| 294 | C2 | — | 388 | 2 | no | — |
| 295 | C2 | — | 194 | 1 | no | — |
| 296 | C2 | D97⋊#99 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D97⋊#100 | C2 | semidirect |
| 2. D97⋊#99 | C2 | semidirect |
| 3. #20261636 | C2 | semidirect |
| 4. #20261635 | C2 | semidirect |
| 9. #15690870 | C22 | semidirect |
| 11. #15690868 | C22 | semidirect |
| 12. C24×D97 | C4 | semidirect |
| 19. #15690778 | C4 | semidirect |
| 21. C22⋊C4×D97 | C4 | semidirect |
| 22. #120×C97 | C4 | semidirect |
| 35. C23×D97 | C2×C4 | semidirect |
| 49. C22⋊Dic97 | C2×C4 | semidirect |
| 53. C24×C97 | C2×C4 | semidirect |
| 57. C22⋊C4×C97 | C2×C4 | semidirect |
| 147. D97 | AΣL(1,𝔽16) | semidirect |
| 149. C97 | C2×AΣL(1,𝔽16) | semidirect |
| 164. #120 | #87604 | semidirect |
| 180. C24 | C2×#87604 | semidirect |
| 206. C22⋊C4 | C2×#87604 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D97⋊#100 | 2 |
| D97⋊#99 | 2 |
| #20261636 | 2 |
| #20261635 | 2 |
| #120×D97 | 2 |
| C2×AΣL(1,𝔽16) | 97 |