Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1630 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 171 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×#90408 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11740604 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11736789 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C7×#1555 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11728553 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #11726146 | C2 | 1 | 8 | yes | maximal, complemented |
| 7 | C22×D4×D7 | — | 2 | 2 | no | — |
| 8 | C2×C7×#127 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#6501 | — | 2 | 2 | no | — |
| 10 | C2×#6500 | — | 2 | 2 | no | — |
| 11 | #90408 | — | 2 | 2 | no | — |
| 12 | #90408 | — | 2 | 4 | no | — |
| 13 | #90408 | C22 | 1 | 4 | yes | — |
| 14 | C2×C4×D28 | C22 | 1 | 2 | yes | — |
| 15 | #90161 | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | C7×#432 | C22 | 1 | 2 | yes | complemented |
| 17 | #89987 | — | 2 | 2 | no | — |
| 18 | #89843 | — | 2 | 2 | no | — |
| 19 | #89842 | C22 | 1 | 4 | yes | complemented |
| 20 | C4×#1136 | C22 | 1 | 1 | yes | characteristic, complemented |
| 21 | #89618 | C22 | 1 | 4 | yes | complemented |
| 22 | #89613 | C22 | 1 | 4 | yes | complemented |
| 23 | #89503 | — | 2 | 2 | no | — |
| 24 | #89494 | — | 2 | 2 | no | — |
| 25 | #89482 | — | 2 | 4 | no | — |
| 26 | C7×#331 | C22 | 1 | 4 | yes | complemented |
| 27 | #89278 | C22 | 1 | 8 | yes | complemented |
| 28 | #18529 | — | 7 | 1 | no | maximal, Sylow |
| 29 | C24×D7 | — | 2 | 2 | no | — |
| 30 | C22×C7×D4 | — | 2 | 2 | no | — |
| 31 | C22×C7×D4 | D4 | 1 | 2 | yes | — |
| 32 | C22×C22⋊D7 | — | 2 | 4 | no | — |
| 33 | C2×D4×D7 | — | 2 | 8 | no | — |
| 34 | C2×D4×D7 | — | 2 | 8 | no | — |
| 35 | C2×D4×D7 | — | 4 | 16 | no | — |
| 36 | C22×D28 | D4 | 1 | 2 | yes | — |
| 37 | C22×C4×D7 | — | 2 | 2 | no | — |
| 38 | C7×#127 | — | 2 | 2 | no | — |
| 39 | C7×#127 | D4 | 1 | 2 | yes | — |
| 40 | C7×#127 | C23 | 1 | 2 | yes | — |
| 41 | C2×C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 42 | C2×C22⋊Dic7 | — | 2 | 2 | no | — |
| 43 | #6501 | — | 2 | 8 | no | — |
| 44 | #6501 | — | 2 | 8 | no | — |
| 45 | #6500 | — | 2 | 8 | no | — |
| 46 | #6500 | — | 2 | 8 | no | — |
| 47 | C2×(C2×D7)⋊C4 | — | 2 | 2 | no | — |
| 48 | #6489 | — | 2 | 1 | no | — |
| 49 | C2×C4⋊Dic7 | D4 | 1 | 2 | yes | — |
| 50 | #6487 | — | 2 | 2 | no | — |
| 51 | C2×#1136 | — | 2 | 1 | no | — |
| 52 | C2×#1132 | — | 2 | 4 | no | — |
| 53 | C4×D28 | — | 2 | 2 | no | — |
| 54 | C4×D28 | — | 2 | 2 | no | — |
| 55 | C4×D28 | D4 | 1 | 4 | yes | — |
| 56 | C4×D28 | C23 | 1 | 4 | yes | — |
| 57 | #6433 | — | 2 | 1 | no | — |
| 58 | #6432 | C23 | 1 | 2 | yes | complemented |
| 59 | C4×Dic14 | — | 2 | 2 | no | — |
| 60 | C7×C4⋊C8 | C23 | 1 | 4 | yes | complemented |
| 61 | C7×C22⋊C8 | — | 2 | 2 | no | — |
| 62 | C7×#97 | — | 2 | 2 | no | — |
| 63 | #6406 | — | 2 | 8 | no | — |
| 64 | #6395 | — | 2 | 4 | no | — |
| 65 | #6392 | — | 2 | 4 | no | — |
| 66 | #6390 | — | 2 | 4 | no | — |
| 67 | #6377 | — | 2 | 1 | no | — |
| 68 | C4×C7⋊C8 | C23 | 1 | 2 | yes | complemented |
| 69 | C2×D42 | — | 7 | 1 | no | — |
| 70 | #3138 | — | 7 | 1 | no | — |
| 71 | #3110 | — | 7 | 2 | no | — |
| 72 | #1639 | — | 7 | 8 | no | — |
| 73 | #1555 | D7 | 1 | 1 | yes | characteristic, complemented |
| 74 | #1490 | — | 7 | 2 | no | — |
| 75 | C24×C7 | — | 2 | 2 | no | — |
| 76 | C23×D7 | — | 2 | 2 | no | — |
| 77 | C23×D7 | — | 2 | 8 | no | — |
| 78 | C23×D7 | — | 4 | 16 | no | — |
| 79 | C23×D7 | — | 8 | 4 | no | — |
| 80 | C2×C7×D4 | — | 2 | 4 | no | — |
| 81 | C2×C7×D4 | — | 2 | 2 | no | — |
| 82 | C2×C7×D4 | — | 2 | 4 | no | — |
| 83 | C2×C7×D4 | — | 4 | 8 | no | — |
| 84 | C2×C7×D4 | — | 4 | 4 | no | — |
| 85 | C2×C7×D4 | C2×D4 | 1 | 4 | yes | — |
| 86 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C22×C4×C7 | C2×D4 | 1 | 2 | yes | — |
| 88 | C2×C22⋊D7 | — | 2 | 16 | no | — |
| 89 | C2×C22⋊D7 | — | 4 | 16 | no | — |
| 90 | C22×Dic7 | — | 2 | 2 | no | — |
| 91 | D4×D7 | — | 4 | 32 | no | — |
| 92 | D4×D7 | — | 4 | 16 | no | — |
| 93 | D4×D7 | — | 4 | 16 | no | — |
| 94 | C2×D28 | — | 2 | 2 | no | — |
| 95 | C2×D28 | — | 2 | 8 | no | — |
| 96 | C2×D28 | C2×D4 | 1 | 4 | yes | — |
| 97 | C2×C4×D7 | — | 2 | 4 | no | — |
| 98 | C2×C4×D7 | — | 2 | 4 | no | — |
| 99 | C2×C4×D7 | — | 4 | 8 | no | — |
| 100 | C2×Dic14 | — | 2 | 2 | no | — |
| 101 | C2×C7×C8 | — | 2 | 4 | no | — |
| 102 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | C42×C7 | C2×D4 | 1 | 2 | yes | — |
| 104 | C42×C7 | C24 | 1 | 1 | yes | characteristic |
| 105 | C22⋊Dic7 | — | 2 | 8 | no | — |
| 106 | C22⋊Dic7 | — | 4 | 1 | no | — |
| 107 | (C2×D7)⋊C4 | — | 2 | 4 | no | — |
| 108 | (C2×D7)⋊C4 | — | 2 | 4 | no | — |
| 109 | C4⋊Dic7 | — | 2 | 2 | no | — |
| 110 | C4⋊Dic7 | — | 2 | 2 | no | — |
| 111 | C4⋊Dic7 | — | 2 | 2 | no | — |
| 112 | C4⋊Dic7 | C2×D4 | 1 | 4 | yes | — |
| 113 | Dic7⋊C4 | — | 4 | 2 | no | — |
| 114 | C4×Dic7 | — | 4 | 1 | no | — |
| 115 | #1136 | — | 4 | 2 | no | — |
| 116 | C2×C7⋊C8 | — | 2 | 2 | no | — |
| 117 | #1132 | — | 4 | 8 | no | — |
| 118 | C23×D4 | — | 14 | 2 | no | — |
| 119 | D42 | — | 7 | 4 | no | — |
| 120 | D42 | — | 14 | 4 | no | — |
| 121 | D42 | — | 14 | 2 | no | — |
| 122 | #533 | — | 7 | 1 | no | — |
| 123 | C2×#127 | C2×D7 | 1 | 1 | yes | characteristic |
| 124 | C2×#121 | — | 14 | 2 | no | — |
| 125 | C2×#120 | — | 14 | 2 | no | — |
| 126 | C2×C4×D4 | — | 7 | 2 | no | — |
| 127 | #490 | — | 14 | 2 | no | — |
| 128 | #486 | — | 7 | 4 | no | — |
| 129 | #484 | — | 14 | 2 | no | — |
| 130 | #474 | — | 7 | 4 | no | — |
| 131 | #461 | — | 7 | 4 | no | — |
| 132 | #450 | — | 14 | 2 | no | — |
| 133 | #440 | — | 14 | 4 | no | — |
| 134 | #432 | C2×D7 | 1 | 2 | yes | complemented |
| 135 | #418 | — | 14 | 2 | no | — |
| 136 | C4×M16 | — | 7 | 1 | no | — |
| 137 | #331 | C2×D7 | 1 | 4 | yes | complemented |
| 138 | D4⋊C8 | — | 7 | 8 | no | — |
| 139 | C23×C7 | — | 2 | 4 | no | — |
| 140 | C23×C7 | — | 4 | 8 | no | — |
| 141 | C23×C7 | — | 4 | 4 | no | — |
| 142 | C23×C7 | #120 | 1 | 1 | yes | characteristic |
| 143 | C22×D7 | — | 2 | 8 | no | — |
| 144 | C22×D7 | — | 2 | 4 | no | — |
| 145 | C22×D7 | — | 4 | 32 | no | — |
| 146 | C22×D7 | — | 4 | 4 | no | — |
| 147 | C22×D7 | — | 4 | 4 | no | — |
| 148 | C22×D7 | — | 8 | 8 | no | — |
| 149 | C22×D7 | — | 8 | 32 | no | — |
| 150 | C7×D4 | — | 4 | 4 | no | — |
| 151 | C7×D4 | — | 4 | 8 | no | — |
| 152 | C7×D4 | — | 4 | 8 | no | — |
| 153 | C7×D4 | — | 8 | 2 | no | — |
| 154 | C2×C4×C7 | — | 2 | 2 | no | — |
| 155 | C2×C4×C7 | — | 2 | 4 | no | — |
| 156 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 157 | C2×C4×C7 | #120 | 1 | 2 | yes | — |
| 158 | C2×C4×C7 | C22×D4 | 1 | 1 | yes | characteristic |
| 159 | C2×C4×C7 | C22×D4 | 1 | 2 | yes | — |
| 160 | C22⋊D7 | — | 4 | 32 | no | — |
| 161 | C2×Dic7 | — | 2 | 4 | no | — |
| 162 | C2×Dic7 | — | 4 | 4 | no | — |
| 163 | C2×Dic7 | — | 4 | 2 | no | — |
| 164 | D28 | — | 2 | 8 | no | — |
| 165 | D28 | — | 2 | 4 | no | — |
| 166 | D28 | AGL(1,ℤ/8) | 1 | 8 | yes | — |
| 167 | C4×D7 | — | 4 | 8 | no | — |
| 168 | C4×D7 | — | 4 | 8 | no | — |
| 169 | Dic14 | — | 4 | 4 | no | — |
| 170 | C7×C8 | — | 4 | 4 | no | — |
| 171 | C7⋊C8 | — | 4 | 2 | no | — |
| 172 | C25 | — | 14 | 2 | no | — |
| 173 | C22×D4 | — | 2 | 2 | no | — |
| 174 | C22×D4 | — | 7 | 2 | no | — |
| 175 | C22×D4 | — | 14 | 8 | no | — |
| 176 | C22×D4 | — | 14 | 8 | no | — |
| 177 | C22×D4 | — | 14 | 4 | no | — |
| 178 | C22×D4 | — | 28 | 16 | no | — |
| 179 | C22×D4 | D28 | 1 | 2 | yes | — |
| 180 | C23×C4 | — | 14 | 2 | no | — |
| 181 | C2×SD16 | — | 14 | 4 | no | — |
| 182 | C2×M16 | — | 14 | 1 | no | — |
| 183 | C4⋊Q8 | — | 7 | 2 | no | — |
| 184 | #127 | — | 2 | 2 | no | — |
| 185 | #127 | D28 | 1 | 2 | yes | — |
| 186 | #127 | C22×D7 | 1 | 2 | yes | — |
| 187 | #125 | — | 14 | 1 | no | — |
| 188 | C22⋊Q8 | — | 14 | 2 | no | — |
| 189 | #121 | — | 14 | 8 | no | — |
| 190 | #121 | — | 14 | 8 | no | — |
| 191 | #120 | — | 14 | 8 | no | — |
| 192 | #120 | — | 14 | 8 | no | — |
| 193 | C4×Q8 | — | 14 | 2 | no | — |
| 194 | C4×D4 | — | 7 | 4 | no | — |
| 195 | C4×D4 | — | 7 | 4 | no | — |
| 196 | C4×D4 | — | 14 | 2 | no | — |
| 197 | C4×D4 | — | 14 | 2 | no | — |
| 198 | #117 | — | 14 | 1 | no | — |
| 199 | C2×C4⋊C4 | — | 7 | 2 | no | — |
| 200 | C2×C22⋊C4 | — | 14 | 2 | no | — |
| 201 | C2×C22⋊C4 | — | 14 | 2 | no | — |
| 202 | C2×C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 203 | #107 | — | 14 | 4 | no | — |
| 204 | C4⋊C8 | C22×D7 | 1 | 4 | yes | complemented |
| 205 | #103 | — | 14 | 4 | no | — |
| 206 | #102 | — | 14 | 4 | no | — |
| 207 | #102 | — | 14 | 8 | no | — |
| 208 | C22⋊C8 | — | 2 | 2 | no | — |
| 209 | #97 | — | 2 | 2 | no | — |
| 210 | #97 | — | 14 | 1 | no | — |
| 211 | C4×C8 | — | 7 | 2 | no | — |
| 212 | C22×C7 | — | 2 | 2 | no | — |
| 213 | C22×C7 | — | 4 | 8 | no | — |
| 214 | C22×C7 | — | 4 | 4 | no | — |
| 215 | C22×C7 | — | 8 | 8 | no | — |
| 216 | C22×C7 | #455 | 1 | 2 | yes | — |
| 217 | C22×C7 | C2×#120 | 1 | 1 | yes | characteristic |
| 218 | C2×D7 | — | 2 | 8 | no | — |
| 219 | C2×D7 | — | 4 | 16 | no | — |
| 220 | C2×D7 | — | 4 | 4 | no | — |
| 221 | C2×D7 | — | 4 | 4 | no | — |
| 222 | C2×D7 | — | 8 | 32 | no | — |
| 223 | C4×C7 | — | 2 | 2 | no | — |
| 224 | C4×C7 | — | 2 | 1 | no | — |
| 225 | C4×C7 | — | 2 | 1 | no | — |
| 226 | C4×C7 | C2×AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 227 | Dic7 | — | 4 | 4 | no | — |
| 228 | Dic7 | — | 8 | 2 | no | — |
| 229 | C24 | — | 2 | 2 | no | — |
| 230 | C24 | — | 14 | 8 | no | — |
| 231 | C24 | — | 14 | 2 | no | — |
| 232 | C24 | — | 28 | 16 | no | — |
| 233 | C24 | — | 56 | 4 | no | — |
| 234 | C2×Q8 | — | 14 | 2 | no | — |
| 235 | C2×D4 | — | 2 | 4 | no | — |
| 236 | C2×D4 | — | 2 | 2 | no | — |
| 237 | C2×D4 | — | 2 | 4 | no | — |
| 238 | C2×D4 | — | 4 | 8 | no | — |
| 239 | C2×D4 | — | 4 | 4 | no | — |
| 240 | C2×D4 | — | 7 | 4 | no | — |
| 241 | C2×D4 | — | 14 | 8 | no | — |
| 242 | C2×D4 | — | 14 | 16 | no | — |
| 243 | C2×D4 | — | 14 | 2 | no | — |
| 244 | C2×D4 | — | 28 | 16 | no | — |
| 245 | C2×D4 | — | 28 | 32 | no | — |
| 246 | C2×D4 | — | 28 | 16 | no | — |
| 247 | C2×D4 | — | 28 | 16 | no | — |
| 248 | C2×D4 | C2×D28 | 1 | 4 | yes | — |
| 249 | C22×C4 | — | 14 | 2 | no | — |
| 250 | C22×C4 | — | 14 | 4 | no | — |
| 251 | C22×C4 | — | 14 | 4 | no | — |
| 252 | C22×C4 | — | 28 | 8 | no | — |
| 253 | C22×C4 | C2×D28 | 1 | 1 | yes | characteristic |
| 254 | C22×C4 | D4×D7 | 1 | 2 | yes | — |
| 255 | SD16 | — | 28 | 8 | no | — |
| 256 | M16 | — | 28 | 2 | no | — |
| 257 | C2×C8 | — | 2 | 4 | no | — |
| 258 | C2×C8 | — | 14 | 2 | no | — |
| 259 | C4⋊C4 | — | 7 | 4 | no | — |
| 260 | C4⋊C4 | — | 14 | 2 | no | — |
| 261 | C4⋊C4 | — | 14 | 2 | no | — |
| 262 | C4⋊C4 | — | 14 | 2 | no | — |
| 263 | C4⋊C4 | — | 28 | 2 | no | — |
| 264 | C22⋊C4 | — | 14 | 8 | no | — |
| 265 | C22⋊C4 | — | 14 | 4 | no | — |
| 266 | C22⋊C4 | — | 14 | 4 | no | — |
| 267 | C22⋊C4 | — | 28 | 1 | no | — |
| 268 | C42 | — | 28 | 1 | no | — |
| 269 | C42 | C2×D28 | 1 | 1 | yes | characteristic |
| 270 | C42 | D4×D7 | 1 | 2 | yes | — |
| 271 | C42 | C23×D7 | 1 | 1 | yes | characteristic |
| 272 | C2×C7 | — | 2 | 2 | no | — |
| 273 | C2×C7 | — | 8 | 4 | no | — |
| 274 | C2×C7 | #3003 | 1 | 2 | yes | — |
| 275 | C2×C7 | C2×#455 | 1 | 1 | yes | characteristic |
| 276 | D7 | — | 4 | 8 | no | — |
| 277 | C23 | — | 2 | 4 | no | — |
| 278 | C23 | — | 4 | 8 | no | — |
| 279 | C23 | — | 4 | 4 | no | — |
| 280 | C23 | — | 14 | 8 | no | — |
| 281 | C23 | — | 14 | 4 | no | — |
| 282 | C23 | — | 28 | 4 | no | — |
| 283 | C23 | — | 28 | 32 | no | — |
| 284 | C23 | — | 28 | 4 | no | — |
| 285 | C23 | — | 56 | 8 | no | — |
| 286 | C23 | — | 56 | 32 | no | — |
| 287 | C23 | #6445 | 1 | 1 | yes | characteristic |
| 288 | Q8 | — | 28 | 4 | no | — |
| 289 | D4 | — | 4 | 4 | no | — |
| 290 | D4 | — | 4 | 8 | no | — |
| 291 | D4 | — | 4 | 8 | no | — |
| 292 | D4 | — | 7 | 8 | no | — |
| 293 | D4 | — | 8 | 2 | no | — |
| 294 | D4 | — | 14 | 8 | no | — |
| 295 | D4 | — | 14 | 4 | no | — |
| 296 | D4 | — | 28 | 32 | no | — |
| 297 | C2×C4 | — | 2 | 2 | no | — |
| 298 | C2×C4 | — | 2 | 4 | no | — |
| 299 | C2×C4 | — | 14 | 4 | no | — |
| 300 | C2×C4 | — | 28 | 8 | no | — |
| 301 | C2×C4 | — | 28 | 4 | no | — |
| 302 | C2×C4 | — | 28 | 8 | no | — |
| 303 | C2×C4 | — | 28 | 2 | no | — |
| 304 | C2×C4 | #6445 | 1 | 1 | yes | characteristic |
| 305 | C2×C4 | #6445 | 1 | 2 | yes | — |
| 306 | C2×C4 | C22×D28 | 1 | 1 | yes | characteristic |
| 307 | C2×C4 | C2×D4×D7 | 1 | 2 | yes | — |
| 308 | C8 | — | 4 | 4 | no | — |
| 309 | C8 | — | 28 | 2 | no | — |
| 310 | C7 | #18529 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 311 | C22 | — | 2 | 2 | no | — |
| 312 | C22 | — | 4 | 8 | no | — |
| 313 | C22 | — | 4 | 4 | no | — |
| 314 | C22 | — | 8 | 8 | no | — |
| 315 | C22 | — | 14 | 8 | no | — |
| 316 | C22 | — | 28 | 16 | no | — |
| 317 | C22 | — | 28 | 4 | no | — |
| 318 | C22 | — | 28 | 4 | no | — |
| 319 | C22 | — | 56 | 32 | no | — |
| 320 | C22 | #89518 | 1 | 2 | yes | — |
| 321 | C22 | C2×#6445 | 1 | 1 | yes | characteristic |
| 322 | C4 | — | 2 | 2 | no | — |
| 323 | C4 | — | 2 | 1 | no | — |
| 324 | C4 | — | 2 | 1 | no | — |
| 325 | C4 | — | 28 | 4 | no | — |
| 326 | C4 | — | 56 | 2 | no | — |
| 327 | C4 | D7×AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 328 | C2 | — | 2 | 2 | no | — |
| 329 | C2 | — | 8 | 4 | no | — |
| 330 | C2 | — | 28 | 8 | no | — |
| 331 | C2 | C2×#89518 | 1 | 1 | yes | characteristic |
| 332 | C2 | #11736371 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11740604 | C2 | semidirect |
| 3. #11736789 | C2 | semidirect |
| 4. C7×#1555 | C2 | semidirect |
| 5. #11728553 | C2 | semidirect |
| 6. #11726146 | C2 | semidirect |
| 15. #90161 | C22 | semidirect |
| 16. C7×#432 | C22 | semidirect |
| 19. #89842 | C22 | semidirect |
| 20. C4×#1136 | C22 | semidirect |
| 21. #89618 | C22 | semidirect |
| 22. #89613 | C22 | semidirect |
| 26. C7×#331 | C22 | semidirect |
| 27. #89278 | C22 | semidirect |
| 58. #6432 | C23 | semidirect |
| 60. C7×C4⋊C8 | C23 | semidirect |
| 68. C4×C7⋊C8 | C23 | semidirect |
| 73. #1555 | D7 | semidirect |
| 134. #432 | C2×D7 | semidirect |
| 137. #331 | C2×D7 | semidirect |
| 204. C4⋊C8 | C22×D7 | semidirect |
| 310. C7 | #18529 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#90408 | 2 |
| #11740604 | 2 |
| #11736789 | 2 |
| C7×#1555 | 2 |
| #11728553 | 2 |
| #11726146 | 2 |
| #18529 | 7 |