Finite Group DatabasebetaFinite Group DatabasebetaOrder 10624.
354 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 41 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 | #19128333 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #19127888 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #19127869 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #19127493 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #19127488 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #19127487 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | B(4,𝔽2)×C83 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×C22⋊D83 | — | 2 | 1 | no | — |
| 9 | C2×D4×D83 | — | 2 | 1 | no | — |
| 10 | #14581817 | — | 2 | 1 | no | — |
| 11 | #14581788 | C22 | 1 | 1 | yes | characteristic |
| 12 | #14581765 | — | 2 | 1 | no | — |
| 13 | #14581764 | — | 2 | 1 | no | — |
| 14 | #14581762 | — | 2 | 1 | no | — |
| 15 | #14581760 | — | 2 | 1 | no | — |
| 16 | #14581760 | C22 | 1 | 1 | yes | characteristic |
| 17 | (C22⋊D83)⋊C4 | — | 2 | 1 | no | — |
| 18 | #14581719 | — | 2 | 1 | no | — |
| 19 | #14581709 | C22 | 1 | 1 | yes | characteristic, complemented |
| 20 | #14581707 | — | 2 | 1 | no | — |
| 21 | #14581707 | C22 | 1 | 1 | yes | characteristic, complemented |
| 22 | H(5,𝔽2)×C83 | C22 | 1 | 1 | yes | characteristic, complemented |
| 23 | #120×C83 | — | 2 | 1 | no | — |
| 24 | #120×C83 | C22 | 1 | 1 | yes | characteristic |
| 25 | #99×C83 | — | 2 | 1 | no | — |
| 26 | #99×C83 | C22 | 1 | 1 | yes | characteristic, complemented |
| 27 | C23×D83 | — | 2 | 1 | no | — |
| 28 | C23×D83 | D4 | 1 | 1 | yes | characteristic, complemented |
| 29 | C2×D332 | — | 2 | 1 | no | — |
| 30 | C2×D332 | D4 | 1 | 1 | yes | characteristic, complemented |
| 31 | C2×C22⋊D83 | — | 2 | 1 | no | — |
| 32 | C2×C22⋊D83 | — | 2 | 1 | no | — |
| 33 | C2×C22⋊D83 | — | 2 | 1 | no | — |
| 34 | C2×C22⋊D83 | — | 2 | 1 | no | — |
| 35 | C2×C22⋊D83 | — | 2 | 1 | no | — |
| 36 | C2×C22⋊D83 | — | 4 | 4 | no | — |
| 37 | C2×C22⋊D83 | — | 4 | 1 | no | — |
| 38 | D4×D83 | — | 4 | 4 | no | — |
| 39 | C22×Dic83 | — | 2 | 1 | no | — |
| 40 | C2×C4×D83 | — | 2 | 1 | no | — |
| 41 | #13050577 | — | 4 | 1 | no | — |
| 42 | #13050575 | — | 4 | 1 | no | — |
| 43 | #13050574 | — | 2 | 1 | no | — |
| 44 | Dic83⋊C4 | — | 2 | 1 | no | — |
| 45 | C22⋊Dic83 | — | 2 | 1 | no | — |
| 46 | C22⋊Dic83 | D4 | 1 | 1 | yes | characteristic, complemented |
| 47 | (C2×D83)⋊C4 | — | 2 | 1 | no | — |
| 48 | (C2×D83)⋊C4 | — | 2 | 1 | no | — |
| 49 | (C2×D83)⋊C4 | — | 4 | 1 | no | — |
| 50 | C4×Dic83 | D4 | 1 | 1 | yes | characteristic, complemented |
| 51 | C24×C83 | D4 | 1 | 1 | yes | characteristic, complemented |
| 52 | #41×C83 | — | 2 | 1 | no | — |
| 53 | #41×C83 | — | 4 | 1 | no | — |
| 54 | C2×D4×C83 | — | 2 | 1 | no | — |
| 55 | C2×D4×C83 | — | 2 | 1 | no | — |
| 56 | C2×D4×C83 | — | 2 | 1 | no | — |
| 57 | C2×D4×C83 | — | 4 | 1 | no | — |
| 58 | C2×D4×C83 | — | 4 | 1 | no | — |
| 59 | C2×D4×C83 | C23 | 1 | 1 | yes | characteristic |
| 60 | C22⋊C4×C83 | — | 2 | 1 | no | — |
| 61 | C22⋊C4×C83 | — | 2 | 1 | no | — |
| 62 | C22⋊C4×C83 | — | 4 | 1 | no | — |
| 63 | C22⋊C4×C83 | D4 | 1 | 1 | yes | characteristic, complemented |
| 64 | C23×C83 | — | 2 | 1 | no | — |
| 65 | C23×C83 | — | 2 | 1 | no | — |
| 66 | C23×C83 | — | 2 | 1 | no | — |
| 67 | C23×C83 | — | 4 | 1 | no | — |
| 68 | C23×C83 | — | 4 | 2 | no | — |
| 69 | C23×C83 | C2×D4 | 1 | 1 | yes | characteristic |
| 70 | C23×C83 | C2×D4 | 1 | 1 | yes | characteristic |
| 71 | C22×D83 | — | 2 | 1 | no | — |
| 72 | C22×D83 | — | 2 | 1 | no | — |
| 73 | C22×D83 | — | 4 | 2 | no | — |
| 74 | C22×D83 | — | 4 | 2 | no | — |
| 75 | C22×D83 | — | 4 | 1 | no | — |
| 76 | C22×D83 | — | 4 | 2 | no | — |
| 77 | C22×D83 | — | 8 | 1 | no | — |
| 78 | Q8×C83 | — | 2 | 1 | no | — |
| 79 | D4×C83 | — | 2 | 1 | no | — |
| 80 | D4×C83 | — | 4 | 1 | no | — |
| 81 | D4×C83 | — | 4 | 1 | no | — |
| 82 | D4×C83 | — | 4 | 1 | no | — |
| 83 | D4×C83 | — | 4 | 2 | no | — |
| 84 | D4×C83 | — | 4 | 1 | no | — |
| 85 | D4×C83 | — | 8 | 2 | no | — |
| 86 | C2×C4×C83 | — | 2 | 1 | no | — |
| 87 | C2×C4×C83 | — | 2 | 1 | no | — |
| 88 | C2×C4×C83 | — | 2 | 1 | no | — |
| 89 | C2×C4×C83 | — | 4 | 1 | no | — |
| 90 | C2×C4×C83 | — | 4 | 1 | no | — |
| 91 | C2×C4×C83 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C22⋊D83 | — | 2 | 4 | no | — |
| 93 | C22⋊D83 | — | 4 | 2 | no | — |
| 94 | C22⋊D83 | — | 4 | 2 | no | — |
| 95 | C22⋊D83 | — | 4 | 2 | no | — |
| 96 | C22⋊D83 | — | 4 | 2 | no | — |
| 97 | C22⋊D83 | — | 8 | 2 | no | — |
| 98 | C2×Dic83 | — | 2 | 1 | no | — |
| 99 | C2×Dic83 | — | 2 | 1 | no | — |
| 100 | C2×Dic83 | — | 4 | 2 | no | — |
| 101 | C2×Dic83 | — | 4 | 1 | no | — |
| 102 | D332 | — | 4 | 1 | no | — |
| 103 | D332 | — | 4 | 2 | no | — |
| 104 | C4×D83 | — | 4 | 2 | no | — |
| 105 | C83⋊C8 | — | 4 | 1 | no | — |
| 106 | C22×C83 | — | 2 | 2 | no | — |
| 107 | C22×C83 | — | 2 | 1 | no | — |
| 108 | C22×C83 | — | 2 | 1 | no | — |
| 109 | C22×C83 | — | 4 | 1 | no | — |
| 110 | C22×C83 | — | 4 | 1 | no | — |
| 111 | C22×C83 | — | 4 | 1 | no | — |
| 112 | C22×C83 | — | 4 | 2 | no | — |
| 113 | C22×C83 | — | 4 | 1 | no | — |
| 114 | C22×C83 | — | 4 | 1 | no | — |
| 115 | C22×C83 | — | 8 | 1 | no | — |
| 116 | C22×C83 | — | 8 | 1 | no | — |
| 117 | C22×C83 | — | 8 | 1 | no | — |
| 118 | C22×C83 | #120 | 1 | 1 | yes | characteristic |
| 119 | C2×D83 | — | 2 | 2 | no | — |
| 120 | C2×D83 | — | 4 | 1 | no | — |
| 121 | C2×D83 | — | 4 | 1 | no | — |
| 122 | C2×D83 | — | 4 | 2 | no | — |
| 123 | C2×D83 | — | 4 | 1 | no | — |
| 124 | C2×D83 | — | 4 | 1 | no | — |
| 125 | C2×D83 | — | 4 | 1 | no | — |
| 126 | C2×D83 | — | 8 | 1 | no | — |
| 127 | C2×D83 | — | 8 | 4 | no | — |
| 128 | C4×C83 | — | 2 | 1 | no | — |
| 129 | C4×C83 | — | 4 | 1 | no | — |
| 130 | C4×C83 | — | 4 | 1 | no | — |
| 131 | C4×C83 | — | 8 | 1 | no | — |
| 132 | Dic83 | — | 2 | 2 | no | — |
| 133 | Dic83 | — | 4 | 1 | no | — |
| 134 | Dic83 | — | 8 | 1 | no | — |
| 135 | C2×C83 | — | 2 | 1 | no | — |
| 136 | C2×C83 | — | 4 | 1 | no | — |
| 137 | C2×C83 | — | 4 | 1 | no | — |
| 138 | C2×C83 | — | 4 | 2 | no | — |
| 139 | C2×C83 | — | 8 | 1 | no | — |
| 140 | C2×C83 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 141 | D83 | — | 4 | 2 | no | — |
| 142 | D83 | — | 8 | 1 | no | — |
| 143 | #2196 | — | 83 | 1 | no | maximal, Sylow |
| 144 | C83 | #2196 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 145 | D42 | — | 83 | 1 | no | — |
| 146 | B(4,𝔽2) | — | 83 | 1 | no | — |
| 147 | B(4,𝔽2) | D83 | 1 | 1 | yes | characteristic, complemented |
| 148 | #453 | — | 83 | 1 | no | — |
| 149 | #353 | — | 83 | 1 | no | — |
| 150 | AΣL(1,𝔽16) | — | 83 | 1 | no | — |
| 151 | AΣL(1,𝔽16) | — | 83 | 1 | no | — |
| 152 | H(5,𝔽2) | C2×D83 | 1 | 1 | yes | characteristic, complemented |
| 153 | C22×D4 | — | 166 | 1 | no | — |
| 154 | C22×D4 | — | 166 | 1 | no | — |
| 155 | AGL(1,ℤ/8) | — | 166 | 1 | no | — |
| 156 | #127 | — | 83 | 1 | no | — |
| 157 | #121 | — | 166 | 1 | no | — |
| 158 | #121 | — | 166 | 1 | no | — |
| 159 | #120 | — | 2 | 1 | no | — |
| 160 | #120 | — | 83 | 1 | no | — |
| 161 | #120 | — | 166 | 1 | no | — |
| 162 | #120 | — | 166 | 1 | no | — |
| 163 | #120 | C2×D83 | 1 | 1 | yes | characteristic |
| 164 | C4×D4 | — | 166 | 1 | no | — |
| 165 | #104 | — | 166 | 1 | no | — |
| 166 | #100 | — | 83 | 1 | no | — |
| 167 | #99 | — | 2 | 1 | no | — |
| 168 | #99 | — | 83 | 1 | no | — |
| 169 | #99 | — | 166 | 1 | no | — |
| 170 | #99 | C2×D83 | 1 | 1 | yes | characteristic, complemented |
| 171 | C24 | — | 83 | 1 | no | — |
| 172 | C24 | — | 166 | 1 | no | — |
| 173 | C24 | C22⋊D83 | 1 | 1 | yes | characteristic, complemented |
| 174 | #41 | — | 2 | 1 | no | — |
| 175 | #41 | — | 4 | 1 | no | — |
| 176 | C2×D4 | — | 2 | 1 | no | — |
| 177 | C2×D4 | — | 2 | 1 | no | — |
| 178 | C2×D4 | — | 2 | 1 | no | — |
| 179 | C2×D4 | — | 4 | 1 | no | — |
| 180 | C2×D4 | — | 4 | 1 | no | — |
| 181 | C2×D4 | — | 332 | 4 | no | — |
| 182 | C2×D4 | — | 332 | 4 | no | — |
| 183 | C2×D4 | — | 332 | 1 | no | — |
| 184 | C2×D4 | — | 83 | 1 | no | — |
| 185 | C2×D4 | — | 166 | 1 | no | — |
| 186 | C2×D4 | — | 166 | 1 | no | — |
| 187 | C2×D4 | — | 166 | 1 | no | — |
| 188 | C2×D4 | — | 166 | 1 | no | — |
| 189 | C2×D4 | — | 166 | 1 | no | — |
| 190 | C2×D4 | — | 166 | 1 | no | — |
| 191 | C2×D4 | C22×D83 | 1 | 1 | yes | characteristic |
| 192 | C22×C4 | — | 166 | 1 | no | — |
| 193 | C22×C4 | — | 166 | 1 | no | — |
| 194 | SD16 | — | 332 | 1 | no | — |
| 195 | D8 | — | 332 | 1 | no | — |
| 196 | M16 | — | 166 | 1 | no | — |
| 197 | C4⋊C4 | — | 166 | 1 | no | — |
| 198 | C22⋊C4 | — | 2 | 1 | no | — |
| 199 | C22⋊C4 | — | 2 | 1 | no | — |
| 200 | C22⋊C4 | — | 4 | 1 | no | — |
| 201 | C22⋊C4 | — | 332 | 1 | no | — |
| 202 | C22⋊C4 | — | 83 | 1 | no | — |
| 203 | C22⋊C4 | — | 166 | 1 | no | — |
| 204 | C22⋊C4 | — | 166 | 1 | no | — |
| 205 | C22⋊C4 | — | 166 | 1 | no | — |
| 206 | C22⋊C4 | C22⋊D83 | 1 | 1 | yes | characteristic, complemented |
| 207 | C42 | — | 83 | 1 | no | — |
| 208 | C23 | — | 2 | 1 | no | — |
| 209 | C23 | — | 2 | 1 | no | — |
| 210 | C23 | — | 2 | 1 | no | — |
| 211 | C23 | — | 4 | 1 | no | — |
| 212 | C23 | — | 4 | 2 | no | — |
| 213 | C23 | — | 332 | 1 | no | — |
| 214 | C23 | — | 332 | 2 | no | — |
| 215 | C23 | — | 332 | 2 | no | — |
| 216 | C23 | — | 332 | 2 | no | — |
| 217 | C23 | — | 664 | 1 | no | — |
| 218 | C23 | — | 166 | 1 | no | — |
| 219 | C23 | — | 166 | 1 | no | — |
| 220 | C23 | D4×D83 | 1 | 1 | yes | characteristic |
| 221 | C23 | C2×C22⋊D83 | 1 | 1 | yes | characteristic |
| 222 | Q8 | — | 2 | 1 | no | — |
| 223 | D4 | — | 2 | 1 | no | — |
| 224 | D4 | — | 4 | 1 | no | — |
| 225 | D4 | — | 4 | 1 | no | — |
| 226 | D4 | — | 4 | 1 | no | — |
| 227 | D4 | — | 4 | 2 | no | — |
| 228 | D4 | — | 4 | 1 | no | — |
| 229 | D4 | — | 8 | 2 | no | — |
| 230 | D4 | — | 332 | 1 | no | — |
| 231 | D4 | — | 332 | 2 | no | — |
| 232 | D4 | — | 332 | 2 | no | — |
| 233 | D4 | — | 332 | 2 | no | — |
| 234 | D4 | — | 332 | 2 | no | — |
| 235 | D4 | — | 332 | 2 | no | — |
| 236 | D4 | — | 664 | 2 | no | — |
| 237 | D4 | — | 166 | 4 | no | — |
| 238 | C2×C4 | — | 2 | 1 | no | — |
| 239 | C2×C4 | — | 2 | 1 | no | — |
| 240 | C2×C4 | — | 2 | 1 | no | — |
| 241 | C2×C4 | — | 4 | 1 | no | — |
| 242 | C2×C4 | — | 4 | 1 | no | — |
| 243 | C2×C4 | — | 332 | 2 | no | — |
| 244 | C2×C4 | — | 332 | 2 | no | — |
| 245 | C2×C4 | — | 332 | 1 | no | — |
| 246 | C2×C4 | — | 166 | 1 | no | — |
| 247 | C2×C4 | — | 166 | 1 | no | — |
| 248 | C2×C4 | D4×D83 | 1 | 1 | yes | characteristic |
| 249 | C8 | — | 332 | 1 | no | — |
| 250 | C22 | — | 2 | 2 | no | — |
| 251 | C22 | — | 2 | 1 | no | — |
| 252 | C22 | — | 2 | 1 | no | — |
| 253 | C22 | — | 4 | 1 | no | — |
| 254 | C22 | — | 4 | 1 | no | — |
| 255 | C22 | — | 4 | 1 | no | — |
| 256 | C22 | — | 4 | 2 | no | — |
| 257 | C22 | — | 4 | 1 | no | — |
| 258 | C22 | — | 4 | 1 | no | — |
| 259 | C22 | — | 8 | 1 | no | — |
| 260 | C22 | — | 8 | 1 | no | — |
| 261 | C22 | — | 8 | 1 | no | — |
| 262 | C22 | — | 332 | 1 | no | — |
| 263 | C22 | — | 332 | 1 | no | — |
| 264 | C22 | — | 332 | 1 | no | — |
| 265 | C22 | — | 332 | 2 | no | — |
| 266 | C22 | — | 332 | 1 | no | — |
| 267 | C22 | — | 332 | 1 | no | — |
| 268 | C22 | — | 664 | 1 | no | — |
| 269 | C22 | — | 664 | 4 | no | — |
| 270 | C22 | — | 166 | 2 | no | — |
| 271 | C22 | #14581760 | 1 | 1 | yes | characteristic |
| 272 | C4 | — | 2 | 1 | no | — |
| 273 | C4 | — | 4 | 1 | no | — |
| 274 | C4 | — | 4 | 1 | no | — |
| 275 | C4 | — | 8 | 1 | no | — |
| 276 | C4 | — | 332 | 1 | no | — |
| 277 | C4 | — | 664 | 1 | no | — |
| 278 | C4 | — | 166 | 2 | no | — |
| 279 | C2 | — | 2 | 1 | no | — |
| 280 | C2 | — | 4 | 1 | no | — |
| 281 | C2 | — | 4 | 1 | no | — |
| 282 | C2 | — | 4 | 2 | no | — |
| 283 | C2 | — | 8 | 1 | no | — |
| 284 | C2 | — | 332 | 2 | no | — |
| 285 | C2 | — | 664 | 1 | no | — |
| 286 | C2 | #19127887 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #19128333 | C2 | semidirect |
| 2. #19127888 | C2 | semidirect |
| 3. #19127869 | C2 | semidirect |
| 4. #19127493 | C2 | semidirect |
| 5. #19127488 | C2 | semidirect |
| 6. #19127487 | C2 | semidirect |
| 7. B(4,𝔽2)×C83 | C2 | semidirect |
| 19. #14581709 | C22 | semidirect |
| 21. #14581707 | C22 | semidirect |
| 22. H(5,𝔽2)×C83 | C22 | semidirect |
| 26. #99×C83 | C22 | semidirect |
| 28. C23×D83 | D4 | semidirect |
| 30. C2×D332 | D4 | semidirect |
| 46. C22⋊Dic83 | D4 | semidirect |
| 50. C4×Dic83 | D4 | semidirect |
| 51. C24×C83 | D4 | semidirect |
| 63. C22⋊C4×C83 | D4 | semidirect |
| 144. C83 | #2196 | semidirect |
| 147. B(4,𝔽2) | D83 | semidirect |
| 152. H(5,𝔽2) | C2×D83 | semidirect |
| 170. #99 | C2×D83 | semidirect |
| 173. C24 | C22⋊D83 | semidirect |
| 206. C22⋊C4 | C22⋊D83 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #19128333 | 2 |
| #19127888 | 2 |
| #19127869 | 2 |
| #19127493 | 2 |
| #19127488 | 2 |
| #19127487 | 2 |
| B(4,𝔽2)×C83 | 2 |
| #2196 | 83 |