← C24×#444 #72286451
Subgroups — C24×#444 #72286451
Order 1024.
111589 conjugacy classes of subgroups.
The subgroup lattice has 266 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C24×#135 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C24×#131 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C23×#444 | C2 | 1 | 120 | yes | maximal, direct product factor |
| 4 | C24×#108 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C24×#104 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C24×#97 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C24×#41 | C22 | 1 | 2 | yes | — |
| 8 | C23×#135 | C4 | 1 | 60 | yes | complemented |
| 9 | C23×#135 | C22 | 1 | 60 | yes | — |
| 10 | C24×Q16 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | C24×SD16 | C4 | 1 | 2 | yes | complemented |
| 12 | C24×D8 | C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | C23×#131 | — | 2 | 60 | no | — |
| 14 | C23×#131 | C22 | 1 | 120 | yes | complemented |
| 15 | C24×M16 | — | 2 | 2 | no | — |
| 16 | C24×M16 | C22 | 1 | 2 | yes | — |
| 17 | C25×C8 | — | 2 | 2 | no | — |
| 18 | C25×C8 | C22 | 1 | 1 | yes | characteristic |
| 19 | C25×C8 | C22 | 1 | 1 | yes | characteristic |
| 20 | C24×C42 | C22 | 1 | 1 | yes | characteristic |
| 21 | C22×#444 | C22 | 1 | 2240 | yes | direct product factor |
| 22 | C23×#108 | C22 | 1 | 60 | yes | complemented |
| 23 | C23×#104 | C22 | 1 | 120 | yes | complemented |
| 24 | C23×#97 | C22 | 1 | 60 | yes | complemented |
| 25 | C23×#41 | — | 2 | 30 | no | — |
| 26 | C23×#41 | — | 2 | 30 | no | — |
| 27 | C23×#41 | C2×C4 | 1 | 60 | yes | — |
| 28 | C23×#41 | C23 | 1 | 60 | yes | — |
| 29 | C24×Q8 | C2×C4 | 1 | 2 | yes | — |
| 30 | C24×D4 | — | 2 | 2 | no | — |
| 31 | C24×D4 | C2×C4 | 1 | 2 | yes | — |
| 32 | C25×C4 | — | 2 | 2 | no | — |
| 33 | C25×C4 | — | 2 | 1 | no | — |
| 34 | C25×C4 | C23 | 1 | 1 | yes | characteristic |
| 35 | C22×#135 | C2×C4 | 1 | 1680 | yes | complemented |
| 36 | C22×#135 | C23 | 1 | 560 | yes | — |
| 37 | C23×Q16 | C2×C4 | 1 | 60 | yes | complemented |
| 38 | C23×SD16 | C2×C4 | 1 | 120 | yes | complemented |
| 39 | C23×D8 | C2×C4 | 1 | 60 | yes | complemented |
| 40 | C22×#131 | — | 2 | 1680 | no | — |
| 41 | C22×#131 | C23 | 1 | 1120 | yes | complemented |
| 42 | C23×M16 | — | 2 | 30 | no | — |
| 43 | C23×M16 | — | 2 | 120 | no | — |
| 44 | C23×M16 | C23 | 1 | 60 | yes | — |
| 45 | C24×C8 | — | 2 | 2 | no | — |
| 46 | C24×C8 | — | 2 | 120 | no | — |
| 47 | C24×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | C24×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C24×C8 | C2×C4 | 1 | 30 | yes | — |
| 50 | C24×C8 | D4 | 1 | 30 | yes | — |
| 51 | C24×C8 | D4 | 1 | 2 | yes | — |
| 52 | C24×C8 | C23 | 1 | 30 | yes | — |
| 53 | C24×C8 | C23 | 1 | 30 | yes | — |
| 54 | C23×C42 | — | 2 | 15 | no | — |
| 55 | C23×C42 | C23 | 1 | 30 | yes | — |
| 56 | C2×#444 | C23 | 1 | 7680 | yes | direct product factor |
| 57 | C22×#108 | C23 | 1 | 560 | yes | complemented |
| 58 | C22×#104 | C23 | 1 | 1120 | yes | complemented |
| 59 | C22×#97 | C23 | 1 | 560 | yes | complemented |
| 60 | C26 | — | 2 | 2 | no | — |
| 61 | C26 | #41 | 1 | 1 | yes | characteristic |
| 62 | C22×#41 | — | 2 | 840 | no | — |
| 63 | C22×#41 | — | 2 | 420 | no | — |
| 64 | C22×#41 | — | 2 | 420 | no | — |
| 65 | C22×#41 | C22×C4 | 1 | 840 | yes | — |
| 66 | C22×#41 | C24 | 1 | 280 | yes | — |
| 67 | C23×Q8 | — | 2 | 30 | no | — |
| 68 | C23×Q8 | C22×C4 | 1 | 60 | yes | — |
| 69 | C23×D4 | — | 2 | 60 | no | — |
| 70 | C23×D4 | — | 2 | 60 | no | — |
| 71 | C23×D4 | — | 2 | 30 | no | — |
| 72 | C23×D4 | C22×C4 | 1 | 60 | yes | — |
| 73 | C24×C4 | — | 2 | 2 | no | — |
| 74 | C24×C4 | — | 2 | 60 | no | — |
| 75 | C24×C4 | — | 2 | 60 | no | — |
| 76 | C24×C4 | — | 2 | 30 | no | — |
| 77 | C24×C4 | — | 4 | 16 | no | — |
| 78 | C24×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 79 | C24×C4 | C22×C4 | 1 | 15 | yes | — |
| 80 | C24×C4 | C2×D4 | 1 | 15 | yes | — |
| 81 | C24×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C24×C4 | #41 | 1 | 15 | yes | — |
| 83 | C24×C4 | C24 | 1 | 15 | yes | — |
| 84 | C2×#135 | C22×C4 | 1 | 6720 | yes | complemented |
| 85 | C2×#135 | C24 | 1 | 960 | yes | — |
| 86 | C22×Q16 | C22×C4 | 1 | 560 | yes | complemented |
| 87 | C22×SD16 | C22×C4 | 1 | 1120 | yes | complemented |
| 88 | C22×D8 | C22×C4 | 1 | 560 | yes | complemented |
| 89 | C2×#131 | — | 2 | 6720 | no | — |
| 90 | C2×#131 | C24 | 1 | 1920 | yes | complemented |
| 91 | C22×M16 | — | 2 | 420 | no | — |
| 92 | C22×M16 | — | 2 | 1120 | no | — |
| 93 | C22×M16 | C24 | 1 | 280 | yes | — |
| 94 | C23×C8 | — | 2 | 60 | no | — |
| 95 | C23×C8 | — | 2 | 1120 | no | — |
| 96 | C23×C8 | C22×C4 | 1 | 30 | yes | — |
| 97 | C23×C8 | C22×C4 | 1 | 30 | yes | — |
| 98 | C23×C8 | C22×C4 | 1 | 420 | yes | — |
| 99 | C23×C8 | C2×D4 | 1 | 60 | yes | — |
| 100 | C23×C8 | C2×D4 | 1 | 420 | yes | — |
| 101 | C23×C8 | C24 | 1 | 140 | yes | — |
| 102 | C23×C8 | C24 | 1 | 140 | yes | — |
| 103 | C22×C42 | — | 2 | 210 | no | — |
| 104 | C22×C42 | C24 | 1 | 140 | yes | — |
| 105 | #444 | C24 | 1 | 4096 | yes | direct product factor |
| 106 | C2×#108 | C24 | 1 | 960 | yes | complemented |
| 107 | C2×#104 | C24 | 1 | 1920 | yes | complemented |
| 108 | C2×#97 | C24 | 1 | 960 | yes | complemented |
| 109 | C25 | — | 2 | 16 | no | — |
| 110 | C25 | — | 2 | 60 | no | — |
| 111 | C25 | — | 4 | 32 | no | — |
| 112 | C25 | C4×D4 | 1 | 15 | yes | — |
| 113 | C25 | C4×D4 | 1 | 1 | yes | characteristic |
| 114 | C25 | C2×#41 | 1 | 15 | yes | — |
| 115 | C2×#41 | — | 2 | 5040 | no | — |
| 116 | C2×#41 | — | 2 | 840 | no | — |
| 117 | C2×#41 | — | 2 | 840 | no | — |
| 118 | C2×#41 | C23×C4 | 1 | 1680 | yes | — |
| 119 | C2×#41 | C25 | 1 | 240 | yes | — |
| 120 | C22×Q8 | — | 2 | 420 | no | — |
| 121 | C22×Q8 | C23×C4 | 1 | 280 | yes | — |
| 122 | C22×D4 | — | 2 | 840 | no | — |
| 123 | C22×D4 | — | 2 | 280 | no | — |
| 124 | C22×D4 | — | 2 | 420 | no | — |
| 125 | C22×D4 | C23×C4 | 1 | 280 | yes | — |
| 126 | C23×C4 | — | 2 | 60 | no | — |
| 127 | C23×C4 | — | 2 | 840 | no | — |
| 128 | C23×C4 | — | 2 | 280 | no | — |
| 129 | C23×C4 | — | 2 | 140 | no | — |
| 130 | C23×C4 | — | 4 | 240 | no | — |
| 131 | C23×C4 | C4×D4 | 1 | 15 | yes | — |
| 132 | C23×C4 | C4×D4 | 1 | 15 | yes | — |
| 133 | C23×C4 | C4×D4 | 1 | 210 | yes | — |
| 134 | C23×C4 | C23×C4 | 1 | 15 | yes | — |
| 135 | C23×C4 | C23×C4 | 1 | 105 | yes | — |
| 136 | C23×C4 | C22×D4 | 1 | 15 | yes | — |
| 137 | C23×C4 | C22×D4 | 1 | 105 | yes | — |
| 138 | C23×C4 | C2×#41 | 1 | 105 | yes | — |
| 139 | C23×C4 | C25 | 1 | 35 | yes | — |
| 140 | #135 | C23×C4 | 1 | 3840 | yes | complemented |
| 141 | #135 | C25 | 1 | 256 | yes | — |
| 142 | C2×Q16 | C23×C4 | 1 | 960 | yes | complemented |
| 143 | C2×SD16 | C23×C4 | 1 | 1920 | yes | complemented |
| 144 | C2×D8 | C23×C4 | 1 | 960 | yes | complemented |
| 145 | #131 | — | 2 | 3840 | no | — |
| 146 | #131 | C25 | 1 | 512 | yes | complemented |
| 147 | C2×M16 | — | 2 | 840 | no | — |
| 148 | C2×M16 | — | 2 | 1920 | no | — |
| 149 | C2×M16 | C25 | 1 | 240 | yes | — |
| 150 | C22×C8 | — | 2 | 280 | no | — |
| 151 | C22×C8 | — | 2 | 1920 | no | — |
| 152 | C22×C8 | C23×C4 | 1 | 140 | yes | — |
| 153 | C22×C8 | C23×C4 | 1 | 140 | yes | — |
| 154 | C22×C8 | C23×C4 | 1 | 840 | yes | — |
| 155 | C22×C8 | C22×D4 | 1 | 280 | yes | — |
| 156 | C22×C8 | C22×D4 | 1 | 840 | yes | — |
| 157 | C22×C8 | C25 | 1 | 120 | yes | — |
| 158 | C22×C8 | C25 | 1 | 120 | yes | — |
| 159 | C2×C42 | — | 2 | 420 | no | — |
| 160 | C2×C42 | C25 | 1 | 120 | yes | — |
| 161 | #108 | C25 | 1 | 256 | yes | complemented |
| 162 | #104 | C25 | 1 | 512 | yes | complemented |
| 163 | #97 | C25 | 1 | 256 | yes | complemented |
| 164 | C24 | — | 2 | 120 | no | — |
| 165 | C24 | — | 2 | 120 | no | — |
| 166 | C24 | — | 2 | 280 | no | — |
| 167 | C24 | — | 4 | 480 | no | — |
| 168 | C24 | #444 | 1 | 16 | yes | direct product factor |
| 169 | C24 | C2×C4×D4 | 1 | 105 | yes | — |
| 170 | C24 | C2×C4×D4 | 1 | 15 | yes | — |
| 171 | C24 | C22×#41 | 1 | 35 | yes | — |
| 172 | #41 | — | 2 | 240 | no | — |
| 173 | #41 | — | 2 | 240 | no | — |
| 174 | #41 | — | 2 | 3360 | no | — |
| 175 | #41 | C24×C4 | 1 | 480 | yes | — |
| 176 | #41 | C26 | 1 | 32 | yes | — |
| 177 | C2×Q8 | — | 2 | 840 | no | — |
| 178 | C2×Q8 | C24×C4 | 1 | 240 | yes | — |
| 179 | C2×D4 | — | 2 | 240 | no | — |
| 180 | C2×D4 | — | 2 | 1680 | no | — |
| 181 | C2×D4 | — | 2 | 840 | no | — |
| 182 | C2×D4 | C24×C4 | 1 | 240 | yes | — |
| 183 | C22×C4 | — | 2 | 280 | no | — |
| 184 | C22×C4 | — | 2 | 240 | no | — |
| 185 | C22×C4 | — | 2 | 1680 | no | — |
| 186 | C22×C4 | — | 2 | 120 | no | — |
| 187 | C22×C4 | — | 4 | 560 | no | — |
| 188 | C22×C4 | C2×C4×D4 | 1 | 105 | yes | — |
| 189 | C22×C4 | C2×C4×D4 | 1 | 105 | yes | — |
| 190 | C22×C4 | C2×C4×D4 | 1 | 630 | yes | — |
| 191 | C22×C4 | C24×C4 | 1 | 35 | yes | — |
| 192 | C22×C4 | C24×C4 | 1 | 105 | yes | — |
| 193 | C22×C4 | C23×D4 | 1 | 35 | yes | — |
| 194 | C22×C4 | C23×D4 | 1 | 105 | yes | — |
| 195 | C22×C4 | C22×#41 | 1 | 105 | yes | — |
| 196 | C22×C4 | C26 | 1 | 15 | yes | — |
| 197 | Q16 | C24×C4 | 1 | 256 | yes | complemented |
| 198 | SD16 | C24×C4 | 1 | 512 | yes | complemented |
| 199 | D8 | C24×C4 | 1 | 256 | yes | complemented |
| 200 | M16 | — | 2 | 240 | no | — |
| 201 | M16 | — | 2 | 512 | no | — |
| 202 | M16 | C26 | 1 | 32 | yes | — |
| 203 | C2×C8 | — | 2 | 240 | no | — |
| 204 | C2×C8 | — | 2 | 512 | no | — |
| 205 | C2×C8 | C24×C4 | 1 | 120 | yes | — |
| 206 | C2×C8 | C24×C4 | 1 | 120 | yes | — |
| 207 | C2×C8 | C24×C4 | 1 | 240 | yes | — |
| 208 | C2×C8 | C23×D4 | 1 | 240 | yes | — |
| 209 | C2×C8 | C23×D4 | 1 | 240 | yes | — |
| 210 | C2×C8 | C26 | 1 | 16 | yes | — |
| 211 | C2×C8 | C26 | 1 | 16 | yes | — |
| 212 | C42 | — | 2 | 120 | no | — |
| 213 | C42 | C26 | 1 | 16 | yes | — |
| 214 | C23 | — | 2 | 140 | no | — |
| 215 | C23 | — | 2 | 420 | no | — |
| 216 | C23 | — | 2 | 240 | no | — |
| 217 | C23 | — | 4 | 1120 | no | — |
| 218 | C23 | C2×#444 | 1 | 120 | yes | direct product factor |
| 219 | C23 | C22×C4×D4 | 1 | 105 | yes | — |
| 220 | C23 | C22×C4×D4 | 1 | 35 | yes | — |
| 221 | C23 | C23×#41 | 1 | 15 | yes | — |
| 222 | Q8 | — | 2 | 240 | no | — |
| 223 | Q8 | C25×C4 | 1 | 32 | yes | — |
| 224 | D4 | — | 2 | 32 | no | — |
| 225 | D4 | — | 2 | 480 | no | — |
| 226 | D4 | — | 2 | 240 | no | — |
| 227 | D4 | C25×C4 | 1 | 32 | yes | — |
| 228 | C2×C4 | — | 2 | 240 | no | — |
| 229 | C2×C4 | — | 2 | 32 | no | — |
| 230 | C2×C4 | — | 2 | 480 | no | — |
| 231 | C2×C4 | — | 2 | 16 | no | — |
| 232 | C2×C4 | — | 4 | 240 | no | — |
| 233 | C2×C4 | C22×C4×D4 | 1 | 105 | yes | — |
| 234 | C2×C4 | C22×C4×D4 | 1 | 105 | yes | — |
| 235 | C2×C4 | C22×C4×D4 | 1 | 210 | yes | — |
| 236 | C2×C4 | C25×C4 | 1 | 15 | yes | — |
| 237 | C2×C4 | C25×C4 | 1 | 15 | yes | — |
| 238 | C2×C4 | C24×D4 | 1 | 15 | yes | — |
| 239 | C2×C4 | C24×D4 | 1 | 15 | yes | — |
| 240 | C2×C4 | C23×#41 | 1 | 15 | yes | — |
| 241 | C2×C4 | C27 | 1 | 1 | yes | characteristic |
| 242 | C8 | — | 2 | 32 | no | — |
| 243 | C8 | C25×C4 | 1 | 16 | yes | — |
| 244 | C8 | C25×C4 | 1 | 16 | yes | — |
| 245 | C8 | C24×D4 | 1 | 32 | yes | — |
| 246 | C22 | — | 2 | 30 | no | — |
| 247 | C22 | — | 2 | 210 | no | — |
| 248 | C22 | — | 2 | 32 | no | — |
| 249 | C22 | — | 4 | 480 | no | — |
| 250 | C22 | C22×#444 | 1 | 140 | yes | direct product factor |
| 251 | C22 | C23×C4×D4 | 1 | 15 | yes | — |
| 252 | C22 | C23×C4×D4 | 1 | 15 | yes | — |
| 253 | C22 | C24×#41 | 1 | 1 | yes | characteristic |
| 254 | C4 | — | 2 | 32 | no | — |
| 255 | C4 | — | 4 | 16 | no | — |
| 256 | C4 | C23×C4×D4 | 1 | 15 | yes | — |
| 257 | C4 | C23×C4×D4 | 1 | 15 | yes | — |
| 258 | C4 | C26×C4 | 1 | 1 | yes | characteristic |
| 259 | C4 | C25×D4 | 1 | 1 | yes | characteristic |
| 260 | C2 | — | 2 | 1 | no | — |
| 261 | C2 | — | 2 | 15 | no | — |
| 262 | C2 | — | 4 | 32 | no | — |
| 263 | C2 | C23×#444 | 1 | 30 | yes | direct product factor |
| 264 | C2 | C24×C4×D4 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta