← #12252006
Subgroups — #12252006
Order 1280.
1114 conjugacy classes of subgroups.
The subgroup lattice has 230 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C5×#3573 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10615511 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#65429 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10602799 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C2×#64781 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | C5×#585 | C4 | 1 | 1 | yes | characteristic |
| 7 | C5×#568 | — | 2 | 4 | no | — |
| 8 | C5×#568 | C4 | 1 | 6 | yes | — |
| 9 | C5×#568 | C4 | 1 | 1 | yes | characteristic |
| 10 | C2×C5×#131 | — | 2 | 4 | no | — |
| 11 | C2×C5×#131 | C22 | 1 | 6 | yes | — |
| 12 | C2×C5×#131 | C22 | 1 | 1 | yes | characteristic |
| 13 | #65429 | — | 2 | 4 | no | — |
| 14 | #65429 | — | 2 | 4 | no | — |
| 15 | C2×#3944 | C22 | 1 | 6 | yes | complemented |
| 16 | C2×#3944 | C22 | 1 | 3 | yes | complemented |
| 17 | C22×#672 | C22 | 1 | 3 | yes | complemented |
| 18 | #64781 | C22 | 1 | 4 | yes | complemented |
| 19 | #64781 | C22 | 1 | 12 | yes | complemented |
| 20 | #33340 | — | 5 | 1 | no | maximal, Sylow |
| 21 | C5×2−1+4 | — | 2 | 1 | no | — |
| 22 | C5×2−1+4 | C8 | 1 | 4 | yes | — |
| 23 | C5×H(5,𝔽2) | — | 2 | 3 | no | — |
| 24 | C5×H(5,𝔽2) | C8 | 1 | 4 | yes | — |
| 25 | C2×C5×#41 | — | 2 | 4 | no | — |
| 26 | C2×C5×#41 | C2×C4 | 1 | 6 | yes | — |
| 27 | C2×C5×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C5×#131 | — | 2 | 4 | no | — |
| 29 | C5×#131 | — | 2 | 12 | no | — |
| 30 | C5×#131 | — | 2 | 6 | no | — |
| 31 | C5×#131 | — | 2 | 6 | no | — |
| 32 | C5×#131 | C2×C4 | 1 | 12 | yes | — |
| 33 | C5×#131 | D4 | 1 | 4 | yes | — |
| 34 | C5×#131 | D4 | 1 | 4 | yes | — |
| 35 | C5×#131 | C23 | 1 | 4 | yes | — |
| 36 | C2×C5×M16 | — | 2 | 12 | no | — |
| 37 | C2×C5×M16 | — | 2 | 6 | no | — |
| 38 | C2×C5×M16 | C2×C4 | 1 | 6 | yes | — |
| 39 | C2×C5×M16 | C2×C4 | 1 | 3 | yes | — |
| 40 | C22×C5×C8 | — | 2 | 6 | no | — |
| 41 | C22×C5×C8 | C23 | 1 | 3 | yes | — |
| 42 | #3944 | — | 2 | 6 | no | — |
| 43 | #3944 | — | 2 | 6 | no | — |
| 44 | #3944 | C23 | 1 | 4 | yes | — |
| 45 | C2×#672 | — | 2 | 6 | no | — |
| 46 | C2×#672 | — | 2 | 6 | no | — |
| 47 | C2×#672 | C23 | 1 | 1 | yes | characteristic |
| 48 | C2×#672 | C23 | 1 | 3 | yes | — |
| 49 | #3573 | D5 | 1 | 1 | yes | characteristic |
| 50 | #3407 | — | 5 | 1 | no | — |
| 51 | C2×#504 | — | 5 | 1 | no | — |
| 52 | #2117 | — | 5 | 6 | no | — |
| 53 | C2×#350 | — | 5 | 6 | no | — |
| 54 | C5×#41 | — | 2 | 4 | no | — |
| 55 | C5×#41 | — | 2 | 12 | no | — |
| 56 | C5×#41 | — | 2 | 6 | no | — |
| 57 | C5×#41 | — | 2 | 6 | no | — |
| 58 | C5×#41 | C22⋊C4 | 1 | 4 | yes | — |
| 59 | C5×#41 | C2×C8 | 1 | 12 | yes | — |
| 60 | C5×#41 | C2×C8 | 1 | 4 | yes | — |
| 61 | C5×#41 | C22×C4 | 1 | 4 | yes | — |
| 62 | C2×C5×Q8 | — | 2 | 4 | no | — |
| 63 | C2×C5×Q8 | C2×C8 | 1 | 6 | yes | — |
| 64 | C2×C5×Q8 | M16 | 1 | 1 | yes | characteristic |
| 65 | C2×C5×D4 | — | 2 | 6 | no | — |
| 66 | C2×C5×D4 | — | 2 | 12 | no | — |
| 67 | C2×C5×D4 | C2×C8 | 1 | 6 | yes | — |
| 68 | C2×C5×D4 | M16 | 1 | 3 | yes | — |
| 69 | C22×C4×C5 | — | 2 | 6 | no | — |
| 70 | C22×C4×C5 | C22×C4 | 1 | 3 | yes | — |
| 71 | C5×M16 | — | 2 | 6 | no | — |
| 72 | C5×M16 | — | 2 | 12 | no | — |
| 73 | C5×M16 | — | 2 | 4 | no | — |
| 74 | C5×M16 | C22⋊C4 | 1 | 12 | yes | — |
| 75 | C5×M16 | C22×C4 | 1 | 4 | yes | — |
| 76 | C2×C5×C8 | — | 2 | 6 | no | — |
| 77 | C2×C5×C8 | — | 2 | 12 | no | — |
| 78 | C2×C5×C8 | — | 2 | 4 | no | — |
| 79 | C2×C5×C8 | C22×C4 | 1 | 3 | yes | — |
| 80 | C2×C5×C8 | C2×D4 | 1 | 6 | yes | — |
| 81 | C2×C5×C8 | C2×D4 | 1 | 6 | yes | — |
| 82 | C2×C5×C8 | C24 | 1 | 1 | yes | characteristic |
| 83 | #672 | — | 2 | 1 | no | — |
| 84 | #672 | — | 2 | 4 | no | — |
| 85 | #672 | — | 2 | 3 | no | — |
| 86 | #585 | Dic5 | 1 | 1 | yes | characteristic |
| 87 | #568 | — | 2 | 4 | no | — |
| 88 | #568 | Dic5 | 1 | 6 | yes | — |
| 89 | #568 | Dic5 | 1 | 1 | yes | characteristic |
| 90 | C2×#131 | — | 2 | 4 | no | — |
| 91 | C2×#131 | C2×D5 | 1 | 6 | yes | — |
| 92 | C2×#131 | C2×D5 | 1 | 1 | yes | characteristic |
| 93 | #504 | — | 10 | 4 | no | — |
| 94 | #504 | — | 10 | 4 | no | — |
| 95 | C2×M32 | — | 5 | 3 | no | — |
| 96 | C2×M32 | — | 5 | 6 | no | — |
| 97 | C22×C16 | — | 5 | 3 | no | — |
| 98 | #350 | — | 5 | 4 | no | — |
| 99 | #350 | — | 5 | 12 | no | — |
| 100 | C23×C5 | — | 2 | 6 | no | — |
| 101 | C23×C5 | C2×M16 | 1 | 3 | yes | — |
| 102 | C5×Q8 | — | 2 | 6 | no | — |
| 103 | C5×Q8 | C22⋊C8 | 1 | 4 | yes | — |
| 104 | C5×Q8 | C22×C8 | 1 | 4 | yes | — |
| 105 | C5×D4 | — | 2 | 4 | no | — |
| 106 | C5×D4 | — | 2 | 12 | no | — |
| 107 | C5×D4 | — | 2 | 6 | no | — |
| 108 | C5×D4 | C22⋊C8 | 1 | 12 | yes | — |
| 109 | C5×D4 | C22×C8 | 1 | 4 | yes | — |
| 110 | C2×C4×C5 | — | 2 | 6 | no | — |
| 111 | C2×C4×C5 | — | 2 | 4 | no | — |
| 112 | C2×C4×C5 | — | 2 | 12 | no | — |
| 113 | C2×C4×C5 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 114 | C2×C4×C5 | C22×C8 | 1 | 6 | yes | — |
| 115 | C2×C4×C5 | C2×M16 | 1 | 3 | yes | — |
| 116 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 117 | C5×C8 | — | 2 | 4 | no | — |
| 118 | C5×C8 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 119 | C5×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 120 | C5×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 121 | 2−1+4 | — | 2 | 1 | no | — |
| 122 | 2−1+4 | #168 | 1 | 4 | yes | — |
| 123 | H(5,𝔽2) | — | 2 | 3 | no | — |
| 124 | H(5,𝔽2) | #168 | 1 | 4 | yes | — |
| 125 | C2×#41 | — | 2 | 4 | no | — |
| 126 | C2×#41 | C2×Dic5 | 1 | 6 | yes | — |
| 127 | C2×#41 | C2×Dic5 | 1 | 1 | yes | characteristic |
| 128 | #131 | — | 2 | 4 | no | — |
| 129 | #131 | — | 2 | 12 | no | — |
| 130 | #131 | — | 2 | 6 | no | — |
| 131 | #131 | — | 2 | 6 | no | — |
| 132 | #131 | C2×Dic5 | 1 | 12 | yes | — |
| 133 | #131 | C22⋊D5 | 1 | 4 | yes | — |
| 134 | #131 | C22⋊D5 | 1 | 4 | yes | — |
| 135 | #131 | C22×D5 | 1 | 4 | yes | — |
| 136 | C2×M16 | — | 2 | 12 | no | — |
| 137 | C2×M16 | — | 2 | 6 | no | — |
| 138 | C2×M16 | C2×Dic5 | 1 | 6 | yes | — |
| 139 | C2×M16 | C2×Dic5 | 1 | 3 | yes | — |
| 140 | C22×C8 | — | 2 | 6 | no | — |
| 141 | C22×C8 | C22×D5 | 1 | 3 | yes | — |
| 142 | M32 | — | 5 | 4 | no | — |
| 143 | M32 | — | 10 | 6 | no | — |
| 144 | M32 | — | 10 | 6 | no | — |
| 145 | C2×C16 | — | 5 | 1 | no | — |
| 146 | C2×C16 | — | 5 | 3 | no | — |
| 147 | C2×C16 | — | 10 | 6 | no | — |
| 148 | C2×C16 | — | 10 | 6 | no | — |
| 149 | C22×C5 | — | 2 | 4 | no | — |
| 150 | C22×C5 | — | 4 | 3 | no | — |
| 151 | C22×C5 | — | 8 | 6 | no | — |
| 152 | C22×C5 | C2×C22⋊C8 | 1 | 6 | yes | — |
| 153 | C22×C5 | C22×M16 | 1 | 1 | yes | characteristic |
| 154 | C4×C5 | — | 2 | 4 | no | — |
| 155 | C4×C5 | C2×C22⋊C8 | 1 | 6 | yes | — |
| 156 | C4×C5 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 157 | C4×C5 | C23×C8 | 1 | 1 | yes | characteristic |
| 158 | #41 | — | 2 | 4 | no | — |
| 159 | #41 | — | 2 | 12 | no | — |
| 160 | #41 | — | 2 | 6 | no | — |
| 161 | #41 | — | 2 | 6 | no | — |
| 162 | #41 | C2×#168 | 1 | 12 | yes | — |
| 163 | #41 | C2×#168 | 1 | 4 | yes | — |
| 164 | #41 | C22⋊Dic5 | 1 | 4 | yes | — |
| 165 | #41 | C22×Dic5 | 1 | 4 | yes | — |
| 166 | C2×Q8 | — | 2 | 4 | no | — |
| 167 | C2×Q8 | C2×#168 | 1 | 6 | yes | — |
| 168 | C2×Q8 | #681 | 1 | 1 | yes | characteristic |
| 169 | C2×D4 | — | 2 | 6 | no | — |
| 170 | C2×D4 | — | 2 | 12 | no | — |
| 171 | C2×D4 | C2×#168 | 1 | 6 | yes | — |
| 172 | C2×D4 | #681 | 1 | 3 | yes | — |
| 173 | C22×C4 | — | 2 | 6 | no | — |
| 174 | C22×C4 | C22×Dic5 | 1 | 3 | yes | — |
| 175 | M16 | — | 2 | 6 | no | — |
| 176 | M16 | — | 2 | 12 | no | — |
| 177 | M16 | — | 2 | 4 | no | — |
| 178 | M16 | C22⋊Dic5 | 1 | 12 | yes | — |
| 179 | M16 | C22×Dic5 | 1 | 4 | yes | — |
| 180 | C2×C8 | — | 2 | 6 | no | — |
| 181 | C2×C8 | — | 2 | 12 | no | — |
| 182 | C2×C8 | — | 2 | 4 | no | — |
| 183 | C2×C8 | C22×Dic5 | 1 | 3 | yes | — |
| 184 | C2×C8 | C2×C22⋊D5 | 1 | 6 | yes | — |
| 185 | C2×C8 | C2×C22⋊D5 | 1 | 6 | yes | — |
| 186 | C2×C8 | C23×D5 | 1 | 1 | yes | characteristic |
| 187 | C16 | — | 10 | 1 | no | — |
| 188 | C16 | — | 10 | 3 | no | — |
| 189 | C16 | — | 10 | 4 | no | — |
| 190 | C2×C5 | — | 2 | 1 | no | — |
| 191 | C2×C5 | — | 2 | 6 | no | — |
| 192 | C2×C5 | — | 4 | 4 | no | — |
| 193 | C2×C5 | C22×C22⋊C8 | 1 | 1 | yes | characteristic |
| 194 | C23 | — | 2 | 6 | no | — |
| 195 | C23 | C2×#681 | 1 | 3 | yes | — |
| 196 | Q8 | — | 2 | 6 | no | — |
| 197 | Q8 | C22⋊#168 | 1 | 4 | yes | — |
| 198 | Q8 | C22×#168 | 1 | 4 | yes | — |
| 199 | D4 | — | 2 | 4 | no | — |
| 200 | D4 | — | 2 | 12 | no | — |
| 201 | D4 | — | 2 | 6 | no | — |
| 202 | D4 | C22⋊#168 | 1 | 12 | yes | — |
| 203 | D4 | C22×#168 | 1 | 4 | yes | — |
| 204 | C2×C4 | — | 2 | 6 | no | — |
| 205 | C2×C4 | — | 2 | 4 | no | — |
| 206 | C2×C4 | — | 2 | 12 | no | — |
| 207 | C2×C4 | C22×#168 | 1 | 6 | yes | — |
| 208 | C2×C4 | C2×#681 | 1 | 3 | yes | — |
| 209 | C2×C4 | C2×C22⋊Dic5 | 1 | 6 | yes | — |
| 210 | C2×C4 | C23×Dic5 | 1 | 1 | yes | characteristic |
| 211 | C8 | — | 2 | 4 | no | — |
| 212 | C8 | C2×C22⋊Dic5 | 1 | 6 | yes | — |
| 213 | C8 | C22×C22⋊D5 | 1 | 1 | yes | characteristic |
| 214 | C8 | C22×C22⋊D5 | 1 | 1 | yes | characteristic |
| 215 | C5 | #33340 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 216 | C22 | — | 2 | 4 | no | — |
| 217 | C22 | — | 4 | 3 | no | — |
| 218 | C22 | — | 8 | 6 | no | — |
| 219 | C22 | C2×C22⋊#168 | 1 | 6 | yes | — |
| 220 | C22 | C22×#681 | 1 | 1 | yes | characteristic |
| 221 | C4 | — | 2 | 4 | no | — |
| 222 | C4 | C2×C22⋊#168 | 1 | 6 | yes | — |
| 223 | C4 | C23×#168 | 1 | 1 | yes | characteristic |
| 224 | C4 | C22×C22⋊Dic5 | 1 | 1 | yes | characteristic |
| 225 | C2 | — | 2 | 1 | no | — |
| 226 | C2 | — | 2 | 6 | no | — |
| 227 | C2 | — | 4 | 4 | no | — |
| 228 | C2 | C22×C22⋊#168 | 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