← #10574082
Subgroups — #10574082
Order 512.
22492 conjugacy classes of subgroups.
The subgroup lattice has 221 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×#585 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#3573 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #60065 | C2 | 1 | 30 | yes | maximal, complemented |
| 4 | #60064 | C2 | 1 | 15 | yes | maximal, complemented |
| 5 | C2×#2880 | C2 | 1 | 15 | yes | maximal, complemented |
| 6 | C2×#585 | — | 2 | 32 | no | — |
| 7 | C2×#585 | C4 | 1 | 6 | yes | — |
| 8 | C2×#585 | C4 | 1 | 10 | yes | — |
| 9 | C2×#585 | C22 | 1 | 15 | yes | — |
| 10 | C2×#585 | C22 | 1 | 1 | yes | characteristic |
| 11 | C22×2−1+4 | C4 | 1 | 6 | yes | — |
| 12 | C22×H(5,𝔽2) | C4 | 1 | 10 | yes | — |
| 13 | C23×#41 | C22 | 1 | 15 | yes | — |
| 14 | #3573 | — | 2 | 32 | no | — |
| 15 | C2×#568 | C22 | 1 | 30 | yes | complemented |
| 16 | C22×#131 | C22 | 1 | 30 | yes | complemented |
| 17 | #2880 | C22 | 1 | 20 | yes | complemented |
| 18 | #2880 | C22 | 1 | 60 | yes | complemented |
| 19 | #2880 | C22 | 1 | 120 | yes | complemented |
| 20 | #2880 | C22 | 1 | 120 | yes | complemented |
| 21 | #2879 | C22 | 1 | 90 | yes | complemented |
| 22 | C2×#408 | C22 | 1 | 90 | yes | complemented |
| 23 | C2×#407 | C22 | 1 | 45 | yes | complemented |
| 24 | C22×C22⋊C8 | C22 | 1 | 15 | yes | complemented |
| 25 | #585 | — | 2 | 96 | no | — |
| 26 | #585 | — | 2 | 160 | no | — |
| 27 | #585 | — | 2 | 240 | no | — |
| 28 | #585 | D4 | 1 | 32 | yes | — |
| 29 | C2×2−1+4 | — | 2 | 96 | no | — |
| 30 | C2×2−1+4 | C2×C4 | 1 | 60 | yes | — |
| 31 | C2×2−1+4 | C2×C4 | 1 | 30 | yes | — |
| 32 | C2×2−1+4 | C2×C4 | 1 | 6 | yes | — |
| 33 | C2×H(5,𝔽2) | — | 2 | 160 | no | — |
| 34 | C2×H(5,𝔽2) | C2×C4 | 1 | 60 | yes | — |
| 35 | C2×H(5,𝔽2) | C2×C4 | 1 | 90 | yes | — |
| 36 | C2×H(5,𝔽2) | C2×C4 | 1 | 10 | yes | — |
| 37 | C22×#41 | — | 2 | 240 | no | — |
| 38 | C22×#41 | C2×C4 | 1 | 60 | yes | — |
| 39 | C22×#41 | C2×C4 | 1 | 60 | yes | — |
| 40 | C22×#41 | C2×C4 | 1 | 15 | yes | — |
| 41 | C22×#41 | C2×C4 | 1 | 45 | yes | — |
| 42 | C22×#41 | C23 | 1 | 20 | yes | — |
| 43 | C22×#41 | C23 | 1 | 45 | yes | — |
| 44 | C22×#41 | C23 | 1 | 60 | yes | — |
| 45 | C22×#41 | C23 | 1 | 15 | yes | — |
| 46 | C23×Q8 | C2×C4 | 1 | 15 | yes | — |
| 47 | C23×D4 | C2×C4 | 1 | 45 | yes | — |
| 48 | C24×C4 | C23 | 1 | 15 | yes | — |
| 49 | #568 | — | 2 | 240 | no | — |
| 50 | C2×#131 | — | 2 | 240 | no | — |
| 51 | C2×#131 | C23 | 1 | 40 | yes | complemented |
| 52 | C2×#131 | C23 | 1 | 120 | yes | complemented |
| 53 | C22×M16 | C23 | 1 | 90 | yes | complemented |
| 54 | C23×C8 | C23 | 1 | 30 | yes | complemented |
| 55 | #408 | C23 | 1 | 120 | yes | complemented |
| 56 | #408 | C23 | 1 | 360 | yes | complemented |
| 57 | #407 | C23 | 1 | 60 | yes | — |
| 58 | #407 | C23 | 1 | 180 | yes | complemented |
| 59 | C2×C22⋊C8 | C23 | 1 | 15 | yes | — |
| 60 | C2×C22⋊C8 | C23 | 1 | 45 | yes | — |
| 61 | C2×C22⋊C8 | C23 | 1 | 90 | yes | complemented |
| 62 | C2×C22⋊C8 | C23 | 1 | 90 | yes | complemented |
| 63 | C25 | C22×C4 | 1 | 15 | yes | — |
| 64 | 2−1+4 | — | 2 | 480 | no | — |
| 65 | 2−1+4 | — | 2 | 240 | no | — |
| 66 | 2−1+4 | C22⋊C4 | 1 | 96 | yes | — |
| 67 | H(5,𝔽2) | — | 2 | 480 | no | — |
| 68 | H(5,𝔽2) | — | 2 | 720 | no | — |
| 69 | H(5,𝔽2) | C22⋊C4 | 1 | 160 | yes | — |
| 70 | C2×#41 | — | 2 | 480 | no | — |
| 71 | C2×#41 | — | 2 | 480 | no | — |
| 72 | C2×#41 | — | 2 | 160 | no | — |
| 73 | C2×#41 | — | 2 | 480 | no | — |
| 74 | C2×#41 | — | 2 | 120 | no | — |
| 75 | C2×#41 | — | 2 | 360 | no | — |
| 76 | C2×#41 | — | 2 | 360 | no | — |
| 77 | C2×#41 | C22×C4 | 1 | 20 | yes | — |
| 78 | C2×#41 | C22×C4 | 1 | 60 | yes | — |
| 79 | C2×#41 | C22×C4 | 1 | 60 | yes | — |
| 80 | C2×#41 | C22×C4 | 1 | 180 | yes | — |
| 81 | C2×#41 | C22×C4 | 1 | 180 | yes | — |
| 82 | C2×#41 | C22×C4 | 1 | 60 | yes | — |
| 83 | C2×#41 | C2×D4 | 1 | 240 | yes | — |
| 84 | C2×#41 | C24 | 1 | 60 | yes | — |
| 85 | C2×#41 | C24 | 1 | 20 | yes | — |
| 86 | C22×Q8 | — | 2 | 120 | no | — |
| 87 | C22×Q8 | C22×C4 | 1 | 20 | yes | — |
| 88 | C22×Q8 | C22×C4 | 1 | 60 | yes | — |
| 89 | C22×Q8 | C22×C4 | 1 | 45 | yes | — |
| 90 | C22×Q8 | C22×C4 | 1 | 15 | yes | — |
| 91 | C22×D4 | — | 2 | 360 | no | — |
| 92 | C22×D4 | C22×C4 | 1 | 60 | yes | — |
| 93 | C22×D4 | C22×C4 | 1 | 90 | yes | — |
| 94 | C22×D4 | C22×C4 | 1 | 180 | yes | — |
| 95 | C22×D4 | C22×C4 | 1 | 45 | yes | — |
| 96 | C22×D4 | C22×C4 | 1 | 45 | yes | — |
| 97 | C23×C4 | — | 2 | 120 | no | — |
| 98 | C23×C4 | C22×C4 | 1 | 45 | yes | — |
| 99 | C23×C4 | C22×C4 | 1 | 15 | yes | — |
| 100 | C23×C4 | C22×C4 | 1 | 45 | yes | — |
| 101 | C23×C4 | C24 | 1 | 15 | yes | — |
| 102 | C23×C4 | C24 | 1 | 45 | yes | — |
| 103 | C23×C4 | C24 | 1 | 15 | yes | — |
| 104 | #131 | — | 2 | 160 | no | — |
| 105 | #131 | — | 2 | 480 | no | — |
| 106 | C2×M16 | — | 2 | 360 | no | — |
| 107 | C2×M16 | C24 | 1 | 120 | yes | — |
| 108 | C22×C8 | — | 2 | 120 | no | — |
| 109 | C22×C8 | C24 | 1 | 30 | yes | — |
| 110 | C22×C8 | C24 | 1 | 90 | yes | — |
| 111 | C22⋊C8 | C24 | 1 | 1 | yes | characteristic |
| 112 | C22⋊C8 | C24 | 1 | 15 | yes | — |
| 113 | C22⋊C8 | C24 | 1 | 30 | yes | — |
| 114 | C22⋊C8 | C24 | 1 | 90 | yes | — |
| 115 | C22⋊C8 | C24 | 1 | 120 | yes | — |
| 116 | C24 | — | 2 | 60 | no | — |
| 117 | C24 | — | 4 | 30 | no | — |
| 118 | C24 | — | 8 | 15 | no | — |
| 119 | C24 | C23×C4 | 1 | 15 | yes | — |
| 120 | C24 | C23×C4 | 1 | 45 | yes | — |
| 121 | C24 | C23×C4 | 1 | 15 | yes | — |
| 122 | #41 | — | 2 | 80 | no | — |
| 123 | #41 | — | 2 | 240 | no | — |
| 124 | #41 | — | 2 | 240 | no | — |
| 125 | #41 | — | 2 | 720 | no | — |
| 126 | #41 | — | 2 | 720 | no | — |
| 127 | #41 | — | 2 | 240 | no | — |
| 128 | #41 | C2×C22⋊C4 | 1 | 480 | yes | — |
| 129 | #41 | C22×D4 | 1 | 160 | yes | — |
| 130 | C2×Q8 | — | 2 | 80 | no | — |
| 131 | C2×Q8 | — | 2 | 240 | no | — |
| 132 | C2×Q8 | — | 2 | 180 | no | — |
| 133 | C2×Q8 | C2×C22⋊C4 | 1 | 120 | yes | — |
| 134 | C2×Q8 | C23×C4 | 1 | 60 | yes | — |
| 135 | C2×Q8 | C23×C4 | 1 | 20 | yes | — |
| 136 | C2×D4 | — | 2 | 360 | no | — |
| 137 | C2×D4 | — | 2 | 240 | no | — |
| 138 | C2×D4 | — | 2 | 720 | no | — |
| 139 | C2×D4 | — | 2 | 180 | no | — |
| 140 | C2×D4 | C2×C22⋊C4 | 1 | 360 | yes | — |
| 141 | C2×D4 | C23×C4 | 1 | 120 | yes | — |
| 142 | C2×D4 | C23×C4 | 1 | 60 | yes | — |
| 143 | C2×D4 | C23×C4 | 1 | 60 | yes | — |
| 144 | C22×C4 | — | 2 | 60 | no | — |
| 145 | C22×C4 | — | 2 | 180 | no | — |
| 146 | C22×C4 | — | 2 | 180 | no | — |
| 147 | C22×C4 | — | 2 | 60 | no | — |
| 148 | C22×C4 | — | 2 | 180 | no | — |
| 149 | C22×C4 | C23×C4 | 1 | 15 | yes | — |
| 150 | C22×C4 | C23×C4 | 1 | 15 | yes | — |
| 151 | C22×C4 | C23×C4 | 1 | 15 | yes | — |
| 152 | C22×C4 | C23×C4 | 1 | 45 | yes | — |
| 153 | C22×C4 | C23×C4 | 1 | 90 | yes | — |
| 154 | C22×C4 | C23×C4 | 1 | 45 | yes | — |
| 155 | C22×C4 | C22×D4 | 1 | 120 | yes | — |
| 156 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 157 | C22×C4 | C25 | 1 | 15 | yes | — |
| 158 | C22×C4 | C25 | 1 | 15 | yes | — |
| 159 | M16 | — | 2 | 240 | no | — |
| 160 | C2×C8 | — | 2 | 60 | no | — |
| 161 | C2×C8 | — | 2 | 180 | no | — |
| 162 | C2×C8 | C25 | 1 | 2 | yes | — |
| 163 | C2×C8 | C25 | 1 | 30 | yes | — |
| 164 | C23 | — | 2 | 30 | no | — |
| 165 | C23 | — | 2 | 30 | no | — |
| 166 | C23 | — | 2 | 90 | no | — |
| 167 | C23 | — | 4 | 15 | no | — |
| 168 | C23 | — | 4 | 15 | no | — |
| 169 | C23 | — | 4 | 15 | no | — |
| 170 | C23 | — | 4 | 45 | no | — |
| 171 | C23 | — | 4 | 45 | no | — |
| 172 | C23 | — | 8 | 120 | no | — |
| 173 | C23 | C22×C22⋊C4 | 1 | 60 | yes | — |
| 174 | C23 | C24×C4 | 1 | 15 | yes | — |
| 175 | C23 | C24×C4 | 1 | 15 | yes | — |
| 176 | C23 | C24×C4 | 1 | 1 | yes | characteristic |
| 177 | Q8 | — | 2 | 120 | no | — |
| 178 | Q8 | C22×C22⋊C4 | 1 | 80 | yes | — |
| 179 | D4 | — | 2 | 240 | no | — |
| 180 | D4 | — | 2 | 120 | no | — |
| 181 | D4 | C22×C22⋊C4 | 1 | 240 | yes | — |
| 182 | C2×C4 | — | 2 | 30 | no | — |
| 183 | C2×C4 | — | 2 | 90 | no | — |
| 184 | C2×C4 | — | 2 | 2 | no | — |
| 185 | C2×C4 | — | 2 | 30 | no | — |
| 186 | C2×C4 | — | 2 | 30 | no | — |
| 187 | C2×C4 | — | 2 | 30 | no | — |
| 188 | C2×C4 | — | 2 | 180 | no | — |
| 189 | C2×C4 | C22×C22⋊C4 | 1 | 180 | yes | — |
| 190 | C2×C4 | C24×C4 | 1 | 1 | yes | characteristic |
| 191 | C2×C4 | C24×C4 | 1 | 15 | yes | — |
| 192 | C2×C4 | C24×C4 | 1 | 15 | yes | — |
| 193 | C2×C4 | C23×D4 | 1 | 60 | yes | — |
| 194 | C2×C4 | C26 | 1 | 1 | yes | characteristic |
| 195 | C8 | — | 2 | 2 | no | — |
| 196 | C8 | — | 2 | 30 | no | — |
| 197 | C22 | — | 2 | 15 | no | — |
| 198 | C22 | — | 2 | 1 | no | — |
| 199 | C22 | — | 2 | 15 | no | — |
| 200 | C22 | — | 2 | 1 | no | — |
| 201 | C22 | — | 2 | 15 | no | — |
| 202 | C22 | — | 2 | 15 | no | — |
| 203 | C22 | — | 2 | 15 | no | — |
| 204 | C22 | — | 4 | 60 | no | — |
| 205 | C22 | — | 4 | 60 | no | — |
| 206 | C22 | — | 8 | 90 | no | — |
| 207 | C22 | C23×C22⋊C4 | 1 | 30 | yes | — |
| 208 | C22 | #3573 | 1 | 2 | yes | complemented |
| 209 | C22 | C25×C4 | 1 | 1 | yes | characteristic |
| 210 | C4 | — | 2 | 1 | no | — |
| 211 | C4 | — | 2 | 15 | no | — |
| 212 | C4 | C23×C22⋊C4 | 1 | 30 | yes | — |
| 213 | C4 | C24×D4 | 1 | 2 | yes | — |
| 214 | C2 | — | 2 | 2 | no | — |
| 215 | C2 | — | 2 | 30 | no | — |
| 216 | C2 | — | 4 | 15 | no | — |
| 217 | C2 | C24×C22⋊C4 | 1 | 1 | yes | characteristic |
| 218 | C2 | C2×#3573 | 1 | 1 | yes | characteristic |
| 219 | C2 | #63076 | 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