← #10587001
Subgroups — #10587001
Order 512.
56113 conjugacy classes of subgroups.
The subgroup lattice has 208 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #63103 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #63093 | C2 | 1 | 16 | yes | maximal, complemented |
| 3 | #63092 | C2 | 1 | 32 | yes | maximal, complemented |
| 4 | #63091 | C2 | 1 | 16 | yes | maximal, complemented |
| 5 | #63090 | C2 | 1 | 30 | yes | maximal, complemented |
| 6 | C2×#3584 | C2 | 1 | 30 | yes | maximal, complemented |
| 7 | C2×#3573 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | 2−1+6 | — | 2 | 12 | no | — |
| 9 | 2−1+6 | C22 | 1 | 32 | yes | complemented |
| 10 | H(7,𝔽2) | — | 2 | 20 | no | — |
| 11 | H(7,𝔽2) | C22 | 1 | 32 | yes | complemented |
| 12 | C2×#585 | — | 2 | 32 | no | — |
| 13 | C2×#585 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#585 | C22 | 1 | 60 | yes | complemented |
| 15 | #3586 | C22 | 1 | 480 | yes | complemented |
| 16 | #3585 | C22 | 1 | 480 | yes | complemented |
| 17 | #3584 | C22 | 1 | 80 | yes | complemented |
| 18 | #3584 | C22 | 1 | 240 | yes | complemented |
| 19 | #3584 | C22 | 1 | 240 | yes | complemented |
| 20 | #3584 | C22 | 1 | 240 | yes | complemented |
| 21 | C2×#578 | C22 | 1 | 120 | yes | complemented |
| 22 | C2×#577 | C22 | 1 | 240 | yes | complemented |
| 23 | C2×#576 | C22 | 1 | 120 | yes | complemented |
| 24 | C2×#575 | C22 | 1 | 180 | yes | complemented |
| 25 | C22×#135 | C22 | 1 | 60 | yes | complemented |
| 26 | #3573 | C22 | 1 | 16 | yes | complemented |
| 27 | #3573 | C22 | 1 | 16 | yes | complemented |
| 28 | C2×#568 | C22 | 1 | 15 | yes | complemented |
| 29 | C22×#131 | C22 | 1 | 15 | yes | complemented |
| 30 | #585 | — | 2 | 192 | no | — |
| 31 | #585 | — | 2 | 320 | no | — |
| 32 | #585 | — | 2 | 240 | no | — |
| 33 | #585 | — | 2 | 240 | no | — |
| 34 | #585 | D4 | 1 | 16 | yes | — |
| 35 | #585 | C23 | 1 | 16 | yes | — |
| 36 | #585 | C23 | 1 | 160 | yes | complemented |
| 37 | #585 | C23 | 1 | 480 | yes | complemented |
| 38 | C2×2−1+4 | — | 2 | 192 | no | — |
| 39 | C2×2−1+4 | — | 2 | 60 | no | — |
| 40 | C2×2−1+4 | D4 | 1 | 6 | yes | — |
| 41 | C2×2−1+4 | C23 | 1 | 240 | yes | complemented |
| 42 | C2×H(5,𝔽2) | — | 2 | 320 | no | — |
| 43 | C2×H(5,𝔽2) | — | 2 | 180 | no | — |
| 44 | C2×H(5,𝔽2) | D4 | 1 | 10 | yes | — |
| 45 | C2×H(5,𝔽2) | C23 | 1 | 240 | yes | complemented |
| 46 | C22×#41 | — | 2 | 240 | no | — |
| 47 | C22×#41 | C23 | 1 | 15 | yes | — |
| 48 | C22×#41 | C23 | 1 | 120 | yes | complemented |
| 49 | #578 | C23 | 1 | 320 | yes | complemented |
| 50 | #578 | C23 | 1 | 960 | yes | complemented |
| 51 | #577 | C23 | 1 | 640 | yes | complemented |
| 52 | #577 | C23 | 1 | 1920 | yes | complemented |
| 53 | #576 | C23 | 1 | 320 | yes | complemented |
| 54 | #576 | C23 | 1 | 960 | yes | complemented |
| 55 | #575 | C23 | 1 | 480 | yes | complemented |
| 56 | #575 | C23 | 1 | 1440 | yes | complemented |
| 57 | C2×#137 | C23 | 1 | 720 | yes | complemented |
| 58 | C2×AGL(1,ℤ/8) | C23 | 1 | 720 | yes | complemented |
| 59 | C2×#135 | C23 | 1 | 120 | yes | complemented |
| 60 | C2×#135 | C23 | 1 | 360 | yes | complemented |
| 61 | C2×#135 | C23 | 1 | 360 | yes | complemented |
| 62 | C2×#135 | C23 | 1 | 360 | yes | complemented |
| 63 | C22×Q16 | C23 | 1 | 60 | yes | complemented |
| 64 | C22×SD16 | C23 | 1 | 120 | yes | complemented |
| 65 | C22×D8 | C23 | 1 | 60 | yes | complemented |
| 66 | #568 | C23 | 1 | 240 | yes | complemented |
| 67 | C2×#131 | C23 | 1 | 20 | yes | complemented |
| 68 | C2×#131 | C23 | 1 | 60 | yes | complemented |
| 69 | C2×#131 | C23 | 1 | 120 | yes | complemented |
| 70 | C2×#131 | C23 | 1 | 120 | yes | complemented |
| 71 | C22×M16 | C23 | 1 | 45 | yes | complemented |
| 72 | C23×C8 | C23 | 1 | 15 | yes | complemented |
| 73 | 2−1+4 | — | 2 | 960 | no | — |
| 74 | 2−1+4 | — | 2 | 480 | no | — |
| 75 | 2−1+4 | — | 2 | 160 | no | — |
| 76 | 2−1+4 | C2×D4 | 1 | 96 | yes | — |
| 77 | 2−1+4 | C24 | 1 | 640 | yes | — |
| 78 | H(5,𝔽2) | — | 2 | 960 | no | — |
| 79 | H(5,𝔽2) | — | 2 | 1440 | no | — |
| 80 | H(5,𝔽2) | — | 2 | 480 | no | — |
| 81 | H(5,𝔽2) | C2×D4 | 1 | 160 | yes | — |
| 82 | H(5,𝔽2) | C24 | 1 | 640 | yes | — |
| 83 | C2×#41 | — | 2 | 640 | no | — |
| 84 | C2×#41 | — | 2 | 1920 | no | — |
| 85 | C2×#41 | — | 2 | 240 | no | — |
| 86 | C2×#41 | — | 2 | 720 | no | — |
| 87 | C2×#41 | — | 2 | 360 | no | — |
| 88 | C2×#41 | — | 2 | 360 | no | — |
| 89 | C2×#41 | C2×D4 | 1 | 120 | yes | — |
| 90 | C2×#41 | C2×D4 | 1 | 60 | yes | — |
| 91 | C2×#41 | C24 | 1 | 240 | yes | — |
| 92 | C2×#41 | C24 | 1 | 720 | yes | — |
| 93 | C2×#41 | C24 | 1 | 20 | yes | — |
| 94 | C2×#41 | C24 | 1 | 120 | yes | — |
| 95 | C22×Q8 | — | 2 | 240 | no | — |
| 96 | C22×Q8 | C2×D4 | 1 | 15 | yes | — |
| 97 | C22×Q8 | C24 | 1 | 120 | yes | — |
| 98 | C22×D4 | — | 2 | 120 | no | — |
| 99 | C22×D4 | — | 2 | 720 | no | — |
| 100 | C22×D4 | C2×D4 | 1 | 45 | yes | — |
| 101 | C22×D4 | C24 | 1 | 120 | yes | — |
| 102 | C23×C4 | — | 2 | 120 | no | — |
| 103 | C23×C4 | C24 | 1 | 15 | yes | — |
| 104 | #137 | C24 | 1 | 1920 | yes | — |
| 105 | AGL(1,ℤ/8) | C24 | 1 | 1920 | yes | — |
| 106 | #135 | C24 | 1 | 16 | yes | — |
| 107 | #135 | C24 | 1 | 240 | yes | — |
| 108 | #135 | C24 | 1 | 240 | yes | — |
| 109 | #135 | C24 | 1 | 240 | yes | — |
| 110 | #135 | C24 | 1 | 1440 | yes | — |
| 111 | C2×Q16 | C24 | 1 | 120 | yes | — |
| 112 | C2×Q16 | C24 | 1 | 360 | yes | — |
| 113 | C2×SD16 | C24 | 1 | 240 | yes | — |
| 114 | C2×SD16 | C24 | 1 | 720 | yes | — |
| 115 | C2×D8 | C24 | 1 | 120 | yes | — |
| 116 | C2×D8 | C24 | 1 | 360 | yes | — |
| 117 | #131 | C24 | 1 | 80 | yes | — |
| 118 | #131 | C24 | 1 | 80 | yes | — |
| 119 | #131 | C24 | 1 | 480 | yes | — |
| 120 | C2×M16 | C24 | 1 | 60 | yes | — |
| 121 | C2×M16 | C24 | 1 | 360 | yes | — |
| 122 | C22×C8 | C24 | 1 | 15 | yes | — |
| 123 | C22×C8 | C24 | 1 | 45 | yes | — |
| 124 | C22×C8 | C24 | 1 | 60 | yes | — |
| 125 | C22×C8 | C24 | 1 | 60 | yes | — |
| 126 | C24 | — | 2 | 120 | no | — |
| 127 | C24 | C22×D4 | 1 | 15 | yes | — |
| 128 | #41 | — | 2 | 640 | no | — |
| 129 | #41 | — | 2 | 1920 | no | — |
| 130 | #41 | — | 2 | 240 | no | — |
| 131 | #41 | — | 2 | 1440 | no | — |
| 132 | #41 | — | 2 | 240 | no | — |
| 133 | #41 | C22×D4 | 1 | 80 | yes | — |
| 134 | #41 | C22×D4 | 1 | 480 | yes | — |
| 135 | #41 | C25 | 1 | 32 | yes | — |
| 136 | #41 | C25 | 1 | 480 | yes | — |
| 137 | #41 | C25 | 1 | 80 | yes | — |
| 138 | C2×Q8 | — | 2 | 640 | no | — |
| 139 | C2×Q8 | — | 2 | 360 | no | — |
| 140 | C2×Q8 | C22×D4 | 1 | 120 | yes | — |
| 141 | C2×Q8 | C22×D4 | 1 | 20 | yes | — |
| 142 | C2×Q8 | C25 | 1 | 240 | yes | — |
| 143 | C2×D4 | — | 2 | 720 | no | — |
| 144 | C2×D4 | — | 2 | 240 | no | — |
| 145 | C2×D4 | — | 2 | 1920 | no | — |
| 146 | C2×D4 | — | 2 | 360 | no | — |
| 147 | C2×D4 | C22×D4 | 1 | 360 | yes | — |
| 148 | C2×D4 | C22×D4 | 1 | 60 | yes | — |
| 149 | C2×D4 | C25 | 1 | 240 | yes | — |
| 150 | C22×C4 | — | 2 | 120 | no | — |
| 151 | C22×C4 | — | 2 | 240 | no | — |
| 152 | C22×C4 | — | 2 | 720 | no | — |
| 153 | C22×C4 | C22×D4 | 1 | 60 | yes | — |
| 154 | C22×C4 | C22×D4 | 1 | 45 | yes | — |
| 155 | C22×C4 | C25 | 1 | 15 | yes | — |
| 156 | C22×C4 | C25 | 1 | 60 | yes | — |
| 157 | Q16 | C25 | 1 | 16 | yes | — |
| 158 | Q16 | C25 | 1 | 240 | yes | — |
| 159 | SD16 | C25 | 1 | 32 | yes | — |
| 160 | SD16 | C25 | 1 | 480 | yes | — |
| 161 | D8 | C25 | 1 | 16 | yes | — |
| 162 | D8 | C25 | 1 | 240 | yes | — |
| 163 | M16 | C25 | 1 | 240 | yes | — |
| 164 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 165 | C2×C8 | C25 | 1 | 15 | yes | — |
| 166 | C2×C8 | C25 | 1 | 30 | yes | — |
| 167 | C2×C8 | C25 | 1 | 30 | yes | — |
| 168 | C2×C8 | C25 | 1 | 180 | yes | — |
| 169 | C23 | — | 2 | 240 | no | — |
| 170 | C23 | — | 8 | 15 | no | — |
| 171 | C23 | — | 16 | 120 | no | — |
| 172 | C23 | C23×D4 | 1 | 60 | yes | — |
| 173 | C23 | C23×D4 | 1 | 15 | yes | — |
| 174 | Q8 | — | 2 | 240 | no | — |
| 175 | Q8 | C23×D4 | 1 | 80 | yes | — |
| 176 | Q8 | C26 | 1 | 32 | yes | — |
| 177 | D4 | — | 2 | 480 | no | — |
| 178 | D4 | — | 2 | 32 | no | — |
| 179 | D4 | — | 2 | 240 | no | — |
| 180 | D4 | C23×D4 | 1 | 240 | yes | — |
| 181 | D4 | C26 | 1 | 32 | yes | — |
| 182 | C2×C4 | — | 2 | 240 | no | — |
| 183 | C2×C4 | — | 2 | 480 | no | — |
| 184 | C2×C4 | — | 2 | 32 | no | — |
| 185 | C2×C4 | C23×D4 | 1 | 180 | yes | — |
| 186 | C2×C4 | C23×D4 | 1 | 15 | yes | — |
| 187 | C2×C4 | C23×D4 | 1 | 30 | yes | — |
| 188 | C2×C4 | C26 | 1 | 30 | yes | — |
| 189 | C2×C4 | C26 | 1 | 1 | yes | characteristic |
| 190 | C8 | C26 | 1 | 1 | yes | characteristic |
| 191 | C8 | C26 | 1 | 1 | yes | characteristic |
| 192 | C8 | C26 | 1 | 30 | yes | — |
| 193 | C22 | — | 2 | 32 | no | — |
| 194 | C22 | — | 4 | 15 | no | — |
| 195 | C22 | — | 4 | 60 | no | — |
| 196 | C22 | — | 8 | 240 | no | — |
| 197 | C22 | C24×D4 | 1 | 30 | yes | — |
| 198 | C22 | C24×D4 | 1 | 1 | yes | characteristic |
| 199 | C4 | — | 2 | 32 | no | — |
| 200 | C4 | C24×D4 | 1 | 30 | yes | — |
| 201 | C4 | C24×D4 | 1 | 1 | yes | characteristic |
| 202 | C4 | C27 | 1 | 1 | yes | characteristic |
| 203 | C2 | — | 2 | 1 | no | — |
| 204 | C2 | — | 2 | 30 | no | — |
| 205 | C2 | — | 4 | 32 | no | — |
| 206 | C2 | C25×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