Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
716 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 716 conjugacy classes of subgroups — too many to draw. Only its 133 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 | #66285 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #65982 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C5×#458 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #65532 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #65342 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #65031 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #65029 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C5×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 9 | #4150 | — | 2 | 3 | no | — |
| 10 | D5×#41 | — | 2 | 3 | no | — |
| 11 | #4144 | — | 2 | 3 | no | — |
| 12 | #4144 | C22 | 1 | 3 | yes | — |
| 13 | C2×#711 | — | 2 | 3 | no | — |
| 14 | C2×#711 | C22 | 1 | 3 | yes | — |
| 15 | C22×Dic10 | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | C5×#123 | C22 | 1 | 3 | yes | complemented |
| 17 | #4081 | C22 | 1 | 3 | yes | complemented |
| 18 | #4075 | C22 | 1 | 3 | yes | complemented |
| 19 | #4042 | — | 2 | 3 | no | — |
| 20 | #4034 | — | 2 | 3 | no | — |
| 21 | #4032 | C22 | 1 | 3 | yes | complemented |
| 22 | #4031 | — | 2 | 3 | no | — |
| 23 | #4024 | — | 2 | 3 | no | — |
| 24 | #4023 | C22 | 1 | 3 | yes | complemented |
| 25 | C5×#99 | C22 | 1 | 3 | yes | complemented |
| 26 | #3966 | C22 | 1 | 3 | yes | complemented |
| 27 | #3938 | C22 | 1 | 6 | yes | complemented |
| 28 | #3028 | — | 5 | 1 | no | maximal, Sylow |
| 29 | C5×#41 | — | 2 | 3 | no | — |
| 30 | C2×C5×D4 | — | 2 | 3 | no | — |
| 31 | C2×C5×D4 | C23 | 1 | 3 | yes | — |
| 32 | C22×C4×C5 | C23 | 1 | 1 | yes | characteristic |
| 33 | C2×C22⋊D5 | — | 2 | 3 | no | — |
| 34 | C2×C22⋊D5 | — | 2 | 3 | no | — |
| 35 | C2×C22⋊D5 | D4 | 1 | 3 | yes | — |
| 36 | C2×C22⋊D5 | C23 | 1 | 3 | yes | — |
| 37 | C22×Dic5 | — | 2 | 1 | no | — |
| 38 | C22×Dic5 | D4 | 1 | 3 | yes | — |
| 39 | C22×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 40 | C22×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 41 | #713 | — | 2 | 1 | no | — |
| 42 | Q8×D5 | — | 2 | 3 | no | — |
| 43 | #711 | — | 2 | 3 | no | — |
| 44 | #711 | — | 2 | 6 | no | — |
| 45 | #711 | — | 2 | 6 | no | — |
| 46 | #711 | — | 2 | 6 | no | — |
| 47 | #711 | — | 2 | 6 | no | — |
| 48 | D4×D5 | — | 2 | 3 | no | — |
| 49 | D4×D5 | — | 2 | 6 | no | — |
| 50 | #709 | — | 2 | 6 | no | — |
| 51 | #709 | — | 2 | 3 | no | — |
| 52 | C2×C4×D5 | — | 2 | 3 | no | — |
| 53 | C2×C4×D5 | D4 | 1 | 3 | yes | — |
| 54 | C2×Dic10 | — | 2 | 3 | no | — |
| 55 | C2×Dic10 | — | 2 | 3 | no | — |
| 56 | C2×Dic10 | — | 2 | 3 | no | — |
| 57 | C2×Dic10 | D4 | 1 | 3 | yes | — |
| 58 | C5×C4⋊C4 | — | 2 | 3 | no | — |
| 59 | C5×C22⋊C4 | — | 2 | 3 | no | — |
| 60 | C5×C22⋊C4 | C23 | 1 | 3 | yes | — |
| 61 | C22⋊Dic5 | — | 2 | 3 | no | — |
| 62 | C22⋊Dic5 | C23 | 1 | 3 | yes | — |
| 63 | #685 | — | 2 | 3 | no | — |
| 64 | #685 | — | 2 | 3 | no | — |
| 65 | C4⋊Dic5 | — | 2 | 3 | no | — |
| 66 | #683 | — | 2 | 3 | no | — |
| 67 | #683 | — | 2 | 3 | no | — |
| 68 | #683 | — | 2 | 3 | no | — |
| 69 | C4×Dic5 | — | 2 | 3 | no | — |
| 70 | #585 | — | 5 | 1 | no | — |
| 71 | #536 | — | 5 | 3 | no | — |
| 72 | #458 | — | 5 | 1 | no | — |
| 73 | #458 | — | 5 | 3 | no | — |
| 74 | #458 | — | 5 | 3 | no | — |
| 75 | #458 | D5 | 1 | 1 | yes | characteristic, complemented |
| 76 | #410 | — | 5 | 3 | no | — |
| 77 | C23×C5 | — | 2 | 1 | no | — |
| 78 | C23×C5 | C2×D4 | 1 | 3 | yes | — |
| 79 | C23×C5 | C24 | 1 | 1 | yes | characteristic |
| 80 | C22×D5 | — | 2 | 3 | no | — |
| 81 | C22×D5 | C2×D4 | 1 | 3 | yes | — |
| 82 | C5×Q8 | — | 2 | 1 | no | — |
| 83 | C5×D4 | — | 2 | 3 | no | — |
| 84 | C5×D4 | — | 2 | 6 | no | — |
| 85 | C2×C4×C5 | — | 2 | 3 | no | — |
| 86 | C2×C4×C5 | — | 2 | 3 | no | — |
| 87 | C2×C4×C5 | — | 2 | 3 | no | — |
| 88 | C2×C4×C5 | C2×D4 | 1 | 3 | yes | — |
| 89 | C22⋊D5 | — | 2 | 6 | no | — |
| 90 | C22⋊D5 | — | 2 | 6 | no | — |
| 91 | C22⋊D5 | — | 2 | 6 | no | — |
| 92 | C2×Dic5 | — | 2 | 3 | no | — |
| 93 | C2×Dic5 | — | 2 | 3 | no | — |
| 94 | C2×Dic5 | — | 2 | 3 | no | — |
| 95 | C2×Dic5 | — | 2 | 3 | no | — |
| 96 | C2×Dic5 | — | 2 | 3 | no | — |
| 97 | C2×Dic5 | C2×D4 | 1 | 3 | yes | — |
| 98 | C2×Dic5 | C2×D4 | 1 | 3 | yes | — |
| 99 | C2×Dic5 | C2×D4 | 1 | 3 | yes | — |
| 100 | D20 | — | 2 | 3 | no | — |
| 101 | C4×D5 | — | 2 | 6 | no | — |
| 102 | C4×D5 | — | 2 | 3 | no | — |
| 103 | C4×D5 | — | 2 | 3 | no | — |
| 104 | Dic10 | — | 2 | 3 | no | — |
| 105 | Dic10 | — | 2 | 6 | no | — |
| 106 | Dic10 | — | 4 | 3 | no | — |
| 107 | Dic10 | — | 4 | 1 | no | — |
| 108 | 2−1+4 | — | 10 | 3 | no | — |
| 109 | H(5,𝔽2) | — | 5 | 3 | no | — |
| 110 | H(5,𝔽2) | — | 10 | 3 | no | — |
| 111 | H(5,𝔽2) | C2×D5 | 1 | 1 | yes | characteristic |
| 112 | C2×#41 | — | 5 | 3 | no | — |
| 113 | C2×#41 | — | 10 | 3 | no | — |
| 114 | C2×#41 | — | 10 | 3 | no | — |
| 115 | C22×Q8 | — | 5 | 1 | no | — |
| 116 | C4⋊Q8 | — | 10 | 3 | no | — |
| 117 | #124 | — | 10 | 3 | no | — |
| 118 | #123 | — | 5 | 3 | no | — |
| 119 | #123 | — | 5 | 3 | no | — |
| 120 | #123 | — | 5 | 3 | no | — |
| 121 | #123 | C2×D5 | 1 | 3 | yes | complemented |
| 122 | C22⋊Q8 | — | 10 | 3 | no | — |
| 123 | C22⋊Q8 | — | 10 | 3 | no | — |
| 124 | #117 | — | 5 | 3 | no | — |
| 125 | #99 | — | 5 | 3 | no | — |
| 126 | #99 | — | 5 | 6 | no | — |
| 127 | #99 | C2×D5 | 1 | 3 | yes | complemented |
| 128 | C22×C5 | — | 2 | 3 | no | — |
| 129 | C22×C5 | — | 4 | 1 | no | — |
| 130 | C22×C5 | — | 4 | 3 | no | — |
| 131 | C22×C5 | — | 8 | 1 | no | — |
| 132 | C22×C5 | C22×D4 | 1 | 3 | yes | — |
| 133 | C2×D5 | — | 2 | 3 | no | — |
| 134 | C2×D5 | — | 4 | 3 | no | — |
| 135 | C2×D5 | — | 8 | 3 | no | — |
| 136 | C4×C5 | — | 2 | 3 | no | — |
| 137 | C4×C5 | — | 4 | 3 | no | — |
| 138 | C4×C5 | — | 4 | 1 | no | — |
| 139 | Dic5 | — | 2 | 3 | no | — |
| 140 | Dic5 | — | 4 | 3 | no | — |
| 141 | Dic5 | — | 4 | 1 | no | — |
| 142 | Dic5 | #120 | 1 | 3 | yes | — |
| 143 | Dic5 | #120 | 1 | 1 | yes | characteristic |
| 144 | #41 | — | 2 | 3 | no | — |
| 145 | #41 | — | 10 | 6 | no | — |
| 146 | #41 | — | 10 | 6 | no | — |
| 147 | #41 | — | 10 | 6 | no | — |
| 148 | #41 | — | 10 | 6 | no | — |
| 149 | #41 | — | 10 | 3 | no | — |
| 150 | #41 | — | 10 | 6 | no | — |
| 151 | #41 | — | 10 | 1 | no | — |
| 152 | #41 | — | 10 | 3 | no | — |
| 153 | C2×Q8 | — | 5 | 3 | no | — |
| 154 | C2×Q8 | — | 10 | 3 | no | — |
| 155 | C2×Q8 | — | 10 | 3 | no | — |
| 156 | C2×Q8 | — | 10 | 3 | no | — |
| 157 | C2×Q8 | — | 10 | 3 | no | — |
| 158 | C2×D4 | — | 2 | 3 | no | — |
| 159 | C2×D4 | — | 5 | 3 | no | — |
| 160 | C2×D4 | — | 5 | 3 | no | — |
| 161 | C2×D4 | — | 10 | 6 | no | — |
| 162 | C2×D4 | — | 10 | 3 | no | — |
| 163 | C2×D4 | — | 10 | 3 | no | — |
| 164 | C2×D4 | — | 10 | 3 | no | — |
| 165 | C2×D4 | C22×D5 | 1 | 3 | yes | — |
| 166 | C22×C4 | — | 5 | 1 | no | — |
| 167 | C22×C4 | — | 5 | 3 | no | — |
| 168 | C22×C4 | — | 5 | 3 | no | — |
| 169 | C22×C4 | — | 5 | 1 | no | — |
| 170 | C22×C4 | — | 10 | 3 | no | — |
| 171 | C22×C4 | — | 10 | 1 | no | — |
| 172 | C22×C4 | C22×D5 | 1 | 1 | yes | characteristic |
| 173 | C4⋊C4 | — | 2 | 3 | no | — |
| 174 | C4⋊C4 | — | 10 | 3 | no | — |
| 175 | C4⋊C4 | — | 10 | 3 | no | — |
| 176 | C4⋊C4 | — | 10 | 3 | no | — |
| 177 | C4⋊C4 | — | 10 | 3 | no | — |
| 178 | C22⋊C4 | — | 2 | 3 | no | — |
| 179 | C22⋊C4 | — | 5 | 3 | no | — |
| 180 | C22⋊C4 | — | 10 | 3 | no | — |
| 181 | C22⋊C4 | — | 10 | 3 | no | — |
| 182 | C22⋊C4 | — | 10 | 3 | no | — |
| 183 | C22⋊C4 | C22×D5 | 1 | 3 | yes | — |
| 184 | C42 | — | 10 | 3 | no | — |
| 185 | C2×C5 | — | 2 | 3 | no | — |
| 186 | C2×C5 | — | 4 | 3 | no | — |
| 187 | C2×C5 | C2×#120 | 1 | 1 | yes | characteristic |
| 188 | D5 | — | 4 | 3 | no | — |
| 189 | C23 | — | 2 | 1 | no | — |
| 190 | C23 | — | 5 | 3 | no | — |
| 191 | C23 | — | 10 | 3 | no | — |
| 192 | C23 | D4×D5 | 1 | 3 | yes | — |
| 193 | C23 | C23×D5 | 1 | 1 | yes | characteristic |
| 194 | Q8 | — | 2 | 1 | no | — |
| 195 | Q8 | — | 10 | 6 | no | — |
| 196 | Q8 | — | 10 | 3 | no | — |
| 197 | Q8 | — | 20 | 1 | no | — |
| 198 | Q8 | — | 20 | 3 | no | — |
| 199 | D4 | — | 2 | 3 | no | — |
| 200 | D4 | — | 2 | 6 | no | — |
| 201 | D4 | — | 10 | 3 | no | — |
| 202 | D4 | — | 10 | 6 | no | — |
| 203 | D4 | — | 10 | 6 | no | — |
| 204 | D4 | — | 10 | 6 | no | — |
| 205 | C2×C4 | — | 2 | 3 | no | — |
| 206 | C2×C4 | — | 2 | 3 | no | — |
| 207 | C2×C4 | — | 2 | 3 | no | — |
| 208 | C2×C4 | — | 5 | 3 | no | — |
| 209 | C2×C4 | — | 5 | 3 | no | — |
| 210 | C2×C4 | — | 5 | 3 | no | — |
| 211 | C2×C4 | — | 10 | 3 | no | — |
| 212 | C2×C4 | — | 10 | 3 | no | — |
| 213 | C2×C4 | — | 10 | 6 | no | — |
| 214 | C2×C4 | — | 10 | 3 | no | — |
| 215 | C2×C4 | — | 10 | 3 | no | — |
| 216 | C2×C4 | — | 10 | 3 | no | — |
| 217 | C2×C4 | — | 10 | 3 | no | — |
| 218 | C2×C4 | — | 10 | 3 | no | — |
| 219 | C2×C4 | D4×D5 | 1 | 3 | yes | — |
| 220 | C5 | #3028 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 221 | C22 | — | 2 | 3 | no | — |
| 222 | C22 | — | 4 | 1 | no | — |
| 223 | C22 | — | 4 | 3 | no | — |
| 224 | C22 | — | 8 | 1 | no | — |
| 225 | C22 | — | 10 | 3 | no | — |
| 226 | C22 | — | 20 | 3 | no | — |
| 227 | C22 | — | 40 | 3 | no | — |
| 228 | C22 | C2×D4×D5 | 1 | 3 | yes | — |
| 229 | C4 | — | 2 | 3 | no | — |
| 230 | C4 | — | 4 | 3 | no | — |
| 231 | C4 | — | 4 | 1 | no | — |
| 232 | C4 | — | 5 | 1 | no | — |
| 233 | C4 | — | 5 | 3 | no | — |
| 234 | C4 | — | 10 | 3 | no | — |
| 235 | C4 | — | 20 | 3 | no | — |
| 236 | C4 | — | 20 | 1 | no | — |
| 237 | C2 | — | 2 | 3 | no | — |
| 238 | C2 | — | 4 | 3 | no | — |
| 239 | C2 | — | 20 | 3 | no | — |
| 240 | C2 | D5×#120 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #65982 | C2 | semidirect |
| 3. C5×#458 | C2 | semidirect |
| 4. #65532 | C2 | semidirect |
| 5. #65342 | C2 | semidirect |
| 6. #65031 | C2 | semidirect |
| 7. #65029 | C2 | semidirect |
| 15. C22×Dic10 | C22 | semidirect |
| 16. C5×#123 | C22 | semidirect |
| 17. #4081 | C22 | semidirect |
| 18. #4075 | C22 | semidirect |
| 21. #4032 | C22 | semidirect |
| 24. #4023 | C22 | semidirect |
| 25. C5×#99 | C22 | semidirect |
| 26. #3966 | C22 | semidirect |
| 27. #3938 | C22 | semidirect |
| 75. #458 | D5 | semidirect |
| 121. #123 | C2×D5 | semidirect |
| 127. #99 | C2×D5 | semidirect |
| 220. C5 | #3028 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #66285 | 2 |
| #65982 | 2 |
| C5×#458 | 2 |
| #65532 | 2 |
| #65342 | 2 |
| #65031 | 2 |
| #65029 | 2 |
| #3028 | 5 |