Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
246 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 41 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 | #66047 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #65366 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #65031 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C5×#353 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #64692 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #4144 | C22 | 1 | 1 | yes | characteristic |
| 7 | C5×#127 | C22 | 1 | 1 | yes | characteristic |
| 8 | #4082 | — | 2 | 2 | no | — |
| 9 | #4081 | — | 2 | 1 | no | — |
| 10 | #4081 | C22 | 1 | 2 | yes | — |
| 11 | D4×Dic5 | — | 2 | 2 | no | — |
| 12 | #4078 | — | 2 | 1 | no | — |
| 13 | #4032 | — | 2 | 2 | no | — |
| 14 | C4×Dic10 | — | 2 | 1 | no | — |
| 15 | C5×#99 | C22 | 1 | 2 | yes | — |
| 16 | #3965 | C22 | 1 | 1 | yes | characteristic |
| 17 | #3938 | — | 2 | 2 | no | — |
| 18 | #3937 | — | 2 | 1 | no | — |
| 19 | #2198 | — | 5 | 1 | no | maximal, Sylow |
| 20 | C2×C5×D4 | — | 2 | 2 | no | — |
| 21 | C2×C5×D4 | D4 | 1 | 1 | yes | characteristic |
| 22 | C2×C5×D4 | C23 | 1 | 1 | yes | characteristic |
| 23 | C2×C22⋊D5 | — | 2 | 2 | no | — |
| 24 | C22×Dic5 | — | 2 | 1 | no | — |
| 25 | C22×Dic5 | D4 | 1 | 2 | yes | — |
| 26 | #711 | — | 4 | 1 | no | — |
| 27 | D4×D5 | — | 4 | 1 | no | — |
| 28 | #709 | — | 2 | 1 | no | — |
| 29 | C2×Dic10 | — | 2 | 1 | no | — |
| 30 | C5×C22⋊C4 | — | 2 | 2 | no | — |
| 31 | C42×C5 | D4 | 1 | 1 | yes | characteristic |
| 32 | C22⋊Dic5 | — | 2 | 2 | no | — |
| 33 | C22⋊Dic5 | — | 2 | 1 | no | — |
| 34 | C22⋊Dic5 | — | 2 | 2 | no | — |
| 35 | C22⋊Dic5 | D4 | 1 | 2 | yes | — |
| 36 | #687 | — | 4 | 1 | no | — |
| 37 | #686 | — | 4 | 1 | no | — |
| 38 | #685 | — | 4 | 2 | no | — |
| 39 | C4⋊Dic5 | — | 2 | 1 | no | — |
| 40 | C4⋊Dic5 | — | 4 | 2 | no | — |
| 41 | #683 | — | 2 | 2 | no | — |
| 42 | C4×Dic5 | — | 2 | 2 | no | — |
| 43 | #681 | — | 2 | 1 | no | — |
| 44 | #559 | — | 5 | 1 | no | — |
| 45 | #458 | — | 5 | 2 | no | — |
| 46 | #453 | — | 5 | 1 | no | — |
| 47 | #353 | D5 | 1 | 1 | yes | characteristic, complemented |
| 48 | #352 | — | 5 | 2 | no | — |
| 49 | C23×C5 | — | 2 | 1 | no | — |
| 50 | C23×C5 | C2×D4 | 1 | 2 | yes | — |
| 51 | C22×D5 | — | 2 | 2 | no | — |
| 52 | C5×D4 | — | 4 | 1 | no | — |
| 53 | C5×D4 | — | 4 | 1 | no | — |
| 54 | C5×D4 | — | 4 | 4 | no | — |
| 55 | C2×C4×C5 | — | 2 | 1 | no | — |
| 56 | C2×C4×C5 | — | 4 | 2 | no | — |
| 57 | C2×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 58 | C22⋊D5 | — | 4 | 2 | no | — |
| 59 | C22⋊D5 | — | 4 | 1 | no | — |
| 60 | C2×Dic5 | — | 2 | 2 | no | — |
| 61 | C2×Dic5 | — | 2 | 2 | no | — |
| 62 | C2×Dic5 | — | 2 | 2 | no | — |
| 63 | C2×Dic5 | — | 4 | 2 | no | — |
| 64 | C2×Dic5 | — | 8 | 1 | no | — |
| 65 | D20 | — | 2 | 1 | no | — |
| 66 | C4×D5 | — | 4 | 1 | no | — |
| 67 | Dic10 | — | 2 | 1 | no | — |
| 68 | Dic10 | — | 4 | 2 | no | — |
| 69 | #168 | — | 4 | 1 | no | — |
| 70 | H(5,𝔽2) | — | 5 | 1 | no | — |
| 71 | AGL(1,ℤ/8) | — | 10 | 1 | no | — |
| 72 | #127 | C2×D5 | 1 | 1 | yes | characteristic |
| 73 | #124 | — | 10 | 2 | no | — |
| 74 | #123 | — | 5 | 2 | no | — |
| 75 | #123 | — | 10 | 2 | no | — |
| 76 | #123 | — | 10 | 1 | no | — |
| 77 | C4×Q8 | — | 10 | 1 | no | — |
| 78 | C4×D4 | — | 10 | 2 | no | — |
| 79 | #104 | — | 10 | 1 | no | — |
| 80 | #100 | — | 5 | 1 | no | — |
| 81 | #99 | — | 10 | 2 | no | — |
| 82 | #99 | C2×D5 | 1 | 2 | yes | — |
| 83 | C22×C5 | — | 2 | 2 | no | — |
| 84 | C22×C5 | — | 4 | 2 | no | — |
| 85 | C22×C5 | — | 4 | 2 | no | — |
| 86 | C22×C5 | — | 4 | 1 | no | — |
| 87 | C22×C5 | #120 | 1 | 1 | yes | characteristic |
| 88 | C2×D5 | — | 4 | 1 | no | — |
| 89 | C2×D5 | — | 8 | 2 | no | — |
| 90 | C4×C5 | — | 2 | 2 | no | — |
| 91 | C4×C5 | — | 2 | 1 | no | — |
| 92 | C4×C5 | — | 8 | 2 | no | — |
| 93 | Dic5 | — | 4 | 2 | no | — |
| 94 | Dic5 | — | 4 | 1 | no | — |
| 95 | Dic5 | — | 4 | 2 | no | — |
| 96 | #41 | — | 10 | 1 | no | — |
| 97 | #41 | — | 20 | 1 | no | — |
| 98 | C2×Q8 | — | 10 | 1 | no | — |
| 99 | C2×D4 | — | 2 | 2 | no | — |
| 100 | C2×D4 | — | 10 | 2 | no | — |
| 101 | C2×D4 | — | 20 | 1 | no | — |
| 102 | C2×D4 | D20 | 1 | 1 | yes | characteristic |
| 103 | C2×D4 | C22×D5 | 1 | 1 | yes | characteristic |
| 104 | C22×C4 | — | 5 | 2 | no | — |
| 105 | C22×C4 | — | 10 | 1 | no | — |
| 106 | SD16 | — | 20 | 1 | no | — |
| 107 | D8 | — | 20 | 1 | no | — |
| 108 | M16 | — | 10 | 1 | no | — |
| 109 | C4⋊C4 | — | 10 | 1 | no | — |
| 110 | C4⋊C4 | — | 10 | 2 | no | — |
| 111 | C4⋊C4 | — | 20 | 2 | no | — |
| 112 | C22⋊C4 | — | 2 | 2 | no | — |
| 113 | C22⋊C4 | — | 5 | 2 | no | — |
| 114 | C22⋊C4 | — | 10 | 2 | no | — |
| 115 | C22⋊C4 | — | 10 | 1 | no | — |
| 116 | C22⋊C4 | — | 10 | 2 | no | — |
| 117 | C22⋊C4 | — | 20 | 2 | no | — |
| 118 | C42 | — | 10 | 2 | no | — |
| 119 | C42 | D20 | 1 | 1 | yes | characteristic |
| 120 | C2×C5 | — | 2 | 1 | no | — |
| 121 | C2×C5 | — | 4 | 2 | no | — |
| 122 | C2×C5 | — | 8 | 1 | no | — |
| 123 | C2×C5 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 124 | D5 | — | 8 | 1 | no | — |
| 125 | C23 | — | 2 | 1 | no | — |
| 126 | C23 | — | 10 | 2 | no | — |
| 127 | C23 | D4×D5 | 1 | 2 | yes | — |
| 128 | Q8 | — | 10 | 1 | no | — |
| 129 | Q8 | — | 20 | 2 | no | — |
| 130 | D4 | — | 4 | 1 | no | — |
| 131 | D4 | — | 4 | 1 | no | — |
| 132 | D4 | — | 4 | 4 | no | — |
| 133 | D4 | — | 10 | 1 | no | — |
| 134 | D4 | — | 20 | 2 | no | — |
| 135 | D4 | — | 20 | 1 | no | — |
| 136 | C2×C4 | — | 2 | 1 | no | — |
| 137 | C2×C4 | — | 4 | 2 | no | — |
| 138 | C2×C4 | — | 10 | 2 | no | — |
| 139 | C2×C4 | — | 10 | 2 | no | — |
| 140 | C2×C4 | — | 10 | 2 | no | — |
| 141 | C2×C4 | — | 20 | 1 | no | — |
| 142 | C2×C4 | — | 20 | 2 | no | — |
| 143 | C2×C4 | — | 40 | 1 | no | — |
| 144 | C2×C4 | C2×D20 | 1 | 1 | yes | characteristic |
| 145 | C8 | — | 20 | 1 | no | — |
| 146 | C5 | #2198 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 147 | C22 | — | 2 | 2 | no | — |
| 148 | C22 | — | 4 | 2 | no | — |
| 149 | C22 | — | 4 | 2 | no | — |
| 150 | C22 | — | 4 | 1 | no | — |
| 151 | C22 | — | 20 | 1 | no | — |
| 152 | C22 | — | 40 | 2 | no | — |
| 153 | C22 | #4028 | 1 | 1 | yes | characteristic |
| 154 | C4 | — | 2 | 2 | no | — |
| 155 | C4 | — | 2 | 1 | no | — |
| 156 | C4 | — | 8 | 2 | no | — |
| 157 | C4 | — | 20 | 2 | no | — |
| 158 | C4 | — | 20 | 1 | no | — |
| 159 | C4 | — | 20 | 2 | no | — |
| 160 | C2 | — | 2 | 1 | no | — |
| 161 | C2 | — | 4 | 2 | no | — |
| 162 | C2 | — | 8 | 1 | no | — |
| 163 | C2 | — | 40 | 1 | no | — |
| 164 | C2 | #65030 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #66047 | C2 | semidirect |
| 3. #65031 | C2 | semidirect |
| 4. C5×#353 | C2 | semidirect |
| 5. #64692 | C2 | semidirect |
| 47. #353 | D5 | semidirect |
| 146. C5 | #2198 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #66047 | 2 |
| #65366 | 2 |
| #65031 | 2 |
| C5×#353 | 2 |
| #64692 | 2 |
| #2198 | 5 |