Finite Group DatabasebetaFinite Group DatabasebetaOrder 21248.
762 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 762 conjugacy classes of subgroups — too many to draw. Only its 77 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 | #24907377 | C2 | 1 | 3 | yes | maximal |
| 2 | C83⋊#1304 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C83×#2840 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C23×C22⋊D83 | C22 | 1 | 1 | yes | characteristic |
| 5 | C2×#14581706 | — | 2 | 6 | no | — |
| 6 | #19127557 | C22 | 1 | 3 | yes | — |
| 7 | #393×C83 | — | 2 | 6 | no | — |
| 8 | #379×C83 | C22 | 1 | 3 | yes | — |
| 9 | C24×D83 | Q8 | 1 | 1 | yes | characteristic, complemented |
| 10 | C22×C22⋊D83 | — | 2 | 3 | no | — |
| 11 | C22×C22⋊D83 | — | 2 | 3 | no | — |
| 12 | C22×C22⋊D83 | — | 4 | 4 | no | — |
| 13 | C23×Dic83 | Q8 | 1 | 1 | yes | characteristic, complemented |
| 14 | C2×C22⋊Dic83 | — | 2 | 3 | no | — |
| 15 | C2×C22⋊Dic83 | — | 2 | 6 | no | — |
| 16 | #14581706 | — | 4 | 12 | no | — |
| 17 | C25×C83 | C23 | 1 | 1 | yes | characteristic |
| 18 | C22×Q8×C83 | — | 2 | 2 | no | — |
| 19 | C2×C22⋊C4×C83 | — | 2 | 6 | no | — |
| 20 | C2×C22⋊C4×C83 | D4 | 1 | 6 | yes | — |
| 21 | #95×C83 | — | 2 | 6 | no | — |
| 22 | C23×D83 | — | 2 | 3 | no | — |
| 23 | C23×D83 | — | 4 | 2 | no | — |
| 24 | C23×D83 | — | 8 | 2 | no | — |
| 25 | C2×C22⋊D83 | — | 2 | 12 | no | — |
| 26 | C2×C22⋊D83 | — | 4 | 6 | no | — |
| 27 | C2×C22⋊D83 | — | 4 | 6 | no | — |
| 28 | C2×C22⋊D83 | — | 4 | 6 | no | — |
| 29 | C2×C22⋊D83 | — | 4 | 1 | no | — |
| 30 | C2×C22⋊D83 | — | 4 | 3 | no | — |
| 31 | C22×Dic83 | — | 2 | 3 | no | — |
| 32 | C22×Dic83 | — | 4 | 2 | no | — |
| 33 | C22×Dic83 | — | 4 | 3 | no | — |
| 34 | C22⋊Dic83 | — | 4 | 6 | no | — |
| 35 | C22⋊Dic83 | — | 8 | 6 | no | — |
| 36 | C24×C83 | — | 4 | 2 | no | — |
| 37 | C24×C83 | — | 8 | 2 | no | — |
| 38 | C24×C83 | C2×D4 | 1 | 3 | yes | — |
| 39 | C24×C83 | C2×Q8 | 1 | 1 | yes | characteristic |
| 40 | C24×C83 | #41 | 1 | 3 | yes | — |
| 41 | C2×Q8×C83 | — | 8 | 4 | no | — |
| 42 | C2×Q8×C83 | — | 8 | 2 | no | — |
| 43 | C22×C4×C83 | — | 2 | 6 | no | — |
| 44 | C22⋊C4×C83 | — | 4 | 6 | no | — |
| 45 | C22⋊C4×C83 | — | 8 | 6 | no | — |
| 46 | C23×C83 | — | 2 | 6 | no | — |
| 47 | C23×C83 | — | 4 | 1 | no | — |
| 48 | C23×C83 | — | 4 | 2 | no | — |
| 49 | C23×C83 | — | 4 | 6 | no | — |
| 50 | C23×C83 | — | 4 | 3 | no | — |
| 51 | C23×C83 | — | 4 | 6 | no | — |
| 52 | C23×C83 | — | 8 | 8 | no | — |
| 53 | C23×C83 | #120 | 1 | 1 | yes | characteristic |
| 54 | C23×C83 | C22⋊Q8 | 1 | 3 | yes | — |
| 55 | C23×C83 | #124 | 1 | 3 | yes | — |
| 56 | C22×D83 | — | 2 | 6 | no | — |
| 57 | C22×D83 | — | 4 | 3 | no | — |
| 58 | C22×D83 | — | 4 | 1 | no | — |
| 59 | C22×D83 | — | 8 | 8 | no | — |
| 60 | C22×D83 | — | 8 | 6 | no | — |
| 61 | Q8×C83 | — | 8 | 8 | no | — |
| 62 | C2×C4×C83 | — | 4 | 12 | no | — |
| 63 | C2×C4×C83 | — | 4 | 6 | no | — |
| 64 | C22⋊D83 | — | 4 | 12 | no | — |
| 65 | C22⋊D83 | — | 4 | 4 | no | — |
| 66 | C2×Dic83 | — | 2 | 6 | no | — |
| 67 | C2×Dic83 | — | 4 | 3 | no | — |
| 68 | C2×Dic83 | — | 4 | 1 | no | — |
| 69 | C2×Dic83 | — | 8 | 3 | no | — |
| 70 | C2×Dic83 | — | 8 | 6 | no | — |
| 71 | C22×C83 | — | 2 | 2 | no | — |
| 72 | C22×C83 | — | 2 | 3 | no | — |
| 73 | C22×C83 | — | 2 | 3 | no | — |
| 74 | C22×C83 | — | 2 | 6 | no | — |
| 75 | C22×C83 | — | 2 | 6 | no | — |
| 76 | C22×C83 | — | 4 | 4 | no | — |
| 77 | C22×C83 | — | 8 | 6 | no | — |
| 78 | C22×C83 | — | 8 | 6 | no | — |
| 79 | C22×C83 | #393 | 1 | 1 | yes | characteristic |
| 80 | C22×C83 | B(4,𝔽2) | 1 | 2 | yes | — |
| 81 | C2×D83 | — | 4 | 2 | no | — |
| 82 | C2×D83 | — | 8 | 6 | no | — |
| 83 | C2×D83 | — | 8 | 2 | no | — |
| 84 | C2×D83 | — | 8 | 6 | no | — |
| 85 | C4×C83 | — | 8 | 6 | no | — |
| 86 | Dic83 | — | 4 | 2 | no | — |
| 87 | Dic83 | — | 16 | 3 | no | — |
| 88 | #10390 | — | 83 | 1 | no | maximal, Sylow |
| 89 | C2×C83 | — | 2 | 2 | no | — |
| 90 | C2×C83 | — | 4 | 3 | no | — |
| 91 | C2×C83 | — | 4 | 3 | no | — |
| 92 | C2×C83 | #2029 | 1 | 1 | yes | characteristic |
| 93 | C2×C83 | #2032 | 1 | 2 | yes | — |
| 94 | D83 | — | 8 | 2 | no | — |
| 95 | #2840 | D83 | 1 | 1 | yes | characteristic, complemented |
| 96 | #1781 | — | 83 | 3 | no | — |
| 97 | #1304 | — | 83 | 3 | no | — |
| 98 | C83 | #10390 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 99 | C23×D4 | — | 83 | 1 | no | — |
| 100 | C2×#99 | — | 166 | 6 | no | — |
| 101 | #393 | — | 2 | 6 | no | — |
| 102 | #379 | — | 83 | 3 | no | — |
| 103 | #379 | C2×D83 | 1 | 3 | yes | — |
| 104 | C25 | — | 83 | 1 | no | — |
| 105 | C25 | C22×D83 | 1 | 1 | yes | characteristic |
| 106 | C22×Q8 | — | 2 | 2 | no | — |
| 107 | C22×D4 | — | 332 | 4 | no | — |
| 108 | C22×D4 | — | 166 | 3 | no | — |
| 109 | C22×D4 | — | 166 | 3 | no | — |
| 110 | C23×C4 | — | 83 | 1 | no | — |
| 111 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 112 | C2×C22⋊C4 | — | 166 | 3 | no | — |
| 113 | C2×C22⋊C4 | — | 166 | 6 | no | — |
| 114 | C2×C22⋊C4 | C22⋊D83 | 1 | 6 | yes | — |
| 115 | #99 | — | 332 | 12 | no | — |
| 116 | #95 | — | 2 | 6 | no | — |
| 117 | C24 | — | 4 | 2 | no | — |
| 118 | C24 | — | 8 | 2 | no | — |
| 119 | C24 | — | 332 | 2 | no | — |
| 120 | C24 | — | 664 | 2 | no | — |
| 121 | C24 | — | 166 | 3 | no | — |
| 122 | C24 | C2×C22⋊D83 | 1 | 3 | yes | — |
| 123 | C24 | Q8×D83 | 1 | 1 | yes | characteristic |
| 124 | C24 | #13050591 | 1 | 3 | yes | — |
| 125 | C2×Q8 | — | 8 | 4 | no | — |
| 126 | C2×Q8 | — | 8 | 2 | no | — |
| 127 | C2×D4 | — | 332 | 6 | no | — |
| 128 | C2×D4 | — | 332 | 6 | no | — |
| 129 | C2×D4 | — | 332 | 1 | no | — |
| 130 | C2×D4 | — | 332 | 3 | no | — |
| 131 | C2×D4 | — | 332 | 6 | no | — |
| 132 | C2×D4 | — | 166 | 12 | no | — |
| 133 | C22×C4 | — | 2 | 6 | no | — |
| 134 | C22×C4 | — | 332 | 2 | no | — |
| 135 | C22×C4 | — | 332 | 3 | no | — |
| 136 | C22×C4 | — | 166 | 3 | no | — |
| 137 | C22⋊C4 | — | 4 | 6 | no | — |
| 138 | C22⋊C4 | — | 8 | 6 | no | — |
| 139 | C22⋊C4 | — | 332 | 6 | no | — |
| 140 | C22⋊C4 | — | 664 | 6 | no | — |
| 141 | C23 | — | 2 | 6 | no | — |
| 142 | C23 | — | 4 | 1 | no | — |
| 143 | C23 | — | 4 | 2 | no | — |
| 144 | C23 | — | 4 | 6 | no | — |
| 145 | C23 | — | 4 | 6 | no | — |
| 146 | C23 | — | 4 | 3 | no | — |
| 147 | C23 | — | 8 | 8 | no | — |
| 148 | C23 | — | 332 | 3 | no | — |
| 149 | C23 | — | 332 | 1 | no | — |
| 150 | C23 | — | 664 | 6 | no | — |
| 151 | C23 | — | 664 | 8 | no | — |
| 152 | C23 | — | 166 | 6 | no | — |
| 153 | C23 | #14581761 | 1 | 1 | yes | characteristic |
| 154 | C23 | #14581768 | 1 | 3 | yes | — |
| 155 | C23 | #14581781 | 1 | 3 | yes | — |
| 156 | Q8 | — | 8 | 8 | no | — |
| 157 | D4 | — | 332 | 12 | no | — |
| 158 | D4 | — | 332 | 4 | no | — |
| 159 | C2×C4 | — | 4 | 12 | no | — |
| 160 | C2×C4 | — | 4 | 6 | no | — |
| 161 | C2×C4 | — | 332 | 3 | no | — |
| 162 | C2×C4 | — | 332 | 1 | no | — |
| 163 | C2×C4 | — | 664 | 3 | no | — |
| 164 | C2×C4 | — | 664 | 6 | no | — |
| 165 | C2×C4 | — | 166 | 6 | no | — |
| 166 | C22 | — | 2 | 2 | no | — |
| 167 | C22 | — | 2 | 3 | no | — |
| 168 | C22 | — | 2 | 3 | no | — |
| 169 | C22 | — | 2 | 6 | no | — |
| 170 | C22 | — | 2 | 6 | no | — |
| 171 | C22 | — | 4 | 4 | no | — |
| 172 | C22 | — | 8 | 6 | no | — |
| 173 | C22 | — | 8 | 6 | no | — |
| 174 | C22 | — | 332 | 2 | no | — |
| 175 | C22 | — | 664 | 6 | no | — |
| 176 | C22 | — | 664 | 6 | no | — |
| 177 | C22 | — | 664 | 2 | no | — |
| 178 | C22 | #19127612 | 1 | 1 | yes | characteristic |
| 179 | C22 | #19127888 | 1 | 2 | yes | — |
| 180 | C4 | — | 8 | 6 | no | — |
| 181 | C4 | — | 1328 | 3 | no | — |
| 182 | C4 | — | 332 | 2 | no | — |
| 183 | C2 | — | 2 | 2 | no | — |
| 184 | C2 | — | 4 | 3 | no | — |
| 185 | C2 | — | 4 | 3 | no | — |
| 186 | C2 | — | 664 | 2 | no | — |
| 187 | C2 | #24908626 | 1 | 1 | yes | characteristic |
| 188 | C2 | #24908639 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C83⋊#1304 | C2 | semidirect |
| 3. C83×#2840 | C2 | semidirect |
| 9. C24×D83 | Q8 | semidirect |
| 13. C23×Dic83 | Q8 | semidirect |
| 95. #2840 | D83 | semidirect |
| 98. C83 | #10390 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #24907377 | 2 |
| C83⋊#1304 | 2 |
| C83×#2840 | 2 |
| #10390 | 83 |