Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
312 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 43 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 | A4×Dic14 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C4×C7⋊S4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C7×A4⋊Q8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | (C7⋊S4)⋊C4 | C2 | 1 | 2 | yes | maximal |
| 5 | #10621477 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #90323 | — | 3 | 1 | no | maximal, Hall |
| 7 | C2×C7⋊S4 | C22 | 1 | 1 | yes | characteristic |
| 8 | C4×C7×A4 | C22 | 1 | 1 | yes | characteristic |
| 9 | D21⋊Q8 | — | 4 | 1 | no | maximal |
| 10 | A4×Dic7 | C22 | 1 | 2 | yes | — |
| 11 | C7⋊SL(2,ℤ/4) | C22 | 1 | 1 | yes | characteristic |
| 12 | C7×SL(2,ℤ/4) | C22 | 1 | 2 | yes | — |
| 13 | C2×Q8×D7 | — | 3 | 1 | no | — |
| 14 | C22×Dic14 | D3 | 1 | 1 | yes | characteristic, complemented |
| 15 | C7×C22⋊Q8 | — | 3 | 1 | no | — |
| 16 | #6507 | — | 3 | 1 | no | — |
| 17 | C4×C22⋊D7 | — | 3 | 1 | no | — |
| 18 | #6460 | — | 3 | 1 | no | — |
| 19 | #6459 | — | 3 | 2 | no | — |
| 20 | #6451 | — | 3 | 2 | no | — |
| 21 | #6450 | — | 3 | 1 | no | — |
| 22 | (C22⋊D7)⋊C4 | — | 3 | 2 | no | — |
| 23 | #6441 | — | 3 | 2 | no | — |
| 24 | Q8×S4 | — | 7 | 1 | no | maximal, Hall |
| 25 | C2×C7×A4 | C23 | 1 | 1 | yes | characteristic |
| 26 | C7⋊S4 | Q8 | 1 | 2 | yes | complemented |
| 27 | C4×D21 | — | 4 | 1 | no | — |
| 28 | C7×Dic6 | — | 4 | 1 | no | — |
| 29 | C3×Dic14 | — | 4 | 1 | no | — |
| 30 | #4248 | — | 4 | 2 | no | — |
| 31 | D21⋊C4 | — | 4 | 2 | no | — |
| 32 | C2×C7×Q8 | — | 3 | 1 | no | — |
| 33 | C22×C4×C7 | C2×D3 | 1 | 1 | yes | characteristic |
| 34 | C2×C22⋊D7 | — | 3 | 1 | no | — |
| 35 | C22×Dic7 | C2×D3 | 1 | 2 | yes | — |
| 36 | Q8×D7 | — | 6 | 1 | no | — |
| 37 | Q8×D7 | — | 6 | 2 | no | — |
| 38 | Q8×D7 | — | 6 | 1 | no | — |
| 39 | C2×C4×D7 | — | 3 | 2 | no | — |
| 40 | C2×C4×D7 | — | 3 | 1 | no | — |
| 41 | C2×Dic14 | — | 3 | 1 | no | — |
| 42 | C2×Dic14 | — | 3 | 2 | no | — |
| 43 | C2×Dic14 | — | 3 | 1 | no | — |
| 44 | C2×Dic14 | — | 3 | 2 | no | — |
| 45 | C7×C4⋊C4 | — | 3 | 1 | no | — |
| 46 | C7×C4⋊C4 | — | 3 | 2 | no | — |
| 47 | C7×C22⋊C4 | — | 3 | 2 | no | — |
| 48 | C22⋊Dic7 | — | 3 | 1 | no | — |
| 49 | (C2×D7)⋊C4 | — | 3 | 2 | no | — |
| 50 | (C2×D7)⋊C4 | — | 3 | 1 | no | — |
| 51 | C4⋊Dic7 | — | 3 | 2 | no | — |
| 52 | Dic7⋊C4 | — | 3 | 2 | no | — |
| 53 | Dic7⋊C4 | — | 3 | 2 | no | — |
| 54 | Dic7⋊C4 | — | 3 | 2 | no | — |
| 55 | Dic7⋊C4 | — | 3 | 1 | no | — |
| 56 | C4×Dic7 | — | 3 | 2 | no | — |
| 57 | C4×Dic7 | — | 3 | 1 | no | — |
| 58 | Q8×A4 | — | 7 | 1 | no | — |
| 59 | C4×S4 | — | 7 | 1 | no | — |
| 60 | C4×S4 | — | 7 | 2 | no | — |
| 61 | A4⋊Q8 | — | 7 | 2 | no | — |
| 62 | A4⋊Q8 | D7 | 1 | 1 | yes | characteristic, complemented |
| 63 | C2×D21 | — | 4 | 1 | no | — |
| 64 | C7×A4 | C2×Q8 | 1 | 1 | yes | characteristic, complemented |
| 65 | C3×C4×C7 | — | 4 | 1 | no | — |
| 66 | Dic21 | — | 4 | 1 | no | — |
| 67 | C3×Dic7 | — | 4 | 2 | no | — |
| 68 | C7×Dic3 | — | 4 | 2 | no | — |
| 69 | D4×Q8 | — | 21 | 1 | no | Sylow |
| 70 | C23×C7 | C22×D3 | 1 | 1 | yes | characteristic |
| 71 | C22×D7 | — | 3 | 1 | no | — |
| 72 | C7×Q8 | — | 6 | 1 | no | — |
| 73 | C7×Q8 | — | 6 | 1 | no | — |
| 74 | C2×C4×C7 | — | 3 | 1 | no | — |
| 75 | C2×C4×C7 | — | 3 | 1 | no | — |
| 76 | C2×C4×C7 | — | 3 | 2 | no | — |
| 77 | C2×C4×C7 | — | 3 | 2 | no | — |
| 78 | C22⋊D7 | — | 3 | 2 | no | — |
| 79 | C22⋊D7 | — | 3 | 2 | no | — |
| 80 | C2×Dic7 | — | 3 | 1 | no | — |
| 81 | C2×Dic7 | — | 3 | 2 | no | — |
| 82 | C2×Dic7 | — | 3 | 2 | no | — |
| 83 | C2×Dic7 | — | 3 | 1 | no | — |
| 84 | C2×Dic7 | — | 3 | 1 | no | — |
| 85 | C4×D7 | — | 6 | 2 | no | — |
| 86 | C4×D7 | — | 6 | 1 | no | — |
| 87 | C4×D7 | — | 6 | 1 | no | — |
| 88 | C4×D7 | — | 6 | 2 | no | — |
| 89 | Dic14 | — | 3 | 1 | no | — |
| 90 | Dic14 | — | 3 | 2 | no | — |
| 91 | Dic14 | — | 6 | 1 | no | — |
| 92 | Dic14 | — | 6 | 2 | no | — |
| 93 | Dic14 | — | 6 | 2 | no | — |
| 94 | Dic14 | S4 | 1 | 1 | yes | characteristic, complemented |
| 95 | C2×S4 | — | 7 | 1 | no | — |
| 96 | D3×Q8 | — | 28 | 1 | no | — |
| 97 | C4×A4 | — | 7 | 2 | no | — |
| 98 | C4×A4 | C2×D7 | 1 | 1 | yes | characteristic |
| 99 | SL(2,ℤ/4) | — | 7 | 1 | no | — |
| 100 | SL(2,ℤ/4) | C2×D7 | 1 | 2 | yes | — |
| 101 | C2×C3×C7 | — | 4 | 1 | no | — |
| 102 | D21 | — | 4 | 2 | no | — |
| 103 | C22×Q8 | — | 7 | 1 | no | — |
| 104 | C22×Q8 | — | 21 | 1 | no | — |
| 105 | C4⋊Q8 | — | 21 | 2 | no | — |
| 106 | C4⋊Q8 | — | 21 | 1 | no | — |
| 107 | C22⋊Q8 | — | 3 | 1 | no | — |
| 108 | C22⋊Q8 | — | 21 | 2 | no | — |
| 109 | C22⋊Q8 | — | 21 | 2 | no | — |
| 110 | C22⋊Q8 | — | 21 | 1 | no | — |
| 111 | C4×Q8 | — | 21 | 1 | no | — |
| 112 | C4×D4 | — | 21 | 2 | no | — |
| 113 | C4×D4 | — | 21 | 1 | no | — |
| 114 | C22×C7 | — | 3 | 1 | no | — |
| 115 | C22×C7 | — | 3 | 1 | no | — |
| 116 | C22×C7 | D3×Q8 | 1 | 1 | yes | characteristic, complemented |
| 117 | C2×D7 | — | 3 | 2 | no | — |
| 118 | C2×D7 | — | 6 | 1 | no | — |
| 119 | C2×D7 | — | 6 | 1 | no | — |
| 120 | C4×C7 | — | 3 | 1 | no | — |
| 121 | C4×C7 | — | 6 | 2 | no | — |
| 122 | C4×C7 | — | 6 | 2 | no | — |
| 123 | C4×C7 | C2×S4 | 1 | 1 | yes | characteristic |
| 124 | Dic7 | — | 3 | 2 | no | — |
| 125 | Dic7 | — | 3 | 2 | no | — |
| 126 | Dic7 | — | 6 | 1 | no | — |
| 127 | Dic7 | — | 6 | 1 | no | — |
| 128 | Dic7 | C2×S4 | 1 | 2 | yes | — |
| 129 | C2×A4 | C22×D7 | 1 | 1 | yes | characteristic |
| 130 | S4 | — | 7 | 2 | no | — |
| 131 | C3×Q8 | — | 28 | 1 | no | — |
| 132 | C4×D3 | — | 28 | 2 | no | — |
| 133 | C4×D3 | — | 28 | 1 | no | — |
| 134 | Dic6 | — | 4 | 1 | no | — |
| 135 | Dic6 | — | 28 | 2 | no | — |
| 136 | C3×C7 | — | 4 | 1 | no | Hall |
| 137 | C2×Q8 | — | 3 | 1 | no | — |
| 138 | C2×Q8 | — | 21 | 1 | no | — |
| 139 | C2×Q8 | — | 21 | 1 | no | — |
| 140 | C2×Q8 | — | 21 | 2 | no | — |
| 141 | C2×Q8 | — | 21 | 2 | no | — |
| 142 | C2×Q8 | — | 42 | 1 | no | — |
| 143 | C2×Q8 | — | 42 | 2 | no | — |
| 144 | C2×Q8 | — | 42 | 1 | no | — |
| 145 | C2×D4 | — | 21 | 1 | no | — |
| 146 | C22×C4 | — | 7 | 2 | no | — |
| 147 | C22×C4 | — | 21 | 1 | no | — |
| 148 | C22×C4 | — | 21 | 2 | no | — |
| 149 | C22×C4 | D3×D7 | 1 | 1 | yes | characteristic |
| 150 | C4⋊C4 | — | 3 | 1 | no | — |
| 151 | C4⋊C4 | — | 3 | 2 | no | — |
| 152 | C4⋊C4 | — | 21 | 1 | no | — |
| 153 | C4⋊C4 | — | 21 | 2 | no | — |
| 154 | C4⋊C4 | — | 21 | 2 | no | — |
| 155 | C4⋊C4 | — | 21 | 2 | no | — |
| 156 | C4⋊C4 | — | 21 | 2 | no | — |
| 157 | C22⋊C4 | — | 3 | 2 | no | — |
| 158 | C22⋊C4 | — | 21 | 1 | no | — |
| 159 | C22⋊C4 | — | 21 | 2 | no | — |
| 160 | C22⋊C4 | — | 21 | 1 | no | — |
| 161 | C42 | — | 21 | 1 | no | — |
| 162 | C42 | — | 21 | 2 | no | — |
| 163 | C2×C7 | — | 3 | 1 | no | — |
| 164 | C2×C7 | — | 3 | 1 | no | — |
| 165 | C2×C7 | C22×S4 | 1 | 1 | yes | characteristic |
| 166 | D7 | — | 6 | 2 | no | — |
| 167 | C2×D3 | — | 28 | 1 | no | — |
| 168 | A4 | Q8×D7 | 1 | 1 | yes | characteristic, complemented |
| 169 | C3×C4 | — | 4 | 1 | no | — |
| 170 | C3×C4 | — | 28 | 2 | no | — |
| 171 | Dic3 | — | 4 | 2 | no | — |
| 172 | Dic3 | — | 28 | 1 | no | — |
| 173 | C23 | — | 21 | 1 | no | — |
| 174 | C23 | C2×D3×D7 | 1 | 1 | yes | characteristic |
| 175 | Q8 | — | 6 | 1 | no | — |
| 176 | Q8 | — | 6 | 1 | no | — |
| 177 | Q8 | — | 7 | 1 | no | — |
| 178 | Q8 | — | 21 | 2 | no | — |
| 179 | Q8 | — | 21 | 1 | no | — |
| 180 | Q8 | — | 42 | 1 | no | — |
| 181 | Q8 | — | 42 | 2 | no | — |
| 182 | Q8 | — | 42 | 2 | no | — |
| 183 | D4 | — | 21 | 2 | no | — |
| 184 | D4 | — | 21 | 2 | no | — |
| 185 | C2×C4 | — | 3 | 1 | no | — |
| 186 | C2×C4 | — | 3 | 1 | no | — |
| 187 | C2×C4 | — | 3 | 2 | no | — |
| 188 | C2×C4 | — | 3 | 2 | no | — |
| 189 | C2×C4 | — | 21 | 2 | no | — |
| 190 | C2×C4 | — | 21 | 1 | no | — |
| 191 | C2×C4 | — | 21 | 2 | no | — |
| 192 | C2×C4 | — | 21 | 1 | no | — |
| 193 | C2×C4 | — | 21 | 1 | no | — |
| 194 | C2×C4 | — | 42 | 1 | no | — |
| 195 | C2×C4 | — | 42 | 2 | no | — |
| 196 | C2×C4 | — | 42 | 2 | no | — |
| 197 | C2×C4 | — | 42 | 1 | no | — |
| 198 | C7 | Q8×S4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 199 | C2×C3 | — | 4 | 1 | no | — |
| 200 | D3 | — | 28 | 2 | no | — |
| 201 | C22 | — | 3 | 1 | no | — |
| 202 | C22 | — | 3 | 1 | no | — |
| 203 | C22 | — | 21 | 2 | no | — |
| 204 | C22 | — | 42 | 1 | no | — |
| 205 | C22 | — | 42 | 1 | no | — |
| 206 | C22 | D21⋊Q8 | 1 | 1 | yes | characteristic, complemented |
| 207 | C4 | — | 3 | 1 | no | — |
| 208 | C4 | — | 6 | 2 | no | — |
| 209 | C4 | — | 6 | 2 | no | — |
| 210 | C4 | — | 7 | 2 | no | — |
| 211 | C4 | — | 21 | 2 | no | — |
| 212 | C4 | — | 21 | 2 | no | — |
| 213 | C4 | — | 42 | 1 | no | — |
| 214 | C4 | — | 42 | 1 | no | — |
| 215 | C4 | D7×S4 | 1 | 1 | yes | characteristic |
| 216 | C3 | — | 4 | 1 | no | Sylow |
| 217 | C2 | — | 3 | 1 | no | — |
| 218 | C2 | — | 3 | 1 | no | — |
| 219 | C2 | — | 42 | 2 | no | — |
| 220 | C2 | C2×D7×S4 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×Dic14 | C2 | semidirect |
| 3. C7×A4⋊Q8 | C2 | semidirect |
| 5. #10621477 | C2 | semidirect |
| 14. C22×Dic14 | D3 | semidirect |
| 26. C7⋊S4 | Q8 | semidirect |
| 62. A4⋊Q8 | D7 | semidirect |
| 64. C7×A4 | C2×Q8 | semidirect |
| 94. Dic14 | S4 | semidirect |
| 116. C22×C7 | D3×Q8 | semidirect |
| 168. A4 | Q8×D7 | semidirect |
| 198. C7 | Q8×S4 | semidirect |
| 206. C22 | D21⋊Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×Dic14 | 2 |
| C4×C7⋊S4 | 2 |
| C7×A4⋊Q8 | 2 |
| (C7⋊S4)⋊C4 | 2 |
| #10621477 | 2 |
| #90323 | 3 |
| D21⋊Q8 | 4 |
| Q8×S4 | 7 |