Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
960 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 960 conjugacy classes of subgroups — too many to draw. Only its 147 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 | C2×C7×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11729480 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×#89335 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C7×H(3,𝔽4) | — | 2 | 6 | no | — |
| 5 | C7×H(3,𝔽4) | C4 | 1 | 4 | yes | complemented |
| 6 | C2×C7×#124 | — | 2 | 3 | no | — |
| 7 | C2×C7×#124 | C22 | 1 | 3 | yes | — |
| 8 | C2×C7×#120 | — | 2 | 3 | no | — |
| 9 | C2×#6409 | — | 2 | 3 | no | — |
| 10 | C2×#6406 | — | 2 | 3 | no | — |
| 11 | #89707 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | C2×#6377 | C22 | 1 | 3 | yes | complemented |
| 13 | #89335 | — | 2 | 6 | no | — |
| 14 | #8506 | — | 7 | 1 | no | maximal, Sylow |
| 15 | C25×C7 | — | 2 | 1 | no | — |
| 16 | C22×C7×Q8 | D4 | 1 | 3 | yes | — |
| 17 | C22×C7×D4 | — | 2 | 3 | no | — |
| 18 | C22×C7×D4 | D4 | 1 | 3 | yes | — |
| 19 | C7×#124 | — | 2 | 12 | no | — |
| 20 | C7×#124 | — | 2 | 12 | no | — |
| 21 | C7×#124 | — | 2 | 6 | no | — |
| 22 | C7×#124 | C2×C4 | 1 | 6 | yes | — |
| 23 | C7×#124 | D4 | 1 | 6 | yes | — |
| 24 | C7×#120 | — | 2 | 12 | no | — |
| 25 | C7×#120 | — | 2 | 12 | no | — |
| 26 | C2×C7×C22⋊C4 | — | 2 | 3 | no | — |
| 27 | C2×C7×C22⋊C4 | — | 2 | 6 | no | — |
| 28 | C2×C42×C7 | — | 2 | 1 | no | — |
| 29 | C2×C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 30 | C2×C4⋊Dic7 | — | 2 | 3 | no | — |
| 31 | C22×C7⋊C8 | — | 2 | 3 | no | — |
| 32 | #6409 | — | 4 | 6 | no | — |
| 33 | #6406 | — | 4 | 6 | no | — |
| 34 | #6377 | — | 2 | 6 | no | — |
| 35 | C2×H(3,𝔽4) | D7 | 1 | 1 | yes | characteristic |
| 36 | #1959 | — | 7 | 3 | no | — |
| 37 | C2×#329 | — | 7 | 3 | no | — |
| 38 | C24×C7 | — | 2 | 4 | no | — |
| 39 | C24×C7 | — | 2 | 3 | no | — |
| 40 | C24×C7 | — | 8 | 3 | no | — |
| 41 | C24×C7 | — | 8 | 3 | no | — |
| 42 | C2×C7×Q8 | — | 2 | 3 | no | — |
| 43 | C2×C7×Q8 | — | 4 | 3 | no | — |
| 44 | C2×C7×Q8 | — | 4 | 3 | no | — |
| 45 | C2×C7×Q8 | C22⋊C4 | 1 | 6 | yes | — |
| 46 | C2×C7×D4 | — | 2 | 12 | no | — |
| 47 | C2×C7×D4 | — | 2 | 3 | no | — |
| 48 | C2×C7×D4 | — | 4 | 3 | no | — |
| 49 | C2×C7×D4 | — | 4 | 3 | no | — |
| 50 | C2×C7×D4 | — | 8 | 3 | no | — |
| 51 | C2×C7×D4 | — | 8 | 3 | no | — |
| 52 | C2×C7×D4 | C22⋊C4 | 1 | 6 | yes | — |
| 53 | C22×C4×C7 | — | 2 | 3 | no | — |
| 54 | C22×C4×C7 | C2×D4 | 1 | 3 | yes | — |
| 55 | C22×Dic7 | — | 4 | 1 | no | — |
| 56 | C7×C22⋊C4 | — | 2 | 12 | no | — |
| 57 | C7×C22⋊C4 | — | 2 | 12 | no | — |
| 58 | C7×C22⋊C4 | — | 2 | 12 | no | — |
| 59 | C42×C7 | — | 2 | 4 | no | — |
| 60 | C42×C7 | C22×C4 | 1 | 1 | yes | characteristic |
| 61 | C42×C7 | C2×D4 | 1 | 3 | yes | — |
| 62 | C4⋊Dic7 | — | 4 | 3 | no | — |
| 63 | C4⋊Dic7 | — | 4 | 3 | no | — |
| 64 | C2×C7⋊C8 | — | 2 | 6 | no | — |
| 65 | C2×C7⋊C8 | — | 4 | 3 | no | — |
| 66 | C2×C7⋊C8 | — | 4 | 3 | no | — |
| 67 | H(3,𝔽4) | — | 2 | 6 | no | — |
| 68 | H(3,𝔽4) | Dic7 | 1 | 4 | yes | complemented |
| 69 | C2×#124 | — | 2 | 3 | no | — |
| 70 | C2×#124 | C2×D7 | 1 | 3 | yes | — |
| 71 | C2×#120 | — | 2 | 3 | no | — |
| 72 | C2×#103 | — | 14 | 3 | no | — |
| 73 | C2×#102 | — | 14 | 3 | no | — |
| 74 | C2×#97 | — | 7 | 3 | no | — |
| 75 | #384 | — | 7 | 1 | no | — |
| 76 | #329 | — | 14 | 6 | no | — |
| 77 | C23×C7 | — | 2 | 6 | no | — |
| 78 | C23×C7 | — | 4 | 3 | no | — |
| 79 | C23×C7 | — | 4 | 6 | no | — |
| 80 | C23×C7 | — | 4 | 3 | no | — |
| 81 | C23×C7 | — | 4 | 6 | no | — |
| 82 | C23×C7 | — | 8 | 12 | no | — |
| 83 | C23×C7 | — | 8 | 4 | no | — |
| 84 | C23×C7 | — | 8 | 12 | no | — |
| 85 | C23×C7 | #120 | 1 | 1 | yes | characteristic |
| 86 | C7×Q8 | — | 4 | 6 | no | — |
| 87 | C7×Q8 | — | 8 | 3 | no | — |
| 88 | C7×D4 | — | 4 | 6 | no | — |
| 89 | C7×D4 | — | 8 | 3 | no | — |
| 90 | C7×D4 | — | 8 | 12 | no | — |
| 91 | C2×C4×C7 | — | 2 | 3 | no | — |
| 92 | C2×C4×C7 | — | 2 | 3 | no | — |
| 93 | C2×C4×C7 | — | 2 | 6 | no | — |
| 94 | C2×C4×C7 | — | 4 | 3 | no | — |
| 95 | C2×C4×C7 | — | 4 | 3 | no | — |
| 96 | C2×C4×C7 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 97 | C2×C4×C7 | #120 | 1 | 3 | yes | — |
| 98 | C2×Dic7 | — | 8 | 3 | no | — |
| 99 | C7⋊C8 | — | 4 | 6 | no | — |
| 100 | C25 | — | 2 | 1 | no | — |
| 101 | C22×Q8 | C22⋊D7 | 1 | 3 | yes | — |
| 102 | C22×D4 | — | 2 | 3 | no | — |
| 103 | C22×D4 | C22⋊D7 | 1 | 3 | yes | — |
| 104 | C22×C8 | — | 14 | 3 | no | — |
| 105 | #124 | — | 2 | 12 | no | — |
| 106 | #124 | — | 2 | 12 | no | — |
| 107 | #124 | — | 2 | 6 | no | — |
| 108 | #124 | C2×Dic7 | 1 | 6 | yes | — |
| 109 | #124 | C22⋊D7 | 1 | 6 | yes | — |
| 110 | #120 | — | 2 | 12 | no | — |
| 111 | #120 | — | 2 | 12 | no | — |
| 112 | C2×C4⋊C4 | — | 14 | 3 | no | — |
| 113 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 114 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 115 | C2×C42 | — | 2 | 1 | no | — |
| 116 | C2×C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 117 | #103 | — | 28 | 6 | no | — |
| 118 | #102 | — | 28 | 6 | no | — |
| 119 | #97 | — | 14 | 6 | no | — |
| 120 | C22×C7 | — | 4 | 6 | no | — |
| 121 | C22×C7 | — | 4 | 12 | no | — |
| 122 | C22×C7 | — | 8 | 16 | no | — |
| 123 | C22×C7 | — | 8 | 3 | no | — |
| 124 | C22×C7 | — | 8 | 3 | no | — |
| 125 | C22×C7 | — | 8 | 6 | no | — |
| 126 | C22×C7 | #379 | 1 | 1 | yes | characteristic |
| 127 | C22×C7 | #453 | 1 | 3 | yes | — |
| 128 | C22×C7 | #455 | 1 | 3 | yes | — |
| 129 | C4×C7 | — | 2 | 3 | no | — |
| 130 | C4×C7 | — | 2 | 3 | no | — |
| 131 | C4×C7 | — | 4 | 6 | no | — |
| 132 | Dic7 | — | 16 | 1 | no | — |
| 133 | C24 | — | 2 | 4 | no | — |
| 134 | C24 | — | 2 | 3 | no | — |
| 135 | C24 | — | 8 | 3 | no | — |
| 136 | C24 | — | 8 | 3 | no | — |
| 137 | C2×Q8 | — | 2 | 3 | no | — |
| 138 | C2×Q8 | — | 4 | 3 | no | — |
| 139 | C2×Q8 | — | 4 | 3 | no | — |
| 140 | C2×Q8 | C22⋊Dic7 | 1 | 6 | yes | — |
| 141 | C2×D4 | — | 2 | 12 | no | — |
| 142 | C2×D4 | — | 2 | 3 | no | — |
| 143 | C2×D4 | — | 4 | 3 | no | — |
| 144 | C2×D4 | — | 4 | 3 | no | — |
| 145 | C2×D4 | — | 8 | 3 | no | — |
| 146 | C2×D4 | — | 8 | 3 | no | — |
| 147 | C2×D4 | C22⋊Dic7 | 1 | 6 | yes | — |
| 148 | C22×C4 | — | 2 | 3 | no | — |
| 149 | C22×C4 | — | 28 | 1 | no | — |
| 150 | C22×C4 | C2×C22⋊D7 | 1 | 3 | yes | — |
| 151 | C2×C8 | — | 14 | 6 | no | — |
| 152 | C2×C8 | — | 28 | 3 | no | — |
| 153 | C2×C8 | — | 28 | 3 | no | — |
| 154 | C4⋊C4 | — | 28 | 3 | no | — |
| 155 | C4⋊C4 | — | 28 | 3 | no | — |
| 156 | C22⋊C4 | — | 2 | 12 | no | — |
| 157 | C22⋊C4 | — | 2 | 12 | no | — |
| 158 | C22⋊C4 | — | 2 | 12 | no | — |
| 159 | C42 | — | 2 | 4 | no | — |
| 160 | C42 | C22×Dic7 | 1 | 1 | yes | characteristic |
| 161 | C42 | C2×C22⋊D7 | 1 | 3 | yes | — |
| 162 | C2×C7 | — | 8 | 6 | no | — |
| 163 | C2×C7 | #1655 | 1 | 1 | yes | characteristic |
| 164 | C2×C7 | #1659 | 1 | 3 | yes | — |
| 165 | C2×C7 | #1792 | 1 | 3 | yes | — |
| 166 | C23 | — | 2 | 6 | no | — |
| 167 | C23 | — | 4 | 3 | no | — |
| 168 | C23 | — | 4 | 6 | no | — |
| 169 | C23 | — | 4 | 3 | no | — |
| 170 | C23 | — | 4 | 6 | no | — |
| 171 | C23 | — | 8 | 12 | no | — |
| 172 | C23 | — | 8 | 4 | no | — |
| 173 | C23 | — | 8 | 12 | no | — |
| 174 | C23 | #6516 | 1 | 1 | yes | characteristic |
| 175 | Q8 | — | 4 | 6 | no | — |
| 176 | Q8 | — | 8 | 3 | no | — |
| 177 | D4 | — | 4 | 6 | no | — |
| 178 | D4 | — | 8 | 3 | no | — |
| 179 | D4 | — | 8 | 12 | no | — |
| 180 | C2×C4 | — | 2 | 3 | no | — |
| 181 | C2×C4 | — | 2 | 3 | no | — |
| 182 | C2×C4 | — | 2 | 6 | no | — |
| 183 | C2×C4 | — | 4 | 3 | no | — |
| 184 | C2×C4 | — | 4 | 3 | no | — |
| 185 | C2×C4 | — | 56 | 3 | no | — |
| 186 | C2×C4 | C2×C22⋊Dic7 | 1 | 3 | yes | — |
| 187 | C2×C4 | #6516 | 1 | 3 | yes | — |
| 188 | C8 | — | 28 | 6 | no | — |
| 189 | C7 | #8506 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 190 | C22 | — | 4 | 6 | no | — |
| 191 | C22 | — | 4 | 12 | no | — |
| 192 | C22 | — | 8 | 16 | no | — |
| 193 | C22 | — | 8 | 3 | no | — |
| 194 | C22 | — | 8 | 3 | no | — |
| 195 | C22 | — | 8 | 6 | no | — |
| 196 | C22 | #90013 | 1 | 3 | yes | — |
| 197 | C22 | #90014 | 1 | 3 | yes | — |
| 198 | C22 | #90019 | 1 | 1 | yes | characteristic |
| 199 | C4 | — | 2 | 3 | no | — |
| 200 | C4 | — | 2 | 3 | no | — |
| 201 | C4 | — | 4 | 6 | no | — |
| 202 | C4 | — | 112 | 1 | no | — |
| 203 | C2 | — | 8 | 6 | no | — |
| 204 | C2 | #11729702 | 1 | 1 | yes | characteristic |
| 205 | C2 | #11729703 | 1 | 3 | yes | — |
| 206 | C2 | #11729894 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11729480 | C2 | semidirect |
| 3. C2×#89335 | C2 | semidirect |
| 5. C7×H(3,𝔽4) | C4 | semidirect |
| 11. #89707 | C22 | semidirect |
| 12. C2×#6377 | C22 | semidirect |
| 68. H(3,𝔽4) | Dic7 | semidirect |
| 189. C7 | #8506 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×H(3,𝔽4) | 2 |
| #11729480 | 2 |
| C2×#89335 | 2 |
| #8506 | 7 |