Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
338 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 39 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 | D7×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10621504 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C7×#986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D7×GL(2,𝔽3) | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10621472 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | D7×AGL(1,ℤ/8) | — | 3 | 1 | no | maximal, Hall |
| 7 | C7×#41⋊C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | D7×D12 | — | 4 | 1 | no | maximal |
| 9 | D7×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | #66662 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | C7⋊GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 12 | C7×GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 13 | D7×#41 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | C2×D4×D7 | — | 3 | 1 | no | — |
| 15 | C7×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 16 | #6512 | — | 3 | 1 | no | — |
| 17 | #6495 | — | 3 | 1 | no | — |
| 18 | #6477 | — | 3 | 2 | no | — |
| 19 | D7×SD16 | — | 3 | 2 | no | — |
| 20 | #6474 | — | 3 | 2 | no | — |
| 21 | D7×D8 | — | 3 | 2 | no | — |
| 22 | #6471 | — | 3 | 1 | no | — |
| 23 | D7×M16 | — | 3 | 1 | no | — |
| 24 | C2×#986 | — | 7 | 1 | no | maximal, Hall |
| 25 | C2×D3×D7 | — | 4 | 2 | no | — |
| 26 | D84 | — | 4 | 1 | no | — |
| 27 | C7×D12 | — | 4 | 1 | no | — |
| 28 | C3×C4×D7 | — | 4 | 1 | no | — |
| 29 | C7×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic, complemented |
| 30 | #4247 | — | 4 | 2 | no | — |
| 31 | C23×D7 | — | 6 | 1 | no | — |
| 32 | C7×#41 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 33 | C2×C7×D4 | — | 3 | 1 | no | — |
| 34 | C2×C22⋊D7 | — | 6 | 1 | no | — |
| 35 | #1161 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 36 | Q8×D7 | C2×D3 | 1 | 1 | yes | characteristic |
| 37 | #1159 | — | 3 | 1 | no | — |
| 38 | D4×D7 | — | 3 | 1 | no | — |
| 39 | D4×D7 | — | 3 | 2 | no | — |
| 40 | D4×D7 | — | 3 | 2 | no | — |
| 41 | D4×D7 | — | 6 | 1 | no | — |
| 42 | D4×D7 | — | 6 | 1 | no | — |
| 43 | #1157 | — | 3 | 1 | no | — |
| 44 | C2×D28 | — | 3 | 1 | no | — |
| 45 | C2×C4×D7 | — | 3 | 1 | no | — |
| 46 | C7×SD16 | — | 3 | 2 | no | — |
| 47 | C7×D8 | — | 3 | 2 | no | — |
| 48 | C7×M16 | — | 3 | 1 | no | — |
| 49 | #1143 | — | 3 | 2 | no | — |
| 50 | #1142 | — | 3 | 2 | no | — |
| 51 | #1141 | — | 3 | 2 | no | — |
| 52 | #1141 | — | 3 | 2 | no | — |
| 53 | #1136 | — | 3 | 1 | no | — |
| 54 | D56 | — | 3 | 2 | no | — |
| 55 | #1132 | — | 3 | 2 | no | — |
| 56 | #1131 | — | 3 | 2 | no | — |
| 57 | C8×D7 | — | 3 | 2 | no | — |
| 58 | C2×#41⋊C3 | — | 7 | 1 | no | — |
| 59 | #986 | — | 7 | 1 | no | — |
| 60 | #986 | — | 7 | 2 | no | — |
| 61 | #986 | D7 | 1 | 1 | yes | characteristic, direct product factor |
| 62 | C2×GL(2,𝔽3) | — | 7 | 2 | no | — |
| 63 | C2×D21 | — | 4 | 2 | no | — |
| 64 | C2×D3×C7 | — | 4 | 2 | no | — |
| 65 | C2×C3×D7 | — | 4 | 1 | no | — |
| 66 | D3×D7 | — | 8 | 4 | no | — |
| 67 | C3×C4×C7 | — | 4 | 1 | no | — |
| 68 | C3×Dic7 | — | 4 | 1 | no | — |
| 69 | C2×AGL(1,ℤ/8) | — | 21 | 1 | no | Sylow |
| 70 | C23×C7 | — | 6 | 1 | no | — |
| 71 | C22×D7 | — | 3 | 1 | no | — |
| 72 | C22×D7 | — | 6 | 2 | no | — |
| 73 | C22×D7 | — | 6 | 1 | no | — |
| 74 | C22×D7 | — | 6 | 2 | no | — |
| 75 | C22×D7 | — | 12 | 4 | no | — |
| 76 | C7×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 77 | C7×D4 | — | 3 | 1 | no | — |
| 78 | C7×D4 | — | 3 | 2 | no | — |
| 79 | C7×D4 | — | 6 | 1 | no | — |
| 80 | C2×C4×C7 | — | 3 | 1 | no | — |
| 81 | C22⋊D7 | — | 6 | 1 | no | — |
| 82 | C22⋊D7 | — | 6 | 2 | no | — |
| 83 | C22⋊D7 | — | 6 | 2 | no | — |
| 84 | C2×Dic7 | — | 3 | 1 | no | — |
| 85 | D28 | — | 3 | 1 | no | — |
| 86 | D28 | — | 3 | 2 | no | — |
| 87 | D28 | — | 6 | 1 | no | — |
| 88 | C4×D7 | — | 3 | 1 | no | — |
| 89 | C4×D7 | — | 3 | 1 | no | — |
| 90 | C4×D7 | — | 3 | 1 | no | — |
| 91 | C4×D7 | S4 | 1 | 1 | yes | characteristic |
| 92 | Dic14 | — | 3 | 1 | no | — |
| 93 | C7×C8 | — | 3 | 2 | no | — |
| 94 | C7⋊C8 | — | 3 | 2 | no | — |
| 95 | C2×D12 | — | 28 | 1 | no | — |
| 96 | #41⋊C3 | — | 7 | 1 | no | — |
| 97 | #41⋊C3 | C2×D7 | 1 | 1 | yes | characteristic, complemented |
| 98 | C2×SL(2,𝔽3) | — | 7 | 1 | no | — |
| 99 | GL(2,𝔽3) | — | 7 | 2 | no | — |
| 100 | GL(2,𝔽3) | C2×D7 | 1 | 2 | yes | complemented |
| 101 | C2×C3×C7 | — | 4 | 1 | no | — |
| 102 | D21 | — | 8 | 2 | no | — |
| 103 | C3×D7 | — | 4 | 2 | no | — |
| 104 | D3×C7 | — | 8 | 2 | no | — |
| 105 | C2×#41 | — | 7 | 1 | no | — |
| 106 | C22×D4 | — | 21 | 1 | no | — |
| 107 | AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 108 | AGL(1,ℤ/8) | — | 21 | 1 | no | — |
| 109 | AGL(1,ℤ/8) | — | 21 | 2 | no | — |
| 110 | AGL(1,ℤ/8) | — | 21 | 2 | no | — |
| 111 | AGL(1,ℤ/8) | — | 21 | 1 | no | — |
| 112 | AGL(1,ℤ/8) | — | 21 | 1 | no | — |
| 113 | C2×SD16 | — | 21 | 2 | no | — |
| 114 | C2×D8 | — | 21 | 2 | no | — |
| 115 | C2×M16 | — | 21 | 1 | no | — |
| 116 | C22×C7 | — | 3 | 1 | no | — |
| 117 | C22×C7 | — | 6 | 2 | no | — |
| 118 | C22×C7 | — | 12 | 2 | no | — |
| 119 | C2×D7 | — | 3 | 1 | no | — |
| 120 | C2×D7 | — | 6 | 2 | no | — |
| 121 | C2×D7 | — | 6 | 2 | no | — |
| 122 | C2×D7 | — | 12 | 4 | no | — |
| 123 | C2×D7 | — | 12 | 2 | no | — |
| 124 | C2×D7 | — | 12 | 4 | no | — |
| 125 | C2×D7 | C2×S4 | 1 | 1 | yes | characteristic |
| 126 | C4×C7 | — | 3 | 1 | no | — |
| 127 | C4×C7 | C2×S4 | 1 | 1 | yes | characteristic |
| 128 | Dic7 | — | 3 | 1 | no | — |
| 129 | Dic7 | C2×S4 | 1 | 1 | yes | characteristic |
| 130 | C22×D3 | — | 28 | 2 | no | — |
| 131 | C2×C3×C4 | — | 28 | 1 | no | — |
| 132 | D12 | — | 4 | 1 | no | — |
| 133 | D12 | — | 28 | 1 | no | — |
| 134 | D12 | — | 28 | 2 | no | — |
| 135 | SL(2,𝔽3) | C22×D7 | 1 | 1 | yes | characteristic, complemented |
| 136 | C3×C7 | — | 4 | 1 | no | Hall |
| 137 | C24 | — | 42 | 1 | no | — |
| 138 | #41 | — | 7 | 1 | no | — |
| 139 | #41 | — | 21 | 1 | no | — |
| 140 | #41 | — | 21 | 1 | no | — |
| 141 | #41 | D3×D7 | 1 | 1 | yes | characteristic, complemented |
| 142 | C2×Q8 | — | 7 | 1 | no | — |
| 143 | C2×D4 | — | 3 | 1 | no | — |
| 144 | C2×D4 | — | 21 | 1 | no | — |
| 145 | C2×D4 | — | 21 | 1 | no | — |
| 146 | C2×D4 | — | 21 | 2 | no | — |
| 147 | C2×D4 | — | 21 | 2 | no | — |
| 148 | C2×D4 | — | 42 | 1 | no | — |
| 149 | C2×D4 | — | 42 | 1 | no | — |
| 150 | C2×D4 | — | 42 | 1 | no | — |
| 151 | C22×C4 | — | 21 | 1 | no | — |
| 152 | SD16 | — | 3 | 2 | no | — |
| 153 | SD16 | — | 21 | 2 | no | — |
| 154 | SD16 | — | 21 | 2 | no | — |
| 155 | SD16 | — | 21 | 2 | no | — |
| 156 | D8 | — | 3 | 2 | no | — |
| 157 | D8 | — | 21 | 2 | no | — |
| 158 | D8 | — | 21 | 2 | no | — |
| 159 | D8 | — | 21 | 2 | no | — |
| 160 | M16 | — | 3 | 1 | no | — |
| 161 | M16 | — | 21 | 2 | no | — |
| 162 | M16 | — | 21 | 1 | no | — |
| 163 | C2×C8 | — | 21 | 2 | no | — |
| 164 | C2×C7 | — | 6 | 1 | no | — |
| 165 | C2×C7 | — | 12 | 2 | no | — |
| 166 | C2×C7 | C22×S4 | 1 | 1 | yes | characteristic |
| 167 | D7 | — | 6 | 1 | no | — |
| 168 | D7 | — | 12 | 2 | no | — |
| 169 | D7 | #986 | 1 | 2 | yes | direct product factor |
| 170 | C22×C3 | — | 28 | 1 | no | — |
| 171 | C2×D3 | — | 4 | 2 | no | — |
| 172 | C2×D3 | — | 28 | 2 | no | — |
| 173 | C2×D3 | — | 56 | 4 | no | — |
| 174 | C3×C4 | — | 4 | 1 | no | — |
| 175 | C3×C4 | — | 28 | 1 | no | — |
| 176 | C23 | — | 6 | 1 | no | — |
| 177 | C23 | — | 21 | 1 | no | — |
| 178 | C23 | — | 42 | 1 | no | — |
| 179 | C23 | — | 42 | 2 | no | — |
| 180 | C23 | — | 42 | 2 | no | — |
| 181 | C23 | — | 84 | 4 | no | — |
| 182 | Q8 | — | 21 | 1 | no | — |
| 183 | Q8 | C2×D3×D7 | 1 | 1 | yes | characteristic |
| 184 | D4 | — | 3 | 1 | no | — |
| 185 | D4 | — | 3 | 2 | no | — |
| 186 | D4 | — | 6 | 1 | no | — |
| 187 | D4 | — | 21 | 1 | no | — |
| 188 | D4 | — | 21 | 2 | no | — |
| 189 | D4 | — | 42 | 2 | no | — |
| 190 | D4 | — | 42 | 1 | no | — |
| 191 | D4 | — | 42 | 2 | no | — |
| 192 | D4 | — | 42 | 1 | no | — |
| 193 | C2×C4 | — | 3 | 1 | no | — |
| 194 | C2×C4 | — | 7 | 1 | no | — |
| 195 | C2×C4 | — | 21 | 1 | no | — |
| 196 | C2×C4 | — | 21 | 1 | no | — |
| 197 | C2×C4 | — | 21 | 1 | no | — |
| 198 | C2×C4 | — | 21 | 1 | no | — |
| 199 | C8 | — | 3 | 2 | no | — |
| 200 | C8 | — | 21 | 2 | no | — |
| 201 | C7 | C2×#986 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 202 | C2×C3 | — | 4 | 1 | no | — |
| 203 | C2×C3 | — | 28 | 2 | no | — |
| 204 | D3 | — | 8 | 2 | no | — |
| 205 | D3 | — | 56 | 2 | no | — |
| 206 | C22 | — | 3 | 1 | no | — |
| 207 | C22 | — | 6 | 2 | no | — |
| 208 | C22 | — | 7 | 1 | no | — |
| 209 | C22 | — | 12 | 2 | no | — |
| 210 | C22 | — | 21 | 1 | no | — |
| 211 | C22 | — | 42 | 2 | no | — |
| 212 | C22 | — | 42 | 2 | no | — |
| 213 | C22 | — | 84 | 2 | no | — |
| 214 | C22 | — | 84 | 4 | no | — |
| 215 | C22 | — | 84 | 4 | no | — |
| 216 | C4 | — | 3 | 1 | no | — |
| 217 | C4 | — | 7 | 1 | no | — |
| 218 | C4 | — | 21 | 1 | no | — |
| 219 | C4 | D7×S4 | 1 | 1 | yes | characteristic |
| 220 | C3 | — | 4 | 1 | no | Sylow |
| 221 | C2 | — | 6 | 1 | no | — |
| 222 | C2 | — | 7 | 2 | no | — |
| 223 | C2 | — | 12 | 2 | no | — |
| 224 | C2 | — | 42 | 1 | no | — |
| 225 | C2 | — | 84 | 2 | no | — |
| 226 | C2 | C2×D7×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7×S4 |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C23 |
| Fitting | C7×#41 | C2×D3 |
| socle | C2×C7 | C22×S4 |
| hypercenter | C4 | D7×S4 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D7×#41⋊C3 | C2 | semidirect |
| 2. #10621504 | C2 | semidirect |
| 3. C7×#986 | C2 | semidirect |
| 4. D7×GL(2,𝔽3) | C2 | semidirect |
| 5. #10621472 | C2 | semidirect |
| 7. C7×#41⋊C3 | C22 | semidirect |
| 9. D7×SL(2,𝔽3) | C22 | semidirect |
| 10. #66662 | C22 | semidirect |
| 11. C7⋊GL(2,𝔽3) | C22 | semidirect |
| 12. C7×GL(2,𝔽3) | C22 | semidirect |
| 13. D7×#41 | D3 | semidirect |
| 29. C7×SL(2,𝔽3) | C23 | semidirect |
| 32. C7×#41 | C2×D3 | semidirect |
| 35. #1161 | C2×D3 | semidirect |
| 61. #986 | D7 | direct (N × H) |
| 97. #41⋊C3 | C2×D7 | semidirect |
| 100. GL(2,𝔽3) | C2×D7 | semidirect |
| 135. SL(2,𝔽3) | C22×D7 | semidirect |
| 141. #41 | D3×D7 | semidirect |
| 169. D7 | #986 | direct (N × H) |
| 201. C7 | C2×#986 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D7×#41⋊C3 | 2 |
| #10621504 | 2 |
| C7×#986 | 2 |
| D7×GL(2,𝔽3) | 2 |
| #10621472 | 2 |
| D7×AGL(1,ℤ/8) | 3 |
| D7×D12 | 4 |
| C2×#986 | 7 |