Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
304 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 304 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 | C7×2−1+4⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10621504 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10621503 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10621501 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C2×C7⋊GL(2,𝔽3) | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #90528 | — | 3 | 1 | no | maximal, Hall |
| 7 | D4×D21 | — | 4 | 1 | no | maximal |
| 8 | C7×#41⋊C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | C2×C7×SL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 10 | C7⋊GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | C7⋊GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 12 | C7⋊#226 | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | C7×2−1+4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | #6553 | — | 3 | 1 | no | — |
| 15 | #6513 | — | 3 | 2 | no | — |
| 16 | #6513 | — | 3 | 1 | no | — |
| 17 | #6512 | — | 3 | 2 | no | — |
| 18 | #6512 | — | 3 | 1 | no | — |
| 19 | #6511 | — | 3 | 1 | no | — |
| 20 | #6505 | — | 3 | 1 | no | — |
| 21 | #6505 | — | 3 | 2 | no | — |
| 22 | C2×#1143 | — | 3 | 1 | no | — |
| 23 | C2×#1143 | — | 3 | 2 | no | — |
| 24 | #5926 | — | 7 | 1 | no | maximal, Hall |
| 25 | C22×D21 | — | 4 | 2 | no | — |
| 26 | C3×C7×D4 | — | 4 | 1 | no | — |
| 27 | #4268 | — | 4 | 2 | no | — |
| 28 | D84 | — | 4 | 1 | no | — |
| 29 | C4×D21 | — | 4 | 1 | no | — |
| 30 | C7×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 31 | C7×#41 | — | 3 | 1 | no | — |
| 32 | C7×#41 | — | 3 | 2 | no | — |
| 33 | C7×#41 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 34 | C2×C7×Q8 | — | 3 | 1 | no | — |
| 35 | C2×C7×Q8 | C2×D3 | 1 | 2 | yes | complemented |
| 36 | #1161 | — | 6 | 1 | no | — |
| 37 | D4×D7 | — | 6 | 2 | no | — |
| 38 | D4×D7 | — | 6 | 1 | no | — |
| 39 | #1157 | — | 3 | 2 | no | — |
| 40 | #1157 | — | 3 | 1 | no | — |
| 41 | C2×D28 | — | 3 | 2 | no | — |
| 42 | C2×D28 | — | 3 | 1 | no | — |
| 43 | #1144 | — | 3 | 2 | no | — |
| 44 | #1144 | — | 3 | 1 | no | — |
| 45 | #1143 | — | 3 | 2 | no | — |
| 46 | #1143 | — | 3 | 2 | no | — |
| 47 | #1143 | — | 3 | 2 | no | — |
| 48 | #1143 | — | 3 | 1 | no | — |
| 49 | #1143 | — | 3 | 2 | no | — |
| 50 | #1142 | — | 3 | 1 | no | — |
| 51 | #1141 | — | 3 | 1 | no | — |
| 52 | #1141 | — | 3 | 2 | no | — |
| 53 | #1136 | — | 3 | 2 | no | — |
| 54 | #1136 | — | 3 | 1 | no | — |
| 55 | C2×C7⋊C8 | — | 3 | 2 | no | — |
| 56 | C2×C7⋊C8 | — | 3 | 1 | no | — |
| 57 | 2−1+4⋊C3 | D7 | 1 | 1 | yes | characteristic, complemented |
| 58 | #986 | — | 7 | 1 | no | — |
| 59 | #985 | — | 7 | 1 | no | — |
| 60 | #983 | — | 7 | 2 | no | — |
| 61 | C2×GL(2,𝔽3) | — | 7 | 2 | no | — |
| 62 | C22×C3×C7 | — | 4 | 2 | no | — |
| 63 | C2×D21 | — | 4 | 1 | no | — |
| 64 | C2×D21 | — | 4 | 2 | no | — |
| 65 | C2×D21 | — | 8 | 4 | no | — |
| 66 | C3×C4×C7 | — | 4 | 1 | no | — |
| 67 | Dic21 | — | 4 | 1 | no | — |
| 68 | #577 | — | 21 | 1 | no | Sylow |
| 69 | C22×D7 | — | 6 | 1 | no | — |
| 70 | C22×D7 | — | 6 | 2 | no | — |
| 71 | C7×Q8 | — | 3 | 2 | no | — |
| 72 | C7×Q8 | — | 3 | 1 | no | — |
| 73 | C7×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 74 | C7×D4 | — | 3 | 1 | no | — |
| 75 | C7×D4 | — | 3 | 2 | no | — |
| 76 | C7×D4 | S4 | 1 | 1 | yes | characteristic |
| 77 | C2×C4×C7 | — | 3 | 2 | no | — |
| 78 | C2×C4×C7 | — | 3 | 1 | no | — |
| 79 | C2×C4×C7 | — | 6 | 1 | no | — |
| 80 | C22⋊D7 | — | 6 | 1 | no | — |
| 81 | C22⋊D7 | — | 6 | 2 | no | — |
| 82 | D28 | — | 3 | 2 | no | — |
| 83 | D28 | — | 3 | 1 | no | — |
| 84 | D28 | — | 6 | 2 | no | — |
| 85 | D28 | — | 6 | 1 | no | — |
| 86 | C4×D7 | — | 6 | 2 | no | — |
| 87 | C4×D7 | — | 6 | 1 | no | — |
| 88 | Dic14 | — | 3 | 1 | no | — |
| 89 | C7⋊C8 | — | 3 | 1 | no | — |
| 90 | C7⋊C8 | — | 3 | 1 | no | — |
| 91 | C7⋊C8 | — | 3 | 2 | no | — |
| 92 | D3×D4 | — | 28 | 1 | no | — |
| 93 | #41⋊C3 | C2×D7 | 1 | 1 | yes | characteristic, complemented |
| 94 | C2×SL(2,𝔽3) | C2×D7 | 1 | 2 | yes | complemented |
| 95 | GL(2,𝔽3) | — | 7 | 1 | no | — |
| 96 | GL(2,𝔽3) | — | 7 | 2 | no | — |
| 97 | #226 | — | 7 | 1 | no | — |
| 98 | C2×C3×C7 | — | 4 | 1 | no | — |
| 99 | C2×C3×C7 | — | 8 | 2 | no | — |
| 100 | D21 | — | 4 | 2 | no | — |
| 101 | D21 | — | 8 | 2 | no | — |
| 102 | 2−1+4 | D21 | 1 | 1 | yes | characteristic, complemented |
| 103 | H(5,𝔽2) | — | 21 | 1 | no | — |
| 104 | #137 | — | 21 | 1 | no | — |
| 105 | #137 | — | 21 | 2 | no | — |
| 106 | AGL(1,ℤ/8) | — | 21 | 2 | no | — |
| 107 | AGL(1,ℤ/8) | — | 21 | 1 | no | — |
| 108 | #135 | — | 21 | 2 | no | — |
| 109 | #135 | — | 21 | 1 | no | — |
| 110 | C2×SD16 | — | 21 | 1 | no | — |
| 111 | C2×SD16 | — | 21 | 2 | no | — |
| 112 | #131 | — | 21 | 1 | no | — |
| 113 | C22×C7 | — | 3 | 1 | no | — |
| 114 | C22×C7 | C2×S4 | 1 | 2 | yes | — |
| 115 | C2×D7 | — | 6 | 1 | no | — |
| 116 | C2×D7 | — | 6 | 2 | no | — |
| 117 | C2×D7 | — | 24 | 2 | no | — |
| 118 | C2×D7 | — | 24 | 1 | no | — |
| 119 | C4×C7 | — | 3 | 1 | no | — |
| 120 | C4×C7 | — | 3 | 2 | no | — |
| 121 | C4×C7 | C2×S4 | 1 | 1 | yes | characteristic |
| 122 | Dic7 | — | 6 | 1 | no | — |
| 123 | C22×D3 | — | 28 | 2 | no | — |
| 124 | C3×D4 | — | 4 | 1 | no | — |
| 125 | #67 | — | 28 | 2 | no | — |
| 126 | D12 | — | 28 | 1 | no | — |
| 127 | C4×D3 | — | 28 | 1 | no | — |
| 128 | SL(2,𝔽3) | C22×D7 | 1 | 1 | yes | characteristic |
| 129 | C3×C7 | — | 4 | 1 | no | Hall |
| 130 | #41 | — | 3 | 1 | no | — |
| 131 | #41 | — | 3 | 2 | no | — |
| 132 | #41 | — | 21 | 2 | no | — |
| 133 | #41 | — | 21 | 1 | no | — |
| 134 | #41 | — | 42 | 1 | no | — |
| 135 | #41 | C2×D21 | 1 | 1 | yes | characteristic, complemented |
| 136 | C2×Q8 | — | 3 | 1 | no | — |
| 137 | C2×Q8 | C2×D21 | 1 | 2 | yes | complemented |
| 138 | C2×D4 | — | 21 | 2 | no | — |
| 139 | C2×D4 | — | 21 | 1 | no | — |
| 140 | C2×D4 | — | 42 | 2 | no | — |
| 141 | C2×D4 | — | 42 | 1 | no | — |
| 142 | Q16 | — | 21 | 2 | no | — |
| 143 | Q16 | — | 21 | 1 | no | — |
| 144 | SD16 | — | 21 | 1 | no | — |
| 145 | SD16 | — | 21 | 2 | no | — |
| 146 | SD16 | — | 21 | 2 | no | — |
| 147 | SD16 | — | 21 | 2 | no | — |
| 148 | SD16 | — | 21 | 1 | no | — |
| 149 | SD16 | — | 21 | 2 | no | — |
| 150 | D8 | — | 21 | 2 | no | — |
| 151 | D8 | — | 21 | 1 | no | — |
| 152 | M16 | — | 21 | 2 | no | — |
| 153 | M16 | — | 21 | 1 | no | — |
| 154 | C2×C8 | — | 21 | 2 | no | — |
| 155 | C2×C8 | — | 21 | 1 | no | — |
| 156 | C2×C7 | — | 2 | 2 | no | — |
| 157 | C2×C7 | — | 6 | 1 | no | — |
| 158 | C2×C7 | C22×S4 | 1 | 1 | yes | characteristic |
| 159 | D7 | — | 12 | 1 | no | — |
| 160 | D7 | — | 12 | 2 | no | — |
| 161 | C22×C3 | — | 4 | 2 | no | — |
| 162 | C2×D3 | — | 28 | 1 | no | — |
| 163 | C2×D3 | — | 28 | 2 | no | — |
| 164 | C2×D3 | — | 56 | 4 | no | — |
| 165 | C3×C4 | — | 4 | 1 | no | — |
| 166 | Dic3 | — | 28 | 1 | no | — |
| 167 | C23 | — | 42 | 1 | no | — |
| 168 | C23 | — | 42 | 2 | no | — |
| 169 | Q8 | — | 3 | 2 | no | — |
| 170 | Q8 | — | 3 | 1 | no | — |
| 171 | Q8 | — | 21 | 1 | no | — |
| 172 | Q8 | C22×D21 | 1 | 1 | yes | characteristic |
| 173 | D4 | — | 3 | 1 | no | — |
| 174 | D4 | — | 3 | 2 | no | — |
| 175 | D4 | — | 21 | 2 | no | — |
| 176 | D4 | — | 21 | 1 | no | — |
| 177 | D4 | — | 42 | 2 | no | — |
| 178 | D4 | — | 42 | 1 | no | — |
| 179 | D4 | — | 42 | 2 | no | — |
| 180 | D4 | — | 42 | 1 | no | — |
| 181 | D4 | C7⋊S4 | 1 | 1 | yes | characteristic |
| 182 | C2×C4 | — | 3 | 2 | no | — |
| 183 | C2×C4 | — | 3 | 1 | no | — |
| 184 | C2×C4 | — | 6 | 1 | no | — |
| 185 | C2×C4 | — | 42 | 2 | no | — |
| 186 | C2×C4 | — | 42 | 1 | no | — |
| 187 | C8 | — | 21 | 1 | no | — |
| 188 | C8 | — | 21 | 1 | no | — |
| 189 | C8 | — | 21 | 2 | no | — |
| 190 | C7 | #5926 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 191 | C2×C3 | — | 4 | 1 | no | — |
| 192 | C2×C3 | — | 8 | 2 | no | — |
| 193 | D3 | — | 28 | 2 | no | — |
| 194 | D3 | — | 56 | 2 | no | — |
| 195 | C22 | — | 3 | 1 | no | — |
| 196 | C22 | — | 42 | 1 | no | — |
| 197 | C22 | — | 42 | 2 | no | — |
| 198 | C22 | — | 168 | 2 | no | — |
| 199 | C22 | — | 168 | 1 | no | — |
| 200 | C22 | C2×C7⋊S4 | 1 | 2 | yes | — |
| 201 | C4 | — | 3 | 1 | no | — |
| 202 | C4 | — | 3 | 2 | no | — |
| 203 | C4 | — | 42 | 1 | no | — |
| 204 | C4 | C2×C7⋊S4 | 1 | 1 | yes | characteristic |
| 205 | C3 | — | 4 | 1 | no | Sylow |
| 206 | C2 | — | 2 | 2 | no | — |
| 207 | C2 | — | 6 | 1 | no | — |
| 208 | C2 | — | 84 | 1 | no | — |
| 209 | C2 | — | 84 | 2 | no | — |
| 210 | C2 | C22×C7⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C7⋊S4 |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C23 |
| Fitting | C7×2−1+4 | D3 |
| socle | C2×C7 | C22×S4 |
| hypercenter | D4 | C7⋊S4 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×2−1+4⋊C3 | C2 | semidirect |
| 2. #10621504 | C2 | semidirect |
| 3. #10621503 | C2 | semidirect |
| 4. #10621501 | C2 | semidirect |
| 5. C2×C7⋊GL(2,𝔽3) | C2 | semidirect |
| 8. C7×#41⋊C3 | C22 | semidirect |
| 9. C2×C7×SL(2,𝔽3) | C22 | semidirect |
| 10. C7⋊GL(2,𝔽3) | C22 | semidirect |
| 11. C7⋊GL(2,𝔽3) | C22 | semidirect |
| 12. C7⋊#226 | C22 | semidirect |
| 13. C7×2−1+4 | D3 | semidirect |
| 33. C7×#41 | C2×D3 | semidirect |
| 35. C2×C7×Q8 | C2×D3 | semidirect |
| 57. 2−1+4⋊C3 | D7 | semidirect |
| 93. #41⋊C3 | C2×D7 | semidirect |
| 94. C2×SL(2,𝔽3) | C2×D7 | semidirect |
| 102. 2−1+4 | D21 | semidirect |
| 135. #41 | C2×D21 | semidirect |
| 137. C2×Q8 | C2×D21 | semidirect |
| 190. C7 | #5926 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×2−1+4⋊C3 | 2 |
| #10621504 | 2 |
| #10621503 | 2 |
| #10621501 | 2 |
| C2×C7⋊GL(2,𝔽3) | 2 |
| #90528 | 3 |
| D4×D21 | 4 |
| #5926 | 7 |