Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
390 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 60 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 | 2−1+4⋊ΓL(1,𝔽8) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D7⋊(#41⋊C3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10621539 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10621538 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C2×#66665 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #90620 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 7 | #41⋊ΓL(1,𝔽8) | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×C7⋊SL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 9 | D7⋊SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | #66665 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | #66665 | C22 | 1 | 2 | yes | complemented |
| 12 | #1159⋊C3 | — | 4 | 1 | no | maximal |
| 13 | C7×2−1+4 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 14 | #6553 | — | 3 | 2 | no | — |
| 15 | #6553 | — | 3 | 1 | no | — |
| 16 | #6553 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 17 | D7×#41 | — | 3 | 1 | no | — |
| 18 | D7×#41 | — | 3 | 2 | no | — |
| 19 | D7×#41 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 20 | #6551 | — | 3 | 1 | no | — |
| 21 | #6551 | C2×C3 | 1 | 2 | yes | complemented |
| 22 | C2×#1161 | — | 3 | 1 | no | — |
| 23 | C2×#1161 | C2×C3 | 1 | 2 | yes | complemented |
| 24 | #5945 | — | 7 | 1 | no | maximal, Hall |
| 25 | C7⋊SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 26 | D4×ΓL(1,𝔽8) | — | 4 | 1 | no | — |
| 27 | (C22⋊D7)⋊C3 | — | 4 | 2 | no | — |
| 28 | C2×ΓL(1,𝔽8)⋊C4 | — | 4 | 2 | no | — |
| 29 | C4×AGL(1,𝔽7) | — | 4 | 1 | no | — |
| 30 | ΓL(1,𝔽8)⋊Q8 | — | 4 | 1 | no | — |
| 31 | C7×#41 | — | 3 | 1 | no | — |
| 32 | C7×#41 | — | 3 | 2 | no | — |
| 33 | C7×#41 | C22×C3 | 1 | 1 | yes | characteristic |
| 34 | C2×C7×Q8 | — | 3 | 1 | no | — |
| 35 | C2×C7×Q8 | C22×C3 | 1 | 2 | yes | — |
| 36 | #1161 | — | 3 | 2 | no | — |
| 37 | #1161 | — | 3 | 2 | no | — |
| 38 | #1161 | — | 3 | 1 | no | — |
| 39 | #1161 | — | 3 | 2 | no | — |
| 40 | #1161 | — | 3 | 2 | no | — |
| 41 | #1161 | C22×C3 | 1 | 1 | yes | characteristic |
| 42 | #1161 | C22×C3 | 1 | 2 | yes | — |
| 43 | Q8×D7 | — | 3 | 1 | no | — |
| 44 | Q8×D7 | — | 3 | 2 | no | — |
| 45 | Q8×D7 | C22×C3 | 1 | 1 | yes | characteristic |
| 46 | #1159 | — | 3 | 2 | no | — |
| 47 | #1159 | — | 3 | 1 | no | — |
| 48 | #1159 | A4 | 1 | 1 | yes | characteristic |
| 49 | D4×D7 | — | 3 | 4 | no | — |
| 50 | D4×D7 | — | 3 | 1 | no | — |
| 51 | D4×D7 | — | 3 | 1 | no | — |
| 52 | D4×D7 | — | 3 | 2 | no | — |
| 53 | D4×D7 | — | 3 | 2 | no | — |
| 54 | #1157 | — | 3 | 4 | no | — |
| 55 | #1157 | — | 3 | 2 | no | — |
| 56 | #1157 | — | 3 | 2 | no | — |
| 57 | #1157 | — | 3 | 1 | no | — |
| 58 | #1157 | — | 3 | 1 | no | — |
| 59 | C2×D28 | — | 3 | 2 | no | — |
| 60 | C2×D28 | — | 3 | 2 | no | — |
| 61 | C2×D28 | — | 3 | 1 | no | — |
| 62 | C2×C4×D7 | — | 3 | 2 | no | — |
| 63 | C2×C4×D7 | — | 3 | 2 | no | — |
| 64 | C2×C4×D7 | — | 3 | 1 | no | — |
| 65 | 2−1+4⋊C3 | — | 7 | 2 | no | — |
| 66 | 2−1+4⋊C3 | — | 7 | 1 | no | — |
| 67 | #994 | — | 7 | 1 | no | — |
| 68 | C2×#41⋊C3 | — | 7 | 2 | no | — |
| 69 | C2×#41⋊C3 | — | 7 | 1 | no | — |
| 70 | C22×ΓL(1,𝔽8) | — | 4 | 2 | no | — |
| 71 | C2×AGL(1,𝔽7) | — | 4 | 1 | no | — |
| 72 | C4×ΓL(1,𝔽8) | — | 4 | 1 | no | — |
| 73 | ΓL(1,𝔽8)⋊C4 | — | 4 | 2 | no | — |
| 74 | ΓL(1,𝔽8)⋊C4 | — | 4 | 1 | no | — |
| 75 | #585 | — | 7 | 1 | no | Sylow |
| 76 | C22×D7 | — | 3 | 2 | no | — |
| 77 | C22×D7 | — | 3 | 2 | no | — |
| 78 | C22×D7 | — | 3 | 1 | no | — |
| 79 | C7×Q8 | — | 3 | 2 | no | — |
| 80 | C7×Q8 | — | 3 | 1 | no | — |
| 81 | C7×Q8 | C23×C3 | 1 | 1 | yes | characteristic |
| 82 | C7×D4 | — | 3 | 1 | no | — |
| 83 | C7×D4 | — | 3 | 2 | no | — |
| 84 | C7×D4 | C2×A4 | 1 | 1 | yes | characteristic |
| 85 | C2×C4×C7 | — | 3 | 2 | no | — |
| 86 | C2×C4×C7 | — | 3 | 2 | no | — |
| 87 | C2×C4×C7 | — | 3 | 1 | no | — |
| 88 | C22⋊D7 | — | 3 | 2 | no | — |
| 89 | C22⋊D7 | — | 3 | 2 | no | — |
| 90 | C22⋊D7 | — | 3 | 2 | no | — |
| 91 | C22⋊D7 | C2×A4 | 1 | 2 | yes | — |
| 92 | C2×Dic7 | — | 3 | 1 | no | — |
| 93 | C2×Dic7 | C2×A4 | 1 | 2 | yes | — |
| 94 | D28 | — | 3 | 1 | no | — |
| 95 | D28 | — | 3 | 2 | no | — |
| 96 | D28 | — | 3 | 4 | no | — |
| 97 | D28 | — | 3 | 2 | no | — |
| 98 | D28 | — | 3 | 1 | no | — |
| 99 | C4×D7 | — | 3 | 4 | no | — |
| 100 | C4×D7 | — | 3 | 1 | no | — |
| 101 | C4×D7 | — | 3 | 1 | no | — |
| 102 | C4×D7 | — | 3 | 2 | no | — |
| 103 | C4×D7 | — | 3 | 1 | no | — |
| 104 | C4×D7 | — | 3 | 2 | no | — |
| 105 | C4×D7 | — | 3 | 2 | no | — |
| 106 | C4×D7 | C2×A4 | 1 | 1 | yes | characteristic |
| 107 | Dic14 | — | 3 | 2 | no | — |
| 108 | Dic14 | — | 3 | 1 | no | — |
| 109 | Dic14 | C2×A4 | 1 | 1 | yes | characteristic |
| 110 | C3×#41 | — | 28 | 1 | no | — |
| 111 | #41⋊C3 | — | 7 | 1 | no | — |
| 112 | #41⋊C3 | — | 7 | 2 | no | — |
| 113 | #41⋊C3 | — | 7 | 1 | no | — |
| 114 | C2×SL(2,𝔽3) | — | 7 | 1 | no | — |
| 115 | C2×SL(2,𝔽3) | — | 7 | 2 | no | — |
| 116 | C2×ΓL(1,𝔽8) | — | 4 | 1 | no | — |
| 117 | C2×ΓL(1,𝔽8) | — | 8 | 2 | no | — |
| 118 | AGL(1,𝔽7) | — | 8 | 1 | no | — |
| 119 | 2−1+4 | — | 7 | 2 | no | — |
| 120 | 2−1+4 | — | 21 | 1 | no | — |
| 121 | 2−1+4 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 122 | H(5,𝔽2) | — | 7 | 1 | no | — |
| 123 | H(5,𝔽2) | — | 21 | 1 | no | — |
| 124 | H(5,𝔽2) | — | 21 | 2 | no | — |
| 125 | C2×#41 | — | 7 | 1 | no | — |
| 126 | C2×#41 | — | 7 | 2 | no | — |
| 127 | C2×#41 | — | 21 | 1 | no | — |
| 128 | C2×#41 | — | 21 | 1 | no | — |
| 129 | C2×#41 | — | 21 | 2 | no | — |
| 130 | C22×C7 | — | 3 | 1 | no | — |
| 131 | C22×C7 | C22×A4 | 1 | 2 | yes | — |
| 132 | C2×D7 | — | 3 | 2 | no | — |
| 133 | C2×D7 | — | 3 | 1 | no | — |
| 134 | C2×D7 | — | 12 | 1 | no | — |
| 135 | C2×D7 | — | 12 | 2 | no | — |
| 136 | C2×D7 | — | 12 | 2 | no | — |
| 137 | C2×D7 | C22×A4 | 1 | 1 | yes | characteristic |
| 138 | C4×C7 | — | 3 | 2 | no | — |
| 139 | C4×C7 | — | 3 | 1 | no | — |
| 140 | C4×C7 | C22×A4 | 1 | 1 | yes | characteristic |
| 141 | Dic7 | — | 3 | 1 | no | — |
| 142 | Dic7 | C22×A4 | 1 | 2 | yes | — |
| 143 | Dic7 | C22×A4 | 1 | 1 | yes | characteristic |
| 144 | C3×Q8 | — | 28 | 1 | no | — |
| 145 | C3×D4 | — | 28 | 2 | no | — |
| 146 | C3×D4 | — | 28 | 1 | no | — |
| 147 | C2×C3×C4 | — | 28 | 1 | no | — |
| 148 | C2×C3×C4 | — | 28 | 2 | no | — |
| 149 | SL(2,𝔽3) | — | 7 | 1 | no | — |
| 150 | ΓL(1,𝔽8) | — | 4 | 1 | no | Hall |
| 151 | #41 | — | 3 | 1 | no | — |
| 152 | #41 | — | 3 | 2 | no | — |
| 153 | #41 | — | 7 | 1 | no | — |
| 154 | #41 | — | 7 | 1 | no | — |
| 155 | #41 | — | 7 | 2 | no | — |
| 156 | #41 | — | 21 | 2 | no | — |
| 157 | #41 | — | 21 | 1 | no | — |
| 158 | #41 | — | 21 | 4 | no | — |
| 159 | #41 | — | 21 | 2 | no | — |
| 160 | #41 | — | 21 | 1 | no | — |
| 161 | #41 | — | 21 | 2 | no | — |
| 162 | #41 | — | 21 | 1 | no | — |
| 163 | #41 | — | 21 | 2 | no | — |
| 164 | #41 | — | 21 | 2 | no | — |
| 165 | #41 | — | 21 | 1 | no | — |
| 166 | #41 | — | 21 | 2 | no | — |
| 167 | #41 | — | 21 | 2 | no | — |
| 168 | #41 | C2×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 169 | C2×Q8 | — | 3 | 1 | no | — |
| 170 | C2×Q8 | — | 7 | 1 | no | — |
| 171 | C2×Q8 | — | 21 | 1 | no | — |
| 172 | C2×Q8 | — | 21 | 2 | no | — |
| 173 | C2×Q8 | C2×AGL(1,𝔽7) | 1 | 2 | yes | — |
| 174 | C2×D4 | — | 21 | 2 | no | — |
| 175 | C2×D4 | — | 21 | 2 | no | — |
| 176 | C2×D4 | — | 21 | 1 | no | — |
| 177 | C2×D4 | — | 21 | 2 | no | — |
| 178 | C2×D4 | — | 21 | 1 | no | — |
| 179 | C2×D4 | — | 21 | 4 | no | — |
| 180 | C2×D4 | — | 21 | 2 | no | — |
| 181 | C2×D4 | — | 21 | 1 | no | — |
| 182 | C22×C4 | — | 21 | 2 | no | — |
| 183 | C22×C4 | — | 21 | 1 | no | — |
| 184 | C22×C4 | — | 21 | 2 | no | — |
| 185 | C2×C7 | — | 2 | 2 | no | — |
| 186 | C2×C7 | — | 6 | 1 | no | — |
| 187 | C2×C7 | C23×A4 | 1 | 1 | yes | characteristic |
| 188 | D7 | — | 2 | 1 | no | — |
| 189 | D7 | — | 6 | 1 | no | — |
| 190 | D7 | — | 6 | 2 | no | — |
| 191 | C22×C3 | — | 28 | 1 | no | — |
| 192 | C22×C3 | — | 28 | 2 | no | — |
| 193 | C3×C4 | — | 28 | 1 | no | — |
| 194 | C3×C4 | — | 28 | 2 | no | — |
| 195 | C3×C4 | — | 28 | 1 | no | — |
| 196 | C23 | — | 21 | 1 | no | — |
| 197 | C23 | — | 21 | 2 | no | — |
| 198 | C23 | — | 21 | 2 | no | — |
| 199 | Q8 | — | 3 | 2 | no | — |
| 200 | Q8 | — | 3 | 1 | no | — |
| 201 | Q8 | — | 7 | 1 | no | — |
| 202 | Q8 | — | 21 | 2 | no | — |
| 203 | Q8 | — | 21 | 1 | no | — |
| 204 | Q8 | C22×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 205 | D4 | — | 3 | 1 | no | — |
| 206 | D4 | — | 3 | 2 | no | — |
| 207 | D4 | — | 7 | 2 | no | — |
| 208 | D4 | — | 21 | 2 | no | — |
| 209 | D4 | — | 21 | 1 | no | — |
| 210 | D4 | — | 21 | 2 | no | — |
| 211 | D4 | — | 21 | 4 | no | — |
| 212 | D4 | — | 21 | 2 | no | — |
| 213 | D4 | — | 21 | 2 | no | — |
| 214 | D4 | — | 21 | 2 | no | — |
| 215 | D4 | — | 21 | 1 | no | — |
| 216 | D4 | D7⋊A4 | 1 | 1 | yes | characteristic |
| 217 | C2×C4 | — | 3 | 2 | no | — |
| 218 | C2×C4 | — | 3 | 2 | no | — |
| 219 | C2×C4 | — | 3 | 1 | no | — |
| 220 | C2×C4 | — | 7 | 2 | no | — |
| 221 | C2×C4 | — | 7 | 1 | no | — |
| 222 | C2×C4 | — | 21 | 1 | no | — |
| 223 | C2×C4 | — | 21 | 2 | no | — |
| 224 | C2×C4 | — | 21 | 2 | no | — |
| 225 | C2×C4 | — | 21 | 2 | no | — |
| 226 | C2×C4 | — | 21 | 1 | no | — |
| 227 | C2×C4 | — | 21 | 4 | no | — |
| 228 | C2×C4 | — | 21 | 1 | no | — |
| 229 | C2×C4 | — | 21 | 1 | no | — |
| 230 | C7 | #5945 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 231 | C2×C3 | — | 28 | 1 | no | — |
| 232 | C2×C3 | — | 56 | 2 | no | — |
| 233 | C2×C3 | — | 56 | 1 | no | — |
| 234 | C22 | — | 3 | 1 | no | — |
| 235 | C22 | — | 7 | 1 | no | — |
| 236 | C22 | — | 21 | 1 | no | — |
| 237 | C22 | — | 21 | 2 | no | — |
| 238 | C22 | — | 84 | 2 | no | — |
| 239 | C22 | — | 84 | 1 | no | — |
| 240 | C22 | — | 84 | 2 | no | — |
| 241 | C22 | C2×D7⋊A4 | 1 | 2 | yes | — |
| 242 | C4 | — | 3 | 1 | no | — |
| 243 | C4 | — | 3 | 2 | no | — |
| 244 | C4 | — | 7 | 1 | no | — |
| 245 | C4 | — | 7 | 2 | no | — |
| 246 | C4 | — | 21 | 1 | no | — |
| 247 | C4 | C2×D7⋊A4 | 1 | 1 | yes | characteristic |
| 248 | C3 | — | 28 | 1 | no | Sylow |
| 249 | C2 | — | 2 | 2 | no | — |
| 250 | C2 | — | 6 | 1 | no | — |
| 251 | C2 | — | 14 | 1 | no | — |
| 252 | C2 | — | 42 | 1 | no | — |
| 253 | C2 | — | 42 | 2 | no | — |
| 254 | C2 | C22×D7⋊A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D7⋊A4 |
| commutator, cosocle, hypocenter | C7×Q8 | C23×C3 |
| Fitting | C7×2−1+4 | C2×C3 |
| socle | C2×C7 | C23×A4 |
| hypercenter | D4 | D7⋊A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. 2−1+4⋊ΓL(1,𝔽8) | C2 | semidirect |
| 2. D7⋊(#41⋊C3) | C2 | semidirect |
| 3. #10621539 | C2 | semidirect |
| 4. #10621538 | C2 | semidirect |
| 5. C2×#66665 | C2 | semidirect |
| 6. #90620 | C3 | semidirect |
| 7. #41⋊ΓL(1,𝔽8) | C22 | semidirect |
| 8. C2×C7⋊SL(2,𝔽3) | C22 | semidirect |
| 9. D7⋊SL(2,𝔽3) | C22 | semidirect |
| 10. #66665 | C22 | semidirect |
| 11. #66665 | C22 | semidirect |
| 13. C7×2−1+4 | C2×C3 | semidirect |
| 16. #6553 | C2×C3 | semidirect |
| 19. D7×#41 | C2×C3 | semidirect |
| 21. #6551 | C2×C3 | semidirect |
| 23. C2×#1161 | C2×C3 | semidirect |
| 121. 2−1+4 | AGL(1,𝔽7) | semidirect |
| 230. C7 | #5945 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 2−1+4⋊ΓL(1,𝔽8) | 2 |
| D7⋊(#41⋊C3) | 2 |
| #10621539 | 2 |
| #10621538 | 2 |
| C2×#66665 | 2 |
| #90620 | 3 |
| #1159⋊C3 | 4 |
| #5945 | 7 |