Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
994 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 175 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 | #81888 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #81761 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C3×#1705 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #71336 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #71262 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #68062 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #68050 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #19745 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×#914 | — | 2 | 6 | no | — |
| 10 | #5725 | C22 | 1 | 3 | yes | — |
| 11 | D3×#127 | C22 | 1 | 1 | yes | characteristic |
| 12 | C3×#498 | C22 | 1 | 3 | yes | complemented |
| 13 | C3×#492 | C22 | 1 | 3 | yes | — |
| 14 | #5176 | — | 2 | 6 | no | — |
| 15 | #5088 | C22 | 1 | 6 | yes | complemented |
| 16 | #5085 | C22 | 1 | 3 | yes | — |
| 17 | #5073 | C22 | 1 | 3 | yes | — |
| 18 | #4869 | — | 2 | 6 | no | — |
| 19 | #4865 | — | 2 | 6 | no | — |
| 20 | #4695 | C22 | 1 | 3 | yes | — |
| 21 | #4695 | C22 | 1 | 3 | yes | complemented |
| 22 | C4×ΓL(1,𝔽25) | C22 | 1 | 3 | yes | — |
| 23 | C3×C82 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | #4455 | C22 | 1 | 3 | yes | complemented |
| 25 | #3184 | — | 3 | 3 | no | — |
| 26 | #3151 | — | 3 | 3 | no | — |
| 27 | #1721 | — | 3 | 3 | no | — |
| 28 | #1715 | — | 3 | 3 | no | — |
| 29 | #1712 | — | 3 | 1 | no | — |
| 30 | #1705 | D3 | 1 | 1 | yes | characteristic, complemented |
| 31 | #1455 | — | 3 | 1 | no | — |
| 32 | C2×#239 | — | 2 | 3 | no | — |
| 33 | C2×D3×D4 | — | 2 | 3 | no | — |
| 34 | C2×C3×SD16 | — | 2 | 6 | no | — |
| 35 | C3×C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 36 | C3×#127 | C23 | 1 | 1 | yes | characteristic |
| 37 | #947 | — | 2 | 3 | no | — |
| 38 | C2×#215 | — | 2 | 6 | no | — |
| 39 | #940 | — | 2 | 3 | no | — |
| 40 | C2×#213 | — | 2 | 6 | no | — |
| 41 | #914 | — | 4 | 24 | no | — |
| 42 | C2×D24 | — | 2 | 6 | no | — |
| 43 | C2×D24 | — | 2 | 6 | no | — |
| 44 | C2×ΓL(1,𝔽25) | D4 | 1 | 6 | yes | — |
| 45 | #895 | — | 2 | 6 | no | — |
| 46 | #893 | — | 2 | 6 | no | — |
| 47 | #874 | C23 | 1 | 1 | yes | characteristic |
| 48 | #874 | C23 | 1 | 3 | yes | — |
| 49 | C42×D3 | C23 | 1 | 1 | yes | characteristic |
| 50 | C3×#106 | — | 2 | 6 | no | — |
| 51 | C3×C4×C8 | C23 | 1 | 3 | yes | — |
| 52 | #815 | D4 | 1 | 6 | yes | — |
| 53 | #809 | — | 2 | 12 | no | — |
| 54 | C4×C3⋊C8 | C23 | 1 | 3 | yes | — |
| 55 | C2×AGL(1,ℤ/8) | — | 6 | 6 | no | — |
| 56 | #532 | — | 3 | 3 | no | — |
| 57 | C2×#127 | — | 3 | 1 | no | — |
| 58 | #498 | C2×D3 | 1 | 3 | yes | complemented |
| 59 | #496 | — | 6 | 6 | no | — |
| 60 | #493 | — | 3 | 3 | no | — |
| 61 | #493 | — | 3 | 3 | no | — |
| 62 | #493 | — | 3 | 3 | no | — |
| 63 | #492 | — | 3 | 3 | no | — |
| 64 | #492 | C2×D3 | 1 | 3 | yes | — |
| 65 | #486 | — | 3 | 6 | no | — |
| 66 | #469 | — | 6 | 6 | no | — |
| 67 | #442 | — | 6 | 6 | no | — |
| 68 | C4×M16 | — | 3 | 3 | no | — |
| 69 | #322 | — | 3 | 3 | no | — |
| 70 | C82 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 71 | C23×D3 | — | 4 | 1 | no | — |
| 72 | C2×C3×Q8 | — | 2 | 3 | no | — |
| 73 | C2×C3×D4 | — | 2 | 3 | no | — |
| 74 | C2×#67 | — | 4 | 3 | no | — |
| 75 | #239 | — | 4 | 12 | no | — |
| 76 | D3×D4 | — | 4 | 12 | no | — |
| 77 | C2×D12 | — | 2 | 6 | no | — |
| 78 | C2×D12 | — | 2 | 3 | no | — |
| 79 | C2×D12 | — | 2 | 3 | no | — |
| 80 | C2×D12 | — | 4 | 3 | no | — |
| 81 | C2×C4×D3 | — | 4 | 3 | no | — |
| 82 | C2×C4×D3 | C2×D4 | 1 | 3 | yes | — |
| 83 | C3×SD16 | — | 4 | 12 | no | — |
| 84 | C2×C3×C8 | C2×D4 | 1 | 6 | yes | — |
| 85 | C3×C4⋊C4 | — | 2 | 6 | no | — |
| 86 | C3×C42 | C24 | 1 | 1 | yes | characteristic |
| 87 | #215 | — | 4 | 12 | no | — |
| 88 | #213 | — | 4 | 12 | no | — |
| 89 | (C2×D3)⋊C4 | — | 4 | 6 | no | — |
| 90 | C4×Dic3 | C2×D4 | 1 | 3 | yes | — |
| 91 | C2×C3⋊C8 | C2×D4 | 1 | 6 | yes | — |
| 92 | D24 | — | 4 | 12 | no | — |
| 93 | D24 | — | 4 | 12 | no | — |
| 94 | ΓL(1,𝔽25) | — | 2 | 12 | no | — |
| 95 | C2×#41 | — | 6 | 3 | no | — |
| 96 | C22×D4 | — | 6 | 3 | no | — |
| 97 | AGL(1,ℤ/8) | — | 12 | 24 | no | — |
| 98 | C2×SD16 | — | 2 | 6 | no | — |
| 99 | C2×SD16 | — | 6 | 6 | no | — |
| 100 | C2×D8 | — | 6 | 6 | no | — |
| 101 | C2×D8 | — | 6 | 6 | no | — |
| 102 | C2×D8 | — | 6 | 6 | no | — |
| 103 | C2×M16 | — | 3 | 6 | no | — |
| 104 | C4⋊Q8 | C22×D3 | 1 | 3 | yes | — |
| 105 | #127 | — | 3 | 1 | no | — |
| 106 | #127 | — | 3 | 3 | no | — |
| 107 | #127 | — | 6 | 3 | no | — |
| 108 | #127 | C22×D3 | 1 | 1 | yes | characteristic |
| 109 | #124 | — | 6 | 3 | no | — |
| 110 | #121 | — | 6 | 6 | no | — |
| 111 | C4×D4 | — | 6 | 6 | no | — |
| 112 | C2×C42 | — | 3 | 1 | no | — |
| 113 | #106 | — | 2 | 6 | no | — |
| 114 | #102 | — | 6 | 12 | no | — |
| 115 | #97 | — | 3 | 6 | no | — |
| 116 | C4×C8 | — | 3 | 3 | no | — |
| 117 | C4×C8 | C22×D3 | 1 | 3 | yes | — |
| 118 | C23×C3 | — | 4 | 1 | no | — |
| 119 | C22×D3 | — | 4 | 3 | no | — |
| 120 | C22×D3 | — | 4 | 1 | no | — |
| 121 | C22×D3 | — | 8 | 6 | no | — |
| 122 | C22×D3 | #127 | 1 | 1 | yes | characteristic |
| 123 | C3×Q8 | — | 4 | 6 | no | — |
| 124 | C3×D4 | — | 4 | 6 | no | — |
| 125 | C2×C3×C4 | — | 4 | 3 | no | — |
| 126 | C2×C3×C4 | C22×D4 | 1 | 3 | yes | — |
| 127 | #67 | — | 8 | 6 | no | — |
| 128 | C2×Dic3 | #127 | 1 | 3 | yes | — |
| 129 | D12 | — | 4 | 6 | no | — |
| 130 | D12 | — | 4 | 6 | no | — |
| 131 | D12 | — | 4 | 12 | no | — |
| 132 | D12 | — | 8 | 6 | no | — |
| 133 | C4×D3 | — | 2 | 6 | no | — |
| 134 | C4×D3 | — | 8 | 6 | no | — |
| 135 | C3×C8 | AGL(1,ℤ/8) | 1 | 12 | yes | complemented |
| 136 | C3⋊C8 | — | 2 | 6 | no | — |
| 137 | C24 | — | 12 | 1 | no | — |
| 138 | #41 | — | 12 | 12 | no | — |
| 139 | C2×Q8 | — | 2 | 3 | no | — |
| 140 | C2×D4 | — | 2 | 3 | no | — |
| 141 | C2×D4 | — | 6 | 6 | no | — |
| 142 | C2×D4 | — | 6 | 3 | no | — |
| 143 | C2×D4 | — | 6 | 3 | no | — |
| 144 | C2×D4 | — | 12 | 3 | no | — |
| 145 | C2×D4 | — | 12 | 3 | no | — |
| 146 | C2×D4 | — | 12 | 12 | no | — |
| 147 | C22×C4 | — | 3 | 3 | no | — |
| 148 | C22×C4 | — | 12 | 3 | no | — |
| 149 | SD16 | — | 4 | 12 | no | — |
| 150 | SD16 | — | 12 | 12 | no | — |
| 151 | D8 | — | 12 | 12 | no | — |
| 152 | D8 | — | 12 | 12 | no | — |
| 153 | D8 | — | 12 | 12 | no | — |
| 154 | M16 | — | 6 | 12 | no | — |
| 155 | C2×C8 | — | 3 | 6 | no | — |
| 156 | C2×C8 | D3×D4 | 1 | 6 | yes | — |
| 157 | C4⋊C4 | — | 2 | 6 | no | — |
| 158 | C22⋊C4 | — | 12 | 6 | no | — |
| 159 | C42 | — | 3 | 3 | no | — |
| 160 | C42 | C23×D3 | 1 | 1 | yes | characteristic |
| 161 | C22×C3 | — | 8 | 3 | no | — |
| 162 | C22×C3 | C2×#127 | 1 | 1 | yes | characteristic |
| 163 | C2×D3 | — | 2 | 3 | no | — |
| 164 | C2×D3 | — | 8 | 3 | no | — |
| 165 | C2×D3 | — | 8 | 3 | no | — |
| 166 | C2×D3 | — | 8 | 6 | no | — |
| 167 | C2×D3 | — | 16 | 4 | no | — |
| 168 | C3×C4 | — | 8 | 3 | no | — |
| 169 | C3×C4 | C2×AGL(1,ℤ/8) | 1 | 6 | yes | — |
| 170 | Dic3 | — | 2 | 3 | no | — |
| 171 | C23 | — | 3 | 1 | no | — |
| 172 | C23 | — | 4 | 1 | no | — |
| 173 | C23 | — | 12 | 3 | no | — |
| 174 | C23 | — | 12 | 1 | no | — |
| 175 | C23 | — | 24 | 6 | no | — |
| 176 | Q8 | — | 4 | 6 | no | — |
| 177 | D4 | — | 4 | 6 | no | — |
| 178 | D4 | — | 12 | 6 | no | — |
| 179 | D4 | — | 12 | 6 | no | — |
| 180 | D4 | — | 12 | 12 | no | — |
| 181 | D4 | — | 24 | 6 | no | — |
| 182 | D4 | — | 24 | 6 | no | — |
| 183 | C2×C4 | — | 3 | 3 | no | — |
| 184 | C2×C4 | — | 4 | 3 | no | — |
| 185 | C2×C4 | — | 6 | 6 | no | — |
| 186 | C2×C4 | — | 24 | 6 | no | — |
| 187 | C2×C4 | C2×D3×D4 | 1 | 3 | yes | — |
| 188 | C8 | — | 6 | 6 | no | — |
| 189 | C8 | #914 | 1 | 12 | yes | complemented |
| 190 | C2×C3 | — | 16 | 1 | no | — |
| 191 | C2×C3 | #3151 | 1 | 3 | yes | — |
| 192 | D3 | — | 4 | 1 | no | — |
| 193 | D3 | — | 16 | 3 | no | — |
| 194 | D3 | — | 16 | 1 | no | — |
| 195 | C22 | — | 6 | 3 | no | — |
| 196 | C22 | — | 8 | 3 | no | — |
| 197 | C22 | — | 24 | 3 | no | — |
| 198 | C22 | — | 24 | 3 | no | — |
| 199 | C22 | — | 24 | 6 | no | — |
| 200 | C22 | — | 48 | 4 | no | — |
| 201 | C22 | D3×#127 | 1 | 1 | yes | characteristic |
| 202 | C4 | — | 6 | 3 | no | — |
| 203 | C4 | — | 8 | 3 | no | — |
| 204 | C4 | C2×#914 | 1 | 6 | yes | — |
| 205 | C3 | #19745 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 206 | C2 | — | 12 | 1 | no | — |
| 207 | C2 | — | 16 | 1 | no | — |
| 208 | C2 | — | 48 | 1 | no | — |
| 209 | C2 | — | 48 | 3 | no | — |
| 210 | C2 | #81761 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D3×#127 |
| commutator, cosocle | C3×C42 | C24 |
| Frattini | C42 | C23×D3 |
| Fitting | C3×#1705 | C2 |
| socle | C22×C3 | C2×#127 |
| hypercenter | #1705 | D3 |
| hypocenter | C3 | #19745 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #1705 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #81888 | C2 | semidirect |
| 2. #81761 | C2 | semidirect |
| 3. C3×#1705 | C2 | semidirect |
| 4. #71336 | C2 | semidirect |
| 5. #71262 | C2 | semidirect |
| 6. #68062 | C2 | semidirect |
| 7. #68050 | C2 | semidirect |
| 12. C3×#498 | C22 | semidirect |
| 15. #5088 | C22 | semidirect |
| 21. #4695 | C22 | semidirect |
| 23. C3×C82 | C22 | semidirect |
| 24. #4455 | C22 | semidirect |
| 30. #1705 | D3 | semidirect |
| 58. #498 | C2×D3 | semidirect |
| 70. C82 | C2×D3 | semidirect |
| 135. C3×C8 | AGL(1,ℤ/8) | semidirect |
| 189. C8 | #914 | semidirect |
| 205. C3 | #19745 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #81888 | 2 |
| #81761 | 2 |
| C3×#1705 | 2 |
| #71336 | 2 |
| #71262 | 2 |
| #68062 | 2 |
| #68050 | 2 |
| #19745 | 3 |