Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1353 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 1353 conjugacy classes of subgroups — too many to draw. Only its 199 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 | C22×AGL(1,ℤ/8) | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C22×#127 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#418 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #2297 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #2014 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #2011 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #1728 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×#41 | C22 | 1 | 1 | yes | characteristic |
| 9 | C23×D4 | — | 2 | 2 | no | — |
| 10 | C23×D4 | C22 | 1 | 2 | yes | — |
| 11 | C2×AGL(1,ℤ/8) | — | 2 | 8 | no | — |
| 12 | C2×AGL(1,ℤ/8) | — | 2 | 8 | no | — |
| 13 | C2×AGL(1,ℤ/8) | — | 2 | 8 | no | — |
| 14 | C22×SD16 | C22 | 1 | 4 | yes | complemented |
| 15 | C22×D8 | C22 | 1 | 4 | yes | complemented |
| 16 | C22×M16 | C22 | 1 | 2 | yes | — |
| 17 | C2×#127 | — | 2 | 8 | no | — |
| 18 | C2×#127 | — | 2 | 2 | no | — |
| 19 | C2×#127 | C22 | 1 | 4 | yes | — |
| 20 | C2×#117 | C22 | 1 | 1 | yes | characteristic |
| 21 | C22×C42 | C22 | 1 | 1 | yes | characteristic |
| 22 | #418 | — | 2 | 8 | no | — |
| 23 | C2×#102 | C22 | 1 | 8 | yes | complemented |
| 24 | #391 | C22 | 1 | 2 | yes | complemented |
| 25 | #390 | C22 | 1 | 2 | yes | complemented |
| 26 | #386 | — | 2 | 2 | no | — |
| 27 | C4×C22⋊C4 | — | 2 | 2 | no | — |
| 28 | #336 | C22 | 1 | 4 | yes | complemented |
| 29 | C25 | — | 2 | 2 | no | — |
| 30 | C2×#41 | — | 2 | 2 | no | — |
| 31 | C2×#41 | — | 2 | 4 | no | — |
| 32 | C2×#41 | — | 2 | 4 | no | — |
| 33 | C2×#41 | D4 | 1 | 4 | yes | complemented |
| 34 | C22×Q8 | C23 | 1 | 1 | yes | characteristic |
| 35 | C22×D4 | — | 2 | 16 | no | — |
| 36 | C22×D4 | — | 2 | 1 | no | — |
| 37 | C22×D4 | — | 2 | 4 | no | — |
| 38 | C22×D4 | — | 2 | 4 | no | — |
| 39 | C22×D4 | — | 2 | 4 | no | — |
| 40 | C22×D4 | — | 2 | 4 | no | — |
| 41 | C22×D4 | — | 2 | 2 | no | — |
| 42 | C22×D4 | — | 2 | 8 | no | — |
| 43 | C22×D4 | — | 2 | 4 | no | — |
| 44 | C22×D4 | — | 4 | 16 | no | — |
| 45 | C22×D4 | D4 | 1 | 8 | yes | — |
| 46 | C22×D4 | C23 | 1 | 4 | yes | — |
| 47 | C22×D4 | C23 | 1 | 1 | yes | characteristic |
| 48 | C23×C4 | — | 2 | 1 | no | — |
| 49 | C23×C4 | D4 | 1 | 2 | yes | — |
| 50 | C23×C4 | C23 | 1 | 1 | yes | characteristic |
| 51 | AGL(1,ℤ/8) | — | 2 | 64 | no | — |
| 52 | C2×SD16 | — | 2 | 8 | no | — |
| 53 | C2×SD16 | — | 2 | 8 | no | — |
| 54 | C2×SD16 | — | 2 | 8 | no | — |
| 55 | C2×D8 | — | 2 | 8 | no | — |
| 56 | C2×D8 | — | 2 | 8 | no | — |
| 57 | C2×D8 | — | 2 | 8 | no | — |
| 58 | C2×M16 | — | 2 | 4 | no | — |
| 59 | C2×M16 | — | 2 | 4 | no | — |
| 60 | C2×M16 | D4 | 1 | 8 | yes | complemented |
| 61 | C22×C8 | C23 | 1 | 4 | yes | — |
| 62 | #127 | — | 2 | 16 | no | — |
| 63 | #127 | — | 2 | 16 | no | — |
| 64 | #117 | — | 2 | 4 | no | — |
| 65 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 66 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 67 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 68 | C2×C42 | — | 2 | 4 | no | — |
| 69 | C2×C42 | — | 2 | 1 | no | — |
| 70 | C2×C42 | D4 | 1 | 2 | yes | — |
| 71 | C2×C42 | C23 | 1 | 2 | yes | — |
| 72 | #102 | — | 2 | 16 | no | — |
| 73 | #95 | — | 2 | 4 | no | — |
| 74 | C24 | — | 2 | 8 | no | — |
| 75 | C24 | — | 2 | 1 | no | — |
| 76 | C24 | — | 2 | 4 | no | — |
| 77 | C24 | — | 4 | 16 | no | — |
| 78 | C24 | — | 4 | 4 | no | — |
| 79 | C24 | — | 4 | 4 | no | — |
| 80 | C24 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | #41 | — | 2 | 16 | no | — |
| 82 | #41 | — | 4 | 8 | no | — |
| 83 | C2×Q8 | — | 2 | 1 | no | — |
| 84 | C2×Q8 | — | 2 | 2 | no | — |
| 85 | C2×Q8 | — | 2 | 2 | no | — |
| 86 | C2×Q8 | C2×D4 | 1 | 2 | yes | complemented |
| 87 | C2×D4 | — | 2 | 32 | no | — |
| 88 | C2×D4 | — | 2 | 4 | no | — |
| 89 | C2×D4 | — | 2 | 16 | no | — |
| 90 | C2×D4 | — | 2 | 8 | no | — |
| 91 | C2×D4 | — | 2 | 16 | no | — |
| 92 | C2×D4 | — | 2 | 4 | no | — |
| 93 | C2×D4 | — | 2 | 2 | no | — |
| 94 | C2×D4 | — | 2 | 2 | no | — |
| 95 | C2×D4 | — | 2 | 4 | no | — |
| 96 | C2×D4 | — | 2 | 4 | no | — |
| 97 | C2×D4 | — | 2 | 4 | no | — |
| 98 | C2×D4 | — | 2 | 4 | no | — |
| 99 | C2×D4 | — | 2 | 1 | no | — |
| 100 | C2×D4 | — | 4 | 64 | no | — |
| 101 | C2×D4 | — | 4 | 16 | no | — |
| 102 | C2×D4 | — | 4 | 2 | no | — |
| 103 | C2×D4 | — | 4 | 2 | no | — |
| 104 | C2×D4 | — | 4 | 8 | no | — |
| 105 | C2×D4 | — | 4 | 8 | no | — |
| 106 | C2×D4 | — | 4 | 4 | no | — |
| 107 | C2×D4 | — | 4 | 4 | no | — |
| 108 | C2×D4 | — | 4 | 16 | no | — |
| 109 | C2×D4 | C2×D4 | 1 | 8 | yes | — |
| 110 | C2×D4 | C2×D4 | 1 | 2 | yes | complemented |
| 111 | C22×C4 | — | 2 | 4 | no | — |
| 112 | C22×C4 | — | 2 | 8 | no | — |
| 113 | C22×C4 | — | 2 | 1 | no | — |
| 114 | C22×C4 | — | 2 | 4 | no | — |
| 115 | C22×C4 | — | 2 | 2 | no | — |
| 116 | C22×C4 | — | 4 | 2 | no | — |
| 117 | C22×C4 | — | 4 | 2 | no | — |
| 118 | C22×C4 | C2×D4 | 1 | 4 | yes | — |
| 119 | C22×C4 | C2×D4 | 1 | 2 | yes | — |
| 120 | C22×C4 | C2×D4 | 1 | 4 | yes | — |
| 121 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 122 | C22×C4 | C2×D4 | 1 | 4 | yes | — |
| 123 | C22×C4 | #41 | 1 | 2 | yes | — |
| 124 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 125 | SD16 | — | 2 | 32 | no | — |
| 126 | D8 | — | 2 | 32 | no | — |
| 127 | M16 | — | 2 | 16 | no | — |
| 128 | C2×C8 | — | 2 | 4 | no | — |
| 129 | C2×C8 | — | 2 | 4 | no | — |
| 130 | C2×C8 | C2×D4 | 1 | 8 | yes | complemented |
| 131 | C4⋊C4 | — | 2 | 4 | no | — |
| 132 | C22⋊C4 | — | 4 | 2 | no | — |
| 133 | C22⋊C4 | — | 4 | 8 | no | — |
| 134 | C42 | — | 2 | 4 | no | — |
| 135 | C42 | — | 2 | 4 | no | — |
| 136 | C42 | — | 4 | 2 | no | — |
| 137 | C23 | — | 2 | 4 | no | — |
| 138 | C23 | — | 2 | 8 | no | — |
| 139 | C23 | — | 2 | 2 | no | — |
| 140 | C23 | — | 4 | 32 | no | — |
| 141 | C23 | — | 4 | 1 | no | — |
| 142 | C23 | — | 4 | 4 | no | — |
| 143 | C23 | — | 4 | 1 | no | — |
| 144 | C23 | — | 4 | 4 | no | — |
| 145 | C23 | — | 4 | 4 | no | — |
| 146 | C23 | — | 4 | 16 | no | — |
| 147 | C23 | — | 4 | 16 | no | — |
| 148 | C23 | — | 8 | 16 | no | — |
| 149 | C23 | — | 8 | 16 | no | — |
| 150 | C23 | #120 | 1 | 2 | yes | — |
| 151 | C23 | #121 | 1 | 2 | yes | — |
| 152 | C23 | #127 | 1 | 2 | yes | — |
| 153 | C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 154 | Q8 | — | 2 | 4 | no | — |
| 155 | Q8 | — | 4 | 2 | no | — |
| 156 | D4 | — | 2 | 4 | no | — |
| 157 | D4 | — | 2 | 8 | no | — |
| 158 | D4 | — | 2 | 8 | no | — |
| 159 | D4 | — | 4 | 64 | no | — |
| 160 | D4 | — | 4 | 8 | no | — |
| 161 | D4 | — | 4 | 4 | no | — |
| 162 | D4 | — | 4 | 4 | no | — |
| 163 | D4 | — | 4 | 2 | no | — |
| 164 | D4 | — | 4 | 16 | no | — |
| 165 | C2×C4 | — | 2 | 16 | no | — |
| 166 | C2×C4 | — | 2 | 4 | no | — |
| 167 | C2×C4 | — | 2 | 4 | no | — |
| 168 | C2×C4 | — | 2 | 2 | no | — |
| 169 | C2×C4 | — | 4 | 1 | no | — |
| 170 | C2×C4 | — | 4 | 1 | no | — |
| 171 | C2×C4 | — | 4 | 8 | no | — |
| 172 | C2×C4 | — | 4 | 2 | no | — |
| 173 | C2×C4 | — | 4 | 4 | no | — |
| 174 | C2×C4 | #120 | 1 | 8 | yes | — |
| 175 | C2×C4 | #120 | 1 | 2 | yes | — |
| 176 | C2×C4 | #121 | 1 | 8 | yes | — |
| 177 | C2×C4 | #121 | 1 | 2 | yes | — |
| 178 | C2×C4 | #127 | 1 | 2 | yes | — |
| 179 | C2×C4 | AGL(1,ℤ/8) | 1 | 8 | yes | complemented |
| 180 | C2×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 181 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 182 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 183 | C2×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 184 | C8 | — | 2 | 8 | no | — |
| 185 | C22 | — | 2 | 4 | no | — |
| 186 | C22 | — | 2 | 4 | no | — |
| 187 | C22 | — | 2 | 4 | no | — |
| 188 | C22 | — | 4 | 2 | no | — |
| 189 | C22 | — | 4 | 8 | no | — |
| 190 | C22 | — | 4 | 16 | no | — |
| 191 | C22 | — | 4 | 4 | no | — |
| 192 | C22 | — | 8 | 32 | no | — |
| 193 | C22 | — | 8 | 2 | no | — |
| 194 | C22 | — | 8 | 8 | no | — |
| 195 | C22 | — | 8 | 2 | no | — |
| 196 | C22 | — | 8 | 8 | no | — |
| 197 | C22 | #392 | 1 | 4 | yes | — |
| 198 | C22 | C2×#120 | 1 | 1 | yes | characteristic |
| 199 | C22 | C2×#121 | 1 | 1 | yes | characteristic |
| 200 | C22 | C2×#127 | 1 | 1 | yes | characteristic |
| 201 | C22 | C2×AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 202 | C4 | — | 2 | 8 | no | — |
| 203 | C4 | — | 4 | 2 | no | — |
| 204 | C4 | — | 8 | 2 | no | — |
| 205 | C4 | #392 | 1 | 4 | yes | — |
| 206 | C4 | C2×#120 | 1 | 2 | yes | — |
| 207 | C4 | C2×#121 | 1 | 2 | yes | — |
| 208 | C2 | — | 2 | 4 | no | — |
| 209 | C2 | — | 8 | 2 | no | — |
| 210 | C2 | — | 8 | 8 | no | — |
| 211 | C2 | C2×#392 | 1 | 1 | yes | characteristic |
| 212 | C2 | #3003 | 1 | 2 | yes | — |
| 213 | C2 | #3038 | 1 | 2 | yes | — |
| 214 | C2 | #3151 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×AGL(1,ℤ/8) | C2 | semidirect |
| 2. C22×#127 | C2 | semidirect |
| 3. C2×#418 | C2 | semidirect |
| 4. #2297 | C2 | semidirect |
| 5. #2014 | C2 | semidirect |
| 6. #2011 | C2 | semidirect |
| 7. #1728 | C2 | semidirect |
| 14. C22×SD16 | C22 | semidirect |
| 15. C22×D8 | C22 | semidirect |
| 23. C2×#102 | C22 | semidirect |
| 24. #391 | C22 | semidirect |
| 25. #390 | C22 | semidirect |
| 28. #336 | C22 | semidirect |
| 33. C2×#41 | D4 | semidirect |
| 60. C2×M16 | D4 | semidirect |
| 86. C2×Q8 | C2×D4 | semidirect |
| 110. C2×D4 | C2×D4 | semidirect |
| 130. C2×C8 | C2×D4 | semidirect |
| 179. C2×C4 | AGL(1,ℤ/8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×AGL(1,ℤ/8) | 2 |
| C22×#127 | 2 |
| C2×#418 | 2 |
| #2297 | 2 |
| #2014 | 2 |
| #2011 | 2 |
| #1728 | 2 |