Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2659 conjugacy classes of subgroups; only the 559 normal subgroups are shown.
This group has 2659 conjugacy classes of subgroups — too many to draw. Only its 559 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×H(5,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#534 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #3026 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 4 | C2×#458 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C2×B(4,𝔽2) | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | C22×#99 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C2×H(5,𝔽2) | — | 2 | 6 | no | — |
| 8 | C2×H(5,𝔽2) | — | 2 | 12 | no | — |
| 9 | C2×H(5,𝔽2) | C22 | 1 | 4 | yes | — |
| 10 | C2×H(5,𝔽2) | C22 | 1 | 8 | yes | complemented |
| 11 | C22×#41 | — | 2 | 3 | no | — |
| 12 | C23×D4 | — | 2 | 3 | no | — |
| 13 | C23×D4 | C22 | 1 | 3 | yes | — |
| 14 | C23×D4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | #534 | — | 2 | 12 | no | — |
| 16 | #534 | C22 | 1 | 24 | yes | complemented |
| 17 | C2×#123 | — | 2 | 3 | no | — |
| 18 | C2×#123 | C22 | 1 | 6 | yes | complemented |
| 19 | C2×#121 | — | 2 | 6 | no | — |
| 20 | C2×#120 | — | 2 | 3 | no | — |
| 21 | C2×#120 | C22 | 1 | 6 | yes | complemented |
| 22 | C22×C22⋊C4 | C22 | 1 | 3 | yes | complemented |
| 23 | #458 | C22 | 1 | 32 | yes | complemented |
| 24 | B(4,𝔽2) | C22 | 1 | 32 | yes | complemented |
| 25 | C2×#99 | C22 | 1 | 12 | yes | complemented |
| 26 | C2×#99 | C22 | 1 | 24 | yes | complemented |
| 27 | C25 | — | 2 | 1 | no | — |
| 28 | C25 | D4 | 1 | 3 | yes | complemented |
| 29 | C25 | C23 | 1 | 1 | yes | characteristic |
| 30 | C25 | C23 | 1 | 1 | yes | characteristic, complemented |
| 31 | H(5,𝔽2) | — | 2 | 48 | no | — |
| 32 | H(5,𝔽2) | — | 2 | 24 | no | — |
| 33 | H(5,𝔽2) | — | 2 | 48 | no | — |
| 34 | H(5,𝔽2) | C23 | 1 | 16 | yes | — |
| 35 | C2×#41 | — | 2 | 24 | no | — |
| 36 | C2×#41 | — | 2 | 12 | no | — |
| 37 | C2×#41 | — | 2 | 12 | no | — |
| 38 | C2×#41 | — | 2 | 24 | no | — |
| 39 | C22×Q8 | — | 2 | 1 | no | — |
| 40 | C22×D4 | — | 2 | 3 | no | — |
| 41 | C22×D4 | — | 2 | 24 | no | — |
| 42 | C22×D4 | — | 2 | 12 | no | — |
| 43 | C22×D4 | — | 2 | 6 | no | — |
| 44 | C22×D4 | — | 2 | 12 | no | — |
| 45 | C22×D4 | — | 2 | 6 | no | — |
| 46 | C22×D4 | — | 2 | 12 | no | — |
| 47 | C22×D4 | — | 2 | 24 | no | — |
| 48 | C22×D4 | — | 2 | 3 | no | — |
| 49 | C22×D4 | — | 2 | 3 | no | — |
| 50 | C22×D4 | — | 2 | 6 | no | — |
| 51 | C22×D4 | — | 2 | 6 | no | — |
| 52 | C22×D4 | D4 | 1 | 6 | yes | complemented |
| 53 | C22×D4 | D4 | 1 | 12 | yes | complemented |
| 54 | C22×D4 | C23 | 1 | 6 | yes | — |
| 55 | C22×D4 | C23 | 1 | 12 | yes | — |
| 56 | C22×D4 | C23 | 1 | 4 | yes | complemented |
| 57 | C23×C4 | — | 2 | 3 | no | — |
| 58 | C23×C4 | — | 2 | 3 | no | — |
| 59 | C23×C4 | D4 | 1 | 3 | yes | complemented |
| 60 | C23×C4 | C23 | 1 | 1 | yes | characteristic, complemented |
| 61 | #123 | — | 2 | 12 | no | — |
| 62 | #123 | — | 2 | 24 | no | — |
| 63 | #123 | C23 | 1 | 24 | yes | complemented |
| 64 | #121 | — | 2 | 48 | no | — |
| 65 | #120 | — | 2 | 24 | no | — |
| 66 | #120 | — | 2 | 12 | no | — |
| 67 | #120 | C23 | 1 | 24 | yes | complemented |
| 68 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 69 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 70 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 71 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 72 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 73 | C2×C22⋊C4 | C23 | 1 | 6 | yes | complemented |
| 74 | C2×C22⋊C4 | C23 | 1 | 12 | yes | complemented |
| 75 | #99 | C23 | 1 | 48 | yes | complemented |
| 76 | C24 | — | 2 | 8 | no | — |
| 77 | C24 | — | 2 | 4 | no | — |
| 78 | C24 | — | 2 | 3 | no | — |
| 79 | C24 | — | 2 | 6 | no | — |
| 80 | C24 | — | 2 | 3 | no | — |
| 81 | C24 | — | 2 | 3 | no | — |
| 82 | C24 | — | 2 | 3 | no | — |
| 83 | C24 | — | 8 | 2 | no | — |
| 84 | C24 | — | 8 | 2 | no | — |
| 85 | C24 | — | 8 | 6 | no | — |
| 86 | C24 | — | 8 | 4 | no | — |
| 87 | C24 | C2×D4 | 1 | 6 | yes | — |
| 88 | C24 | C2×D4 | 1 | 3 | yes | — |
| 89 | C24 | C2×D4 | 1 | 3 | yes | — |
| 90 | C24 | C2×D4 | 1 | 6 | yes | — |
| 91 | C24 | C2×D4 | 1 | 12 | yes | complemented |
| 92 | C24 | C24 | 1 | 1 | yes | characteristic |
| 93 | C24 | C24 | 1 | 2 | yes | — |
| 94 | C24 | C24 | 1 | 2 | yes | complemented |
| 95 | #41 | — | 2 | 48 | no | — |
| 96 | #41 | — | 2 | 96 | no | — |
| 97 | #41 | — | 2 | 48 | no | — |
| 98 | C2×Q8 | — | 2 | 8 | no | — |
| 99 | C2×Q8 | — | 2 | 4 | no | — |
| 100 | C2×D4 | — | 2 | 24 | no | — |
| 101 | C2×D4 | — | 2 | 12 | no | — |
| 102 | C2×D4 | — | 2 | 48 | no | — |
| 103 | C2×D4 | — | 2 | 96 | no | — |
| 104 | C2×D4 | — | 2 | 24 | no | — |
| 105 | C2×D4 | — | 2 | 12 | no | — |
| 106 | C2×D4 | — | 2 | 12 | no | — |
| 107 | C2×D4 | — | 2 | 24 | no | — |
| 108 | C2×D4 | — | 2 | 24 | no | — |
| 109 | C2×D4 | — | 2 | 12 | no | — |
| 110 | C2×D4 | — | 2 | 48 | no | — |
| 111 | C2×D4 | — | 2 | 12 | no | — |
| 112 | C2×D4 | — | 2 | 24 | no | — |
| 113 | C2×D4 | — | 2 | 12 | no | — |
| 114 | C2×D4 | — | 2 | 12 | no | — |
| 115 | C2×D4 | — | 2 | 24 | no | — |
| 116 | C2×D4 | — | 4 | 6 | no | — |
| 117 | C2×D4 | — | 4 | 6 | no | — |
| 118 | C2×D4 | — | 4 | 1 | no | — |
| 119 | C2×D4 | — | 4 | 3 | no | — |
| 120 | C2×D4 | — | 8 | 12 | no | — |
| 121 | C2×D4 | C2×D4 | 1 | 12 | yes | — |
| 122 | C2×D4 | C2×D4 | 1 | 24 | yes | complemented |
| 123 | C2×D4 | C24 | 1 | 12 | yes | — |
| 124 | C22×C4 | — | 2 | 24 | no | — |
| 125 | C22×C4 | — | 2 | 12 | no | — |
| 126 | C22×C4 | — | 2 | 3 | no | — |
| 127 | C22×C4 | — | 2 | 6 | no | — |
| 128 | C22×C4 | — | 2 | 3 | no | — |
| 129 | C22×C4 | — | 2 | 6 | no | — |
| 130 | C22×C4 | — | 2 | 3 | no | — |
| 131 | C22×C4 | — | 2 | 3 | no | — |
| 132 | C22×C4 | — | 2 | 12 | no | — |
| 133 | C22×C4 | — | 8 | 6 | no | — |
| 134 | C22×C4 | C2×D4 | 1 | 6 | yes | — |
| 135 | C22×C4 | C2×D4 | 1 | 12 | yes | complemented |
| 136 | C22×C4 | C24 | 1 | 2 | yes | complemented |
| 137 | C4⋊C4 | — | 2 | 24 | no | — |
| 138 | C22⋊C4 | — | 2 | 6 | no | — |
| 139 | C22⋊C4 | — | 2 | 12 | no | — |
| 140 | C22⋊C4 | — | 2 | 24 | no | — |
| 141 | C22⋊C4 | — | 2 | 24 | no | — |
| 142 | C22⋊C4 | C24 | 1 | 12 | yes | complemented |
| 143 | C23 | — | 2 | 16 | no | — |
| 144 | C23 | — | 2 | 12 | no | — |
| 145 | C23 | — | 2 | 6 | no | — |
| 146 | C23 | — | 2 | 6 | no | — |
| 147 | C23 | — | 2 | 12 | no | — |
| 148 | C23 | — | 2 | 6 | no | — |
| 149 | C23 | — | 4 | 6 | no | — |
| 150 | C23 | — | 4 | 6 | no | — |
| 151 | C23 | — | 4 | 2 | no | — |
| 152 | C23 | — | 4 | 12 | no | — |
| 153 | C23 | — | 4 | 12 | no | — |
| 154 | C23 | — | 4 | 12 | no | — |
| 155 | C23 | — | 4 | 3 | no | — |
| 156 | C23 | — | 4 | 1 | no | — |
| 157 | C23 | — | 8 | 6 | no | — |
| 158 | C23 | — | 8 | 6 | no | — |
| 159 | C23 | — | 8 | 2 | no | — |
| 160 | C23 | — | 8 | 2 | no | — |
| 161 | C23 | — | 8 | 12 | no | — |
| 162 | C23 | — | 8 | 12 | no | — |
| 163 | C23 | — | 8 | 6 | no | — |
| 164 | C23 | — | 8 | 16 | no | — |
| 165 | C23 | — | 8 | 8 | no | — |
| 166 | C23 | #120 | 1 | 1 | yes | characteristic |
| 167 | C23 | #120 | 1 | 3 | yes | — |
| 168 | C23 | #120 | 1 | 6 | yes | — |
| 169 | C23 | #120 | 1 | 6 | yes | — |
| 170 | C23 | C22×D4 | 1 | 3 | yes | — |
| 171 | C23 | C22×D4 | 1 | 12 | yes | — |
| 172 | C23 | C22×D4 | 1 | 3 | yes | — |
| 173 | C23 | C22×D4 | 1 | 6 | yes | — |
| 174 | C23 | C22×D4 | 1 | 6 | yes | — |
| 175 | C23 | C25 | 1 | 1 | yes | characteristic |
| 176 | Q8 | — | 2 | 16 | no | — |
| 177 | D4 | — | 2 | 48 | no | — |
| 178 | D4 | — | 2 | 24 | no | — |
| 179 | D4 | — | 2 | 48 | no | — |
| 180 | D4 | — | 2 | 24 | no | — |
| 181 | D4 | — | 4 | 4 | no | — |
| 182 | D4 | — | 4 | 12 | no | — |
| 183 | D4 | — | 8 | 24 | no | — |
| 184 | C2×C4 | — | 2 | 48 | no | — |
| 185 | C2×C4 | — | 2 | 12 | no | — |
| 186 | C2×C4 | — | 2 | 6 | no | — |
| 187 | C2×C4 | — | 2 | 12 | no | — |
| 188 | C2×C4 | — | 2 | 6 | no | — |
| 189 | C2×C4 | — | 2 | 12 | no | — |
| 190 | C2×C4 | — | 2 | 6 | no | — |
| 191 | C2×C4 | — | 4 | 12 | no | — |
| 192 | C2×C4 | — | 4 | 3 | no | — |
| 193 | C2×C4 | — | 4 | 1 | no | — |
| 194 | C2×C4 | — | 8 | 12 | no | — |
| 195 | C2×C4 | C22×D4 | 1 | 12 | yes | — |
| 196 | C22 | — | 2 | 6 | no | — |
| 197 | C22 | — | 2 | 2 | no | — |
| 198 | C22 | — | 2 | 6 | no | — |
| 199 | C22 | — | 2 | 12 | no | — |
| 200 | C22 | — | 4 | 6 | no | — |
| 201 | C22 | — | 4 | 6 | no | — |
| 202 | C22 | — | 4 | 6 | no | — |
| 203 | C22 | — | 4 | 3 | no | — |
| 204 | C22 | — | 4 | 1 | no | — |
| 205 | C22 | — | 4 | 6 | no | — |
| 206 | C22 | — | 4 | 12 | no | — |
| 207 | C22 | — | 4 | 24 | no | — |
| 208 | C22 | — | 4 | 12 | no | — |
| 209 | C22 | — | 4 | 12 | no | — |
| 210 | C22 | — | 4 | 2 | no | — |
| 211 | C22 | — | 8 | 6 | no | — |
| 212 | C22 | — | 8 | 12 | no | — |
| 213 | C22 | — | 8 | 6 | no | — |
| 214 | C22 | — | 8 | 16 | no | — |
| 215 | C22 | — | 8 | 2 | no | — |
| 216 | C22 | C2×#120 | 1 | 2 | yes | — |
| 217 | C22 | C2×#120 | 1 | 3 | yes | — |
| 218 | C22 | C2×#120 | 1 | 6 | yes | — |
| 219 | C22 | C2×#120 | 1 | 1 | yes | characteristic |
| 220 | C22 | C23×D4 | 1 | 3 | yes | — |
| 221 | C4 | — | 2 | 12 | no | — |
| 222 | C4 | — | 4 | 12 | no | — |
| 223 | C4 | — | 4 | 2 | no | — |
| 224 | C2 | — | 2 | 3 | no | — |
| 225 | C2 | — | 2 | 2 | no | — |
| 226 | C2 | — | 2 | 3 | no | — |
| 227 | C2 | — | 2 | 6 | no | — |
| 228 | C2 | — | 4 | 12 | no | — |
| 229 | C2 | — | 8 | 2 | no | — |
| 230 | C2 | #3026 | 1 | 2 | yes | direct product factor |
| 231 | C2 | C22×#120 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×H(5,𝔽2) | C2 | semidirect |
| 2. C2×#534 | C2 | semidirect |
| 3. #3026 | C2 | direct (N × H) |
| 4. C2×#458 | C2 | semidirect |
| 5. C2×B(4,𝔽2) | C2 | semidirect |
| 6. C22×#99 | C2 | semidirect |
| 10. C2×H(5,𝔽2) | C22 | semidirect |
| 14. C23×D4 | C22 | semidirect |
| 16. #534 | C22 | semidirect |
| 18. C2×#123 | C22 | semidirect |
| 21. C2×#120 | C22 | semidirect |
| 22. C22×C22⋊C4 | C22 | semidirect |
| 23. #458 | C22 | semidirect |
| 24. B(4,𝔽2) | C22 | semidirect |
| 25. C2×#99 | C22 | semidirect |
| 26. C2×#99 | C22 | semidirect |
| 28. C25 | D4 | semidirect |
| 30. C25 | C23 | semidirect |
| 52. C22×D4 | D4 | semidirect |
| 53. C22×D4 | D4 | semidirect |
| 56. C22×D4 | C23 | semidirect |
| 59. C23×C4 | D4 | semidirect |
| 60. C23×C4 | C23 | semidirect |
| 63. #123 | C23 | semidirect |
| 67. #120 | C23 | semidirect |
| 73. C2×C22⋊C4 | C23 | semidirect |
| 74. C2×C22⋊C4 | C23 | semidirect |
| 75. #99 | C23 | semidirect |
| 91. C24 | C2×D4 | semidirect |
| 94. C24 | C24 | semidirect |
| 122. C2×D4 | C2×D4 | semidirect |
| 135. C22×C4 | C2×D4 | semidirect |
| 136. C22×C4 | C24 | semidirect |
| 142. C22⋊C4 | C24 | semidirect |
| 230. C2 | #3026 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×H(5,𝔽2) | 2 |
| C2×#534 | 2 |
| #3026 | 2 |
| C2×#458 | 2 |
| C2×B(4,𝔽2) | 2 |
| C22×#99 | 2 |