Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1300 conjugacy classes of subgroups; only the 485 normal subgroups are shown.
This group has 1300 conjugacy classes of subgroups — too many to draw. Only its 485 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 | #3586 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #3293 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #3291 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #3289 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | #3152 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #3069 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #3018 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | C2×#456 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #2956 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #585 | C22 | 1 | 2 | yes | complemented |
| 11 | C2×2−1+4 | C22 | 1 | 2 | yes | — |
| 12 | #578 | — | 2 | 8 | no | — |
| 13 | #578 | C22 | 1 | 16 | yes | complemented |
| 14 | #577 | — | 2 | 8 | no | — |
| 15 | #577 | C22 | 1 | 16 | yes | complemented |
| 16 | #575 | — | 2 | 8 | no | — |
| 17 | #575 | C22 | 1 | 4 | yes | — |
| 18 | #575 | C22 | 1 | 2 | yes | — |
| 19 | C2×#137 | — | 2 | 8 | no | — |
| 20 | C2×#137 | C22 | 1 | 4 | yes | complemented |
| 21 | C2×#137 | C22 | 1 | 4 | yes | complemented |
| 22 | #568 | C22 | 1 | 4 | yes | complemented |
| 23 | #532 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 25 | #497 | C22 | 1 | 4 | yes | complemented |
| 26 | #494 | C22 | 1 | 2 | yes | complemented |
| 27 | #492 | C22 | 1 | 2 | yes | complemented |
| 28 | #473 | C22 | 1 | 16 | yes | complemented |
| 29 | #471 | C22 | 1 | 16 | yes | complemented |
| 30 | #456 | C22 | 1 | 8 | yes | complemented |
| 31 | #456 | C22 | 1 | 8 | yes | complemented |
| 32 | #456 | C22 | 1 | 4 | yes | complemented |
| 33 | #455 | C22 | 1 | 4 | yes | complemented |
| 34 | #454 | C22 | 1 | 4 | yes | complemented |
| 35 | #453 | C22 | 1 | 4 | yes | complemented |
| 36 | #444 | C22 | 1 | 8 | yes | complemented |
| 37 | #443 | C22 | 1 | 8 | yes | complemented |
| 38 | #430 | C22 | 1 | 2 | yes | complemented |
| 39 | C2×#104 | C22 | 1 | 4 | yes | complemented |
| 40 | #413 | C22 | 1 | 2 | yes | complemented |
| 41 | C2×#101 | C22 | 1 | 2 | yes | complemented |
| 42 | C4×M16 | C22 | 1 | 1 | yes | characteristic, complemented |
| 43 | 2−1+4 | — | 2 | 8 | no | — |
| 44 | 2−1+4 | — | 2 | 4 | no | — |
| 45 | 2−1+4 | — | 4 | 2 | no | — |
| 46 | 2−1+4 | C23 | 1 | 8 | yes | — |
| 47 | 2−1+4 | C23 | 1 | 4 | yes | — |
| 48 | H(5,𝔽2) | — | 2 | 4 | no | — |
| 49 | H(5,𝔽2) | — | 4 | 2 | no | — |
| 50 | H(5,𝔽2) | C23 | 1 | 4 | yes | — |
| 51 | C2×#41 | — | 2 | 8 | no | — |
| 52 | C2×#41 | — | 2 | 4 | no | — |
| 53 | C2×#41 | — | 4 | 2 | no | — |
| 54 | C2×#41 | — | 4 | 2 | no | — |
| 55 | C2×#41 | C23 | 1 | 4 | yes | — |
| 56 | C2×#41 | C23 | 1 | 2 | yes | — |
| 57 | C22×Q8 | — | 2 | 4 | no | — |
| 58 | C22×Q8 | C23 | 1 | 2 | yes | — |
| 59 | #137 | — | 2 | 8 | no | — |
| 60 | #137 | — | 2 | 4 | no | — |
| 61 | #137 | — | 2 | 16 | no | — |
| 62 | #137 | — | 2 | 16 | no | — |
| 63 | #137 | — | 2 | 16 | no | — |
| 64 | #137 | — | 2 | 8 | no | — |
| 65 | #137 | — | 2 | 8 | no | — |
| 66 | #137 | D4 | 1 | 8 | yes | complemented |
| 67 | #137 | C23 | 1 | 16 | yes | — |
| 68 | #137 | C23 | 1 | 8 | yes | — |
| 69 | AGL(1,ℤ/8) | — | 2 | 4 | no | — |
| 70 | AGL(1,ℤ/8) | — | 2 | 16 | no | — |
| 71 | AGL(1,ℤ/8) | D4 | 1 | 8 | yes | complemented |
| 72 | AGL(1,ℤ/8) | C23 | 1 | 8 | yes | — |
| 73 | #135 | — | 2 | 8 | no | — |
| 74 | #135 | — | 2 | 16 | no | — |
| 75 | #135 | — | 2 | 16 | no | — |
| 76 | #135 | C23 | 1 | 8 | yes | — |
| 77 | C2×Q16 | — | 2 | 4 | no | — |
| 78 | C2×Q16 | — | 2 | 16 | no | — |
| 79 | C2×Q16 | C23 | 1 | 8 | yes | — |
| 80 | C2×SD16 | — | 2 | 4 | no | — |
| 81 | C2×SD16 | — | 2 | 16 | no | — |
| 82 | C2×SD16 | C23 | 1 | 8 | yes | — |
| 83 | #131 | — | 2 | 8 | no | — |
| 84 | #131 | C23 | 1 | 16 | yes | — |
| 85 | C2×M16 | — | 2 | 8 | no | — |
| 86 | C2×M16 | C23 | 1 | 4 | yes | — |
| 87 | C2×M16 | C23 | 1 | 2 | yes | — |
| 88 | C4⋊Q8 | — | 4 | 1 | no | — |
| 89 | C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 90 | C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 91 | C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 92 | #127 | C23 | 1 | 1 | yes | characteristic |
| 93 | #124 | C23 | 1 | 2 | yes | — |
| 94 | #121 | — | 4 | 1 | no | — |
| 95 | C4×D4 | — | 4 | 1 | no | — |
| 96 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 97 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 98 | #108 | C23 | 1 | 8 | yes | — |
| 99 | #104 | C23 | 1 | 8 | yes | — |
| 100 | #104 | C23 | 1 | 8 | yes | — |
| 101 | #101 | C23 | 1 | 8 | yes | — |
| 102 | #101 | C23 | 1 | 4 | yes | — |
| 103 | #100 | C23 | 1 | 4 | yes | — |
| 104 | #97 | C23 | 1 | 2 | yes | — |
| 105 | C4×C8 | C23 | 1 | 2 | yes | — |
| 106 | #41 | — | 2 | 4 | no | — |
| 107 | #41 | — | 2 | 4 | no | — |
| 108 | #41 | — | 2 | 8 | no | — |
| 109 | #41 | — | 2 | 16 | no | — |
| 110 | #41 | — | 2 | 8 | no | — |
| 111 | #41 | — | 2 | 4 | no | — |
| 112 | #41 | — | 2 | 4 | no | — |
| 113 | #41 | — | 4 | 4 | no | — |
| 114 | #41 | — | 4 | 4 | no | — |
| 115 | #41 | — | 4 | 8 | no | — |
| 116 | #41 | — | 4 | 8 | no | — |
| 117 | #41 | — | 4 | 8 | no | — |
| 118 | #41 | — | 4 | 8 | no | — |
| 119 | #41 | — | 4 | 4 | no | — |
| 120 | #41 | C2×D4 | 1 | 8 | yes | — |
| 121 | #41 | C24 | 1 | 8 | yes | — |
| 122 | C2×Q8 | — | 2 | 2 | no | — |
| 123 | C2×Q8 | — | 2 | 16 | no | — |
| 124 | C2×Q8 | — | 2 | 8 | no | — |
| 125 | C2×Q8 | — | 2 | 4 | no | — |
| 126 | C2×Q8 | — | 2 | 4 | no | — |
| 127 | C2×Q8 | — | 4 | 2 | no | — |
| 128 | C2×Q8 | — | 4 | 8 | no | — |
| 129 | C2×Q8 | — | 4 | 2 | no | — |
| 130 | C2×Q8 | C2×D4 | 1 | 4 | yes | — |
| 131 | C2×Q8 | C24 | 1 | 2 | yes | — |
| 132 | C2×Q8 | C24 | 1 | 4 | yes | — |
| 133 | C2×D4 | — | 2 | 4 | no | — |
| 134 | C2×D4 | — | 2 | 8 | no | — |
| 135 | C2×D4 | — | 2 | 2 | no | — |
| 136 | C2×D4 | — | 4 | 1 | no | — |
| 137 | C2×D4 | — | 4 | 4 | no | — |
| 138 | C2×D4 | — | 4 | 2 | no | — |
| 139 | C2×D4 | — | 4 | 8 | no | — |
| 140 | C2×D4 | — | 4 | 2 | no | — |
| 141 | C2×D4 | C2×D4 | 1 | 4 | yes | — |
| 142 | C2×D4 | C24 | 1 | 2 | yes | — |
| 143 | C22×C4 | — | 2 | 4 | no | — |
| 144 | C22×C4 | — | 2 | 1 | no | — |
| 145 | C22×C4 | — | 2 | 2 | no | — |
| 146 | C22×C4 | — | 2 | 2 | no | — |
| 147 | C22×C4 | — | 4 | 4 | no | — |
| 148 | C22×C4 | — | 4 | 4 | no | — |
| 149 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 150 | Q16 | — | 2 | 16 | no | — |
| 151 | Q16 | — | 2 | 8 | no | — |
| 152 | Q16 | — | 2 | 16 | no | — |
| 153 | Q16 | C2×D4 | 1 | 8 | yes | — |
| 154 | SD16 | — | 2 | 8 | no | — |
| 155 | SD16 | — | 2 | 16 | no | — |
| 156 | SD16 | — | 2 | 8 | no | — |
| 157 | SD16 | — | 2 | 16 | no | — |
| 158 | SD16 | C2×D4 | 1 | 16 | yes | — |
| 159 | D8 | — | 2 | 8 | no | — |
| 160 | D8 | C2×D4 | 1 | 8 | yes | — |
| 161 | M16 | — | 2 | 4 | no | — |
| 162 | M16 | — | 2 | 16 | no | — |
| 163 | M16 | C2×D4 | 1 | 8 | yes | — |
| 164 | M16 | C24 | 1 | 8 | yes | — |
| 165 | C2×C8 | — | 2 | 4 | no | — |
| 166 | C2×C8 | — | 2 | 16 | no | — |
| 167 | C2×C8 | C24 | 1 | 4 | yes | — |
| 168 | C4⋊C4 | — | 4 | 2 | no | — |
| 169 | C4⋊C4 | — | 4 | 2 | no | — |
| 170 | C4⋊C4 | — | 4 | 1 | no | — |
| 171 | C22⋊C4 | — | 4 | 2 | no | — |
| 172 | C42 | — | 2 | 1 | no | — |
| 173 | C42 | C24 | 1 | 1 | yes | characteristic |
| 174 | C42 | C24 | 1 | 1 | yes | characteristic |
| 175 | C23 | — | 2 | 4 | no | — |
| 176 | C23 | — | 2 | 2 | no | — |
| 177 | C23 | — | 4 | 4 | no | — |
| 178 | C23 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 179 | Q8 | — | 2 | 4 | no | — |
| 180 | Q8 | — | 2 | 8 | no | — |
| 181 | Q8 | — | 2 | 4 | no | — |
| 182 | Q8 | — | 4 | 8 | no | — |
| 183 | Q8 | — | 4 | 4 | no | — |
| 184 | Q8 | — | 4 | 4 | no | — |
| 185 | Q8 | C22×D4 | 1 | 8 | yes | — |
| 186 | D4 | — | 2 | 8 | no | — |
| 187 | D4 | — | 2 | 8 | no | — |
| 188 | D4 | — | 2 | 4 | no | — |
| 189 | D4 | — | 2 | 4 | no | — |
| 190 | D4 | — | 4 | 8 | no | — |
| 191 | D4 | — | 4 | 4 | no | — |
| 192 | D4 | — | 4 | 8 | no | — |
| 193 | D4 | — | 4 | 4 | no | — |
| 194 | D4 | — | 8 | 2 | no | — |
| 195 | D4 | C22×D4 | 1 | 8 | yes | — |
| 196 | C2×C4 | — | 2 | 4 | no | — |
| 197 | C2×C4 | — | 2 | 1 | no | — |
| 198 | C2×C4 | — | 2 | 8 | no | — |
| 199 | C2×C4 | — | 2 | 8 | no | — |
| 200 | C2×C4 | — | 2 | 1 | no | — |
| 201 | C2×C4 | — | 2 | 2 | no | — |
| 202 | C2×C4 | — | 2 | 4 | no | — |
| 203 | C2×C4 | — | 4 | 4 | no | — |
| 204 | C2×C4 | — | 4 | 8 | no | — |
| 205 | C2×C4 | — | 4 | 16 | no | — |
| 206 | C2×C4 | — | 4 | 2 | no | — |
| 207 | C2×C4 | — | 4 | 2 | no | — |
| 208 | C2×C4 | — | 4 | 2 | no | — |
| 209 | C2×C4 | — | 4 | 2 | no | — |
| 210 | C2×C4 | — | 4 | 2 | no | — |
| 211 | C2×C4 | C22×D4 | 1 | 4 | yes | — |
| 212 | C2×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | C25 | 1 | 1 | yes | characteristic |
| 214 | C8 | — | 2 | 8 | no | — |
| 215 | C8 | C22×D4 | 1 | 8 | yes | — |
| 216 | C22 | — | 2 | 8 | no | — |
| 217 | C22 | — | 4 | 1 | no | — |
| 218 | C22 | — | 4 | 2 | no | — |
| 219 | C22 | — | 8 | 4 | no | — |
| 220 | C22 | — | 8 | 2 | no | — |
| 221 | C22 | — | 16 | 4 | no | — |
| 222 | C22 | D42 | 1 | 2 | yes | — |
| 223 | C22 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 224 | C4 | — | 2 | 2 | no | — |
| 225 | C4 | — | 2 | 8 | no | — |
| 226 | C4 | — | 4 | 1 | no | — |
| 227 | C4 | — | 4 | 2 | no | — |
| 228 | C4 | — | 4 | 4 | no | — |
| 229 | C4 | D42 | 1 | 2 | yes | — |
| 230 | C4 | C23×D4 | 1 | 2 | yes | — |
| 231 | C2 | — | 2 | 1 | no | — |
| 232 | C2 | — | 2 | 2 | no | — |
| 233 | C2 | — | 4 | 8 | no | — |
| 234 | C2 | — | 8 | 2 | no | — |
| 235 | C2 | C2×D42 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D42 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3586 | C2 | semidirect |
| 2. #3293 | C2 | semidirect |
| 3. #3291 | C2 | semidirect |
| 4. #3289 | C2 | semidirect |
| 5. #3152 | C2 | semidirect |
| 6. #3069 | C2 | semidirect |
| 7. #3018 | C2 | semidirect |
| 8. C2×#456 | C2 | semidirect |
| 9. #2956 | C2 | semidirect |
| 10. #585 | C22 | semidirect |
| 13. #578 | C22 | semidirect |
| 15. #577 | C22 | semidirect |
| 20. C2×#137 | C22 | semidirect |
| 21. C2×#137 | C22 | semidirect |
| 22. #568 | C22 | semidirect |
| 24. C2×C4⋊Q8 | C22 | semidirect |
| 25. #497 | C22 | semidirect |
| 26. #494 | C22 | semidirect |
| 27. #492 | C22 | semidirect |
| 28. #473 | C22 | semidirect |
| 29. #471 | C22 | semidirect |
| 30. #456 | C22 | semidirect |
| 31. #456 | C22 | semidirect |
| 32. #456 | C22 | semidirect |
| 33. #455 | C22 | semidirect |
| 34. #454 | C22 | semidirect |
| 35. #453 | C22 | semidirect |
| 36. #444 | C22 | semidirect |
| 37. #443 | C22 | semidirect |
| 38. #430 | C22 | semidirect |
| 39. C2×#104 | C22 | semidirect |
| 40. #413 | C22 | semidirect |
| 41. C2×#101 | C22 | semidirect |
| 42. C4×M16 | C22 | semidirect |
| 66. #137 | D4 | semidirect |
| 71. AGL(1,ℤ/8) | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3586 | 2 |
| #3293 | 2 |
| #3291 | 2 |
| #3289 | 2 |
| #3152 | 2 |
| #3069 | 2 |
| #3018 | 2 |
| C2×#456 | 2 |
| #2956 | 2 |