Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
408 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 109 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 | #2420 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #2101 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #2076 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #2067 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #1999 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #1893 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #1879 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #1850 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #1804 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | C22×D8 | C22 | 1 | 1 | yes | characteristic |
| 11 | C23×C8 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#121 | C22 | 1 | 2 | yes | — |
| 13 | C2×C4×D4 | C22 | 1 | 2 | yes | — |
| 14 | C2×#107 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×C4⋊C8 | C22 | 1 | 2 | yes | — |
| 16 | C2×#102 | C22 | 1 | 2 | yes | — |
| 17 | C2×#102 | C22 | 1 | 2 | yes | — |
| 18 | C2×#102 | C22 | 1 | 2 | yes | — |
| 19 | C2×C22⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 20 | #397 | — | 2 | 2 | no | — |
| 21 | #394 | — | 2 | 2 | no | — |
| 22 | #390 | C22 | 1 | 2 | yes | — |
| 23 | #389 | C22 | 1 | 2 | yes | — |
| 24 | #386 | — | 2 | 2 | no | — |
| 25 | #382 | C22 | 1 | 2 | yes | — |
| 26 | #381 | — | 2 | 2 | no | — |
| 27 | #380 | C22 | 1 | 2 | yes | — |
| 28 | C4×C22⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 29 | #340 | C22 | 1 | 1 | yes | characteristic |
| 30 | #340 | C22 | 1 | 2 | yes | — |
| 31 | #340 | C22 | 1 | 2 | yes | — |
| 32 | #340 | C22 | 1 | 2 | yes | — |
| 33 | #340 | C22 | 1 | 1 | yes | characteristic |
| 34 | #336 | C22 | 1 | 2 | yes | — |
| 35 | C22×D4 | — | 2 | 2 | no | — |
| 36 | C22×D4 | C23 | 1 | 2 | yes | — |
| 37 | C23×C4 | — | 2 | 2 | no | — |
| 38 | C23×C4 | C23 | 1 | 1 | yes | characteristic |
| 39 | C2×D8 | — | 2 | 4 | no | — |
| 40 | C2×D8 | — | 4 | 1 | no | — |
| 41 | C22×C8 | — | 4 | 1 | no | — |
| 42 | C22×C8 | — | 4 | 2 | no | — |
| 43 | C22×C8 | C23 | 1 | 2 | yes | — |
| 44 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 45 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 46 | #121 | — | 2 | 4 | no | — |
| 47 | #121 | — | 4 | 2 | no | — |
| 48 | C4×D4 | — | 2 | 4 | no | — |
| 49 | C4×D4 | — | 4 | 2 | no | — |
| 50 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 51 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 52 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 53 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 54 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 55 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 56 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 57 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 58 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 59 | C2×C22⋊C4 | D4 | 1 | 1 | yes | characteristic |
| 60 | C2×C22⋊C4 | D4 | 1 | 1 | yes | characteristic |
| 61 | C2×C42 | C23 | 1 | 2 | yes | — |
| 62 | #107 | — | 4 | 1 | no | — |
| 63 | C4⋊C8 | — | 4 | 2 | no | — |
| 64 | #102 | — | 4 | 2 | no | — |
| 65 | #102 | — | 4 | 2 | no | — |
| 66 | #102 | — | 4 | 2 | no | — |
| 67 | C22⋊C8 | — | 4 | 1 | no | — |
| 68 | #95 | — | 2 | 2 | no | — |
| 69 | #95 | — | 2 | 2 | no | — |
| 70 | #95 | — | 2 | 2 | no | — |
| 71 | #95 | — | 2 | 2 | no | — |
| 72 | #95 | D4 | 1 | 2 | yes | — |
| 73 | C24 | — | 2 | 2 | no | — |
| 74 | C24 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C2×D4 | — | 2 | 2 | no | — |
| 76 | C2×D4 | — | 2 | 2 | no | — |
| 77 | C2×D4 | — | 2 | 2 | no | — |
| 78 | C2×D4 | — | 2 | 2 | no | — |
| 79 | C2×D4 | — | 2 | 2 | no | — |
| 80 | C2×D4 | — | 4 | 4 | no | — |
| 81 | C2×D4 | — | 4 | 4 | no | — |
| 82 | C2×D4 | — | 8 | 2 | no | — |
| 83 | C2×D4 | #41 | 1 | 4 | yes | — |
| 84 | C22×C4 | — | 2 | 2 | no | — |
| 85 | C22×C4 | — | 2 | 1 | no | — |
| 86 | C22×C4 | — | 2 | 2 | no | — |
| 87 | C22×C4 | — | 2 | 1 | no | — |
| 88 | C22×C4 | — | 2 | 2 | no | — |
| 89 | C22×C4 | — | 2 | 2 | no | — |
| 90 | C22×C4 | — | 2 | 2 | no | — |
| 91 | C22×C4 | — | 4 | 4 | no | — |
| 92 | C22×C4 | — | 4 | 4 | no | — |
| 93 | C22×C4 | — | 8 | 2 | no | — |
| 94 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C22×C4 | C2×D4 | 1 | 2 | yes | — |
| 96 | C22×C4 | C2×D4 | 1 | 2 | yes | — |
| 97 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 98 | D8 | — | 4 | 4 | no | — |
| 99 | C2×C8 | — | 2 | 2 | no | — |
| 100 | C2×C8 | — | 2 | 1 | no | — |
| 101 | C2×C8 | — | 2 | 1 | no | — |
| 102 | C2×C8 | — | 2 | 2 | no | — |
| 103 | C2×C8 | — | 2 | 2 | no | — |
| 104 | C2×C8 | — | 2 | 2 | no | — |
| 105 | C2×C8 | — | 2 | 2 | no | — |
| 106 | C2×C8 | — | 4 | 4 | no | — |
| 107 | C4⋊C4 | — | 2 | 2 | no | — |
| 108 | C4⋊C4 | — | 2 | 2 | no | — |
| 109 | C4⋊C4 | — | 2 | 2 | no | — |
| 110 | C4⋊C4 | — | 2 | 2 | no | — |
| 111 | C4⋊C4 | — | 2 | 2 | no | — |
| 112 | C4⋊C4 | — | 2 | 2 | no | — |
| 113 | C4⋊C4 | — | 8 | 2 | no | — |
| 114 | C22⋊C4 | — | 2 | 2 | no | — |
| 115 | C22⋊C4 | — | 4 | 1 | no | — |
| 116 | C22⋊C4 | — | 4 | 4 | no | — |
| 117 | C22⋊C4 | — | 4 | 4 | no | — |
| 118 | C22⋊C4 | — | 8 | 2 | no | — |
| 119 | C22⋊C4 | — | 8 | 2 | no | — |
| 120 | C22⋊C4 | — | 8 | 2 | no | — |
| 121 | C42 | — | 2 | 2 | no | — |
| 122 | C42 | — | 2 | 2 | no | — |
| 123 | C23 | — | 2 | 1 | no | — |
| 124 | C23 | — | 2 | 2 | no | — |
| 125 | C23 | — | 2 | 1 | no | — |
| 126 | C23 | — | 2 | 2 | no | — |
| 127 | C23 | — | 2 | 4 | no | — |
| 128 | C23 | — | 4 | 2 | no | — |
| 129 | C23 | — | 4 | 2 | no | — |
| 130 | C23 | — | 4 | 2 | no | — |
| 131 | C23 | — | 4 | 2 | no | — |
| 132 | C23 | — | 4 | 2 | no | — |
| 133 | C23 | — | 4 | 2 | no | — |
| 134 | C23 | #135 | 1 | 2 | yes | — |
| 135 | C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 136 | D4 | — | 2 | 4 | no | — |
| 137 | D4 | — | 2 | 4 | no | — |
| 138 | D4 | — | 4 | 2 | no | — |
| 139 | D4 | — | 4 | 2 | no | — |
| 140 | D4 | — | 8 | 8 | no | — |
| 141 | C2×C4 | — | 2 | 1 | no | — |
| 142 | C2×C4 | — | 2 | 2 | no | — |
| 143 | C2×C4 | — | 2 | 2 | no | — |
| 144 | C2×C4 | — | 2 | 2 | no | — |
| 145 | C2×C4 | — | 2 | 2 | no | — |
| 146 | C2×C4 | — | 2 | 2 | no | — |
| 147 | C2×C4 | — | 2 | 2 | no | — |
| 148 | C2×C4 | — | 2 | 2 | no | — |
| 149 | C2×C4 | — | 4 | 2 | no | — |
| 150 | C2×C4 | — | 4 | 1 | no | — |
| 151 | C2×C4 | — | 4 | 1 | no | — |
| 152 | C2×C4 | — | 4 | 2 | no | — |
| 153 | C2×C4 | — | 4 | 2 | no | — |
| 154 | C2×C4 | — | 4 | 2 | no | — |
| 155 | C2×C4 | — | 4 | 2 | no | — |
| 156 | C2×C4 | — | 4 | 2 | no | — |
| 157 | C2×C4 | — | 4 | 2 | no | — |
| 158 | C2×C4 | — | 4 | 2 | no | — |
| 159 | C2×C4 | — | 4 | 2 | no | — |
| 160 | C2×C4 | — | 4 | 2 | no | — |
| 161 | C2×C4 | — | 8 | 8 | no | — |
| 162 | C2×C4 | C2×#41 | 1 | 2 | yes | — |
| 163 | C2×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 164 | C2×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 165 | C2×C4 | 2−1+4 | 1 | 1 | yes | characteristic |
| 166 | C8 | — | 4 | 1 | no | — |
| 167 | C8 | — | 4 | 1 | no | — |
| 168 | C8 | — | 4 | 2 | no | — |
| 169 | C22 | — | 2 | 2 | no | — |
| 170 | C22 | — | 2 | 4 | no | — |
| 171 | C22 | — | 4 | 1 | no | — |
| 172 | C22 | — | 4 | 1 | no | — |
| 173 | C22 | — | 4 | 2 | no | — |
| 174 | C22 | — | 4 | 4 | no | — |
| 175 | C22 | — | 4 | 4 | no | — |
| 176 | C22 | — | 4 | 4 | no | — |
| 177 | C22 | — | 8 | 2 | no | — |
| 178 | C22 | — | 8 | 2 | no | — |
| 179 | C22 | — | 8 | 2 | no | — |
| 180 | C22 | — | 8 | 2 | no | — |
| 181 | C22 | #537 | 1 | 1 | yes | characteristic |
| 182 | C22 | #546 | 1 | 2 | yes | — |
| 183 | C22 | C2×#135 | 1 | 1 | yes | characteristic |
| 184 | C22 | #575 | 1 | 1 | yes | characteristic |
| 185 | C22 | #576 | 1 | 2 | yes | — |
| 186 | C4 | — | 4 | 1 | no | — |
| 187 | C4 | — | 4 | 2 | no | — |
| 188 | C4 | — | 4 | 2 | no | — |
| 189 | C4 | — | 8 | 2 | no | — |
| 190 | C4 | — | 8 | 2 | no | — |
| 191 | C4 | — | 8 | 2 | no | — |
| 192 | C4 | #539 | 1 | 2 | yes | — |
| 193 | C4 | #551 | 1 | 1 | yes | characteristic |
| 194 | C4 | #560 | 1 | 1 | yes | characteristic |
| 195 | C2 | — | 4 | 2 | no | — |
| 196 | C2 | — | 8 | 4 | no | — |
| 197 | C2 | #2693 | 1 | 1 | yes | characteristic |
| 198 | C2 | #3250 | 1 | 1 | yes | characteristic |
| 199 | C2 | #3259 | 1 | 1 | yes | characteristic |
| 200 | C2 | #3297 | 1 | 2 | yes | — |
| 201 | C2 | #3319 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #2420 | C2 | semidirect |
| 2. #2101 | C2 | semidirect |
| 3. #2076 | C2 | semidirect |
| 4. #2067 | C2 | semidirect |
| 6. #1893 | C2 | semidirect |
| 7. #1879 | C2 | semidirect |
| 8. #1850 | C2 | semidirect |
| 9. #1804 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2420 | 2 |
| #2101 | 2 |
| #2076 | 2 |
| #2067 | 2 |
| #1999 | 2 |
| #1893 | 2 |
| #1879 | 2 |
| #1850 | 2 |
| #1804 | 2 |