Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
172 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 35 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 | #3263 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #2191 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #2190 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #2053 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #1340 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #1339 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #1328 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #575 | C22 | 1 | 1 | yes | characteristic |
| 9 | #542 | — | 2 | 1 | no | — |
| 10 | #510 | — | 2 | 1 | no | — |
| 11 | #509 | — | 2 | 1 | no | — |
| 12 | #476 | — | 2 | 1 | no | — |
| 13 | #475 | — | 2 | 1 | no | — |
| 14 | #470 | C22 | 1 | 1 | yes | characteristic |
| 15 | #469 | C22 | 1 | 1 | yes | characteristic |
| 16 | #465 | — | 2 | 1 | no | — |
| 17 | #464 | — | 2 | 1 | no | — |
| 18 | #463 | — | 2 | 1 | no | — |
| 19 | #424 | C22 | 1 | 1 | yes | characteristic |
| 20 | #416 | — | 2 | 1 | no | — |
| 21 | #410 | — | 2 | 1 | no | — |
| 22 | #360 | — | 2 | 1 | no | — |
| 23 | #360 | — | 2 | 1 | no | — |
| 24 | #349 | C22 | 1 | 1 | yes | characteristic |
| 25 | #349 | C22 | 1 | 1 | yes | characteristic |
| 26 | H(3,ℤ/4) | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×#41 | — | 2 | 1 | no | — |
| 28 | #137 | — | 4 | 1 | no | — |
| 29 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 30 | #135 | — | 2 | 1 | no | — |
| 31 | #135 | — | 2 | 1 | no | — |
| 32 | C2×Q16 | D4 | 1 | 1 | yes | characteristic |
| 33 | C2×SD16 | — | 2 | 1 | no | — |
| 34 | C2×D8 | D4 | 1 | 1 | yes | characteristic |
| 35 | C2×M16 | C23 | 1 | 1 | yes | characteristic |
| 36 | C22×C8 | D4 | 1 | 1 | yes | characteristic |
| 37 | C4⋊Q8 | — | 2 | 1 | no | — |
| 38 | #126 | — | 4 | 1 | no | — |
| 39 | #124 | — | 2 | 1 | no | — |
| 40 | #124 | — | 4 | 1 | no | — |
| 41 | #123 | — | 4 | 1 | no | — |
| 42 | C22⋊Q8 | — | 2 | 1 | no | — |
| 43 | C22⋊Q8 | — | 4 | 1 | no | — |
| 44 | #121 | — | 2 | 1 | no | — |
| 45 | C4×Q8 | — | 2 | 1 | no | — |
| 46 | C4×D4 | — | 2 | 1 | no | — |
| 47 | #117 | — | 2 | 1 | no | — |
| 48 | #117 | D4 | 1 | 1 | yes | characteristic |
| 49 | Q32 | — | 4 | 1 | no | — |
| 50 | SD32 | — | 4 | 1 | no | — |
| 51 | SD32 | — | 4 | 1 | no | — |
| 52 | D16 | — | 4 | 1 | no | — |
| 53 | M32 | — | 2 | 1 | no | — |
| 54 | M32 | — | 2 | 1 | no | — |
| 55 | #106 | — | 2 | 1 | no | — |
| 56 | #106 | D4 | 1 | 1 | yes | characteristic |
| 57 | #106 | D4 | 1 | 1 | yes | characteristic |
| 58 | C4⋊C8 | — | 2 | 1 | no | — |
| 59 | C4⋊C8 | — | 2 | 1 | no | — |
| 60 | #103 | — | 2 | 1 | no | — |
| 61 | #103 | — | 2 | 1 | no | — |
| 62 | #103 | — | 4 | 1 | no | — |
| 63 | #102 | — | 2 | 1 | no | — |
| 64 | #102 | — | 2 | 1 | no | — |
| 65 | #102 | — | 4 | 1 | no | — |
| 66 | #99 | — | 4 | 1 | no | — |
| 67 | #99 | — | 4 | 1 | no | — |
| 68 | #41 | — | 4 | 1 | no | — |
| 69 | #41 | — | 4 | 1 | no | — |
| 70 | #41 | — | 4 | 1 | no | — |
| 71 | #41 | — | 4 | 1 | no | — |
| 72 | C2×Q8 | — | 2 | 1 | no | — |
| 73 | C2×Q8 | — | 2 | 1 | no | — |
| 74 | C2×Q8 | — | 4 | 1 | no | — |
| 75 | C2×D4 | — | 2 | 1 | no | — |
| 76 | C2×D4 | — | 2 | 1 | no | — |
| 77 | C2×D4 | — | 4 | 1 | no | — |
| 78 | C2×D4 | — | 4 | 1 | no | — |
| 79 | C22×C4 | — | 4 | 1 | no | — |
| 80 | C22×C4 | — | 4 | 1 | no | — |
| 81 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | Q16 | — | 2 | 1 | no | — |
| 83 | Q16 | — | 2 | 1 | no | — |
| 84 | Q16 | — | 4 | 1 | no | — |
| 85 | SD16 | — | 4 | 1 | no | — |
| 86 | SD16 | — | 4 | 1 | no | — |
| 87 | SD16 | — | 8 | 1 | no | — |
| 88 | D8 | — | 2 | 1 | no | — |
| 89 | D8 | — | 2 | 1 | no | — |
| 90 | D8 | — | 4 | 1 | no | — |
| 91 | M16 | — | 4 | 1 | no | — |
| 92 | C2×C8 | — | 2 | 1 | no | — |
| 93 | C2×C8 | — | 4 | 1 | no | — |
| 94 | C2×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C2×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C4⋊C4 | — | 2 | 1 | no | — |
| 97 | C4⋊C4 | — | 2 | 1 | no | — |
| 98 | C4⋊C4 | — | 2 | 1 | no | — |
| 99 | C4⋊C4 | — | 4 | 1 | no | — |
| 100 | C4⋊C4 | — | 4 | 1 | no | — |
| 101 | C4⋊C4 | — | 4 | 1 | no | — |
| 102 | C4⋊C4 | — | 4 | 1 | no | — |
| 103 | C22⋊C4 | — | 4 | 1 | no | — |
| 104 | C22⋊C4 | — | 4 | 1 | no | — |
| 105 | C22⋊C4 | — | 4 | 1 | no | — |
| 106 | C22⋊C4 | — | 4 | 1 | no | — |
| 107 | C22⋊C4 | — | 4 | 1 | no | — |
| 108 | C22⋊C4 | — | 4 | 1 | no | — |
| 109 | C22⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 110 | C22⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 111 | C42 | — | 2 | 1 | no | — |
| 112 | C42 | — | 4 | 1 | no | — |
| 113 | C42 | — | 4 | 1 | no | — |
| 114 | C16 | — | 4 | 1 | no | — |
| 115 | C16 | — | 4 | 1 | no | — |
| 116 | C23 | — | 4 | 1 | no | — |
| 117 | C23 | — | 4 | 1 | no | — |
| 118 | C23 | C2×D8 | 1 | 1 | yes | characteristic |
| 119 | Q8 | — | 4 | 1 | no | — |
| 120 | Q8 | — | 4 | 1 | no | — |
| 121 | Q8 | — | 4 | 1 | no | — |
| 122 | Q8 | — | 4 | 1 | no | — |
| 123 | Q8 | — | 16 | 1 | no | — |
| 124 | D4 | — | 4 | 1 | no | — |
| 125 | D4 | — | 4 | 1 | no | — |
| 126 | D4 | — | 4 | 1 | no | — |
| 127 | D4 | — | 4 | 1 | no | — |
| 128 | D4 | — | 8 | 1 | no | — |
| 129 | D4 | — | 8 | 1 | no | — |
| 130 | D4 | — | 16 | 1 | no | — |
| 131 | C2×C4 | — | 2 | 1 | no | — |
| 132 | C2×C4 | — | 2 | 1 | no | — |
| 133 | C2×C4 | — | 2 | 1 | no | — |
| 134 | C2×C4 | — | 2 | 1 | no | — |
| 135 | C2×C4 | — | 4 | 1 | no | — |
| 136 | C2×C4 | — | 4 | 1 | no | — |
| 137 | C2×C4 | — | 4 | 1 | no | — |
| 138 | C2×C4 | — | 4 | 1 | no | — |
| 139 | C2×C4 | — | 8 | 1 | no | — |
| 140 | C2×C4 | — | 8 | 1 | no | — |
| 141 | C2×C4 | — | 8 | 1 | no | — |
| 142 | C2×C4 | — | 16 | 1 | no | — |
| 143 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 144 | C2×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 145 | C8 | — | 2 | 1 | no | — |
| 146 | C8 | — | 2 | 1 | no | — |
| 147 | C8 | — | 4 | 1 | no | — |
| 148 | C22 | — | 2 | 1 | no | — |
| 149 | C22 | — | 4 | 1 | no | — |
| 150 | C22 | — | 8 | 1 | no | — |
| 151 | C22 | — | 8 | 1 | no | — |
| 152 | C22 | — | 8 | 1 | no | — |
| 153 | C22 | — | 8 | 1 | no | — |
| 154 | C22 | — | 16 | 1 | no | — |
| 155 | C22 | #447 | 1 | 1 | yes | characteristic |
| 156 | C4 | — | 2 | 1 | no | — |
| 157 | C4 | — | 4 | 1 | no | — |
| 158 | C4 | — | 4 | 1 | no | — |
| 159 | C4 | — | 8 | 1 | no | — |
| 160 | C4 | — | 8 | 1 | no | — |
| 161 | C4 | — | 8 | 1 | no | — |
| 162 | C4 | — | 8 | 1 | no | — |
| 163 | C4 | — | 16 | 1 | no | — |
| 164 | C4 | #454 | 1 | 1 | yes | characteristic |
| 165 | C4 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 166 | C2 | — | 2 | 1 | no | — |
| 167 | C2 | — | 4 | 1 | no | — |
| 168 | C2 | — | 16 | 1 | no | — |
| 169 | C2 | — | 16 | 1 | no | — |
| 170 | C2 | #1598 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1598 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| prime radical | #2190 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3263 | C2 | semidirect |
| 2. #2191 | C2 | semidirect |
| 4. #2053 | C2 | semidirect |
| 5. #1340 | C2 | semidirect |
| 6. #1339 | C2 | semidirect |
| 7. #1328 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3263 | 2 |
| #2191 | 2 |
| #2190 | 2 |
| #2053 | 2 |
| #1340 | 2 |
| #1339 | 2 |
| #1328 | 2 |