Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2021 conjugacy classes of subgroups; only the 303 normal subgroups are shown.
This group has 2021 conjugacy classes of subgroups — too many to draw. Only its 303 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 |
| 2 | C2×#410 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #2309 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #1781 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | C2×H(5,𝔽2) | — | 2 | 16 | no | — |
| 6 | C2×H(5,𝔽2) | — | 2 | 8 | no | — |
| 7 | C22×#41 | C4 | 1 | 4 | yes | complemented |
| 8 | C22×#41 | C22 | 1 | 2 | yes | — |
| 9 | C23×D4 | C4 | 1 | 4 | yes | complemented |
| 10 | C23×D4 | C22 | 1 | 4 | yes | — |
| 11 | C23×D4 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#117 | C22 | 1 | 4 | yes | complemented |
| 13 | #410 | — | 2 | 16 | no | — |
| 14 | C2×#99 | C22 | 1 | 16 | yes | complemented |
| 15 | #390 | — | 2 | 8 | no | — |
| 16 | #379 | C22 | 1 | 8 | yes | complemented |
| 17 | C25 | C2×C4 | 1 | 4 | yes | — |
| 18 | C25 | C23 | 1 | 2 | yes | — |
| 19 | H(5,𝔽2) | — | 2 | 64 | no | — |
| 20 | H(5,𝔽2) | — | 2 | 64 | no | — |
| 21 | C2×#41 | — | 2 | 16 | no | — |
| 22 | C2×#41 | — | 2 | 16 | no | — |
| 23 | C2×#41 | — | 2 | 16 | no | — |
| 24 | C2×#41 | — | 2 | 8 | no | — |
| 25 | C2×#41 | — | 2 | 8 | no | — |
| 26 | C2×#41 | D4 | 1 | 16 | yes | — |
| 27 | C22×Q8 | C2×C4 | 1 | 2 | yes | — |
| 28 | C22×D4 | — | 2 | 16 | no | — |
| 29 | C22×D4 | — | 2 | 16 | no | — |
| 30 | C22×D4 | — | 2 | 16 | no | — |
| 31 | C22×D4 | — | 2 | 32 | no | — |
| 32 | C22×D4 | — | 2 | 16 | no | — |
| 33 | C22×D4 | — | 2 | 8 | no | — |
| 34 | C22×D4 | C2×C4 | 1 | 8 | yes | — |
| 35 | C22×D4 | C2×C4 | 1 | 4 | yes | — |
| 36 | C22×D4 | C2×C4 | 1 | 2 | yes | — |
| 37 | C22×D4 | D4 | 1 | 8 | yes | — |
| 38 | C22×D4 | C23 | 1 | 4 | yes | — |
| 39 | C23×C4 | C2×C4 | 1 | 4 | yes | — |
| 40 | C23×C4 | C2×C4 | 1 | 4 | yes | — |
| 41 | C23×C4 | C23 | 1 | 1 | yes | characteristic |
| 42 | #117 | — | 2 | 16 | no | — |
| 43 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 44 | C2×C22⋊C4 | — | 2 | 16 | no | — |
| 45 | C2×C22⋊C4 | C23 | 1 | 8 | yes | — |
| 46 | C2×C42 | — | 2 | 4 | no | — |
| 47 | #99 | — | 2 | 32 | no | — |
| 48 | C24 | — | 2 | 16 | no | — |
| 49 | C24 | — | 2 | 8 | no | — |
| 50 | C24 | — | 2 | 8 | no | — |
| 51 | C24 | — | 8 | 4 | no | — |
| 52 | C24 | — | 8 | 8 | no | — |
| 53 | C24 | C22×C4 | 1 | 4 | yes | — |
| 54 | C24 | C22×C4 | 1 | 4 | yes | — |
| 55 | C24 | C2×D4 | 1 | 8 | yes | — |
| 56 | C24 | C24 | 1 | 1 | yes | characteristic |
| 57 | #41 | — | 2 | 128 | no | — |
| 58 | #41 | — | 2 | 32 | no | — |
| 59 | #41 | — | 2 | 32 | no | — |
| 60 | C2×Q8 | — | 2 | 4 | no | — |
| 61 | C2×Q8 | — | 2 | 4 | no | — |
| 62 | C2×Q8 | C22⋊C4 | 1 | 8 | yes | complemented |
| 63 | C2×D4 | — | 2 | 128 | no | — |
| 64 | C2×D4 | — | 2 | 128 | no | — |
| 65 | C2×D4 | — | 2 | 16 | no | — |
| 66 | C2×D4 | — | 2 | 16 | no | — |
| 67 | C2×D4 | — | 2 | 8 | no | — |
| 68 | C2×D4 | — | 2 | 8 | no | — |
| 69 | C2×D4 | — | 2 | 16 | no | — |
| 70 | C2×D4 | — | 2 | 8 | no | — |
| 71 | C2×D4 | — | 2 | 8 | no | — |
| 72 | C2×D4 | — | 2 | 4 | no | — |
| 73 | C2×D4 | — | 2 | 4 | no | — |
| 74 | C2×D4 | — | 2 | 16 | no | — |
| 75 | C2×D4 | — | 2 | 16 | no | — |
| 76 | C2×D4 | C22⋊C4 | 1 | 16 | yes | — |
| 77 | C2×D4 | C22⋊C4 | 1 | 8 | yes | complemented |
| 78 | C2×D4 | C2×D4 | 1 | 16 | yes | — |
| 79 | C22×C4 | — | 2 | 8 | no | — |
| 80 | C22×C4 | — | 2 | 16 | no | — |
| 81 | C22×C4 | — | 2 | 8 | no | — |
| 82 | C22×C4 | — | 2 | 8 | no | — |
| 83 | C22×C4 | — | 2 | 8 | no | — |
| 84 | C22×C4 | C22⋊C4 | 1 | 16 | yes | complemented |
| 85 | C22×C4 | C22×C4 | 1 | 4 | yes | — |
| 86 | C22×C4 | C22×C4 | 1 | 2 | yes | — |
| 87 | C22×C4 | C2×D4 | 1 | 4 | yes | — |
| 88 | C22×C4 | C2×D4 | 1 | 8 | yes | — |
| 89 | C4⋊C4 | — | 4 | 8 | no | — |
| 90 | C22⋊C4 | — | 2 | 16 | no | — |
| 91 | C22⋊C4 | — | 8 | 16 | no | — |
| 92 | C42 | — | 4 | 8 | no | — |
| 93 | C23 | — | 2 | 32 | no | — |
| 94 | C23 | — | 2 | 8 | no | — |
| 95 | C23 | — | 2 | 4 | no | — |
| 96 | C23 | — | 2 | 4 | no | — |
| 97 | C23 | — | 2 | 16 | no | — |
| 98 | C23 | — | 4 | 8 | no | — |
| 99 | C23 | — | 4 | 2 | no | — |
| 100 | C23 | — | 4 | 16 | no | — |
| 101 | C23 | — | 4 | 8 | no | — |
| 102 | C23 | — | 8 | 16 | no | — |
| 103 | C23 | — | 8 | 16 | no | — |
| 104 | C23 | — | 8 | 32 | no | — |
| 105 | C23 | C2×C22⋊C4 | 1 | 8 | yes | — |
| 106 | C23 | C2×C22⋊C4 | 1 | 8 | yes | — |
| 107 | C23 | C23×C4 | 1 | 1 | yes | characteristic |
| 108 | C23 | C22×D4 | 1 | 2 | yes | — |
| 109 | C23 | C22×D4 | 1 | 4 | yes | — |
| 110 | Q8 | — | 2 | 16 | no | — |
| 111 | D4 | — | 2 | 64 | no | — |
| 112 | D4 | — | 2 | 32 | no | — |
| 113 | D4 | — | 2 | 32 | no | — |
| 114 | D4 | — | 2 | 16 | no | — |
| 115 | C2×C4 | — | 2 | 32 | no | — |
| 116 | C2×C4 | — | 2 | 4 | no | — |
| 117 | C2×C4 | — | 2 | 4 | no | — |
| 118 | C2×C4 | — | 2 | 32 | no | — |
| 119 | C2×C4 | — | 4 | 16 | no | — |
| 120 | C2×C4 | — | 4 | 8 | no | — |
| 121 | C2×C4 | C2×C22⋊C4 | 1 | 8 | yes | — |
| 122 | C2×C4 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 123 | C2×C4 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 124 | C2×C4 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 125 | C2×C4 | #120 | 1 | 8 | yes | — |
| 126 | C2×C4 | #120 | 1 | 8 | yes | — |
| 127 | C22 | — | 2 | 4 | no | — |
| 128 | C22 | — | 2 | 2 | no | — |
| 129 | C22 | — | 2 | 16 | no | — |
| 130 | C22 | — | 2 | 8 | no | — |
| 131 | C22 | — | 2 | 8 | no | — |
| 132 | C22 | — | 4 | 4 | no | — |
| 133 | C22 | — | 4 | 8 | no | — |
| 134 | C22 | — | 4 | 16 | no | — |
| 135 | C22 | — | 4 | 8 | no | — |
| 136 | C22 | — | 4 | 8 | no | — |
| 137 | C22 | — | 8 | 16 | no | — |
| 138 | C22 | — | 8 | 32 | no | — |
| 139 | C22 | C22×C22⋊C4 | 1 | 2 | yes | — |
| 140 | C22 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 141 | C22 | C2×#120 | 1 | 4 | yes | — |
| 142 | C4 | — | 2 | 8 | no | — |
| 143 | C4 | — | 8 | 8 | no | — |
| 144 | C4 | #379 | 1 | 8 | yes | — |
| 145 | C2 | — | 2 | 4 | no | — |
| 146 | C2 | — | 2 | 2 | no | — |
| 147 | C2 | — | 4 | 8 | no | — |
| 148 | C2 | — | 4 | 8 | no | — |
| 149 | C2 | C2×#379 | 1 | 1 | yes | characteristic |
| 150 | C2 | #2883 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C22×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#410 | C2 | semidirect |
| 3. #2309 | C2 | semidirect |
| 4. #1781 | C2 | semidirect |
| 7. C22×#41 | C4 | semidirect |
| 9. C23×D4 | C4 | semidirect |
| 12. C2×#117 | C22 | semidirect |
| 14. C2×#99 | C22 | semidirect |
| 16. #379 | C22 | semidirect |
| 62. C2×Q8 | C22⋊C4 | semidirect |
| 77. C2×D4 | C22⋊C4 | semidirect |
| 84. C22×C4 | C22⋊C4 | semidirect |
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×#410 | 2 |
| #2309 | 2 |
| #1781 | 2 |