Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3507 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 3507 conjugacy classes of subgroups — too many to draw. Only its 567 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 | C22×#120 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×B(4,𝔽2) | C2 | 1 | 24 | yes | maximal, direct product factor |
| 4 | C22×#99 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C26 | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×H(5,𝔽2) | — | 2 | 18 | no | — |
| 7 | C2×H(5,𝔽2) | C22 | 1 | 12 | yes | complemented |
| 8 | C22×#41 | — | 2 | 3 | no | — |
| 9 | C23×D4 | — | 2 | 3 | no | — |
| 10 | C23×D4 | — | 2 | 3 | no | — |
| 11 | C23×D4 | C22 | 1 | 3 | yes | — |
| 12 | C2×#120 | — | 2 | 18 | no | — |
| 13 | C2×#120 | C22 | 1 | 36 | yes | complemented |
| 14 | C22×C22⋊C4 | — | 2 | 3 | no | — |
| 15 | C22×C22⋊C4 | C22 | 1 | 3 | yes | complemented |
| 16 | B(4,𝔽2) | C22 | 1 | 64 | yes | direct product factor |
| 17 | C2×#99 | C22 | 1 | 36 | yes | complemented |
| 18 | C25 | — | 2 | 1 | no | — |
| 19 | C25 | — | 2 | 3 | no | — |
| 20 | C25 | — | 2 | 9 | no | — |
| 21 | C25 | — | 4 | 8 | no | — |
| 22 | C25 | D4 | 1 | 3 | yes | complemented |
| 23 | C25 | C23 | 1 | 1 | yes | characteristic |
| 24 | C25 | C23 | 1 | 6 | yes | complemented |
| 25 | H(5,𝔽2) | — | 2 | 48 | no | — |
| 26 | H(5,𝔽2) | — | 2 | 72 | no | — |
| 27 | H(5,𝔽2) | C23 | 1 | 16 | yes | complemented |
| 28 | C2×#41 | — | 2 | 36 | no | — |
| 29 | C2×#41 | — | 2 | 36 | no | — |
| 30 | C22×Q8 | — | 2 | 1 | no | — |
| 31 | C22×D4 | — | 2 | 3 | no | — |
| 32 | C22×D4 | — | 2 | 36 | no | — |
| 33 | C22×D4 | — | 2 | 6 | no | — |
| 34 | C22×D4 | — | 2 | 18 | no | — |
| 35 | C22×D4 | — | 2 | 36 | no | — |
| 36 | C22×D4 | — | 2 | 36 | no | — |
| 37 | C22×D4 | — | 4 | 24 | no | — |
| 38 | C22×D4 | D4 | 1 | 18 | yes | complemented |
| 39 | C22×D4 | C23 | 1 | 18 | yes | — |
| 40 | C23×C4 | — | 2 | 3 | no | — |
| 41 | C23×C4 | — | 2 | 3 | no | — |
| 42 | C23×C4 | D4 | 1 | 3 | yes | complemented |
| 43 | #120 | — | 2 | 72 | no | — |
| 44 | #120 | C23 | 1 | 48 | yes | complemented |
| 45 | C2×C22⋊C4 | — | 2 | 9 | no | — |
| 46 | C2×C22⋊C4 | — | 2 | 36 | no | — |
| 47 | C2×C22⋊C4 | C23 | 1 | 18 | yes | complemented |
| 48 | #99 | C23 | 1 | 48 | yes | complemented |
| 49 | C24 | — | 2 | 24 | no | — |
| 50 | C24 | — | 2 | 12 | no | — |
| 51 | C24 | — | 2 | 3 | no | — |
| 52 | C24 | — | 2 | 18 | no | — |
| 53 | C24 | — | 2 | 9 | no | — |
| 54 | C24 | — | 2 | 18 | no | — |
| 55 | C24 | — | 4 | 4 | no | — |
| 56 | C24 | — | 4 | 24 | no | — |
| 57 | C24 | — | 4 | 12 | no | — |
| 58 | C24 | — | 4 | 72 | no | — |
| 59 | C24 | — | 4 | 12 | no | — |
| 60 | C24 | — | 4 | 9 | no | — |
| 61 | C24 | — | 4 | 6 | no | — |
| 62 | C24 | — | 4 | 1 | no | — |
| 63 | C24 | — | 8 | 4 | no | — |
| 64 | C24 | C2×D4 | 1 | 3 | yes | — |
| 65 | C24 | C2×D4 | 1 | 9 | yes | — |
| 66 | C24 | C2×D4 | 1 | 18 | yes | complemented |
| 67 | C24 | C24 | 1 | 3 | yes | — |
| 68 | C24 | C24 | 1 | 4 | yes | complemented |
| 69 | #41 | — | 2 | 144 | no | — |
| 70 | #41 | — | 2 | 48 | no | — |
| 71 | C2×Q8 | — | 2 | 12 | no | — |
| 72 | C2×D4 | — | 2 | 36 | no | — |
| 73 | C2×D4 | — | 2 | 144 | no | — |
| 74 | C2×D4 | — | 2 | 36 | no | — |
| 75 | C2×D4 | — | 2 | 36 | no | — |
| 76 | C2×D4 | — | 2 | 36 | no | — |
| 77 | C2×D4 | — | 2 | 36 | no | — |
| 78 | C2×D4 | — | 2 | 48 | no | — |
| 79 | C2×D4 | — | 2 | 48 | no | — |
| 80 | C2×D4 | — | 4 | 144 | no | — |
| 81 | C2×D4 | C2×D4 | 1 | 36 | yes | complemented |
| 82 | C2×D4 | C24 | 1 | 12 | yes | — |
| 83 | C22×C4 | — | 2 | 36 | no | — |
| 84 | C22×C4 | — | 2 | 9 | no | — |
| 85 | C22×C4 | — | 2 | 3 | no | — |
| 86 | C22×C4 | — | 2 | 18 | no | — |
| 87 | C22×C4 | — | 4 | 12 | no | — |
| 88 | C22×C4 | C2×D4 | 1 | 18 | yes | complemented |
| 89 | C22⋊C4 | — | 2 | 18 | no | — |
| 90 | C22⋊C4 | — | 2 | 48 | no | — |
| 91 | C22⋊C4 | C24 | 1 | 12 | yes | complemented |
| 92 | C23 | — | 2 | 12 | no | — |
| 93 | C23 | — | 2 | 72 | no | — |
| 94 | C23 | — | 2 | 16 | no | — |
| 95 | C23 | — | 2 | 18 | no | — |
| 96 | C23 | — | 2 | 12 | no | — |
| 97 | C23 | — | 2 | 18 | no | — |
| 98 | C23 | — | 4 | 18 | no | — |
| 99 | C23 | — | 4 | 6 | no | — |
| 100 | C23 | — | 4 | 8 | no | — |
| 101 | C23 | — | 4 | 36 | no | — |
| 102 | C23 | — | 4 | 72 | no | — |
| 103 | C23 | — | 4 | 72 | no | — |
| 104 | C23 | — | 4 | 36 | no | — |
| 105 | C23 | — | 4 | 48 | no | — |
| 106 | C23 | — | 4 | 18 | no | — |
| 107 | C23 | — | 4 | 6 | no | — |
| 108 | C23 | — | 4 | 8 | no | — |
| 109 | C23 | — | 4 | 12 | no | — |
| 110 | C23 | — | 8 | 18 | no | — |
| 111 | C23 | — | 8 | 24 | no | — |
| 112 | C23 | #120 | 1 | 9 | yes | — |
| 113 | C23 | #120 | 1 | 6 | yes | — |
| 114 | C23 | #120 | 1 | 1 | yes | characteristic |
| 115 | C23 | C22×D4 | 1 | 9 | yes | — |
| 116 | C23 | C22×D4 | 1 | 9 | yes | — |
| 117 | C23 | C22×D4 | 1 | 12 | yes | complemented |
| 118 | C23 | C25 | 1 | 1 | yes | characteristic |
| 119 | Q8 | — | 2 | 16 | no | — |
| 120 | D4 | — | 2 | 48 | no | — |
| 121 | D4 | — | 2 | 72 | no | — |
| 122 | D4 | — | 2 | 24 | no | — |
| 123 | D4 | — | 4 | 96 | no | — |
| 124 | C2×C4 | — | 2 | 48 | no | — |
| 125 | C2×C4 | — | 2 | 18 | no | — |
| 126 | C2×C4 | — | 2 | 18 | no | — |
| 127 | C2×C4 | — | 2 | 12 | no | — |
| 128 | C2×C4 | — | 4 | 36 | no | — |
| 129 | C2×C4 | C22×D4 | 1 | 12 | yes | complemented |
| 130 | C22 | — | 2 | 18 | no | — |
| 131 | C22 | — | 2 | 18 | no | — |
| 132 | C22 | — | 2 | 24 | no | — |
| 133 | C22 | — | 2 | 12 | no | — |
| 134 | C22 | — | 4 | 6 | no | — |
| 135 | C22 | — | 4 | 9 | no | — |
| 136 | C22 | — | 4 | 1 | no | — |
| 137 | C22 | — | 4 | 12 | no | — |
| 138 | C22 | — | 4 | 12 | no | — |
| 139 | C22 | — | 4 | 72 | no | — |
| 140 | C22 | — | 4 | 4 | no | — |
| 141 | C22 | — | 4 | 24 | no | — |
| 142 | C22 | — | 4 | 36 | no | — |
| 143 | C22 | — | 8 | 18 | no | — |
| 144 | C22 | — | 8 | 16 | no | — |
| 145 | C22 | B(4,𝔽2) | 1 | 4 | yes | direct product factor |
| 146 | C22 | C2×#120 | 1 | 9 | yes | — |
| 147 | C22 | C2×#120 | 1 | 3 | yes | — |
| 148 | C22 | C23×D4 | 1 | 3 | yes | — |
| 149 | C4 | — | 2 | 12 | no | — |
| 150 | C4 | — | 4 | 12 | no | — |
| 151 | C2 | — | 2 | 3 | no | — |
| 152 | C2 | — | 2 | 9 | no | — |
| 153 | C2 | — | 4 | 8 | no | — |
| 154 | C2 | — | 4 | 12 | no | — |
| 155 | C2 | C2×B(4,𝔽2) | 1 | 6 | yes | direct product factor |
| 156 | C2 | C22×#120 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ 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. C22×#120 | C2 | semidirect |
| 3. C2×B(4,𝔽2) | C2 | direct (N × H) |
| 4. C22×#99 | C2 | semidirect |
| 5. C26 | C22 | semidirect |
| 7. C2×H(5,𝔽2) | C22 | semidirect |
| 13. C2×#120 | C22 | semidirect |
| 15. C22×C22⋊C4 | C22 | semidirect |
| 16. B(4,𝔽2) | C22 | direct (N × H) |
| 17. C2×#99 | C22 | semidirect |
| 22. C25 | D4 | semidirect |
| 24. C25 | C23 | semidirect |
| 27. H(5,𝔽2) | C23 | semidirect |
| 38. C22×D4 | D4 | semidirect |
| 42. C23×C4 | D4 | semidirect |
| 44. #120 | C23 | semidirect |
| 47. C2×C22⋊C4 | C23 | semidirect |
| 48. #99 | C23 | semidirect |
| 66. C24 | C2×D4 | semidirect |
| 68. C24 | C24 | semidirect |
| 81. C2×D4 | C2×D4 | semidirect |
| 88. C22×C4 | C2×D4 | semidirect |
| 91. C22⋊C4 | C24 | semidirect |
| 117. C23 | C22×D4 | semidirect |
| 129. C2×C4 | C22×D4 | semidirect |
| 145. C22 | B(4,𝔽2) | direct (N × H) |
| 155. C2 | C2×B(4,𝔽2) | 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 |
| C22×#120 | 2 |
| C2×B(4,𝔽2) | 2 |
| C22×#99 | 2 |