Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
356 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 115 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 | #3227 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #3210 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #3184 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #3139 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #1720 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #1717 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #1716 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #1708 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #1453 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #533 | C22 | 1 | 1 | yes | characteristic |
| 11 | #532 | C22 | 1 | 2 | yes | — |
| 12 | C2×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 13 | #498 | C22 | 1 | 1 | yes | characteristic |
| 14 | #498 | C22 | 1 | 2 | yes | — |
| 15 | #498 | C22 | 1 | 2 | yes | — |
| 16 | #494 | C22 | 1 | 2 | yes | — |
| 17 | #493 | C22 | 1 | 1 | yes | characteristic |
| 18 | #492 | C22 | 1 | 2 | yes | — |
| 19 | #491 | — | 2 | 2 | no | — |
| 20 | #487 | C22 | 1 | 2 | yes | — |
| 21 | #487 | C22 | 1 | 2 | yes | — |
| 22 | #487 | C22 | 1 | 2 | yes | — |
| 23 | #486 | C22 | 1 | 2 | yes | — |
| 24 | #486 | C22 | 1 | 2 | yes | — |
| 25 | #486 | C22 | 1 | 2 | yes | — |
| 26 | #486 | C22 | 1 | 2 | yes | — |
| 27 | #481 | — | 2 | 4 | no | — |
| 28 | #476 | — | 2 | 4 | no | — |
| 29 | #469 | — | 2 | 2 | no | — |
| 30 | #464 | — | 2 | 4 | no | — |
| 31 | #452 | — | 2 | 4 | no | — |
| 32 | #442 | — | 2 | 2 | no | — |
| 33 | #431 | C22 | 1 | 2 | yes | — |
| 34 | #419 | — | 2 | 2 | no | — |
| 35 | C4×M16 | C22 | 1 | 1 | yes | characteristic |
| 36 | #322 | C22 | 1 | 2 | yes | — |
| 37 | #322 | C22 | 1 | 1 | yes | characteristic |
| 38 | C82 | C22 | 1 | 1 | yes | characteristic |
| 39 | C2×#41 | — | 2 | 2 | no | — |
| 40 | C22×Q8 | — | 2 | 1 | no | — |
| 41 | C2×Q16 | — | 2 | 4 | no | — |
| 42 | C2×SD16 | — | 2 | 4 | no | — |
| 43 | C2×D8 | — | 2 | 2 | no | — |
| 44 | C2×M16 | D4 | 1 | 2 | yes | — |
| 45 | C4⋊Q8 | — | 2 | 2 | no | — |
| 46 | C4⋊Q8 | — | 2 | 1 | no | — |
| 47 | C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 48 | C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 49 | C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 50 | #127 | C23 | 1 | 2 | yes | — |
| 51 | #125 | — | 2 | 1 | no | — |
| 52 | #124 | — | 2 | 2 | no | — |
| 53 | C22⋊Q8 | — | 2 | 2 | no | — |
| 54 | #121 | — | 2 | 2 | no | — |
| 55 | #121 | — | 2 | 2 | no | — |
| 56 | C4×Q8 | — | 2 | 2 | no | — |
| 57 | C4×D4 | — | 2 | 2 | no | — |
| 58 | C4×D4 | — | 2 | 2 | no | — |
| 59 | #117 | — | 2 | 1 | no | — |
| 60 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 61 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 62 | #106 | — | 2 | 2 | no | — |
| 63 | #106 | — | 2 | 4 | no | — |
| 64 | #106 | — | 2 | 4 | no | — |
| 65 | C4⋊C8 | D4 | 1 | 4 | yes | — |
| 66 | #103 | — | 2 | 4 | no | — |
| 67 | #103 | — | 2 | 4 | no | — |
| 68 | #103 | — | 2 | 4 | no | — |
| 69 | #102 | — | 2 | 4 | no | — |
| 70 | #102 | — | 2 | 4 | no | — |
| 71 | #102 | — | 2 | 4 | no | — |
| 72 | #102 | — | 2 | 4 | no | — |
| 73 | C22⋊C8 | D4 | 1 | 4 | yes | — |
| 74 | #97 | D4 | 1 | 2 | yes | — |
| 75 | C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 76 | C4×C8 | C23 | 1 | 2 | yes | — |
| 77 | C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 78 | C4×C8 | C23 | 1 | 2 | yes | — |
| 79 | #41 | — | 8 | 4 | no | — |
| 80 | C2×Q8 | — | 2 | 2 | no | — |
| 81 | C2×Q8 | — | 2 | 2 | no | — |
| 82 | C2×Q8 | — | 2 | 2 | no | — |
| 83 | C2×Q8 | — | 4 | 4 | no | — |
| 84 | C2×Q8 | — | 4 | 1 | no | — |
| 85 | C2×D4 | — | 2 | 2 | no | — |
| 86 | C2×D4 | — | 2 | 2 | no | — |
| 87 | C2×D4 | — | 2 | 2 | no | — |
| 88 | C2×D4 | — | 4 | 2 | no | — |
| 89 | C22×C4 | — | 4 | 1 | no | — |
| 90 | C22×C4 | — | 4 | 2 | no | — |
| 91 | C22×C4 | C2×D4 | 1 | 2 | yes | — |
| 92 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | Q16 | — | 8 | 4 | no | — |
| 94 | SD16 | — | 8 | 4 | no | — |
| 95 | D8 | — | 4 | 4 | no | — |
| 96 | M16 | — | 4 | 2 | no | — |
| 97 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 98 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 99 | C2×C8 | C2×D4 | 1 | 4 | yes | — |
| 100 | C2×C8 | C2×D4 | 1 | 4 | yes | — |
| 101 | C4⋊C4 | — | 2 | 2 | no | — |
| 102 | C4⋊C4 | — | 2 | 2 | no | — |
| 103 | C4⋊C4 | — | 2 | 2 | no | — |
| 104 | C4⋊C4 | — | 2 | 2 | no | — |
| 105 | C4⋊C4 | — | 2 | 2 | no | — |
| 106 | C4⋊C4 | — | 2 | 2 | no | — |
| 107 | C4⋊C4 | — | 4 | 2 | no | — |
| 108 | C4⋊C4 | — | 4 | 2 | no | — |
| 109 | C22⋊C4 | — | 4 | 1 | no | — |
| 110 | C22⋊C4 | — | 4 | 2 | no | — |
| 111 | C22⋊C4 | — | 4 | 2 | no | — |
| 112 | C42 | — | 4 | 1 | no | — |
| 113 | C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 114 | C42 | C2×D4 | 1 | 2 | yes | — |
| 115 | C42 | C24 | 1 | 1 | yes | characteristic |
| 116 | C23 | — | 4 | 2 | no | — |
| 117 | C23 | #127 | 1 | 1 | yes | characteristic |
| 118 | Q8 | — | 4 | 4 | no | — |
| 119 | Q8 | — | 4 | 4 | no | — |
| 120 | Q8 | — | 4 | 4 | no | — |
| 121 | Q8 | — | 8 | 2 | no | — |
| 122 | D4 | — | 4 | 4 | no | — |
| 123 | D4 | — | 4 | 4 | no | — |
| 124 | D4 | — | 4 | 4 | no | — |
| 125 | D4 | — | 8 | 4 | no | — |
| 126 | C2×C4 | — | 2 | 2 | no | — |
| 127 | C2×C4 | — | 2 | 4 | no | — |
| 128 | C2×C4 | — | 4 | 2 | no | — |
| 129 | C2×C4 | — | 4 | 2 | no | — |
| 130 | C2×C4 | — | 4 | 2 | no | — |
| 131 | C2×C4 | — | 8 | 4 | no | — |
| 132 | C2×C4 | — | 8 | 2 | no | — |
| 133 | C2×C4 | #127 | 1 | 1 | yes | characteristic |
| 134 | C2×C4 | #127 | 1 | 2 | yes | — |
| 135 | C2×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 136 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 137 | C8 | — | 2 | 2 | no | — |
| 138 | C8 | — | 2 | 2 | no | — |
| 139 | C8 | — | 2 | 4 | no | — |
| 140 | C8 | — | 2 | 4 | no | — |
| 141 | C22 | — | 2 | 1 | no | — |
| 142 | C22 | — | 2 | 2 | no | — |
| 143 | C22 | — | 8 | 2 | no | — |
| 144 | C22 | — | 8 | 2 | no | — |
| 145 | C22 | — | 8 | 2 | no | — |
| 146 | C22 | C2×#127 | 1 | 1 | yes | characteristic |
| 147 | C4 | — | 2 | 1 | no | — |
| 148 | C4 | — | 2 | 2 | no | — |
| 149 | C4 | — | 8 | 2 | no | — |
| 150 | C4 | — | 8 | 2 | no | — |
| 151 | C4 | — | 8 | 2 | no | — |
| 152 | C4 | #575 | 1 | 2 | yes | — |
| 153 | C4 | #577 | 1 | 4 | yes | — |
| 154 | C2 | — | 4 | 1 | no | — |
| 155 | C2 | — | 16 | 2 | no | — |
| 156 | C2 | #3153 | 1 | 1 | yes | characteristic |
| 157 | C2 | #3154 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3184 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3227 | C2 | semidirect |
| 2. #3210 | C2 | semidirect |
| 4. #3139 | C2 | semidirect |
| 5. #1720 | C2 | semidirect |
| 6. #1717 | C2 | semidirect |
| 7. #1716 | C2 | semidirect |
| 8. #1708 | C2 | semidirect |
| 9. #1453 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3227 | 2 |
| #3210 | 2 |
| #3184 | 2 |
| #3139 | 2 |
| #1720 | 2 |
| #1717 | 2 |
| #1716 | 2 |
| #1708 | 2 |
| #1453 | 2 |