Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
979 conjugacy classes of subgroups; only the 481 normal subgroups are shown.
This group has 979 conjugacy classes of subgroups — too many to draw. Only its 481 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 | #3497 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#557 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #3392 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | #3359 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | #3206 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #3103 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #2982 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×2−1+4 | — | 2 | 1 | no | — |
| 9 | #557 | — | 2 | 4 | no | — |
| 10 | #557 | C22 | 1 | 8 | yes | — |
| 11 | #550 | C22 | 1 | 4 | yes | — |
| 12 | D4×Q8 | C22 | 1 | 4 | yes | — |
| 13 | #548 | — | 2 | 4 | no | — |
| 14 | #537 | — | 2 | 3 | no | — |
| 15 | #532 | C22 | 1 | 6 | yes | — |
| 16 | C2×C4⋊Q8 | C22 | 1 | 3 | yes | — |
| 17 | C2×#125 | — | 2 | 3 | no | — |
| 18 | C2×C4×Q8 | — | 2 | 1 | no | — |
| 19 | C2×C4×Q8 | C22 | 1 | 3 | yes | — |
| 20 | C2×C4×Q8 | C22 | 1 | 1 | yes | characteristic |
| 21 | #492 | C22 | 1 | 12 | yes | — |
| 22 | #487 | C22 | 1 | 12 | yes | — |
| 23 | #481 | C22 | 1 | 4 | yes | — |
| 24 | #475 | C22 | 1 | 12 | yes | — |
| 25 | #461 | C22 | 1 | 24 | yes | — |
| 26 | #460 | C22 | 1 | 12 | yes | — |
| 27 | #448 | C22 | 1 | 12 | yes | — |
| 28 | C8×Q8 | C22 | 1 | 8 | yes | — |
| 29 | C4×SD16 | C22 | 1 | 24 | yes | — |
| 30 | #431 | C22 | 1 | 6 | yes | — |
| 31 | 2−1+4 | — | 4 | 8 | no | — |
| 32 | C2×#41 | — | 2 | 4 | no | — |
| 33 | C2×#41 | — | 2 | 6 | no | — |
| 34 | C22×Q8 | — | 2 | 4 | no | — |
| 35 | C22×Q8 | D4 | 1 | 1 | yes | characteristic |
| 36 | C22×Q8 | C23 | 1 | 3 | yes | — |
| 37 | C2×SD16 | C23 | 1 | 24 | yes | — |
| 38 | C4⋊Q8 | — | 2 | 6 | no | — |
| 39 | C4⋊Q8 | C23 | 1 | 12 | yes | — |
| 40 | C4⋊Q8 | C23 | 1 | 6 | yes | — |
| 41 | #127 | C23 | 1 | 6 | yes | — |
| 42 | #125 | — | 2 | 24 | no | — |
| 43 | #124 | — | 2 | 6 | no | — |
| 44 | C22⋊Q8 | — | 2 | 12 | no | — |
| 45 | #121 | — | 2 | 6 | no | — |
| 46 | #121 | — | 2 | 12 | no | — |
| 47 | #121 | C23 | 1 | 4 | yes | — |
| 48 | C4×Q8 | — | 2 | 8 | no | — |
| 49 | C4×Q8 | — | 2 | 6 | no | — |
| 50 | C4×Q8 | D4 | 1 | 4 | yes | — |
| 51 | C4×Q8 | C23 | 1 | 12 | yes | — |
| 52 | C4×Q8 | C23 | 1 | 4 | yes | — |
| 53 | C4×D4 | — | 2 | 6 | no | — |
| 54 | C4×D4 | C23 | 1 | 12 | yes | — |
| 55 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 56 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 57 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 58 | C2×C4⋊C4 | D4 | 1 | 3 | yes | — |
| 59 | C2×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 60 | C2×C42 | — | 2 | 3 | no | — |
| 61 | C2×C42 | C23 | 1 | 3 | yes | — |
| 62 | #106 | C23 | 1 | 8 | yes | — |
| 63 | C4⋊C8 | C23 | 1 | 12 | yes | — |
| 64 | #103 | C23 | 1 | 24 | yes | — |
| 65 | #102 | C23 | 1 | 8 | yes | — |
| 66 | C22⋊C8 | C23 | 1 | 4 | yes | — |
| 67 | C4×C8 | C23 | 1 | 12 | yes | — |
| 68 | #41 | — | 4 | 16 | no | — |
| 69 | #41 | — | 4 | 24 | no | — |
| 70 | C2×Q8 | — | 2 | 3 | no | — |
| 71 | C2×Q8 | — | 2 | 6 | no | — |
| 72 | C2×Q8 | — | 4 | 6 | no | — |
| 73 | C2×Q8 | — | 4 | 16 | no | — |
| 74 | C2×Q8 | SD16 | 1 | 8 | yes | complemented |
| 75 | C2×Q8 | C2×D4 | 1 | 4 | yes | — |
| 76 | C2×Q8 | C24 | 1 | 6 | yes | — |
| 77 | C2×D4 | — | 2 | 4 | no | — |
| 78 | C2×D4 | — | 2 | 6 | no | — |
| 79 | C2×D4 | — | 2 | 6 | no | — |
| 80 | C2×D4 | C24 | 1 | 4 | yes | — |
| 81 | C22×C4 | — | 2 | 1 | no | — |
| 82 | C22×C4 | — | 2 | 3 | no | — |
| 83 | C22×C4 | — | 2 | 12 | no | — |
| 84 | C22×C4 | C2×D4 | 1 | 3 | yes | — |
| 85 | C22×C4 | C2×D4 | 1 | 3 | yes | — |
| 86 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 87 | SD16 | — | 4 | 24 | no | — |
| 88 | C2×C8 | C24 | 1 | 8 | yes | — |
| 89 | C4⋊C4 | — | 2 | 12 | no | — |
| 90 | C4⋊C4 | — | 2 | 12 | no | — |
| 91 | C4⋊C4 | — | 2 | 12 | no | — |
| 92 | C4⋊C4 | — | 2 | 1 | no | — |
| 93 | C4⋊C4 | — | 2 | 6 | no | — |
| 94 | C4⋊C4 | C2×D4 | 1 | 12 | yes | — |
| 95 | C4⋊C4 | C24 | 1 | 2 | yes | — |
| 96 | C4⋊C4 | C24 | 1 | 4 | yes | — |
| 97 | C22⋊C4 | — | 2 | 4 | no | — |
| 98 | C22⋊C4 | — | 2 | 12 | no | — |
| 99 | C42 | — | 2 | 12 | no | — |
| 100 | C42 | C2×D4 | 1 | 6 | yes | — |
| 101 | C42 | C24 | 1 | 6 | yes | — |
| 102 | C23 | — | 2 | 4 | no | — |
| 103 | C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 104 | Q8 | — | 2 | 12 | no | — |
| 105 | Q8 | — | 4 | 6 | no | — |
| 106 | Q8 | — | 4 | 12 | no | — |
| 107 | Q8 | C2×SD16 | 1 | 16 | yes | — |
| 108 | D4 | — | 2 | 8 | no | — |
| 109 | D4 | — | 4 | 8 | no | — |
| 110 | D4 | — | 4 | 12 | no | — |
| 111 | D4 | — | 8 | 6 | no | — |
| 112 | C2×C4 | — | 2 | 2 | no | — |
| 113 | C2×C4 | — | 2 | 2 | no | — |
| 114 | C2×C4 | — | 2 | 6 | no | — |
| 115 | C2×C4 | — | 2 | 4 | no | — |
| 116 | C2×C4 | — | 4 | 3 | no | — |
| 117 | C2×C4 | — | 4 | 2 | no | — |
| 118 | C2×C4 | — | 4 | 6 | no | — |
| 119 | C2×C4 | — | 4 | 24 | no | — |
| 120 | C2×C4 | C2×SD16 | 1 | 12 | yes | — |
| 121 | C2×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 122 | C2×C4 | C22×D4 | 1 | 6 | yes | — |
| 123 | C2×C4 | C22×D4 | 1 | 6 | yes | — |
| 124 | C2×C4 | C25 | 1 | 1 | yes | characteristic |
| 125 | C8 | — | 2 | 8 | no | — |
| 126 | C22 | — | 2 | 1 | no | — |
| 127 | C22 | — | 2 | 1 | no | — |
| 128 | C22 | — | 4 | 4 | no | — |
| 129 | C22 | — | 4 | 4 | no | — |
| 130 | C22 | — | 4 | 4 | no | — |
| 131 | C22 | C22×SD16 | 1 | 2 | yes | — |
| 132 | C22 | C23×D4 | 1 | 1 | yes | characteristic |
| 133 | C4 | — | 2 | 6 | no | — |
| 134 | C4 | — | 2 | 1 | no | — |
| 135 | C4 | — | 4 | 2 | no | — |
| 136 | C4 | — | 4 | 6 | no | — |
| 137 | C4 | — | 4 | 4 | no | — |
| 138 | C4 | C22×SD16 | 1 | 12 | yes | — |
| 139 | C4 | #585 | 1 | 2 | yes | — |
| 140 | C2 | — | 2 | 2 | no | — |
| 141 | C2 | — | 8 | 4 | no | — |
| 142 | C2 | #3487 | 1 | 1 | yes | characteristic |
| 143 | C2 | C23×SD16 | 1 | 1 | yes | characteristic |
| 144 | C2 | #3586 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3497 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#557 | C2 | semidirect |
| 3. #3392 | C2 | semidirect |
| 4. #3359 | C2 | semidirect |
| 5. #3206 | C2 | semidirect |
| 6. #3103 | C2 | semidirect |
| 7. #2982 | C2 | semidirect |
| 74. C2×Q8 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3497 | 2 |
| C2×#557 | 2 |
| #3392 | 2 |
| #3359 | 2 |
| #3206 | 2 |
| #3103 | 2 |
| #2982 | 2 |