Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
463 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 463 conjugacy classes of subgroups — too many to draw. Only its 135 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 | C2×#441 | C2 | 1 | 2 | yes | maximal |
| 2 | C2×C4×Q16 | C2 | 1 | 2 | yes | maximal |
| 3 | #2327 | C2 | 1 | 2 | yes | maximal |
| 4 | #2097 | C2 | 1 | 4 | yes | maximal |
| 5 | #2016 | C2 | 1 | 4 | yes | maximal |
| 6 | #1746 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C22×Q16 | C22 | 1 | 2 | yes | — |
| 8 | C22×Q16 | C22 | 1 | 2 | yes | — |
| 9 | C2×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 10 | C2×C4×Q8 | C22 | 1 | 4 | yes | — |
| 11 | #441 | — | 2 | 8 | no | — |
| 12 | C4×Q16 | — | 2 | 8 | no | — |
| 13 | C2×#107 | C22 | 1 | 2 | yes | — |
| 14 | C2×#106 | C22 | 1 | 2 | yes | — |
| 15 | C2×#103 | C22 | 1 | 4 | yes | — |
| 16 | C2×#103 | C22 | 1 | 4 | yes | — |
| 17 | C2×#97 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 19 | #391 | C22 | 1 | 4 | yes | — |
| 20 | #387 | — | 2 | 8 | no | — |
| 21 | #382 | C22 | 1 | 2 | yes | — |
| 22 | #376 | C22 | 1 | 1 | yes | characteristic |
| 23 | #336 | C22 | 1 | 4 | yes | — |
| 24 | C22×Q8 | C23 | 1 | 4 | yes | — |
| 25 | C2×Q16 | — | 2 | 8 | no | — |
| 26 | C2×Q16 | — | 2 | 8 | no | — |
| 27 | C2×Q16 | — | 2 | 4 | no | — |
| 28 | C2×Q16 | — | 2 | 4 | no | — |
| 29 | C22×C8 | C23 | 1 | 2 | yes | — |
| 30 | C22×C8 | C23 | 1 | 2 | yes | — |
| 31 | C4⋊Q8 | — | 2 | 8 | no | — |
| 32 | C4×Q8 | — | 2 | 8 | no | — |
| 33 | C4×Q8 | — | 2 | 8 | no | — |
| 34 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 35 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 36 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 37 | C2×C42 | — | 2 | 4 | no | — |
| 38 | C2×C42 | C23 | 1 | 2 | yes | — |
| 39 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 40 | #107 | — | 2 | 4 | no | — |
| 41 | #106 | — | 2 | 4 | no | — |
| 42 | #103 | — | 2 | 8 | no | — |
| 43 | #103 | — | 2 | 8 | no | — |
| 44 | #97 | D4 | 1 | 4 | yes | — |
| 45 | C4×C8 | D4 | 1 | 4 | yes | — |
| 46 | #95 | — | 2 | 4 | no | — |
| 47 | #95 | — | 2 | 8 | no | — |
| 48 | #95 | D4 | 1 | 4 | yes | — |
| 49 | C2×Q8 | — | 2 | 4 | no | — |
| 50 | C2×Q8 | — | 2 | 4 | no | — |
| 51 | C2×Q8 | — | 2 | 4 | no | — |
| 52 | C2×Q8 | — | 2 | 4 | no | — |
| 53 | C2×Q8 | — | 2 | 4 | no | — |
| 54 | C2×Q8 | #41 | 1 | 8 | yes | — |
| 55 | C22×C4 | — | 2 | 4 | no | — |
| 56 | C22×C4 | — | 2 | 4 | no | — |
| 57 | C22×C4 | C2×D4 | 1 | 4 | yes | — |
| 58 | C22×C4 | C2×D4 | 1 | 2 | yes | — |
| 59 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 60 | Q16 | — | 2 | 16 | no | — |
| 61 | Q16 | — | 2 | 16 | no | — |
| 62 | C2×C8 | — | 2 | 4 | no | — |
| 63 | C2×C8 | — | 2 | 4 | no | — |
| 64 | C2×C8 | C2×D4 | 1 | 4 | yes | — |
| 65 | C2×C8 | C2×D4 | 1 | 4 | yes | — |
| 66 | C4⋊C4 | — | 2 | 16 | no | — |
| 67 | C4⋊C4 | — | 2 | 4 | no | — |
| 68 | C4⋊C4 | — | 2 | 4 | no | — |
| 69 | C4⋊C4 | — | 4 | 8 | no | — |
| 70 | C42 | — | 2 | 4 | no | — |
| 71 | C42 | — | 4 | 8 | no | — |
| 72 | C42 | C2×D4 | 1 | 2 | yes | — |
| 73 | C42 | C2×D4 | 1 | 2 | yes | — |
| 74 | C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 75 | Q8 | — | 2 | 8 | no | — |
| 76 | Q8 | — | 2 | 8 | no | — |
| 77 | Q8 | — | 4 | 4 | no | — |
| 78 | Q8 | — | 4 | 4 | no | — |
| 79 | C2×C4 | — | 2 | 4 | no | — |
| 80 | C2×C4 | — | 2 | 4 | no | — |
| 81 | C2×C4 | — | 2 | 8 | no | — |
| 82 | C2×C4 | — | 2 | 8 | no | — |
| 83 | C2×C4 | — | 4 | 4 | no | — |
| 84 | C2×C4 | — | 4 | 4 | no | — |
| 85 | C2×C4 | — | 4 | 4 | no | — |
| 86 | C2×C4 | — | 4 | 8 | no | — |
| 87 | C2×C4 | #127 | 1 | 4 | yes | — |
| 88 | C2×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 89 | C2×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 90 | C2×C4 | C2×#41 | 1 | 4 | yes | — |
| 91 | C8 | — | 2 | 4 | no | — |
| 92 | C8 | — | 2 | 4 | no | — |
| 93 | C22 | #525 | 1 | 2 | yes | — |
| 94 | C22 | C2×#127 | 1 | 1 | yes | characteristic |
| 95 | C22 | #578 | 1 | 4 | yes | — |
| 96 | C4 | — | 2 | 4 | no | — |
| 97 | C4 | — | 4 | 4 | no | — |
| 98 | C4 | — | 4 | 8 | no | — |
| 99 | C4 | — | 8 | 4 | no | — |
| 100 | C4 | #525 | 1 | 2 | yes | — |
| 101 | C4 | #525 | 1 | 2 | yes | — |
| 102 | C2 | #2397 | 1 | 1 | yes | characteristic |
| 103 | C2 | #3127 | 1 | 4 | yes | — |
| 104 | C2 | #3156 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#441 | 2 |
| C2×C4×Q16 | 2 |
| #2327 | 2 |
| #2097 | 2 |
| #2016 | 2 |
| #1746 | 2 |