Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
667 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 667 conjugacy classes of subgroups — too many to draw. Only its 145 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 | #3573 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #3058 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #3027 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #1910 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #1812 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #585 | C22 | 1 | 1 | yes | characteristic |
| 7 | C22×D8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | #568 | — | 2 | 6 | no | — |
| 9 | #568 | C22 | 1 | 3 | yes | — |
| 10 | C2×#131 | — | 2 | 6 | no | — |
| 11 | C2×#131 | C22 | 1 | 3 | yes | — |
| 12 | #535 | C22 | 1 | 6 | yes | — |
| 13 | #469 | — | 2 | 6 | no | — |
| 14 | #466 | — | 2 | 6 | no | — |
| 15 | B(4,𝔽2) | — | 2 | 8 | no | — |
| 16 | #427 | C22 | 1 | 3 | yes | complemented |
| 17 | #418 | C22 | 1 | 6 | yes | complemented |
| 18 | #410 | C22 | 1 | 6 | yes | — |
| 19 | H(3,ℤ/4) | C22 | 1 | 6 | yes | complemented |
| 20 | 2−1+4 | — | 2 | 3 | no | — |
| 21 | H(5,𝔽2) | — | 2 | 3 | no | — |
| 22 | H(5,𝔽2) | D4 | 1 | 4 | yes | — |
| 23 | C2×#41 | — | 2 | 6 | no | — |
| 24 | C2×#41 | C23 | 1 | 3 | yes | — |
| 25 | C22×D4 | C23 | 1 | 2 | yes | — |
| 26 | C2×D8 | — | 2 | 3 | no | — |
| 27 | C2×D8 | — | 2 | 3 | no | — |
| 28 | #131 | — | 2 | 4 | no | — |
| 29 | #131 | — | 2 | 12 | no | — |
| 30 | #131 | — | 2 | 6 | no | — |
| 31 | #131 | — | 2 | 12 | no | — |
| 32 | #131 | — | 2 | 6 | no | — |
| 33 | C2×M16 | — | 2 | 12 | no | — |
| 34 | C2×M16 | — | 2 | 6 | no | — |
| 35 | C2×M16 | D4 | 1 | 6 | yes | complemented |
| 36 | C2×M16 | C23 | 1 | 3 | yes | — |
| 37 | C22×C8 | — | 2 | 4 | no | — |
| 38 | C22×C8 | D4 | 1 | 6 | yes | complemented |
| 39 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 40 | #127 | — | 2 | 6 | no | — |
| 41 | #124 | — | 2 | 6 | no | — |
| 42 | #121 | — | 2 | 12 | no | — |
| 43 | #120 | — | 2 | 12 | no | — |
| 44 | #117 | C23 | 1 | 6 | yes | — |
| 45 | #107 | — | 2 | 3 | no | — |
| 46 | #106 | — | 2 | 3 | no | — |
| 47 | #102 | — | 2 | 12 | no | — |
| 48 | #99 | — | 2 | 12 | no | — |
| 49 | C24 | — | 2 | 2 | no | — |
| 50 | #41 | — | 2 | 4 | no | — |
| 51 | #41 | — | 2 | 12 | no | — |
| 52 | #41 | — | 2 | 12 | no | — |
| 53 | #41 | — | 2 | 6 | no | — |
| 54 | #41 | — | 2 | 6 | no | — |
| 55 | C2×Q8 | — | 2 | 6 | no | — |
| 56 | C2×Q8 | #41 | 1 | 3 | yes | — |
| 57 | C2×D4 | — | 2 | 6 | no | — |
| 58 | C2×D4 | — | 2 | 12 | no | — |
| 59 | C2×D4 | — | 2 | 6 | no | — |
| 60 | C2×D4 | — | 2 | 6 | no | — |
| 61 | C2×D4 | — | 4 | 12 | no | — |
| 62 | C2×D4 | C2×D4 | 1 | 6 | yes | — |
| 63 | C2×D4 | #41 | 1 | 3 | yes | — |
| 64 | C22×C4 | — | 2 | 4 | no | — |
| 65 | C22×C4 | C2×D4 | 1 | 6 | yes | — |
| 66 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 67 | D8 | — | 4 | 1 | no | — |
| 68 | D8 | — | 4 | 3 | no | — |
| 69 | M16 | — | 2 | 12 | no | — |
| 70 | M16 | — | 2 | 6 | no | — |
| 71 | M16 | — | 2 | 12 | no | — |
| 72 | C2×C8 | — | 2 | 12 | no | — |
| 73 | C2×C8 | — | 2 | 6 | no | — |
| 74 | C2×C8 | — | 2 | 6 | no | — |
| 75 | C2×C8 | C2×D4 | 1 | 6 | yes | — |
| 76 | C2×C8 | C2×D4 | 1 | 3 | yes | — |
| 77 | C2×C8 | C2×D4 | 1 | 3 | yes | — |
| 78 | C4⋊C4 | — | 2 | 6 | no | — |
| 79 | C22⋊C4 | — | 2 | 6 | no | — |
| 80 | C22⋊C4 | — | 4 | 12 | no | — |
| 81 | C42 | — | 2 | 6 | no | — |
| 82 | C23 | — | 2 | 4 | no | — |
| 83 | C23 | — | 4 | 6 | no | — |
| 84 | C23 | — | 16 | 2 | no | — |
| 85 | C23 | #121 | 1 | 6 | yes | — |
| 86 | C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 87 | Q8 | — | 2 | 4 | no | — |
| 88 | Q8 | — | 2 | 6 | no | — |
| 89 | D4 | — | 2 | 12 | no | — |
| 90 | D4 | — | 2 | 12 | no | — |
| 91 | D4 | — | 2 | 6 | no | — |
| 92 | D4 | — | 4 | 2 | no | — |
| 93 | D4 | — | 4 | 6 | no | — |
| 94 | D4 | — | 16 | 12 | no | — |
| 95 | C2×C4 | — | 2 | 12 | no | — |
| 96 | C2×C4 | — | 2 | 6 | no | — |
| 97 | C2×C4 | — | 2 | 6 | no | — |
| 98 | C2×C4 | — | 4 | 6 | no | — |
| 99 | C2×C4 | #121 | 1 | 6 | yes | — |
| 100 | C2×C4 | C22×D4 | 1 | 3 | yes | — |
| 101 | C2×C4 | C2×#41 | 1 | 3 | yes | — |
| 102 | C8 | — | 2 | 6 | no | — |
| 103 | C8 | #120 | 1 | 1 | yes | characteristic |
| 104 | C8 | #120 | 1 | 3 | yes | — |
| 105 | C22 | — | 2 | 6 | no | — |
| 106 | C22 | — | 4 | 1 | no | — |
| 107 | C22 | — | 4 | 6 | no | — |
| 108 | C22 | — | 8 | 12 | no | — |
| 109 | C22 | — | 8 | 4 | no | — |
| 110 | C22 | — | 8 | 2 | no | — |
| 111 | C22 | C2×#121 | 1 | 3 | yes | — |
| 112 | C4 | — | 2 | 6 | no | — |
| 113 | C4 | — | 8 | 6 | no | — |
| 114 | C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 115 | C4 | #525 | 1 | 3 | yes | — |
| 116 | C2 | — | 2 | 3 | no | — |
| 117 | C2 | — | 4 | 6 | no | — |
| 118 | C2 | — | 16 | 2 | no | — |
| 119 | C2 | #2847 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3573 | C2 | semidirect |
| 2. #3058 | C2 | semidirect |
| 3. #3027 | C2 | semidirect |
| 4. #1910 | C2 | semidirect |
| 5. #1812 | C2 | semidirect |
| 7. C22×D8 | C22 | semidirect |
| 16. #427 | C22 | semidirect |
| 17. #418 | C22 | semidirect |
| 19. H(3,ℤ/4) | C22 | semidirect |
| 35. C2×M16 | D4 | semidirect |
| 38. C22×C8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3573 | 2 |
| #3058 | 2 |
| #3027 | 2 |
| #1910 | 2 |
| #1812 | 2 |