Finite Group Databasebeta

← #62635

Subgroups — #62635

Order 256.

1019 conjugacy classes of subgroups; only the 477 normal subgroups are shown.

normal subgroups only

This group has 1019 conjugacy classes of subgroups — too many to draw. Only its 477 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3584C212yesmaximal, complemented
2#3417C214yesmaximal, complemented
3#3416C218yesmaximal, complemented
4#3415C214yesmaximal, complemented
5#3414C216yesmaximal, complemented
6C2×#508C216yesmaximal, complemented
7C2×#504C211yesmaximal, characteristic, complemented
8#585—22no—
9#578C2218yescomplemented
10#577—28no—
11#576C2218yescomplemented
12#575—26no—
13C2×#135C22112yescomplemented
14C2×#131C2211yescharacteristic
15#510C22124yescomplemented
16#509C22124yescomplemented
17#508C2214yescomplemented
18#508C22112yescomplemented
19#508C22112yescomplemented
20#508C22112yescomplemented
21C2×Q32C2216yescomplemented
22C2×SD32C22112yescomplemented
23C2×D16C2216yescomplemented
24#504C2214yescomplemented
25#504C2214yescomplemented
26C2×M32C2213yescomplemented
27C22×C16C2213yescomplemented
282−1+4—28no—
292−1+4—42no—
30H(5,𝔽2)—28no—
31H(5,𝔽2)—46no—
32C2×#41—212no—
33C2×#41—48no—
34C2×#41D411yescharacteristic
35#137—224no—
36AGL(1,ℤ/8)—224no—
37#135—212no—
38#135—212no—
39#135C2318yes—
40#135C23124yes—
41C2×Q16C23112yes—
42C2×SD16—212no—
43C2×D8C23112yes—
44#131D414yes—
45#131C2314yes—
46C2×M16D413yes—
47C22×C8C2313yes—
48Q32C2314yes—
49Q32C23112yes—
50SD32C2318yes—
51SD32C23124yes—
52D16C2314yes—
53D16C23112yes—
54M32C23112yes—
55C2×C16C2311yescharacteristic
56C2×C16C2313yes—
57C2×C16C2316yes—
58C2×C16C2316yes—
59#41—28no—
60#41—224no—
61#41—48no—
62#41—424no—
63#41—412no—
64#41—412no—
65#41D814yes—
66#41C2×D414yes—
67C2×Q8—212no—
68C2×Q8—48no—
69C2×Q8D811yescharacteristic
70C2×D4—212no—
71C2×D4—412no—
72C2×D4—424no—
73C2×D4D813yes—
74C22×C4—412no—
75C22×C4C2×D413yes—
76Q16—212no—
77Q16C2418yes—
78SD16—28no—
79SD16—224no—
80D8—212no—
81D8C2418yes—
82M16C2×D4112yes—
83C2×C8C2×D413yes—
84C2×C8C2×D416yes—
85C2×C8C2411yescharacteristic
86C2×C8C2416yes—
87C16C2411yescharacteristic
88C16C2411yescharacteristic
89C16C2416yes—
90C23—412no—
91C23C2×D813yes—
92Q8—28no—
93Q8—412no—
94Q8C2×D814yes—
95D4—28no—
96D4—424no—
97D4—48no—
98D4—412no—
99D4C2×D8112yes—
100C2×C4—412no—
101C2×C4—424no—
102C2×C4—48no—
103C2×C4C2×D813yes—
104C2×C4C2×D816yes—
105C2×C4C22×D416yes—
106C2×C4C22×D411yescharacteristic
107C8C22×D411yescharacteristic
108C8C22×D416yes—
109C8C2511yescharacteristic
110C22—43no—
111C22—48no—
112C22—1612no—
113C22C22×D816yes—
114C22C22×D811yescharacteristic
115C4—48no—
116C4C22×D816yes—
117C4C22×D811yescharacteristic
118C4C23×D411yescharacteristic
119C2—21no—
120C2—26no—
121C2—88no—
122C2C23×D811yescharacteristic
subgroupG/H
centerC4C22×D8
commutator, Frattini, cosocleC8C25
socleC2C23×D8
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×#41 ◁ C2×#131 ◁ G
LECSG ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3584C2semidirect
2. #3417C2semidirect
3. #3416C2semidirect
4. #3415C2semidirect
5. #3414C2semidirect
6. C2×#508C2semidirect
7. C2×#504C2semidirect
9. #578C22semidirect
11. #576C22semidirect
13. C2×#135C22semidirect
15. #510C22semidirect
16. #509C22semidirect
17. #508C22semidirect
18. #508C22semidirect
19. #508C22semidirect
20. #508C22semidirect
21. C2×Q32C22semidirect
22. C2×SD32C22semidirect
23. C2×D16C22semidirect
24. #504C22semidirect
25. #504C22semidirect
26. C2×M32C22semidirect
27. C22×C16C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#35842
#34172
#34162
#34152
#34142
C2×#5082
C2×#5042