Finite Group Databasebeta

← #63092

Subgroups — #63092

Order 256.

5398 conjugacy classes of subgroups; only the 3133 normal subgroups are shown.

normal subgroups only

This group has 5398 conjugacy classes of subgroups — too many to draw. Only its 3133 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
12−1+6C211yesmaximal, characteristic, complemented
2H(7,𝔽2)C211yesmaximal, characteristic, complemented
3#3586C2115yesmaximal, complemented
4#3585C2115yesmaximal, complemented
5#3584C2115yesmaximal, complemented
6C2×#577C2115yesmaximal, complemented
7#3573C211yesmaximal, characteristic, complemented
8#585—26no—
9#585—210no—
10#585C2211yescharacteristic
11#585C22115yescomplemented
12#585C22115yescomplemented
13C2×2−1+4—26no—
14C2×2−1+4C22115yescomplemented
15C2×H(5,𝔽2)—210no—
16C2×H(5,𝔽2)C22115yescomplemented
17#578C22160yescomplemented
18#577C22160yescomplemented
19#577C221120yescomplemented
20#577C22120yescomplemented
21#576C22160yescomplemented
22#575C22190yescomplemented
23C2×#137C22145yescomplemented
24C2×AGL(1,ℤ/8)C22145yescomplemented
25C2×#135C22145yescomplemented
26C22×SD16C22115yescomplemented
27#568C22115yescomplemented
28C2×#131C22115yescomplemented
292−1+4—260no—
302−1+4—215no—
312−1+4—215no—
322−1+4D416yes—
332−1+4C23120yes—
342−1+4C23160yescomplemented
35H(5,𝔽2)—260no—
36H(5,𝔽2)—245no—
37H(5,𝔽2)—245no—
38H(5,𝔽2)D4110yes—
39H(5,𝔽2)C23160yes—
40H(5,𝔽2)C23120yescomplemented
41C2×#41—260no—
42C2×#41—260no—
43C2×#41—215no—
44C2×#41—245no—
45C2×#41C23145yes—
46C2×#41C23115yes—
47C2×#41C23145yescomplemented
48C22×Q8—215no—
49C22×Q8C23115yescomplemented
50C22×D4—245no—
51C22×D4C23115yes—
52#137C231180yescomplemented
53#137C23160yescomplemented
54AGL(1,ℤ/8)C231180yescomplemented
55AGL(1,ℤ/8)C23160yescomplemented
56#135C23160yescomplemented
57#135C231180yescomplemented
58C2×Q16C23145yescomplemented
59C2×SD16C23145yescomplemented
60C2×SD16C23190yescomplemented
61C2×SD16C23115yescomplemented
62C2×D8C23145yescomplemented
63#131C23120yescomplemented
64#131C23160yescomplemented
65C2×M16C23145yescomplemented
66C22×C8C23115yescomplemented
67C24—215no—
68#41—260no—
69#41—220no—
70#41—260no—
71#41—2180no—
72#41—290no—
73#41—290no—
74#41C2×D4160yes—
75#41C24160yes—
76#41C24160yes—
77#41C24120yes—
78C2×Q8—260no—
79C2×Q8—220no—
80C2×Q8—245no—
81C2×Q8C2×D4115yes—
82C2×Q8C24145yes—
83C2×Q8C24115yes—
84C2×D4—245no—
85C2×D4—245no—
86C2×D4—2180no—
87C2×D4—260no—
88C2×D4—245no—
89C2×D4C2×D4145yes—
90C2×D4C24115yes—
91C2×D4C24145yes—
92C22×C4—215no—
93C22×C4—245no—
94C22×C4—245no—
95C22×C4C24115yes—
96Q16C24160yes—
97SD16C24110yes—
98SD16C2416yes—
99SD16C24190yes—
100SD16C24130yes—
101D8C24160yes—
102M16C24160yes—
103C2×C8C24115yes—
104C2×C8C24145yes—
105C23—215no—
106C23—245no—
107C23—1615no—
108C23C22×D4115yes—
109Q8—245no—
110Q8—215no—
111Q8C22×D4120yes—
112Q8C25110yes—
113Q8C2516yes—
114D4—260no—
115D4—260no—
116D4—245no—
117D4—215no—
118D4C22×D4160yes—
119D4C2516yes—
120D4C25110yes—
121C2×C4—215no—
122C2×C4—245no—
123C2×C4—260no—
124C2×C4—260no—
125C2×C4C22×D4145yes—
126C2×C4C25115yes—
127C8C2511yescharacteristic
128C8C25115yes—
129C22—26no—
130C22—210no—
131C22—415no—
132C22—815no—
133C22—845no—
134C22C23×D4115yes—
135C4—26no—
136C4—210no—
137C4C23×D4115yes—
138C4C2611yescharacteristic
139C2—215no—
140C2—46no—
141C2—410no—
142C2C24×D411yescharacteristic
subgroupG/H
center, socleC2C24×D4
commutator, Frattini, cosocleC4C26
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ #585 ◁ G
LECSG ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. 2−1+6C2semidirect
2. H(7,𝔽2)C2semidirect
3. #3586C2semidirect
4. #3585C2semidirect
5. #3584C2semidirect
6. C2×#577C2semidirect
7. #3573C2semidirect
11. #585C22semidirect
12. #585C22semidirect
14. C2×2−1+4C22semidirect
16. C2×H(5,𝔽2)C22semidirect
17. #578C22semidirect
18. #577C22semidirect
19. #577C22semidirect
20. #577C22semidirect
21. #576C22semidirect
22. #575C22semidirect
23. C2×#137C22semidirect
24. C2×AGL(1,ℤ/8)C22semidirect
25. C2×#135C22semidirect
26. C22×SD16C22semidirect
27. #568C22semidirect
28. C2×#131C22semidirect
34. 2−1+4C23semidirect
40. H(5,𝔽2)C23semidirect
47. C2×#41C23semidirect
49. C22×Q8C23semidirect
52. #137C23semidirect
53. #137C23semidirect
54. AGL(1,ℤ/8)C23semidirect
55. AGL(1,ℤ/8)C23semidirect
56. #135C23semidirect
57. #135C23semidirect
58. C2×Q16C23semidirect
59. C2×SD16C23semidirect
60. C2×SD16C23semidirect
61. C2×SD16C23semidirect
62. C2×D8C23semidirect
63. #131C23semidirect
64. #131C23semidirect
65. C2×M16C23semidirect
66. C22×C8C23semidirect

Maximal subgroups

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

subgroupindex
2−1+62
H(7,𝔽2)2
#35862
#35852
#35842
C2×#5772
#35732