Finite Group Databasebeta

← #45656042

Subgroups — #45656042

Order 32576.

268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 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#30654261C211yesmaximal, characteristic, complemented
2#30654260C211yesmaximal, characteristic, complemented
3#30654243C211yesmaximal, characteristic, complemented
4#30654242C212yesmaximal, complemented
5#30654239C212yesmaximal, complemented
6#30654236C212yesmaximal, complemented
7#30654235C211yesmaximal, characteristic, complemented
8C2×#22550846C211yesmaximal, characteristic, complemented
9D8×D509C212yesmaximal, complemented
10#30654223C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C509C211yesmaximal, characteristic, complemented
12#22550862C2212yescomplemented
13#22550861—21no—
14#22550861C2211yescharacteristic
15#22550861C2211yescharacteristic, complemented
16#22550860—21no—
17#22550860C2212yescomplemented
18C2×D2036—21no—
19C2×D2036C2211yescharacteristic, complemented
20C2×C22⋊D509—22no—
21D4×D509—21no—
22D4×D509—22no—
23D4×D509C2212yescomplemented
24D4×D509C2212yescomplemented
25#22550851C2211yescharacteristic, complemented
26#22550850C2212yescomplemented
27#22550849C2211yescharacteristic, complemented
28#22550848C2211yescharacteristic, complemented
29D4072C2212yescomplemented
30#22550846C2211yescharacteristic, complemented
31#22550846C2212yescomplemented
32#22550846C2212yescomplemented
33#22550844C2212yescomplemented
34C2×#16861325C2211yescharacteristic, complemented
35C8×D509C2212yescomplemented
36#41×C509C2211yescharacteristic, complemented
37C2×D4×C509C2211yescharacteristic, complemented
38SD16×C509C2212yescomplemented
39D8×C509C2212yescomplemented
40M16×C509C2211yescharacteristic, complemented
41C22×D509—21no—
42C22×D509—22no—
43C22×D509—22no—
44Dic1018D411yescharacteristic
45Dic1018C2311yescharacteristic
46D2036—22no—
47D2036—21no—
48D2036D411yescharacteristic
49D2036C2312yes—
50D2036C2311yescharacteristic
51C22⋊D509—21no—
52C22⋊D509—22no—
53C22⋊D509—22no—
54C22⋊D509—22no—
55C22⋊D509D412yes—
56C2×Dic509—22no—
57C4×D509—21no—
58C4×D509—22no—
59C4×D509C2312yes—
60#16861325C2311yescharacteristic
61#16861325C2311yescharacteristic
62C23×C509—21no—
63Q8×C509C2311yescharacteristic
64D4×C509—21no—
65D4×C509—21no—
66D4×C509C2311yescharacteristic
67D4×C509C2312yes—
68C2×C4×C509—21no—
69C2×C4×C509C2311yescharacteristic
70C8×C509C2312yes—
71C22×C509—21no—
72C22×C509—22no—
73C22×C509—81no—
74C22×C509C2×D411yescharacteristic
75C2×D509—21no—
76C2×D509—22no—
77C2×D509—82no—
78C2×D509—82no—
79C2×D509—81no—
80C2×D509C2×D412yes—
81C4×C509—21no—
82C4×C509C2×D411yescharacteristic
83C4×C509C2411yescharacteristic
84Dic509—21no—
85Dic509C2×D412yes—
86C2×C509—21no—
87C2×C509—41no—
88C2×C509—42no—
89C2×C509C22×D411yescharacteristic
90D509—22no—
91D509—41no—
92D509—42no—
93C509#57611yescharacteristic, complemented, Sylow
94#576—5091nomaximal, Sylow
95H(5,𝔽2)—5091no—
96H(5,𝔽2)—5091no—
97AGL(1,ℤ/8)—5092no—
98AGL(1,ℤ/8)—5092no—
99AGL(1,ℤ/8)—5091no—
100AGL(1,ℤ/8)D50911yescharacteristic, complemented
101#135—5091no—
102#135—5092no—
103C2×D8—5091no—
104C2×D8—5092no—
105#131—5091no—
106#41—10181no—
107#41—10181no—
108#41—5092no—
109#41—5092no—
110#41—5091no—
111#41—5091no—
112#41C2×D50911yescharacteristic, complemented
113C2×D4—10181no—
114C2×D4—10182no—
115C2×D4—10181no—
116C2×D4—10182no—
117C2×D4—5092no—
118C2×D4—5092no—
119C2×D4—5091no—
120C2×D4C2×D50911yescharacteristic, complemented
121Q16—5091no—
122SD16—5091no—
123SD16—5091no—
124SD16—5092no—
125SD16C2×D50912yescomplemented
126D8—5092no—
127D8—5091no—
128D8—5092no—
129D8—5092no—
130D8C2×D50912yescomplemented
131M16—5092no—
132M16C2×D50911yescharacteristic, complemented
133C2×C8—5091no—
134C2×C8—5092no—
135C23—21no—
136C23—10182no—
137C23—10182no—
138C23—10181no—
139Q8—5091no—
140Q8—5091no—
141Q8C22×D50911yescharacteristic
142D4—21no—
143D4—21no—
144D4—10182no—
145D4—10182no—
146D4—10181no—
147D4—10182no—
148D4—10182no—
149D4—10181no—
150D4—5092no—
151D4—5091no—
152D4—5092no—
153D4—5091no—
154D4C22×D50912yes—
155D4C22×D50911yescharacteristic
156C2×C4—21no—
157C2×C4—10182no—
158C2×C4—10182no—
159C2×C4—10181no—
160C2×C4—5092no—
161C2×C4C22×D50911yescharacteristic
162C8—5091no—
163C8—5091no—
164C8C22×D50912yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—40721no—
169C22—40722no—
170C22—40722no—
171C22—10182no—
172C22—10181no—
173C22—5092no—
174C22D4×D50911yescharacteristic
175C4—21no—
176C4—10181no—
177C4—5092no—
178C4D4×D50911yescharacteristic
179C4C23×D50911yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—20362no—
184C2—20361no—
185C2—10182no—
186C2C2×D4×D50911yescharacteristic
subgroupG/H
centerC2C2×D4×D509
commutator, cosocleC4×C509C24
FrattiniC4C23×D509
FittingAGL(1,ℤ/8)×C509C2
socleC2×C509C22×D4
hypercenterAGL(1,ℤ/8)D509
hypocenterC509#576
LCSG ▷ C4×C509 ▷ C2×C509 ▷ C509
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #30654261C2semidirect
2. #30654260C2semidirect
3. #30654243C2semidirect
4. #30654242C2semidirect
5. #30654239C2semidirect
6. #30654236C2semidirect
7. #30654235C2semidirect
8. C2×#22550846C2semidirect
9. D8×D509C2semidirect
10. #30654223C2semidirect
11. AGL(1,ℤ/8)×C509C2semidirect
12. #22550862C22semidirect
15. #22550861C22semidirect
17. #22550860C22semidirect
19. C2×D2036C22semidirect
23. D4×D509C22semidirect
24. D4×D509C22semidirect
25. #22550851C22semidirect
26. #22550850C22semidirect
27. #22550849C22semidirect
28. #22550848C22semidirect
29. D4072C22semidirect
30. #22550846C22semidirect
31. #22550846C22semidirect
32. #22550846C22semidirect
33. #22550844C22semidirect
34. C2×#16861325C22semidirect
35. C8×D509C22semidirect
36. #41×C509C22semidirect
37. C2×D4×C509C22semidirect
38. SD16×C509C22semidirect
39. D8×C509C22semidirect
40. M16×C509C22semidirect
93. C509#576semidirect
100. AGL(1,ℤ/8)D509semidirect
112. #41C2×D509semidirect
120. C2×D4C2×D509semidirect
125. SD16C2×D509semidirect
130. D8C2×D509semidirect
132. M16C2×D509semidirect

Maximal subgroups

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

subgroupindex
#306542612
#306542602
#306542432
#306542422
#306542392
#306542362
#306542352
C2×#225508462
D8×D5092
#306542232
AGL(1,ℤ/8)×C5092
#576509