Finite Group Databasebeta

← #39910051

Subgroups — #39910051

Order 26176.

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#27189763C211yesmaximal, characteristic, complemented
2#27189762C211yesmaximal, characteristic, complemented
3#27189745C211yesmaximal, characteristic, complemented
4#27189744C212yesmaximal, complemented
5#27189741C212yesmaximal, complemented
6#27189738C212yesmaximal, complemented
7#27189737C211yesmaximal, characteristic, complemented
8C2×#20266973C211yesmaximal, characteristic, complemented
9D8×D409C212yesmaximal, complemented
10#27189725C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C409C211yesmaximal, characteristic, complemented
12#20266989C2212yescomplemented
13#20266988—21no—
14#20266988C2211yescharacteristic
15#20266988C2211yescharacteristic, complemented
16#20266987—21no—
17#20266987C2212yescomplemented
18C2×D1636—21no—
19C2×D1636C2211yescharacteristic, complemented
20C2×C22⋊D409—22no—
21D4×D409—21no—
22D4×D409—22no—
23D4×D409C2212yescomplemented
24D4×D409C2212yescomplemented
25#20266978C2211yescharacteristic, complemented
26#20266977C2212yescomplemented
27#20266976C2211yescharacteristic, complemented
28#20266975C2211yescharacteristic, complemented
29D3272C2212yescomplemented
30#20266973C2211yescharacteristic, complemented
31#20266973C2212yescomplemented
32#20266973C2212yescomplemented
33#20266971C2212yescomplemented
34C2×#15692543C2211yescharacteristic, complemented
35C8×D409C2212yescomplemented
36#41×C409C2211yescharacteristic, complemented
37C2×D4×C409C2211yescharacteristic, complemented
38SD16×C409C2212yescomplemented
39D8×C409C2212yescomplemented
40M16×C409C2211yescharacteristic, complemented
41C22×D409—21no—
42C22×D409—22no—
43C22×D409—22no—
44Dic818D411yescharacteristic
45Dic818C2311yescharacteristic
46D1636—22no—
47D1636—21no—
48D1636D411yescharacteristic
49D1636C2312yes—
50D1636C2311yescharacteristic
51C22⋊D409—21no—
52C22⋊D409—22no—
53C22⋊D409—22no—
54C22⋊D409—22no—
55C22⋊D409D412yes—
56C2×Dic409—22no—
57C4×D409—21no—
58C4×D409—22no—
59C4×D409C2312yes—
60#15692543C2311yescharacteristic
61#15692543C2311yescharacteristic
62C23×C409—21no—
63Q8×C409C2311yescharacteristic
64D4×C409—21no—
65D4×C409—21no—
66D4×C409C2311yescharacteristic
67D4×C409C2312yes—
68C2×C4×C409—21no—
69C2×C4×C409C2311yescharacteristic
70C8×C409C2312yes—
71C22×C409—22no—
72C22×C409—21no—
73C22×C409—81no—
74C22×C409C2×D411yescharacteristic
75C2×D409—21no—
76C2×D409—22no—
77C2×D409—82no—
78C2×D409—82no—
79C2×D409—81no—
80C2×D409C2×D412yes—
81C4×C409—21no—
82C4×C409C2×D411yescharacteristic
83C4×C409C2411yescharacteristic
84Dic409—21no—
85Dic409C2×D412yes—
86C2×C409—21no—
87C2×C409—41no—
88C2×C409—42no—
89C2×C409C22×D411yescharacteristic
90D409—22no—
91D409—41no—
92D409—42no—
93C409#57611yescharacteristic, complemented, Sylow
94#576—4091nomaximal, Sylow
95H(5,𝔽2)—4091no—
96H(5,𝔽2)—4091no—
97AGL(1,ℤ/8)—4092no—
98AGL(1,ℤ/8)—4092no—
99AGL(1,ℤ/8)—4091no—
100AGL(1,ℤ/8)D40911yescharacteristic, complemented
101#135—4091no—
102#135—4092no—
103C2×D8—4091no—
104C2×D8—4092no—
105#131—4091no—
106#41—8181no—
107#41—8181no—
108#41—4092no—
109#41—4092no—
110#41—4091no—
111#41—4091no—
112#41C2×D40911yescharacteristic, complemented
113C2×D4—8181no—
114C2×D4—8182no—
115C2×D4—8181no—
116C2×D4—8182no—
117C2×D4—4092no—
118C2×D4—4092no—
119C2×D4—4091no—
120C2×D4C2×D40911yescharacteristic, complemented
121Q16—4091no—
122SD16—4091no—
123SD16—4091no—
124SD16—4092no—
125SD16C2×D40912yescomplemented
126D8—4092no—
127D8—4091no—
128D8—4092no—
129D8—4092no—
130D8C2×D40912yescomplemented
131M16—4092no—
132M16C2×D40911yescharacteristic, complemented
133C2×C8—4091no—
134C2×C8—4092no—
135C23—21no—
136C23—8182no—
137C23—8182no—
138C23—8181no—
139Q8—4091no—
140Q8—4091no—
141Q8C22×D40911yescharacteristic
142D4—21no—
143D4—21no—
144D4—8182no—
145D4—8182no—
146D4—8181no—
147D4—8182no—
148D4—8182no—
149D4—8181no—
150D4—4092no—
151D4—4091no—
152D4—4092no—
153D4—4091no—
154D4C22×D40912yes—
155D4C22×D40911yescharacteristic
156C2×C4—21no—
157C2×C4—8182no—
158C2×C4—8182no—
159C2×C4—8181no—
160C2×C4—4092no—
161C2×C4C22×D40911yescharacteristic
162C8—4091no—
163C8—4091no—
164C8C22×D40912yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—8182no—
169C22—8181no—
170C22—4092no—
171C22—32721no—
172C22—32722no—
173C22—32722no—
174C22D4×D40911yescharacteristic
175C4—21no—
176C4—8181no—
177C4—4092no—
178C4D4×D40911yescharacteristic
179C4C23×D40911yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—8182no—
184C2—16362no—
185C2—16361no—
186C2C2×D4×D40911yescharacteristic
subgroupG/H
centerC2C2×D4×D409
commutator, cosocleC4×C409C24
FrattiniC4C23×D409
FittingAGL(1,ℤ/8)×C409C2
socleC2×C409C22×D4
hypercenterAGL(1,ℤ/8)D409
hypocenterC409#576
LCSG ▷ C4×C409 ▷ C2×C409 ▷ C409
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27189763C2semidirect
2. #27189762C2semidirect
3. #27189745C2semidirect
4. #27189744C2semidirect
5. #27189741C2semidirect
6. #27189738C2semidirect
7. #27189737C2semidirect
8. C2×#20266973C2semidirect
9. D8×D409C2semidirect
10. #27189725C2semidirect
11. AGL(1,ℤ/8)×C409C2semidirect
12. #20266989C22semidirect
15. #20266988C22semidirect
17. #20266987C22semidirect
19. C2×D1636C22semidirect
23. D4×D409C22semidirect
24. D4×D409C22semidirect
25. #20266978C22semidirect
26. #20266977C22semidirect
27. #20266976C22semidirect
28. #20266975C22semidirect
29. D3272C22semidirect
30. #20266973C22semidirect
31. #20266973C22semidirect
32. #20266973C22semidirect
33. #20266971C22semidirect
34. C2×#15692543C22semidirect
35. C8×D409C22semidirect
36. #41×C409C22semidirect
37. C2×D4×C409C22semidirect
38. SD16×C409C22semidirect
39. D8×C409C22semidirect
40. M16×C409C22semidirect
93. C409#576semidirect
100. AGL(1,ℤ/8)D409semidirect
112. #41C2×D409semidirect
120. C2×D4C2×D409semidirect
125. SD16C2×D409semidirect
130. D8C2×D409semidirect
132. M16C2×D409semidirect

Maximal subgroups

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

subgroupindex
#271897632
#271897622
#271897452
#271897442
#271897412
#271897382
#271897372
C2×#202669732
D8×D4092
#271897252
AGL(1,ℤ/8)×C4092
#576409