Finite Group Databasebeta

← #38763326

Subgroups — #38763326

Order 25408.

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#27163142C211yesmaximal, characteristic, complemented
2#27163141C211yesmaximal, characteristic, complemented
3#27163124C211yesmaximal, characteristic, complemented
4#27163123C212yesmaximal, complemented
5#27163120C212yesmaximal, complemented
6#27163117C212yesmaximal, complemented
7#27163116C211yesmaximal, characteristic, complemented
8C2×#20263716C211yesmaximal, characteristic, complemented
9D8×D397C212yesmaximal, complemented
10#27163104C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C397C211yesmaximal, characteristic, complemented
12#20263732C2212yescomplemented
13#20263731—21no—
14#20263731C2211yescharacteristic
15#20263731C2211yescharacteristic, complemented
16#20263730—21no—
17#20263730C2212yescomplemented
18C2×D1588—21no—
19C2×D1588C2211yescharacteristic, complemented
20C2×C22⋊D397—22no—
21D4×D397—21no—
22D4×D397—22no—
23D4×D397C2212yescomplemented
24D4×D397C2212yescomplemented
25#20263721C2211yescharacteristic, complemented
26#20263720C2212yescomplemented
27#20263719C2211yescharacteristic, complemented
28#20263718C2211yescharacteristic, complemented
29D3176C2212yescomplemented
30#20263716C2211yescharacteristic, complemented
31#20263716C2212yescomplemented
32#20263716C2212yescomplemented
33#20263714C2212yescomplemented
34C2×#15691787C2211yescharacteristic, complemented
35C8×D397C2212yescomplemented
36#41×C397C2211yescharacteristic, complemented
37C2×D4×C397C2211yescharacteristic, complemented
38SD16×C397C2212yescomplemented
39D8×C397C2212yescomplemented
40M16×C397C2211yescharacteristic, complemented
41C22×D397—21no—
42C22×D397—22no—
43C22×D397—22no—
44Dic794D411yescharacteristic
45Dic794C2311yescharacteristic
46D1588—22no—
47D1588—21no—
48D1588D411yescharacteristic
49D1588C2312yes—
50D1588C2311yescharacteristic
51C22⋊D397—21no—
52C22⋊D397—22no—
53C22⋊D397—22no—
54C22⋊D397—22no—
55C22⋊D397D412yes—
56C2×Dic397—22no—
57C4×D397—21no—
58C4×D397—22no—
59C4×D397C2312yes—
60#15691787C2311yescharacteristic
61#15691787C2311yescharacteristic
62C23×C397—21no—
63Q8×C397C2311yescharacteristic
64D4×C397—21no—
65D4×C397—21no—
66D4×C397C2311yescharacteristic
67D4×C397C2312yes—
68C2×C4×C397—21no—
69C2×C4×C397C2311yescharacteristic
70C8×C397C2312yes—
71C22×C397—21no—
72C22×C397—22no—
73C22×C397—81no—
74C22×C397C2×D411yescharacteristic
75C2×D397—21no—
76C2×D397—22no—
77C2×D397—82no—
78C2×D397—82no—
79C2×D397—81no—
80C2×D397C2×D412yes—
81C4×C397—21no—
82C4×C397C2×D411yescharacteristic
83C4×C397C2411yescharacteristic
84Dic397—21no—
85Dic397C2×D412yes—
86C2×C397—21no—
87C2×C397—41no—
88C2×C397—42no—
89C2×C397C22×D411yescharacteristic
90D397—22no—
91D397—41no—
92D397—42no—
93C397#57611yescharacteristic, complemented, Sylow
94#576—3971nomaximal, Sylow
95H(5,𝔽2)—3971no—
96H(5,𝔽2)—3971no—
97AGL(1,ℤ/8)—3972no—
98AGL(1,ℤ/8)—3972no—
99AGL(1,ℤ/8)—3971no—
100AGL(1,ℤ/8)D39711yescharacteristic, complemented
101#135—3971no—
102#135—3972no—
103C2×D8—3971no—
104C2×D8—3972no—
105#131—3971no—
106#41—7941no—
107#41—7941no—
108#41—3972no—
109#41—3972no—
110#41—3971no—
111#41—3971no—
112#41C2×D39711yescharacteristic, complemented
113C2×D4—7941no—
114C2×D4—7942no—
115C2×D4—7941no—
116C2×D4—7942no—
117C2×D4—3972no—
118C2×D4—3972no—
119C2×D4—3971no—
120C2×D4C2×D39711yescharacteristic, complemented
121Q16—3971no—
122SD16—3971no—
123SD16—3971no—
124SD16—3972no—
125SD16C2×D39712yescomplemented
126D8—3972no—
127D8—3971no—
128D8—3972no—
129D8—3972no—
130D8C2×D39712yescomplemented
131M16—3972no—
132M16C2×D39711yescharacteristic, complemented
133C2×C8—3971no—
134C2×C8—3972no—
135C23—21no—
136C23—7942no—
137C23—7942no—
138C23—7941no—
139Q8—3971no—
140Q8—3971no—
141Q8C22×D39711yescharacteristic
142D4—21no—
143D4—21no—
144D4—7942no—
145D4—7942no—
146D4—7941no—
147D4—7942no—
148D4—7942no—
149D4—7941no—
150D4—3972no—
151D4—3971no—
152D4—3972no—
153D4—3971no—
154D4C22×D39712yes—
155D4C22×D39711yescharacteristic
156C2×C4—21no—
157C2×C4—7942no—
158C2×C4—7942no—
159C2×C4—7941no—
160C2×C4—3972no—
161C2×C4C22×D39711yescharacteristic
162C8—3971no—
163C8—3971no—
164C8C22×D39712yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—7942no—
169C22—7941no—
170C22—31761no—
171C22—31762no—
172C22—31762no—
173C22—3972no—
174C22D4×D39711yescharacteristic
175C4—21no—
176C4—7941no—
177C4—3972no—
178C4D4×D39711yescharacteristic
179C4C23×D39711yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—7942no—
184C2—15882no—
185C2—15881no—
186C2C2×D4×D39711yescharacteristic
subgroupG/H
centerC2C2×D4×D397
commutator, cosocleC4×C397C24
FrattiniC4C23×D397
FittingAGL(1,ℤ/8)×C397C2
socleC2×C397C22×D4
hypercenterAGL(1,ℤ/8)D397
hypocenterC397#576
LCSG ▷ C4×C397 ▷ C2×C397 ▷ C397
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27163142C2semidirect
2. #27163141C2semidirect
3. #27163124C2semidirect
4. #27163123C2semidirect
5. #27163120C2semidirect
6. #27163117C2semidirect
7. #27163116C2semidirect
8. C2×#20263716C2semidirect
9. D8×D397C2semidirect
10. #27163104C2semidirect
11. AGL(1,ℤ/8)×C397C2semidirect
12. #20263732C22semidirect
15. #20263731C22semidirect
17. #20263730C22semidirect
19. C2×D1588C22semidirect
23. D4×D397C22semidirect
24. D4×D397C22semidirect
25. #20263721C22semidirect
26. #20263720C22semidirect
27. #20263719C22semidirect
28. #20263718C22semidirect
29. D3176C22semidirect
30. #20263716C22semidirect
31. #20263716C22semidirect
32. #20263716C22semidirect
33. #20263714C22semidirect
34. C2×#15691787C22semidirect
35. C8×D397C22semidirect
36. #41×C397C22semidirect
37. C2×D4×C397C22semidirect
38. SD16×C397C22semidirect
39. D8×C397C22semidirect
40. M16×C397C22semidirect
93. C397#576semidirect
100. AGL(1,ℤ/8)D397semidirect
112. #41C2×D397semidirect
120. C2×D4C2×D397semidirect
125. SD16C2×D397semidirect
130. D8C2×D397semidirect
132. M16C2×D397semidirect

Maximal subgroups

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

subgroupindex
#271631422
#271631412
#271631242
#271631232
#271631202
#271631172
#271631162
C2×#202637162
D8×D3972
#271631042
AGL(1,ℤ/8)×C3972
#576397