Finite Group Databasebeta

← #27161883

Subgroups — #27161883

Order 12608.

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#20263372C211yesmaximal, characteristic, complemented
2#20263371C211yesmaximal, characteristic, complemented
3#20263354C211yesmaximal, characteristic, complemented
4#20263353C212yesmaximal, complemented
5#20263350C212yesmaximal, complemented
6#20263347C212yesmaximal, complemented
7#20263346C211yesmaximal, characteristic, complemented
8C2×#15691343C211yesmaximal, characteristic, complemented
9D8×D197C212yesmaximal, complemented
10#20263334C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C197C211yesmaximal, characteristic, complemented
12#15691359C2212yescomplemented
13#15691358—21no—
14#15691358C2211yescharacteristic
15#15691358C2211yescharacteristic, complemented
16#15691357—21no—
17#15691357C2212yescomplemented
18C2×D788—21no—
19C2×D788C2211yescharacteristic, complemented
20C2×C22⋊D197—22no—
21D4×D197—21no—
22D4×D197—22no—
23D4×D197C2212yescomplemented
24D4×D197C2212yescomplemented
25#15691348C2211yescharacteristic, complemented
26#15691347C2212yescomplemented
27#15691346C2211yescharacteristic, complemented
28#15691345C2211yescharacteristic, complemented
29D1576C2212yescomplemented
30#15691343C2211yescharacteristic, complemented
31#15691343C2212yescomplemented
32#15691343C2212yescomplemented
33#15691341C2212yescomplemented
34C2×#13096802C2211yescharacteristic, complemented
35C8×D197C2212yescomplemented
36#41×C197C2211yescharacteristic, complemented
37C2×D4×C197C2211yescharacteristic, complemented
38SD16×C197C2212yescomplemented
39D8×C197C2212yescomplemented
40M16×C197C2211yescharacteristic, complemented
41C22×D197—21no—
42C22×D197—22no—
43C22×D197—22no—
44Dic394D411yescharacteristic
45Dic394C2311yescharacteristic
46D788—22no—
47D788—21no—
48D788D411yescharacteristic
49D788C2312yes—
50D788C2311yescharacteristic
51C22⋊D197—21no—
52C22⋊D197—22no—
53C22⋊D197—22no—
54C22⋊D197—22no—
55C22⋊D197D412yes—
56C2×Dic197—22no—
57C4×D197—21no—
58C4×D197—22no—
59C4×D197C2312yes—
60#13096802C2311yescharacteristic
61#13096802C2311yescharacteristic
62C23×C197—21no—
63Q8×C197C2311yescharacteristic
64D4×C197—21no—
65D4×C197—21no—
66D4×C197C2311yescharacteristic
67D4×C197C2312yes—
68C2×C4×C197—21no—
69C2×C4×C197C2311yescharacteristic
70C8×C197C2312yes—
71C22×C197—21no—
72C22×C197—22no—
73C22×C197—81no—
74C22×C197C2×D411yescharacteristic
75C2×D197—21no—
76C2×D197—22no—
77C2×D197—82no—
78C2×D197—82no—
79C2×D197—81no—
80C2×D197C2×D412yes—
81C4×C197—21no—
82C4×C197C2×D411yescharacteristic
83C4×C197C2411yescharacteristic
84Dic197—21no—
85Dic197C2×D412yes—
86C2×C197—21no—
87C2×C197—41no—
88C2×C197—42no—
89C2×C197C22×D411yescharacteristic
90D197—22no—
91D197—41no—
92D197—42no—
93C197#57611yescharacteristic, complemented, Sylow
94#576—1971nomaximal, Sylow
95H(5,𝔽2)—1971no—
96H(5,𝔽2)—1971no—
97AGL(1,ℤ/8)—1972no—
98AGL(1,ℤ/8)—1972no—
99AGL(1,ℤ/8)—1971no—
100AGL(1,ℤ/8)D19711yescharacteristic, complemented
101#135—1971no—
102#135—1972no—
103C2×D8—1971no—
104C2×D8—1972no—
105#131—1971no—
106#41—3941no—
107#41—3941no—
108#41—1972no—
109#41—1972no—
110#41—1971no—
111#41—1971no—
112#41C2×D19711yescharacteristic, complemented
113C2×D4—3941no—
114C2×D4—3942no—
115C2×D4—3941no—
116C2×D4—3942no—
117C2×D4—1972no—
118C2×D4—1972no—
119C2×D4—1971no—
120C2×D4C2×D19711yescharacteristic, complemented
121Q16—1971no—
122SD16—1971no—
123SD16—1971no—
124SD16—1972no—
125SD16C2×D19712yescomplemented
126D8—1972no—
127D8—1971no—
128D8—1972no—
129D8—1972no—
130D8C2×D19712yescomplemented
131M16—1972no—
132M16C2×D19711yescharacteristic, complemented
133C2×C8—1971no—
134C2×C8—1972no—
135C23—21no—
136C23—3942no—
137C23—3942no—
138C23—3941no—
139Q8—1971no—
140Q8—1971no—
141Q8C22×D19711yescharacteristic
142D4—21no—
143D4—21no—
144D4—3942no—
145D4—3942no—
146D4—3941no—
147D4—3942no—
148D4—3942no—
149D4—3941no—
150D4—1972no—
151D4—1971no—
152D4—1972no—
153D4—1971no—
154D4C22×D19712yes—
155D4C22×D19711yescharacteristic
156C2×C4—21no—
157C2×C4—3942no—
158C2×C4—3942no—
159C2×C4—3941no—
160C2×C4—1972no—
161C2×C4C22×D19711yescharacteristic
162C8—1971no—
163C8—1971no—
164C8C22×D19712yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—15761no—
169C22—15762no—
170C22—15762no—
171C22—3942no—
172C22—3941no—
173C22—1972no—
174C22D4×D19711yescharacteristic
175C4—21no—
176C4—3941no—
177C4—1972no—
178C4D4×D19711yescharacteristic
179C4C23×D19711yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—7882no—
184C2—7881no—
185C2—3942no—
186C2C2×D4×D19711yescharacteristic
subgroupG/H
centerC2C2×D4×D197
commutator, cosocleC4×C197C24
FrattiniC4C23×D197
FittingAGL(1,ℤ/8)×C197C2
socleC2×C197C22×D4
hypercenterAGL(1,ℤ/8)D197
hypocenterC197#576
LCSG ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20263372C2semidirect
2. #20263371C2semidirect
3. #20263354C2semidirect
4. #20263353C2semidirect
5. #20263350C2semidirect
6. #20263347C2semidirect
7. #20263346C2semidirect
8. C2×#15691343C2semidirect
9. D8×D197C2semidirect
10. #20263334C2semidirect
11. AGL(1,ℤ/8)×C197C2semidirect
12. #15691359C22semidirect
15. #15691358C22semidirect
17. #15691357C22semidirect
19. C2×D788C22semidirect
23. D4×D197C22semidirect
24. D4×D197C22semidirect
25. #15691348C22semidirect
26. #15691347C22semidirect
27. #15691346C22semidirect
28. #15691345C22semidirect
29. D1576C22semidirect
30. #15691343C22semidirect
31. #15691343C22semidirect
32. #15691343C22semidirect
33. #15691341C22semidirect
34. C2×#13096802C22semidirect
35. C8×D197C22semidirect
36. #41×C197C22semidirect
37. C2×D4×C197C22semidirect
38. SD16×C197C22semidirect
39. D8×C197C22semidirect
40. M16×C197C22semidirect
93. C197#576semidirect
100. AGL(1,ℤ/8)D197semidirect
112. #41C2×D197semidirect
120. C2×D4C2×D197semidirect
125. SD16C2×D197semidirect
130. D8C2×D197semidirect
132. M16C2×D197semidirect

Maximal subgroups

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

subgroupindex
#202633722
#202633712
#202633542
#202633532
#202633502
#202633472
#202633462
C2×#156913432
D8×D1972
#202633342
AGL(1,ℤ/8)×C1972
#576197