Finite Group Databasebeta

← #27164822

Subgroups — #27164822

Order 12736.

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#20264001C211yesmaximal, characteristic, complemented
2#20264000C211yesmaximal, characteristic, complemented
3#20263983C211yesmaximal, characteristic, complemented
4#20263982C212yesmaximal, complemented
5#20263979C212yesmaximal, complemented
6#20263976C212yesmaximal, complemented
7#20263975C211yesmaximal, characteristic, complemented
8C2×#15691871C211yesmaximal, characteristic, complemented
9D8×D199C212yesmaximal, complemented
10#20263963C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C199C211yesmaximal, characteristic, complemented
12#15691887C2212yescomplemented
13#15691886—21no—
14#15691886C2211yescharacteristic
15#15691886C2211yescharacteristic, complemented
16#15691885—21no—
17#15691885C2212yescomplemented
18C2×D796—21no—
19C2×D796C2211yescharacteristic, complemented
20C2×C22⋊D199—22no—
21D4×D199—21no—
22D4×D199—22no—
23D4×D199C2212yescomplemented
24D4×D199C2212yescomplemented
25#15691876C2211yescharacteristic, complemented
26#15691875C2212yescomplemented
27#15691874C2211yescharacteristic, complemented
28#15691873C2211yescharacteristic, complemented
29D1592C2212yescomplemented
30#15691871C2211yescharacteristic, complemented
31#15691871C2212yescomplemented
32#15691871C2212yescomplemented
33#15691869C2212yescomplemented
34C2×C199⋊C8C2211yescharacteristic, complemented
35C8×D199C2212yescomplemented
36#41×C199C2211yescharacteristic, complemented
37C2×D4×C199C2211yescharacteristic, complemented
38SD16×C199C2212yescomplemented
39D8×C199C2212yescomplemented
40M16×C199C2211yescharacteristic, complemented
41C22×D199—21no—
42C22×D199—22no—
43C22×D199—22no—
44Dic398D411yescharacteristic
45Dic398C2311yescharacteristic
46D796—22no—
47D796—21no—
48D796D411yescharacteristic
49D796C2312yes—
50D796C2311yescharacteristic
51C22⋊D199—21no—
52C22⋊D199—22no—
53C22⋊D199—22no—
54C22⋊D199—22no—
55C22⋊D199D412yes—
56C2×Dic199—22no—
57C4×D199—21no—
58C4×D199—22no—
59C4×D199C2312yes—
60C199⋊C8C2311yescharacteristic
61C199⋊C8C2311yescharacteristic
62C23×C199—21no—
63Q8×C199C2311yescharacteristic
64D4×C199—21no—
65D4×C199—21no—
66D4×C199C2311yescharacteristic
67D4×C199C2312yes—
68C2×C4×C199—21no—
69C2×C4×C199C2311yescharacteristic
70C8×C199C2312yes—
71C22×C199—21no—
72C22×C199—22no—
73C22×C199—81no—
74C22×C199C2×D411yescharacteristic
75C2×D199—21no—
76C2×D199—22no—
77C2×D199—82no—
78C2×D199—82no—
79C2×D199—81no—
80C2×D199C2×D412yes—
81C4×C199—21no—
82C4×C199C2×D411yescharacteristic
83C4×C199C2411yescharacteristic
84Dic199—21no—
85Dic199C2×D412yes—
86C2×C199—21no—
87C2×C199—41no—
88C2×C199—42no—
89C2×C199C22×D411yescharacteristic
90D199—22no—
91D199—41no—
92D199—42no—
93C199#57611yescharacteristic, complemented, Sylow
94#576—1991nomaximal, Sylow
95H(5,𝔽2)—1991no—
96H(5,𝔽2)—1991no—
97AGL(1,ℤ/8)—1992no—
98AGL(1,ℤ/8)—1992no—
99AGL(1,ℤ/8)—1991no—
100AGL(1,ℤ/8)D19911yescharacteristic, complemented
101#135—1991no—
102#135—1992no—
103C2×D8—1991no—
104C2×D8—1992no—
105#131—1991no—
106#41—3981no—
107#41—3981no—
108#41—1992no—
109#41—1992no—
110#41—1991no—
111#41—1991no—
112#41C2×D19911yescharacteristic, complemented
113C2×D4—3981no—
114C2×D4—3982no—
115C2×D4—3981no—
116C2×D4—3982no—
117C2×D4—1992no—
118C2×D4—1992no—
119C2×D4—1991no—
120C2×D4C2×D19911yescharacteristic, complemented
121Q16—1991no—
122SD16—1991no—
123SD16—1991no—
124SD16—1992no—
125SD16C2×D19912yescomplemented
126D8—1992no—
127D8—1991no—
128D8—1992no—
129D8—1992no—
130D8C2×D19912yescomplemented
131M16—1992no—
132M16C2×D19911yescharacteristic, complemented
133C2×C8—1991no—
134C2×C8—1992no—
135C23—21no—
136C23—3982no—
137C23—3982no—
138C23—3981no—
139Q8—1991no—
140Q8—1991no—
141Q8C22×D19911yescharacteristic
142D4—21no—
143D4—21no—
144D4—3982no—
145D4—3982no—
146D4—3981no—
147D4—3982no—
148D4—3982no—
149D4—3981no—
150D4—1992no—
151D4—1991no—
152D4—1992no—
153D4—1991no—
154D4C22×D19912yes—
155D4C22×D19911yescharacteristic
156C2×C4—21no—
157C2×C4—3982no—
158C2×C4—3982no—
159C2×C4—3981no—
160C2×C4—1992no—
161C2×C4C22×D19911yescharacteristic
162C8—1991no—
163C8—1991no—
164C8C22×D19912yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—15921no—
169C22—15922no—
170C22—15922no—
171C22—3982no—
172C22—3981no—
173C22—1992no—
174C22D4×D19911yescharacteristic
175C4—21no—
176C4—3981no—
177C4—1992no—
178C4D4×D19911yescharacteristic
179C4C23×D19911yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—7962no—
184C2—7961no—
185C2—3982no—
186C2C2×D4×D19911yescharacteristic
subgroupG/H
centerC2C2×D4×D199
commutator, cosocleC4×C199C24
FrattiniC4C23×D199
FittingAGL(1,ℤ/8)×C199C2
socleC2×C199C22×D4
hypercenterAGL(1,ℤ/8)D199
hypocenterC199#576
LCSG ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20264001C2semidirect
2. #20264000C2semidirect
3. #20263983C2semidirect
4. #20263982C2semidirect
5. #20263979C2semidirect
6. #20263976C2semidirect
7. #20263975C2semidirect
8. C2×#15691871C2semidirect
9. D8×D199C2semidirect
10. #20263963C2semidirect
11. AGL(1,ℤ/8)×C199C2semidirect
12. #15691887C22semidirect
15. #15691886C22semidirect
17. #15691885C22semidirect
19. C2×D796C22semidirect
23. D4×D199C22semidirect
24. D4×D199C22semidirect
25. #15691876C22semidirect
26. #15691875C22semidirect
27. #15691874C22semidirect
28. #15691873C22semidirect
29. D1592C22semidirect
30. #15691871C22semidirect
31. #15691871C22semidirect
32. #15691871C22semidirect
33. #15691869C22semidirect
34. C2×C199⋊C8C22semidirect
35. C8×D199C22semidirect
36. #41×C199C22semidirect
37. C2×D4×C199C22semidirect
38. SD16×C199C22semidirect
39. D8×C199C22semidirect
40. M16×C199C22semidirect
93. C199#576semidirect
100. AGL(1,ℤ/8)D199semidirect
112. #41C2×D199semidirect
120. C2×D4C2×D199semidirect
125. SD16C2×D199semidirect
130. D8C2×D199semidirect
132. M16C2×D199semidirect

Maximal subgroups

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

subgroupindex
#202640012
#202640002
#202639832
#202639822
#202639792
#202639762
#202639752
C2×#156918712
D8×D1992
#202639632
AGL(1,ℤ/8)×C1992
#576199