Finite Group Databasebeta

← #27161885

Subgroups — #27161885

Order 12608.

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

normal subgroups only

This group has 258 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#20263374C211yesmaximal, characteristic, complemented
2#20263371C211yesmaximal, characteristic, complemented
3#20263357C211yesmaximal, characteristic, complemented
4#20263355C212yesmaximal, complemented
5#20263353C212yesmaximal, complemented
6#20263352C211yesmaximal, characteristic, complemented
7#20263348C211yesmaximal, characteristic, complemented
8#20263347C212yesmaximal, complemented
9SD16×D197C212yesmaximal, complemented
10#20263332C211yesmaximal, characteristic, complemented
11C2×SD16×C197C211yesmaximal, characteristic, complemented
12#15691359—21no—
13#15691359C2212yescomplemented
14#15691358—22no—
15#15691358C2211yescharacteristic
16#15691358C2211yescharacteristic
17#15691358C2211yescharacteristic, complemented
18#15691357—21no—
19#15691357C2212yes—
20Q8×D197—21no—
21Q8×D197C2212yescomplemented
22C2×C22⋊D197—22no—
23D4×D197—21no—
24D4×D197C2212yes—
25Dic788C2211yescharacteristic, complemented
26#15691348C2212yescomplemented
27#15691347C2211yescharacteristic, complemented
28#15691347C2211yescharacteristic, complemented
29#15691346C2212yescomplemented
30#15691345C2212yescomplemented
31D1576C2211yescharacteristic, complemented
32#15691343C2212yescomplemented
33#15691342C2211yescharacteristic, complemented
34#15691341C2212yescomplemented
35C8×D197C2212yescomplemented
36C2×Q8×C197C2211yescharacteristic, complemented
37C2×D4×C197C2211yescharacteristic
38SD16×C197C2212yescomplemented
39SD16×C197C2212yescomplemented
40C2×C8×C197C2211yescharacteristic, complemented
41C22×D197—22no—
42Dic394—22no—
43Dic394D411yescharacteristic
44Dic394C2311yescharacteristic
45Dic394C2311yescharacteristic
46D788—22no—
47D788D411yescharacteristic
48D788C2311yescharacteristic
49D788C2311yescharacteristic
50C22⋊D197—21no—
51C22⋊D197—21no—
52C22⋊D197—22no—
53C22⋊D197—22no—
54C22⋊D197D412yes—
55C2×Dic197—22no—
56C4×D197—22no—
57C4×D197—21no—
58C4×D197—21no—
59C4×D197—22no—
60C4×D197C2312yes—
61#13096802C2312yes—
62C23×C197—21no—
63Q8×C197—21no—
64Q8×C197C2312yes—
65D4×C197—21no—
66D4×C197C2312yes—
67C2×C4×C197—21no—
68C2×C4×C197C2311yescharacteristic
69C8×C197C2311yescharacteristic
70C8×C197C2311yescharacteristic
71C22×C197—22no—
72C22×C197—81no—
73C22×C197C2×D411yescharacteristic
74C2×D197—21no—
75C2×D197—21no—
76C2×D197—82no—
77C2×D197C2×D412yes—
78C4×C197—22no—
79C4×C197C2×D411yescharacteristic
80C4×C197C2411yescharacteristic
81Dic197—21no—
82Dic197—21no—
83Dic197C2×D412yes—
84C2×C197—21no—
85C2×C197—42no—
86C2×C197C22×D411yescharacteristic
87D197—22no—
88D197—41no—
89D197—41no—
90C197#57711yescharacteristic, complemented, Sylow
91#577—1971nomaximal, Sylow
922−1+4—1971no—
93H(5,𝔽2)—1971no—
94#137—1971no—
95#137—1972no—
96AGL(1,ℤ/8)—1971no—
97AGL(1,ℤ/8)—1972no—
98#135—1972no—
99#135—1971no—
100C2×SD16—1972no—
101C2×SD16D19711yescharacteristic, complemented
102#131—1971no—
103#41—3942no—
104#41—3941no—
105#41—3941no—
106#41—1972no—
107#41—1972no—
108#41—1971no—
109#41—1971no—
110#41—1971no—
111C2×Q8—3941no—
112C2×Q8—1972no—
113C2×Q8C2×D19711yescharacteristic, complemented
114C2×D4—3941no—
115C2×D4—3942no—
116C2×D4—1972no—
117C2×D4C2×D19711yescharacteristic
118Q16—1972no—
119Q16—1971no—
120SD16—1972no—
121SD16—1972no—
122SD16—1971no—
123SD16—1971no—
124SD16C2×D19712yescomplemented
125SD16C2×D19712yescomplemented
126D8—1972no—
127D8—1971no—
128M16—1971no—
129M16—1972no—
130C2×C8—1972no—
131C2×C8C2×D19711yescharacteristic, complemented
132C23—21no—
133C23—3942no—
134Q8—21no—
135Q8—3942no—
136Q8—1971no—
137Q8—1971no—
138Q8—1971no—
139Q8C22×D19712yes—
140D4—21no—
141D4—3942no—
142D4—3942no—
143D4—3941no—
144D4—3941no—
145D4—3942no—
146D4—1972no—
147D4—1971no—
148D4—1971no—
149D4—1971no—
150D4C22×D19712yes—
151C2×C4—21no—
152C2×C4—3942no—
153C2×C4—3942no—
154C2×C4—3942no—
155C2×C4—3941no—
156C2×C4—3941no—
157C2×C4—1972no—
158C2×C4C22×D19711yescharacteristic
159C8—1972no—
160C8C22×D19711yescharacteristic
161C8C22×D19711yescharacteristic
162C22—22no—
163C22—81no—
164C22—15762no—
165C22—3941no—
166C22—3941no—
167C22—1972no—
168C22D4×D19711yescharacteristic
169C4—22no—
170C4—3941no—
171C4—3941no—
172C4—1972no—
173C4D4×D19711yescharacteristic
174C4C23×D19711yescharacteristic
175C2—21no—
176C2—42no—
177C2—7881no—
178C2—7881no—
179C2—3942no—
180C2C2×D4×D19711yescharacteristic
subgroupG/H
centerC2C2×D4×D197
commutator, cosocleC4×C197C24
FrattiniC4C23×D197
FittingC2×SD16×C197C2
socleC2×C197C22×D4
hypercenterC2×SD16D197
hypocenterC197#577
LCSG ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #20263374C2semidirect
2. #20263371C2semidirect
3. #20263357C2semidirect
4. #20263355C2semidirect
5. #20263353C2semidirect
6. #20263352C2semidirect
7. #20263348C2semidirect
8. #20263347C2semidirect
9. SD16×D197C2semidirect
10. #20263332C2semidirect
11. C2×SD16×C197C2semidirect
13. #15691359C22semidirect
17. #15691358C22semidirect
21. Q8×D197C22semidirect
25. Dic788C22semidirect
26. #15691348C22semidirect
27. #15691347C22semidirect
28. #15691347C22semidirect
29. #15691346C22semidirect
30. #15691345C22semidirect
31. D1576C22semidirect
32. #15691343C22semidirect
33. #15691342C22semidirect
34. #15691341C22semidirect
35. C8×D197C22semidirect
36. C2×Q8×C197C22semidirect
38. SD16×C197C22semidirect
39. SD16×C197C22semidirect
40. C2×C8×C197C22semidirect
90. C197#577semidirect
101. C2×SD16D197semidirect
113. C2×Q8C2×D197semidirect
124. SD16C2×D197semidirect
125. SD16C2×D197semidirect
131. C2×C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#202633742
#202633712
#202633572
#202633552
#202633532
#202633522
#202633482
#202633472
SD16×D1972
#202633322
C2×SD16×C1972
#577197