Finite Group Databasebeta

← #27161878

Subgroups — #27161878

Order 12608.

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

normal subgroups only

This group has 262 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
1C2×#15691359C211yesmaximal, characteristic, complemented
2#41×D197C211yesmaximal, characteristic, complemented
3#20263358C212yesmaximal, complemented
4#20263357C211yesmaximal, characteristic, complemented
5#20263354C211yesmaximal, characteristic, complemented
6#20263353C212yesmaximal, complemented
7#20263351C211yesmaximal, characteristic, complemented
8#20263349C212yesmaximal, complemented
9#20263347C212yesmaximal, complemented
10M16×D197C211yesmaximal, characteristic, complemented
11#137×C197C211yesmaximal, characteristic, complemented
12#15691359—21no—
13#15691359—21no—
14#15691359C2212yes—
15#15691359C2212yescomplemented
16#15691359C2211yescharacteristic, complemented
17#15691358—21no—
18#15691358C2211yescharacteristic
19#15691357—21no—
20#15691357C2211yescharacteristic, complemented
21Q8×D197C2211yescharacteristic, complemented
22C2×D788—21no—
23C2×D788C2211yescharacteristic, complemented
24D4×D197—21no—
25D4×D197C2211yescharacteristic, complemented
26C2×C4×D197—21no—
27C2×C4×D197—21no—
28C2×C4×D197C2211yescharacteristic
29#15691348C2212yescomplemented
30#15691347C2212yescomplemented
31#15691345C2212yescomplemented
32#15691345C2212yescomplemented
33D1576C2212yescomplemented
34#15691343C2212yescomplemented
35#15691342C2211yescharacteristic, complemented
36#15691341C2212yescomplemented
37C8×D197C2212yescomplemented
38#41×C197C2211yescharacteristic
39C2×Q8×C197C2211yescharacteristic, complemented
40Q16×C197C2212yescomplemented
41SD16×C197C2212yescomplemented
42M16×C197C2211yescharacteristic, complemented
43C22×D197—21no—
44C22×D197—21no—
45C22×D197D411yescharacteristic
46Dic394—21no—
47Dic394C2311yescharacteristic
48D788—21no—
49D788—22no—
50D788—22no—
51D788—21no—
52D788C2312yes—
53D788C2311yescharacteristic
54C22⋊D197—21no—
55C22⋊D197—21no—
56C22⋊D197—21no—
57C2×Dic197—21no—
58C2×Dic197D411yescharacteristic
59C4×D197—21no—
60C4×D197—21no—
61C4×D197—21no—
62C4×D197—22no—
63C4×D197—22no—
64C4×D197D411yescharacteristic
65C4×D197D411yescharacteristic
66C4×D197C2311yescharacteristic
67C4×D197C2311yescharacteristic
68#13096802C2312yes—
69Q8×C197—21no—
70Q8×C197C2312yes—
71Q8×C197C2311yescharacteristic
72D4×C197—21no—
73D4×C197C2311yescharacteristic
74C2×C4×C197—21no—
75C2×C4×C197—21no—
76C2×C4×C197C2311yescharacteristic
77C8×C197C2312yes—
78C22×C197—21no—
79C22×C197C2×D411yescharacteristic
80C2×D197—21no—
81C2×D197—22no—
82C2×D197—41no—
83C2×D197—42no—
84C2×D197—42no—
85C2×D197C2×D411yescharacteristic
86C2×D197C2×D411yescharacteristic
87C4×C197—21no—
88C4×C197—22no—
89C4×C197C2×D411yescharacteristic
90C4×C197C2411yescharacteristic
91Dic197—21no—
92Dic197C2×D411yescharacteristic
93Dic197C2×D411yescharacteristic
94C2×C197—21no—
95C2×C197—41no—
96C2×C197C22×D411yescharacteristic
97D197—21no—
98D197—21no—
99D197—41no—
100D197—42no—
101C197#57511yescharacteristic, complemented, Sylow
102#575—1971nomaximal, Sylow
103C2×#41—1971no—
104C2×#41—1971no—
105#137—1971no—
106#137—1972no—
107#137D19711yescharacteristic, complemented
108AGL(1,ℤ/8)—1971no—
109AGL(1,ℤ/8)—1972no—
110AGL(1,ℤ/8)—1971no—
111#135—1972no—
112#135—1972no—
113C2×M16—1971no—
114#41—3941no—
115#41—3941no—
116#41—3941no—
117#41—3941no—
118#41—1972no—
119#41—1971no—
120#41—1972no—
121#41—1971no—
122#41—1971no—
123#41C2×D19711yescharacteristic
124C2×Q8—1971no—
125C2×Q8C2×D19711yescharacteristic, complemented
126C2×D4—3941no—
127C2×D4—3941no—
128C2×D4—1971no—
129C2×D4—1971no—
130C22×C4—3941no—
131C22×C4—3941no—
132C22×C4—1971no—
133Q16—1972no—
134Q16C2×D19712yescomplemented
135SD16—1972no—
136SD16—1972no—
137SD16—1972no—
138SD16C2×D19712yescomplemented
139D8—1972no—
140D8—1972no—
141M16—1971no—
142M16—1972no—
143M16C2×D19711yescharacteristic, complemented
144C2×C8—1972no—
145C23—3941no—
146C23—3941no—
147C23—1971no—
148Q8—21no—
149Q8—3941no—
150Q8—1971no—
151Q8C22×D19711yescharacteristic
152Q8C22×D19712yes—
153D4—21no—
154D4—3941no—
155D4—3942no—
156D4—3942no—
157D4—3941no—
158D4—3941no—
159D4—3941no—
160D4—3941no—
161D4—1972no—
162D4—1971no—
163D4C22×D19711yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—3941no—
167C2×C4—3941no—
168C2×C4—3942no—
169C2×C4—3942no—
170C2×C4—3941no—
171C2×C4—3941no—
172C2×C4—1971no—
173C2×C4—1971no—
174C2×C4—1971no—
175C2×C4—1971no—
176C2×C4—1971no—
177C2×C4C22×D19711yescharacteristic
178C8—1972no—
179C8C22×D19712yes—
180C22—21no—
181C22—7882no—
182C22—7881no—
183C22—7882no—
184C22—3942no—
185C22—3941no—
186C22—1971no—
187C22—1971no—
188C22D4×D19711yescharacteristic
189C4—21no—
190C4—22no—
191C4—3941no—
192C4—1971no—
193C4—1971no—
194C4D4×D19711yescharacteristic
195C4C23×D19711yescharacteristic
196C2—21no—
197C2—41no—
198C2—7882no—
199C2—7881no—
200C2—3941no—
201C2—3941no—
202C2C2×D4×D19711yescharacteristic
subgroupG/H
centerC2C2×D4×D197
commutator, cosocleC4×C197C24
FrattiniC4C23×D197
Fitting#137×C197C2
socleC2×C197C22×D4
hypercenter#137D197
hypocenterC197#575
LCSG ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#15691359C2semidirect
2. #41×D197C2semidirect
3. #20263358C2semidirect
4. #20263357C2semidirect
5. #20263354C2semidirect
6. #20263353C2semidirect
7. #20263351C2semidirect
8. #20263349C2semidirect
9. #20263347C2semidirect
10. M16×D197C2semidirect
11. #137×C197C2semidirect
15. #15691359C22semidirect
16. #15691359C22semidirect
20. #15691357C22semidirect
21. Q8×D197C22semidirect
23. C2×D788C22semidirect
25. D4×D197C22semidirect
29. #15691348C22semidirect
30. #15691347C22semidirect
31. #15691345C22semidirect
32. #15691345C22semidirect
33. D1576C22semidirect
34. #15691343C22semidirect
35. #15691342C22semidirect
36. #15691341C22semidirect
37. C8×D197C22semidirect
39. C2×Q8×C197C22semidirect
40. Q16×C197C22semidirect
41. SD16×C197C22semidirect
42. M16×C197C22semidirect
101. C197#575semidirect
107. #137D197semidirect
125. C2×Q8C2×D197semidirect
134. Q16C2×D197semidirect
138. SD16C2×D197semidirect
143. M16C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#156913592
#41×D1972
#202633582
#202633572
#202633542
#202633532
#202633512
#202633492
#202633472
M16×D1972
#137×C1972
#575197