Finite Group Databasebeta

← #38763328

Subgroups — #38763328

Order 25408.

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#27163144C211yesmaximal, characteristic, complemented
2#27163141C211yesmaximal, characteristic, complemented
3#27163127C211yesmaximal, characteristic, complemented
4#27163125C212yesmaximal, complemented
5#27163123C212yesmaximal, complemented
6#27163122C211yesmaximal, characteristic, complemented
7#27163118C211yesmaximal, characteristic, complemented
8#27163117C212yesmaximal, complemented
9SD16×D397C212yesmaximal, complemented
10#27163102C211yesmaximal, characteristic, complemented
11C2×SD16×C397C211yesmaximal, characteristic, complemented
12#20263732—21no—
13#20263732C2212yescomplemented
14#20263731—22no—
15#20263731C2211yescharacteristic
16#20263731C2211yescharacteristic
17#20263731C2211yescharacteristic, complemented
18#20263730—21no—
19#20263730C2212yes—
20Q8×D397—21no—
21Q8×D397C2212yescomplemented
22C2×C22⋊D397—22no—
23D4×D397—21no—
24D4×D397C2212yes—
25Dic1588C2211yescharacteristic, complemented
26#20263721C2212yescomplemented
27#20263720C2211yescharacteristic, complemented
28#20263720C2211yescharacteristic, complemented
29#20263719C2212yescomplemented
30#20263718C2212yescomplemented
31D3176C2211yescharacteristic, complemented
32#20263716C2212yescomplemented
33#20263715C2211yescharacteristic, complemented
34#20263714C2212yescomplemented
35C8×D397C2212yescomplemented
36C2×Q8×C397C2211yescharacteristic, complemented
37C2×D4×C397C2211yescharacteristic
38SD16×C397C2212yescomplemented
39SD16×C397C2212yescomplemented
40C2×C8×C397C2211yescharacteristic, complemented
41C22×D397—22no—
42Dic794—22no—
43Dic794D411yescharacteristic
44Dic794C2311yescharacteristic
45Dic794C2311yescharacteristic
46D1588—22no—
47D1588D411yescharacteristic
48D1588C2311yescharacteristic
49D1588C2311yescharacteristic
50C22⋊D397—21no—
51C22⋊D397—21no—
52C22⋊D397—22no—
53C22⋊D397—22no—
54C22⋊D397D412yes—
55C2×Dic397—22no—
56C4×D397—22no—
57C4×D397—21no—
58C4×D397—21no—
59C4×D397—22no—
60C4×D397C2312yes—
61#15691787C2312yes—
62C23×C397—21no—
63Q8×C397—21no—
64Q8×C397C2312yes—
65D4×C397—21no—
66D4×C397C2312yes—
67C2×C4×C397—21no—
68C2×C4×C397C2311yescharacteristic
69C8×C397C2311yescharacteristic
70C8×C397C2311yescharacteristic
71C22×C397—22no—
72C22×C397—81no—
73C22×C397C2×D411yescharacteristic
74C2×D397—21no—
75C2×D397—21no—
76C2×D397—82no—
77C2×D397C2×D412yes—
78C4×C397—22no—
79C4×C397C2×D411yescharacteristic
80C4×C397C2411yescharacteristic
81Dic397—21no—
82Dic397—21no—
83Dic397C2×D412yes—
84C2×C397—21no—
85C2×C397—42no—
86C2×C397C22×D411yescharacteristic
87D397—22no—
88D397—41no—
89D397—41no—
90C397#57711yescharacteristic, complemented, Sylow
91#577—3971nomaximal, Sylow
922−1+4—3971no—
93H(5,𝔽2)—3971no—
94#137—3971no—
95#137—3972no—
96AGL(1,ℤ/8)—3971no—
97AGL(1,ℤ/8)—3972no—
98#135—3972no—
99#135—3971no—
100C2×SD16—3972no—
101C2×SD16D39711yescharacteristic, complemented
102#131—3971no—
103#41—7942no—
104#41—7941no—
105#41—7941no—
106#41—3972no—
107#41—3972no—
108#41—3971no—
109#41—3971no—
110#41—3971no—
111C2×Q8—7941no—
112C2×Q8—3972no—
113C2×Q8C2×D39711yescharacteristic, complemented
114C2×D4—7941no—
115C2×D4—7942no—
116C2×D4—3972no—
117C2×D4C2×D39711yescharacteristic
118Q16—3972no—
119Q16—3971no—
120SD16—3972no—
121SD16—3972no—
122SD16—3971no—
123SD16—3971no—
124SD16C2×D39712yescomplemented
125SD16C2×D39712yescomplemented
126D8—3972no—
127D8—3971no—
128M16—3971no—
129M16—3972no—
130C2×C8—3972no—
131C2×C8C2×D39711yescharacteristic, complemented
132C23—21no—
133C23—7942no—
134Q8—21no—
135Q8—7942no—
136Q8—3971no—
137Q8—3971no—
138Q8—3971no—
139Q8C22×D39712yes—
140D4—21no—
141D4—7942no—
142D4—7942no—
143D4—7941no—
144D4—7941no—
145D4—7942no—
146D4—3972no—
147D4—3971no—
148D4—3971no—
149D4—3971no—
150D4C22×D39712yes—
151C2×C4—21no—
152C2×C4—7942no—
153C2×C4—7942no—
154C2×C4—7942no—
155C2×C4—7941no—
156C2×C4—7941no—
157C2×C4—3972no—
158C2×C4C22×D39711yescharacteristic
159C8—3972no—
160C8C22×D39711yescharacteristic
161C8C22×D39711yescharacteristic
162C22—22no—
163C22—81no—
164C22—7941no—
165C22—7941no—
166C22—31762no—
167C22—3972no—
168C22D4×D39711yescharacteristic
169C4—22no—
170C4—7941no—
171C4—7941no—
172C4—3972no—
173C4D4×D39711yescharacteristic
174C4C23×D39711yescharacteristic
175C2—21no—
176C2—42no—
177C2—7942no—
178C2—15881no—
179C2—15881no—
180C2C2×D4×D39711yescharacteristic
subgroupG/H
centerC2C2×D4×D397
commutator, cosocleC4×C397C24
FrattiniC4C23×D397
FittingC2×SD16×C397C2
socleC2×C397C22×D4
hypercenterC2×SD16D397
hypocenterC397#577
LCSG ▷ C4×C397 ▷ C2×C397 ▷ C397
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #27163144C2semidirect
2. #27163141C2semidirect
3. #27163127C2semidirect
4. #27163125C2semidirect
5. #27163123C2semidirect
6. #27163122C2semidirect
7. #27163118C2semidirect
8. #27163117C2semidirect
9. SD16×D397C2semidirect
10. #27163102C2semidirect
11. C2×SD16×C397C2semidirect
13. #20263732C22semidirect
17. #20263731C22semidirect
21. Q8×D397C22semidirect
25. Dic1588C22semidirect
26. #20263721C22semidirect
27. #20263720C22semidirect
28. #20263720C22semidirect
29. #20263719C22semidirect
30. #20263718C22semidirect
31. D3176C22semidirect
32. #20263716C22semidirect
33. #20263715C22semidirect
34. #20263714C22semidirect
35. C8×D397C22semidirect
36. C2×Q8×C397C22semidirect
38. SD16×C397C22semidirect
39. SD16×C397C22semidirect
40. C2×C8×C397C22semidirect
90. C397#577semidirect
101. C2×SD16D397semidirect
113. C2×Q8C2×D397semidirect
124. SD16C2×D397semidirect
125. SD16C2×D397semidirect
131. C2×C8C2×D397semidirect

Maximal subgroups

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

subgroupindex
#271631442
#271631412
#271631272
#271631252
#271631232
#271631222
#271631182
#271631172
SD16×D3972
#271631022
C2×SD16×C3972
#577397