Finite Group Databasebeta

← #38763332

Subgroups — #38763332

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#27163143C211yesmaximal, characteristic, complemented
2#27163141C211yesmaximal, characteristic, complemented
3#27163129C211yesmaximal, characteristic, complemented
4#27163125C212yesmaximal, complemented
5#27163120C212yesmaximal, complemented
6#27163116C211yesmaximal, characteristic, complemented
7#27163115C212yesmaximal, complemented
8C2×#20263719C211yesmaximal, characteristic, complemented
9SD16×D397C212yesmaximal, complemented
10#27163104C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C397C211yesmaximal, characteristic, complemented
12#20263731—21no—
13#20263731C2211yescharacteristic
14#20263731C2211yescharacteristic
15#20263730—21no—
16#20263730—22no—
17#20263730C2212yes—
18#20263730C2212yescomplemented
19C2×Dic794—21no—
20C2×Dic794C2211yescharacteristic, complemented
21Q8×D397C2212yescomplemented
22C2×C22⋊D397—22no—
23D4×D397—21no—
24D4×D397C2212yes—
25Dic1588C2212yescomplemented
26#20263721C2211yescharacteristic, complemented
27#20263720C2212yescomplemented
28#20263719C2212yescomplemented
29#20263719C2212yescomplemented
30#20263719C2211yescharacteristic, complemented
31#20263718C2211yescharacteristic, complemented
32#20263716C2211yescharacteristic, complemented
33#20263714C2212yescomplemented
34C2×#15691787C2211yescharacteristic, complemented
35C8×D397C2212yescomplemented
36#41×C397C2211yescharacteristic, complemented
37C2×D4×C397C2211yescharacteristic
38SD16×C397C2212yescomplemented
39D8×C397C2212yescomplemented
40M16×C397C2211yescharacteristic, complemented
41C22×D397—22no—
42Dic794—21no—
43Dic794—22no—
44Dic794D411yescharacteristic
45Dic794C2312yes—
46Dic794C2311yescharacteristic
47D1588D411yescharacteristic
48D1588C2311yescharacteristic
49C22⋊D397—21no—
50C22⋊D397—22no—
51C22⋊D397—22no—
52C22⋊D397—22no—
53C22⋊D397D412yes—
54C2×Dic397—21no—
55C2×Dic397—22no—
56C2×Dic397—22no—
57C4×D397—22no—
58C4×D397—21no—
59C4×D397C2312yes—
60#15691787C2311yescharacteristic
61#15691787C2311yescharacteristic
62C23×C397—21no—
63Q8×C397C2311yescharacteristic
64D4×C397—21no—
65D4×C397—21no—
66D4×C397C2311yescharacteristic
67D4×C397C2312yes—
68C2×C4×C397—21no—
69C2×C4×C397C2311yescharacteristic
70C8×C397C2312yes—
71C22×C397—21no—
72C22×C397—22no—
73C22×C397—81no—
74C22×C397C2×D411yescharacteristic
75C2×D397—21no—
76C2×D397—82no—
77C2×D397C2×D412yes—
78C4×C397—21no—
79C4×C397C2×D411yescharacteristic
80C4×C397C2411yescharacteristic
81Dic397—21no—
82Dic397—22no—
83Dic397C2×D412yes—
84C2×C397—21no—
85C2×C397—42no—
86C2×C397—41no—
87C2×C397C22×D411yescharacteristic
88D397—22no—
89D397—41no—
90C397#57711yescharacteristic, complemented, Sylow
91#577—3971nomaximal, Sylow
922−1+4—3971no—
93H(5,𝔽2)—3971no—
94#137—3972no—
95#137—3971no—
96AGL(1,ℤ/8)—3972no—
97AGL(1,ℤ/8)D39711yescharacteristic, complemented
98#135—3971no—
99#135—3972no—
100C2×SD16—3971no—
101C2×SD16—3972no—
102#131—3971no—
103#41—7941no—
104#41—7941no—
105#41—7942no—
106#41—3972no—
107#41—3972no—
108#41—3971no—
109#41—3971no—
110#41C2×D39711yescharacteristic, complemented
111C2×Q8—7941no—
112C2×Q8—3972no—
113C2×Q8—3971no—
114C2×D4—7941no—
115C2×D4—7942no—
116C2×D4—3972no—
117C2×D4C2×D39711yescharacteristic
118Q16—3971no—
119Q16—3972no—
120SD16—3971no—
121SD16—3972no—
122SD16—3972no—
123SD16—3971no—
124SD16—3972no—
125SD16C2×D39712yescomplemented
126D8—3971no—
127D8C2×D39712yescomplemented
128M16—3972no—
129M16C2×D39711yescharacteristic, complemented
130C2×C8—3971no—
131C2×C8—3972no—
132C23—21no—
133C23—7942no—
134Q8—7942no—
135Q8—7941no—
136Q8—3972no—
137Q8—3971no—
138Q8—3971no—
139Q8C22×D39711yescharacteristic
140D4—21no—
141D4—21no—
142D4—7942no—
143D4—7942no—
144D4—7941no—
145D4—7942no—
146D4—3972no—
147D4—3971no—
148D4—3971no—
149D4C22×D39711yescharacteristic
150D4C22×D39712yes—
151C2×C4—21no—
152C2×C4—7942no—
153C2×C4—7941no—
154C2×C4—7942no—
155C2×C4—7942no—
156C2×C4—7941no—
157C2×C4—3972no—
158C2×C4C22×D39711yescharacteristic
159C8—3971no—
160C8—3971no—
161C8C22×D39712yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—7941no—
166C22—31762no—
167C22—3972no—
168C22D4×D39711yescharacteristic
169C4—21no—
170C4—7941no—
171C4—7942no—
172C4—3972no—
173C4D4×D39711yescharacteristic
174C4C23×D39711yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—7942no—
179C2—15881no—
180C2C2×D4×D39711yescharacteristic
subgroupG/H
centerC2C2×D4×D397
commutator, cosocleC4×C397C24
FrattiniC4C23×D397
FittingAGL(1,ℤ/8)×C397C2
socleC2×C397C22×D4
hypercenterAGL(1,ℤ/8)D397
hypocenterC397#577
LCSG ▷ C4×C397 ▷ C2×C397 ▷ C397
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27163143C2semidirect
2. #27163141C2semidirect
3. #27163129C2semidirect
4. #27163125C2semidirect
5. #27163120C2semidirect
6. #27163116C2semidirect
7. #27163115C2semidirect
8. C2×#20263719C2semidirect
9. SD16×D397C2semidirect
10. #27163104C2semidirect
11. AGL(1,ℤ/8)×C397C2semidirect
18. #20263730C22semidirect
20. C2×Dic794C22semidirect
21. Q8×D397C22semidirect
25. Dic1588C22semidirect
26. #20263721C22semidirect
27. #20263720C22semidirect
28. #20263719C22semidirect
29. #20263719C22semidirect
30. #20263719C22semidirect
31. #20263718C22semidirect
32. #20263716C22semidirect
33. #20263714C22semidirect
34. C2×#15691787C22semidirect
35. C8×D397C22semidirect
36. #41×C397C22semidirect
38. SD16×C397C22semidirect
39. D8×C397C22semidirect
40. M16×C397C22semidirect
90. C397#577semidirect
97. AGL(1,ℤ/8)D397semidirect
110. #41C2×D397semidirect
125. SD16C2×D397semidirect
127. D8C2×D397semidirect
129. M16C2×D397semidirect

Maximal subgroups

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

subgroupindex
#271631432
#271631412
#271631292
#271631252
#271631202
#271631162
#271631152
C2×#202637192
SD16×D3972
#271631042
AGL(1,ℤ/8)×C3972
#577397