Finite Group Databasebeta

← #36342111

Subgroups — #36342111

Order 21568.

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#24925007C211yesmaximal, characteristic, complemented
2#24925004C211yesmaximal, characteristic, complemented
3#24924990C211yesmaximal, characteristic, complemented
4#24924988C212yesmaximal, complemented
5#24924986C212yesmaximal, complemented
6#24924985C211yesmaximal, characteristic, complemented
7#24924981C211yesmaximal, characteristic, complemented
8#24924980C212yesmaximal, complemented
9SD16×D337C212yesmaximal, complemented
10#24924965C211yesmaximal, characteristic, complemented
11C2×SD16×C337C211yesmaximal, characteristic, complemented
12#19129186—21no—
13#19129186C2212yescomplemented
14#19129185—22no—
15#19129185C2211yescharacteristic
16#19129185C2211yescharacteristic
17#19129185C2211yescharacteristic, complemented
18#19129184—21no—
19#19129184C2212yes—
20Q8×D337—21no—
21Q8×D337C2212yescomplemented
22C2×C22⋊D337—22no—
23D4×D337—21no—
24D4×D337C2212yes—
25Dic1348C2211yescharacteristic, complemented
26#19129175C2212yescomplemented
27#19129174C2211yescharacteristic, complemented
28#19129174C2211yescharacteristic, complemented
29#19129173C2212yescomplemented
30#19129172C2212yescomplemented
31D2696C2211yescharacteristic, complemented
32#19129170C2212yescomplemented
33#19129169C2211yescharacteristic, complemented
34#19129168C2212yescomplemented
35C8×D337C2212yescomplemented
36C2×Q8×C337C2211yescharacteristic, complemented
37C2×D4×C337C2211yescharacteristic
38SD16×C337C2212yescomplemented
39SD16×C337C2212yescomplemented
40C2×C8×C337C2211yescharacteristic, complemented
41C22×D337—22no—
42Dic674—22no—
43Dic674D411yescharacteristic
44Dic674C2311yescharacteristic
45Dic674C2311yescharacteristic
46D1348—22no—
47D1348D411yescharacteristic
48D1348C2311yescharacteristic
49D1348C2311yescharacteristic
50C22⋊D337—21no—
51C22⋊D337—21no—
52C22⋊D337—22no—
53C22⋊D337—22no—
54C22⋊D337D412yes—
55C2×Dic337—22no—
56C4×D337—22no—
57C4×D337—21no—
58C4×D337—21no—
59C4×D337—22no—
60C4×D337C2312yes—
61#14582106C2312yes—
62C23×C337—21no—
63Q8×C337—21no—
64Q8×C337C2312yes—
65D4×C337—21no—
66D4×C337C2312yes—
67C2×C4×C337—21no—
68C2×C4×C337C2311yescharacteristic
69C8×C337C2311yescharacteristic
70C8×C337C2311yescharacteristic
71C22×C337—22no—
72C22×C337—81no—
73C22×C337C2×D411yescharacteristic
74C2×D337—21no—
75C2×D337—21no—
76C2×D337—82no—
77C2×D337C2×D412yes—
78C4×C337—22no—
79C4×C337C2×D411yescharacteristic
80C4×C337C2411yescharacteristic
81Dic337—21no—
82Dic337—21no—
83Dic337C2×D412yes—
84C2×C337—21no—
85C2×C337—42no—
86C2×C337C22×D411yescharacteristic
87D337—22no—
88D337—41no—
89D337—41no—
90C337#57711yescharacteristic, complemented, Sylow
91#577—3371nomaximal, Sylow
922−1+4—3371no—
93H(5,𝔽2)—3371no—
94#137—3371no—
95#137—3372no—
96AGL(1,ℤ/8)—3371no—
97AGL(1,ℤ/8)—3372no—
98#135—3372no—
99#135—3371no—
100C2×SD16—3372no—
101C2×SD16D33711yescharacteristic, complemented
102#131—3371no—
103#41—3371no—
104#41—3372no—
105#41—3372no—
106#41—3371no—
107#41—3371no—
108#41—6742no—
109#41—6741no—
110#41—6741no—
111C2×Q8—3372no—
112C2×Q8—6741no—
113C2×Q8C2×D33711yescharacteristic, complemented
114C2×D4—3372no—
115C2×D4—6741no—
116C2×D4—6742no—
117C2×D4C2×D33711yescharacteristic
118Q16—3372no—
119Q16—3371no—
120SD16—3372no—
121SD16—3372no—
122SD16—3371no—
123SD16—3371no—
124SD16C2×D33712yescomplemented
125SD16C2×D33712yescomplemented
126D8—3372no—
127D8—3371no—
128M16—3371no—
129M16—3372no—
130C2×C8—3372no—
131C2×C8C2×D33711yescharacteristic, complemented
132C23—21no—
133C23—6742no—
134Q8—21no—
135Q8—3371no—
136Q8—3371no—
137Q8—3371no—
138Q8—6742no—
139Q8C22×D33712yes—
140D4—21no—
141D4—3372no—
142D4—3371no—
143D4—3371no—
144D4—3371no—
145D4—6742no—
146D4—6742no—
147D4—6742no—
148D4—6741no—
149D4—6741no—
150D4C22×D33712yes—
151C2×C4—21no—
152C2×C4—3372no—
153C2×C4—6742no—
154C2×C4—6742no—
155C2×C4—6742no—
156C2×C4—6741no—
157C2×C4—6741no—
158C2×C4C22×D33711yescharacteristic
159C8—3372no—
160C8C22×D33711yescharacteristic
161C8C22×D33711yescharacteristic
162C22—22no—
163C22—81no—
164C22—3372no—
165C22—26962no—
166C22—6741no—
167C22—6741no—
168C22D4×D33711yescharacteristic
169C4—22no—
170C4—3372no—
171C4—6741no—
172C4—6741no—
173C4D4×D33711yescharacteristic
174C4C23×D33711yescharacteristic
175C2—21no—
176C2—42no—
177C2—13481no—
178C2—13481no—
179C2—6742no—
180C2C2×D4×D33711yescharacteristic
subgroupG/H
centerC2C2×D4×D337
commutator, cosocleC4×C337C24
FrattiniC4C23×D337
FittingC2×SD16×C337C2
socleC2×C337C22×D4
hypercenterC2×SD16D337
hypocenterC337#577
LCSG ▷ C4×C337 ▷ C2×C337 ▷ C337
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #24925007C2semidirect
2. #24925004C2semidirect
3. #24924990C2semidirect
4. #24924988C2semidirect
5. #24924986C2semidirect
6. #24924985C2semidirect
7. #24924981C2semidirect
8. #24924980C2semidirect
9. SD16×D337C2semidirect
10. #24924965C2semidirect
11. C2×SD16×C337C2semidirect
13. #19129186C22semidirect
17. #19129185C22semidirect
21. Q8×D337C22semidirect
25. Dic1348C22semidirect
26. #19129175C22semidirect
27. #19129174C22semidirect
28. #19129174C22semidirect
29. #19129173C22semidirect
30. #19129172C22semidirect
31. D2696C22semidirect
32. #19129170C22semidirect
33. #19129169C22semidirect
34. #19129168C22semidirect
35. C8×D337C22semidirect
36. C2×Q8×C337C22semidirect
38. SD16×C337C22semidirect
39. SD16×C337C22semidirect
40. C2×C8×C337C22semidirect
90. C337#577semidirect
101. C2×SD16D337semidirect
113. C2×Q8C2×D337semidirect
124. SD16C2×D337semidirect
125. SD16C2×D337semidirect
131. C2×C8C2×D337semidirect

Maximal subgroups

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

subgroupindex
#249250072
#249250042
#249249902
#249249882
#249249862
#249249852
#249249812
#249249802
SD16×D3372
#249249652
C2×SD16×C3372
#577337