Finite Group Databasebeta

← #14580977

Subgroups — #14580977

Order 2624.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#10620310C211yesmaximal, characteristic, complemented
2C22×D164C211yesmaximal, characteristic, complemented
3#13050153C211yesmaximal, characteristic, complemented
4#13050149C212yesmaximal, complemented
5#13050139C211yesmaximal, characteristic, complemented
6#13050139C212yesmaximal, complemented
7#13050134C212yesmaximal, complemented
8#13050126C211yesmaximal, characteristic, complemented
9C4×C22⋊D41C211yesmaximal, characteristic, complemented
10C22⋊C4×D41C212yesmaximal, complemented
11C22⋊Q8×C41C211yesmaximal, characteristic, complemented
12C23×D41C2212yes—
13C2×Q8×C41C2211yescharacteristic, complemented
14C22×C4×C41C2211yescharacteristic
15C2×C22⋊D41C2211yescharacteristic
16#10620310—24no—
17C2×D164—24no—
18C2×D164C2212yes—
19C2×D164C2211yescharacteristic
20C2×D164C2211yescharacteristic
21C2×D164C2212yescomplemented
22C2×D164C2212yescomplemented
23C2×C4×D41C2212yescomplemented
24C2×C4×D41C2211yescharacteristic, complemented
25C2×C4×D41C2212yescomplemented
26C4⋊C4×C41C2211yescharacteristic, complemented
27C4⋊C4×C41C2212yescomplemented
28C22⋊C4×C41C2212yes—
29C22⋊Dic41C2211yescharacteristic
30#10620281C2212yescomplemented
31#10620281C2212yescomplemented
32#10620281C2211yescharacteristic, complemented
33#10620281C2212yescomplemented
34#10620281C2212yescomplemented
35#10620279C2211yescharacteristic, complemented
36C4×Dic41C2211yescharacteristic, complemented
37C23×C41C2311yescharacteristic
38C22×D41—24no—
39C22×D41—24no—
40C22×D41—42no—
41C22×D41C2312yes—
42C22×D41C2311yescharacteristic
43C22×D41C2312yes—
44Q8×C41—22no—
45C2×C4×C41—22no—
46C2×C4×C41C2311yescharacteristic
47C2×C4×C41C2312yes—
48C2×C4×C41C2312yes—
49C2×C4×C41C2311yescharacteristic
50C22⋊D41—22no—
51C2×Dic41C2311yescharacteristic
52C2×Dic41C2311yescharacteristic
53C2×Dic41C2311yescharacteristic
54D164—24no—
55D164—24no—
56D164—22no—
57D164—42no—
58D164D414yescomplemented
59C4×D41—24no—
60C4×D41—22no—
61C4×D41—42no—
62C22×C41—21no—
63C22×C41—21no—
64C22×C41#4112yescomplemented
65C22×C41C2411yescharacteristic
66C2×D41—22no—
67C2×D41—22no—
68C2×D41—21no—
69C2×D41—22no—
70C2×D41—21no—
71C2×D41—22no—
72C2×D41—21no—
73C2×D41—22no—
74C2×D41—48no—
75C2×D41C2×D414yes—
76C4×C41—22no—
77C4×C41—21no—
78C4×C41—22no—
79C4×C41C2×D412yes—
80Dic41—21no—
81Dic41—21no—
82Dic41—21no—
83C2×C41—22no—
84C2×C41C22×D411yescharacteristic
85C2×C41C2×#4111yescharacteristic
86C2×C41H(5,𝔽2)11yescharacteristic
87D41—24no—
88D41—41no—
89D41—42no—
90#546—411nomaximal, Sylow
91C41#54611yescharacteristic, complemented, Sylow
92C2×#41—411no—
93C22×D4—411no—
94#124—411no—
95#123—412no—
96C22⋊Q8D4111yescharacteristic, complemented
97#121—411no—
98#121—412no—
99#120—412no—
100C4×D4—411no—
101C4×D4—411no—
102C2×C22⋊C4—412no—
103C24—412no—
104#41—824no—
105C2×Q8C2×D4111yescharacteristic, complemented
106C2×D4—411no—
107C2×D4—411no—
108C2×D4—412no—
109C2×D4—412no—
110C2×D4—411no—
111C2×D4—412no—
112C2×D4—824no—
113C22×C4—412no—
114C22×C4—411no—
115C22×C4—412no—
116C22×C4C2×D4111yescharacteristic
117C4⋊C4—411no—
118C4⋊C4C2×D4111yescharacteristic, complemented
119C4⋊C4C2×D4112yescomplemented
120C22⋊C4—411no—
121C22⋊C4—412no—
122C22⋊C4—412no—
123C22⋊C4—412no—
124C22⋊C4—411no—
125C22⋊C4—412no—
126C22⋊C4C2×D4112yes—
127C42—411no—
128C23—412no—
129C23—412no—
130C23—411no—
131C23—824no—
132C23—824no—
133C23—1642no—
134C23C22×D4111yescharacteristic
135Q8—22no—
136D4—414no—
137D4—824no—
138D4—824no—
139D4—822no—
140D4—822no—
141D4—1642no—
142C2×C4—22no—
143C2×C4—411no—
144C2×C4—411no—
145C2×C4—411no—
146C2×C4—824no—
147C2×C4—822no—
148C2×C4—1642no—
149C2×C4C22×D4112yes—
150C2×C4C22×D4112yes—
151C2×C4C22×D4111yescharacteristic
152C2×C4C22×D4111yescharacteristic
153C22—21no—
154C22—21no—
155C22—414no—
156C22—822no—
157C22—822no—
158C22—821no—
159C22—822no—
160C22—821no—
161C22—822no—
162C22—821no—
163C22—822no—
164C22—1648no—
165C22#1062031012yescomplemented
166C22C23×D4111yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—821no—
171C4—821no—
172C4—821no—
173C4D4×D4112yes—
174C2—22no—
175C2—824no—
176C2—1642no—
177C2—1641no—
178C2C2×D4×D4111yescharacteristic
179C2C2×#1062031011yescharacteristic
180C2#1305021011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D41
commutator, socle, cosocleC22×C41C24
FittingC22⋊Q8×C41C2
hypercenterC22⋊Q8D41
hypocenterC41#546
LCSG ▷ C22×C41 ▷ C41
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#10620310C2semidirect
2. C22×D164C2semidirect
3. #13050153C2semidirect
4. #13050149C2semidirect
5. #13050139C2semidirect
6. #13050139C2semidirect
7. #13050134C2semidirect
8. #13050126C2semidirect
9. C4×C22⋊D41C2semidirect
10. C22⋊C4×D41C2semidirect
11. C22⋊Q8×C41C2semidirect
13. C2×Q8×C41C22semidirect
21. C2×D164C22semidirect
22. C2×D164C22semidirect
23. C2×C4×D41C22semidirect
24. C2×C4×D41C22semidirect
25. C2×C4×D41C22semidirect
26. C4⋊C4×C41C22semidirect
27. C4⋊C4×C41C22semidirect
30. #10620281C22semidirect
31. #10620281C22semidirect
32. #10620281C22semidirect
33. #10620281C22semidirect
34. #10620281C22semidirect
35. #10620279C22semidirect
36. C4×Dic41C22semidirect
58. D164D4semidirect
64. C22×C41#41semidirect
91. C41#546semidirect
96. C22⋊Q8D41semidirect
105. C2×Q8C2×D41semidirect
118. C4⋊C4C2×D41semidirect
119. C4⋊C4C2×D41semidirect
165. C22#10620310semidirect

Maximal subgroups

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

subgroupindex
C2×#106203102
C22×D1642
#130501532
#130501492
#130501392
#130501342
#130501262
C4×C22⋊D412
C22⋊C4×D412
C22⋊Q8×C412
#54641