Finite Group Databasebeta

← #41003485

Subgroups — #41003485

Order 26944.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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×#20269770C211yesmaximal, characteristic, complemented
2C2×D4×D421C211yesmaximal, characteristic, complemented
3#27212354C211yesmaximal, characteristic, complemented
4#27212350C212yesmaximal, complemented
5#27212346C211yesmaximal, characteristic, complemented
6#27212341C211yesmaximal, characteristic, complemented
7#27212340C212yesmaximal, complemented
8#27212336C212yesmaximal, complemented
9C4×D1684C211yesmaximal, characteristic, complemented
10C2×#20269748C212yesmaximal, complemented
11C4×D4×C421C211yesmaximal, characteristic, complemented
12C23×D421C2212yes—
13#20269770—24no—
14C2×Dic842C2211yescharacteristic, complemented
15C2×D1684C2211yescharacteristic
16C2×D1684C2211yescharacteristic
17C2×D1684C2212yescomplemented
18C2×C22⋊D421C2212yes—
19C2×C22⋊D421C2212yescomplemented
20D4×D421—24no—
21C22×Dic421C2212yescomplemented
22C2×C4×D421C2211yescharacteristic
23C2×C4×D421C2211yescharacteristic, complemented
24C4⋊Dic421C2211yescharacteristic, complemented
25C4⋊Dic421C2212yescomplemented
26#20269748C2211yescharacteristic
27#20269748C2212yes—
28#20269748C2212yescomplemented
29#20269748C2212yescomplemented
30#20269748C2211yescharacteristic, complemented
31#20269748C2212yescomplemented
32C2×D4×C421C2211yescharacteristic
33C22×C4×C421C2212yescomplemented
34C4⋊C4×C421C2211yescharacteristic, complemented
35C22⋊C4×C421C2212yescomplemented
36C42×C421C2211yescharacteristic, complemented
37C22×D421—24no—
38C22×D421—24no—
39C22×D421—42no—
40C22×D421C2312yes—
41C22×D421C2311yescharacteristic
42C22×D421C2311yescharacteristic
43Dic842—22no—
44D1684—22no—
45D1684—22no—
46D1684—42no—
47C22⋊D421—24no—
48C22⋊D421—24no—
49C2×Dic421—24no—
50C2×Dic421C2311yescharacteristic
51C2×Dic421C2312yes—
52C2×Dic421C2311yescharacteristic
53C4×D421—22no—
54C4×D421—22no—
55C23×C421C2312yes—
56D4×C421D414yescomplemented
57C2×C4×C421—42no—
58C2×C4×C421C2311yescharacteristic
59C2×C4×C421C2311yescharacteristic
60C2×C4×C421C2312yes—
61C2×C4×C421C2311yescharacteristic
62C22×C421—22no—
63C22×C421—22no—
64C22×C421C2×D414yes—
65C22×C421C2411yescharacteristic
66C2×D421—21no—
67C2×D421—21no—
68C2×D421—21no—
69C2×D421—22no—
70C2×D421—21no—
71C2×D421—22no—
72C2×D421—21no—
73C2×D421—22no—
74C2×D421—48no—
75C2×D421#4112yescomplemented
76C4×C421—21no—
77C4×C421—21no—
78C4×C421—22no—
79C4×C421C2×D412yes—
80Dic421—21no—
81Dic421—21no—
82Dic421—22no—
83C2×C421—24no—
84C2×C421C22×D411yescharacteristic
85C2×C421C2×#4111yescharacteristic
86C2×C421H(5,𝔽2)11yescharacteristic
87D421—22no—
88D421—41no—
89D421—42no—
90C421#54611yescharacteristic, complemented, Sylow
91#546—4211nomaximal, Sylow
92C2×#41—4211no—
93C22×D4—4211no—
94#124—4211no—
95#123—4212no—
96C22⋊Q8—4211no—
97#121—4212no—
98#121—4211no—
99#120—4212no—
100C4×D4—4211no—
101C4×D4D42111yescharacteristic, complemented
102C2×C22⋊C4—4212no—
103C24—4212no—
104#41—8424no—
105C2×Q8—4211no—
106C2×D4—8424no—
107C2×D4—4212no—
108C2×D4—4212no—
109C2×D4—4211no—
110C2×D4—4211no—
111C2×D4—4212no—
112C2×D4C2×D42111yescharacteristic
113C22×C4—4212no—
114C22×C4—4211no—
115C22×C4—4211no—
116C22×C4C2×D42112yescomplemented
117C4⋊C4—4211no—
118C4⋊C4—4212no—
119C4⋊C4C2×D42111yescharacteristic, complemented
120C22⋊C4—4211no—
121C22⋊C4—4212no—
122C22⋊C4—4212no—
123C22⋊C4—4212no—
124C22⋊C4—4211no—
125C22⋊C4—4212no—
126C22⋊C4C2×D42112yescomplemented
127C42C2×D42111yescharacteristic, complemented
128C23—8424no—
129C23—8424no—
130C23—16842no—
131C23—4212no—
132C23—4211no—
133C23—4211no—
134C23C22×D42112yes—
135Q8—8422no—
136D4—8422no—
137D4—8424no—
138D4—8424no—
139D4—8422no—
140D4—16842no—
141D4D168414yescomplemented
142C2×C4—42no—
143C2×C4—8424no—
144C2×C4—8422no—
145C2×C4—8422no—
146C2×C4—4211no—
147C2×C4—4212no—
148C2×C4—4211no—
149C2×C4C22×D42111yescharacteristic
150C2×C4C22×D42111yescharacteristic
151C2×C4C22×D42112yes—
152C2×C4C22×D42111yescharacteristic
153C22—22no—
154C22—22no—
155C22—8421no—
156C22—8421no—
157C22—8421no—
158C22—8422no—
159C22—8421no—
160C22—8422no—
161C22—8421no—
162C22—8422no—
163C22—16848no—
164C22—4212no—
165C22C2×D168414yes—
166C22C23×D42111yescharacteristic
167C4—21no—
168C4—21no—
169C4—22no—
170C4—8421no—
171C4—8421no—
172C4—8422no—
173C4C2×D168412yes—
174C2—24no—
175C2—8422no—
176C2—16842no—
177C2—16841no—
178C2C22×D168411yescharacteristic
179C2#41×D42111yescharacteristic
180C2#2721241111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
FittingC4×D4×C421C2
hypercenterC4×D4D421
hypocenterC421#546
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#20269770C2semidirect
2. C2×D4×D421C2semidirect
3. #27212354C2semidirect
4. #27212350C2semidirect
5. #27212346C2semidirect
6. #27212341C2semidirect
7. #27212340C2semidirect
8. #27212336C2semidirect
9. C4×D1684C2semidirect
10. C2×#20269748C2semidirect
11. C4×D4×C421C2semidirect
14. C2×Dic842C22semidirect
17. C2×D1684C22semidirect
19. C2×C22⋊D421C22semidirect
21. C22×Dic421C22semidirect
23. C2×C4×D421C22semidirect
24. C4⋊Dic421C22semidirect
25. C4⋊Dic421C22semidirect
28. #20269748C22semidirect
29. #20269748C22semidirect
30. #20269748C22semidirect
31. #20269748C22semidirect
33. C22×C4×C421C22semidirect
34. C4⋊C4×C421C22semidirect
35. C22⋊C4×C421C22semidirect
36. C42×C421C22semidirect
56. D4×C421D4semidirect
75. C2×D421#41semidirect
90. C421#546semidirect
101. C4×D4D421semidirect
116. C22×C4C2×D421semidirect
119. C4⋊C4C2×D421semidirect
126. C22⋊C4C2×D421semidirect
127. C42C2×D421semidirect
141. D4D1684semidirect

Maximal subgroups

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

subgroupindex
C2×#202697702
C2×D4×D4212
#272123542
#272123502
#272123462
#272123412
#272123402
#272123362
C4×D16842
C2×#202697482
C4×D4×C4212
#546421