Finite Group Databasebeta

← #41003312

Subgroups — #41003312

Order 26944.

338 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 151 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×D4×D421C211yesmaximal, characteristic
2C4×D1684C211yesmaximal, characteristic, complemented
3D4×Dic421C211yesmaximal, characteristic, complemented
4#27212328C211yesmaximal, characteristic, complemented
5C4×C22⋊D421C212yesmaximal, complemented
6#27212326C212yesmaximal, complemented
7#27212324C211yesmaximal, characteristic, complemented
8#27212323C211yesmaximal, characteristic, complemented
9C2×#20269748C212yesmaximal, complemented
10C22⋊C4×D421C212yesmaximal, complemented
11C4×D4×C421C211yesmaximal, characteristic, complemented
12C23×D421C2212yes—
13C2×D1684C2211yescharacteristic
14C2×C22⋊D421C2212yes—
15D4×D421C418yescomplemented
16C22×Dic421C2212yescomplemented
17C2×C4×D421C2211yescharacteristic
18C2×C4×D421C2212yescomplemented
19C2×C4×D421C2212yescomplemented
20C4⋊Dic421C2211yescharacteristic, complemented
21#20269750C2211yescharacteristic, complemented
22#20269750C2211yescharacteristic, complemented
23C22⋊Dic421C2212yescomplemented
24#20269748C2211yescharacteristic
25#20269748C2212yes—
26#20269748C2211yescharacteristic
27#20269748C2212yescomplemented
28#20269748C2212yescomplemented
29C4×Dic421C2211yescharacteristic, complemented
30C4×Dic421C2212yescomplemented
31C2×D4×C421C2211yescharacteristic
32C22×C4×C421C2212yescomplemented
33C4⋊C4×C421C2211yescharacteristic, complemented
34C22⋊C4×C421C2212yescomplemented
35C42×C421C2211yescharacteristic, complemented
36C22×D421—42no—
37C22×D421—42no—
38C22×D421C2×C418yescomplemented
39C22×D421C2311yescharacteristic
40C22×D421C2312yes—
41D1684C2×C414yescomplemented
42C22⋊D421C2×C418yescomplemented
43C2×Dic421—42no—
44C2×Dic421C2311yescharacteristic
45C2×Dic421C2312yes—
46C2×Dic421C2311yescharacteristic
47C2×Dic421C2311yescharacteristic
48C4×D421—42no—
49C4×D421—42no—
50C4×D421C2×C414yescomplemented
51C23×C421C2312yes—
52D4×C421C2×C414yescomplemented
53C2×C4×C421—42no—
54C2×C4×C421C2311yescharacteristic
55C2×C4×C421C2311yescharacteristic
56C2×C4×C421C2312yes—
57C2×C4×C421C2311yescharacteristic
58C22×C421—22no—
59C22×C421—22no—
60C22×C421C22×C414yes—
61C22×C421C2411yescharacteristic
62C2×D421—22no—
63C2×D421—22no—
64C2×D421—21no—
65C2×D421—21no—
66C2×D421—48no—
67C2×D421C22×C412yes—
68C2×D421C22×C414yes—
69C4×C421—21no—
70C4×C421—22no—
71C4×C421—21no—
72C4×C421C22×C412yes—
73Dic421—21no—
74Dic421—22no—
75Dic421—21no—
76Dic421C22×C412yes—
77C2×C421—24no—
78C2×C421C23×C411yescharacteristic
79C2×C421H(5,𝔽2)11yescharacteristic
80C2×C421H(5,𝔽2)11yescharacteristic
81D421—22no—
82D421—24no—
83C421#51811yescharacteristic, complemented, Sylow
84#518—4211nomaximal, Sylow
85C22×D4—4211no—
86C4×D4—4211no—
87C4×D4—4212no—
88C4×D4—4211no—
89C4×D4—4212no—
90C4×D4—4211no—
91C4×D4D42111yescharacteristic, complemented
92#117—4211no—
93#117—4211no—
94C2×C22⋊C4—4212no—
95C2×C22⋊C4—4212no—
96C24—4212no—
97C2×D4—4212no—
98C2×D4—4218no—
99C2×D4—4211no—
100C2×D4C2×D42111yescharacteristic
101C22×C4—4212no—
102C22×C4—4212no—
103C22×C4—4212no—
104C22×C4—4211no—
105C22×C4C2×D42112yescomplemented
106C4⋊C4—4211no—
107C4⋊C4—4211no—
108C4⋊C4—4211no—
109C4⋊C4C2×D42111yescharacteristic, complemented
110C22⋊C4—4212no—
111C22⋊C4—4212no—
112C22⋊C4—4211no—
113C22⋊C4—4211no—
114C22⋊C4—4212no—
115C22⋊C4—4212no—
116C22⋊C4C2×D42112yescomplemented
117C42—4211no—
118C42—4212no—
119C42C2×D42111yescharacteristic, complemented
120C23—16842no—
121C23—16842no—
122C23—4211no—
123C23—4212no—
124C23—4218no—
125C23C22×D42112yes—
126D4—4218no—
127D4—4214no—
128D4C4×D42114yescomplemented
129C2×C4—42no—
130C2×C4—16842no—
131C2×C4—16842no—
132C2×C4—16842no—
133C2×C4—4211no—
134C2×C4—4212no—
135C2×C4—4211no—
136C2×C4—4211no—
137C2×C4—4214no—
138C2×C4C22×D42112yes—
139C2×C4C22×D42111yescharacteristic
140C2×C4C22×D42111yescharacteristic
141C2×C4C22×D42111yescharacteristic
142C22—22no—
143C22—22no—
144C22—8422no—
145C22—8421no—
146C22—8422no—
147C22—8421no—
148C22—16848no—
149C22—4212no—
150C22—4214no—
151C22C2×C4×D42114yes—
152C22C23×D42111yescharacteristic
153C4—21no—
154C4—22no—
155C4—21no—
156C4—8421no—
157C4—8422no—
158C4—8421no—
159C4—4212no—
160C4C2×C4×D42112yes—
161C2—24no—
162C2—8422no—
163C2—8424no—
164C2C22×C4×D42111yescharacteristic
165C2#2721241111yescharacteristic
166C2#2721241211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D421
commutatorC2×C421C23×C4
FittingC4×D4×C421C2
socle, cosocleC22×C421C24
prime radicalC2×D4×D421C2
hypercenterC4×D4D421
hypocenterC421#518
LCSG ▷ C2×C421 ▷ C421
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×D1684C2semidirect
3. D4×Dic421C2semidirect
4. #27212328C2semidirect
5. C4×C22⋊D421C2semidirect
6. #27212326C2semidirect
7. #27212324C2semidirect
8. #27212323C2semidirect
9. C2×#20269748C2semidirect
10. C22⋊C4×D421C2semidirect
11. C4×D4×C421C2semidirect
15. D4×D421C4semidirect
16. C22×Dic421C22semidirect
18. C2×C4×D421C22semidirect
19. C2×C4×D421C22semidirect
20. C4⋊Dic421C22semidirect
21. #20269750C22semidirect
22. #20269750C22semidirect
23. C22⋊Dic421C22semidirect
27. #20269748C22semidirect
28. #20269748C22semidirect
29. C4×Dic421C22semidirect
30. C4×Dic421C22semidirect
32. C22×C4×C421C22semidirect
33. C4⋊C4×C421C22semidirect
34. C22⋊C4×C421C22semidirect
35. C42×C421C22semidirect
38. C22×D421C2×C4semidirect
41. D1684C2×C4semidirect
42. C22⋊D421C2×C4semidirect
50. C4×D421C2×C4semidirect
52. D4×C421C2×C4semidirect
83. C421#518semidirect
91. C4×D4D421semidirect
105. C22×C4C2×D421semidirect
109. C4⋊C4C2×D421semidirect
116. C22⋊C4C2×D421semidirect
119. C42C2×D421semidirect
128. D4C4×D421semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D4212
C4×D16842
D4×Dic4212
#272123282
C4×C22⋊D4212
#272123262
#272123242
#272123232
C2×#202697482
C22⋊C4×D4212
C4×D4×C4212
#518421