Finite Group Databasebeta

← #36378259

Subgroups — #36378259

Order 22592.

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×D353C211yesmaximal, characteristic
2C4×D1412C211yesmaximal, characteristic, complemented
3D4×Dic353C211yesmaximal, characteristic, complemented
4#26014954C211yesmaximal, characteristic, complemented
5C4×C22⋊D353C212yesmaximal, complemented
6#26014952C212yesmaximal, complemented
7#26014950C211yesmaximal, characteristic, complemented
8#26014949C211yesmaximal, characteristic, complemented
9C2×#19150714C212yesmaximal, complemented
10C22⋊C4×D353C212yesmaximal, complemented
11C4×D4×C353C211yesmaximal, characteristic, complemented
12C23×D353C2212yes—
13C2×D1412C2211yescharacteristic
14C2×C22⋊D353C2212yes—
15D4×D353C418yescomplemented
16C22×Dic353C2212yescomplemented
17C2×C4×D353C2211yescharacteristic
18C2×C4×D353C2212yescomplemented
19C2×C4×D353C2212yescomplemented
20C4⋊Dic353C2211yescharacteristic, complemented
21#19150716C2211yescharacteristic, complemented
22#19150716C2211yescharacteristic, complemented
23C22⋊Dic353C2212yescomplemented
24#19150714C2211yescharacteristic
25#19150714C2212yes—
26#19150714C2211yescharacteristic
27#19150714C2212yescomplemented
28#19150714C2212yescomplemented
29C4×Dic353C2211yescharacteristic, complemented
30C4×Dic353C2212yescomplemented
31C2×D4×C353C2211yescharacteristic
32C22×C4×C353C2212yescomplemented
33C4⋊C4×C353C2211yescharacteristic, complemented
34C22⋊C4×C353C2212yescomplemented
35C42×C353C2211yescharacteristic, complemented
36C22×D353—42no—
37C22×D353—42no—
38C22×D353C2×C418yescomplemented
39C22×D353C2311yescharacteristic
40C22×D353C2312yes—
41D1412C2×C414yescomplemented
42C22⋊D353C2×C418yescomplemented
43C2×Dic353—42no—
44C2×Dic353C2311yescharacteristic
45C2×Dic353C2312yes—
46C2×Dic353C2311yescharacteristic
47C2×Dic353C2311yescharacteristic
48C4×D353—42no—
49C4×D353—42no—
50C4×D353C2×C414yescomplemented
51C23×C353C2312yes—
52D4×C353C2×C414yescomplemented
53C2×C4×C353—42no—
54C2×C4×C353C2311yescharacteristic
55C2×C4×C353C2311yescharacteristic
56C2×C4×C353C2312yes—
57C2×C4×C353C2311yescharacteristic
58C22×C353—22no—
59C22×C353—22no—
60C22×C353C22×C414yes—
61C22×C353C2411yescharacteristic
62C2×D353—22no—
63C2×D353—22no—
64C2×D353—21no—
65C2×D353—21no—
66C2×D353—48no—
67C2×D353C22×C412yes—
68C2×D353C22×C414yes—
69C4×C353—21no—
70C4×C353—22no—
71C4×C353—21no—
72C4×C353C22×C412yes—
73Dic353—21no—
74Dic353—22no—
75Dic353—21no—
76Dic353C22×C412yes—
77C2×C353—24no—
78C2×C353C23×C411yescharacteristic
79C2×C353H(5,𝔽2)11yescharacteristic
80C2×C353H(5,𝔽2)11yescharacteristic
81D353—22no—
82D353—24no—
83C353#51811yescharacteristic, complemented, Sylow
84#518—3531nomaximal, Sylow
85C22×D4—3531no—
86C4×D4—3531no—
87C4×D4—3532no—
88C4×D4—3531no—
89C4×D4—3532no—
90C4×D4—3531no—
91C4×D4D35311yescharacteristic, complemented
92#117—3531no—
93#117—3531no—
94C2×C22⋊C4—3532no—
95C2×C22⋊C4—3532no—
96C24—3532no—
97C2×D4—3532no—
98C2×D4—3538no—
99C2×D4—3531no—
100C2×D4C2×D35311yescharacteristic
101C22×C4—3531no—
102C22×C4—3532no—
103C22×C4—3532no—
104C22×C4—3532no—
105C22×C4C2×D35312yescomplemented
106C4⋊C4—3531no—
107C4⋊C4—3531no—
108C4⋊C4—3531no—
109C4⋊C4C2×D35311yescharacteristic, complemented
110C22⋊C4—3532no—
111C22⋊C4—3532no—
112C22⋊C4—3531no—
113C22⋊C4—3531no—
114C22⋊C4—3532no—
115C22⋊C4—3532no—
116C22⋊C4C2×D35312yescomplemented
117C42—3532no—
118C42—3531no—
119C42C2×D35311yescharacteristic, complemented
120C23—3538no—
121C23—3531no—
122C23—3532no—
123C23—14122no—
124C23—14122no—
125C23C22×D35312yes—
126D4—3538no—
127D4—3534no—
128D4C4×D35314yescomplemented
129C2×C4—42no—
130C2×C4—3531no—
131C2×C4—3534no—
132C2×C4—3531no—
133C2×C4—3531no—
134C2×C4—3532no—
135C2×C4—14122no—
136C2×C4—14122no—
137C2×C4—14122no—
138C2×C4C22×D35312yes—
139C2×C4C22×D35311yescharacteristic
140C2×C4C22×D35311yescharacteristic
141C2×C4C22×D35311yescharacteristic
142C22—22no—
143C22—22no—
144C22—3532no—
145C22—3534no—
146C22—14128no—
147C22—7062no—
148C22—7061no—
149C22—7062no—
150C22—7061no—
151C22C2×C4×D35314yes—
152C22C23×D35311yescharacteristic
153C4—21no—
154C4—22no—
155C4—21no—
156C4—3532no—
157C4—7061no—
158C4—7062no—
159C4—7061no—
160C4C2×C4×D35312yes—
161C2—24no—
162C2—7062no—
163C2—7064no—
164C2C22×C4×D35311yescharacteristic
165C2#2601503711yescharacteristic
166C2#2601503811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D353
commutatorC2×C353C23×C4
FittingC4×D4×C353C2
socle, cosocleC22×C353C24
prime radicalC2×D4×D353C2
hypercenterC4×D4D353
hypocenterC353#518
LCSG ▷ C2×C353 ▷ C353
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×D1412C2semidirect
3. D4×Dic353C2semidirect
4. #26014954C2semidirect
5. C4×C22⋊D353C2semidirect
6. #26014952C2semidirect
7. #26014950C2semidirect
8. #26014949C2semidirect
9. C2×#19150714C2semidirect
10. C22⋊C4×D353C2semidirect
11. C4×D4×C353C2semidirect
15. D4×D353C4semidirect
16. C22×Dic353C22semidirect
18. C2×C4×D353C22semidirect
19. C2×C4×D353C22semidirect
20. C4⋊Dic353C22semidirect
21. #19150716C22semidirect
22. #19150716C22semidirect
23. C22⋊Dic353C22semidirect
27. #19150714C22semidirect
28. #19150714C22semidirect
29. C4×Dic353C22semidirect
30. C4×Dic353C22semidirect
32. C22×C4×C353C22semidirect
33. C4⋊C4×C353C22semidirect
34. C22⋊C4×C353C22semidirect
35. C42×C353C22semidirect
38. C22×D353C2×C4semidirect
41. D1412C2×C4semidirect
42. C22⋊D353C2×C4semidirect
50. C4×D353C2×C4semidirect
52. D4×C353C2×C4semidirect
83. C353#518semidirect
91. C4×D4D353semidirect
105. C22×C4C2×D353semidirect
109. C4⋊C4C2×D353semidirect
116. C22⋊C4C2×D353semidirect
119. C42C2×D353semidirect
128. D4C4×D353semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D3532
C4×D14122
D4×Dic3532
#260149542
C4×C22⋊D3532
#260149522
#260149502
#260149492
C2×#191507142
C22⋊C4×D3532
C4×D4×C3532
#518353