Finite Group Databasebeta

← #28409590

Subgroups — #28409590

Order 14528.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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×D227C212yesmaximal, complemented
2#20300273C212yesmaximal, complemented
3#20300270C212yesmaximal, complemented
4#20300262C211yesmaximal, characteristic, complemented
5#20300260C212yesmaximal, complemented
6#20300258C212yesmaximal, complemented
7#20300256C212yesmaximal, complemented
8C2×(C2×D227)⋊C4C211yesmaximal, characteristic, complemented
9#121×C227C211yesmaximal, characteristic, complemented
10C23×D227C2211yescharacteristic
11C23×D227C2212yescomplemented
12C2×D908C2212yes—
13C2×D908C2211yescharacteristic, complemented
14C2×C22⋊D227C2212yes—
15C2×C22⋊D227C2212yescomplemented
16C2×C22⋊D227C2212yescomplemented
17D4×D227—28no—
18C22×Dic227C2211yescharacteristic, complemented
19C2×C4×D227C2212yescomplemented
20C4⋊Dic227C2211yescharacteristic, complemented
21Dic227⋊C4C2212yescomplemented
22C22⋊Dic227C2212yescomplemented
23(C2×D227)⋊C4C2212yescomplemented
24(C2×D227)⋊C4C2212yescomplemented
25(C2×D227)⋊C4C2212yescomplemented
26(C2×D227)⋊C4C2212yescomplemented
27C2×D4×C227C2212yescomplemented
28C2×D4×C227C2211yescharacteristic, complemented
29C22×C4×C227C2211yescharacteristic, complemented
30C4⋊C4×C227C2211yescharacteristic, complemented
31C22⋊C4×C227C2212yescomplemented
32C22×D227—24no—
33C22×D227—41no—
34C22×D227—41no—
35C22×D227—42no—
36C22×D227—42no—
37C22×D227D414yescomplemented
38C22×D227C2312yes—
39C22×D227C2312yes—
40D908—24no—
41D908—41no—
42C22⋊D227—24no—
43C22⋊D227—24no—
44C22⋊D227—42no—
45C2×Dic227—41no—
46C2×Dic227C2312yes—
47C2×Dic227C2312yes—
48C4×D227—24no—
49C23×C227C2312yes—
50C23×C227C2311yescharacteristic
51D4×C227—24no—
52D4×C227—41no—
53C2×C4×C227—41no—
54C2×C4×C227C2311yescharacteristic
55C2×C4×C227C2312yes—
56C2×C4×C227C2311yescharacteristic
57C22×C227—21no—
58C22×C227—21no—
59C22×C227—22no—
60C22×C227—22no—
61C22×C227—22no—
62C22×C227C2×D412yes—
63C22×C227C2411yescharacteristic
64C2×D227—22no—
65C2×D227—22no—
66C2×D227—22no—
67C2×D227—22no—
68C2×D227—22no—
69C2×D227—44no—
70C2×D227—48no—
71C2×D227C2×D414yes—
72C4×C227—22no—
73C4×C227—21no—
74C4×C227—21no—
75Dic227—22no—
76Dic227—22no—
77C2×C227—22no—
78C2×C227—42no—
79C2×C227C22×D411yescharacteristic
80C2×C227H(5,𝔽2)11yescharacteristic
81C2×C227H(5,𝔽2)11yescharacteristic
82D227—24no—
83D227—42no—
84C227#53411yescharacteristic, complemented, Sylow
85#534—2271nomaximal, Sylow
86C22×D4—2272no—
87#123—2272no—
88#123—2272no—
89#121—2272no—
90#121—2271no—
91#121D22711yescharacteristic, complemented
92#120—2272no—
93#120—2272no—
94C2×C22⋊C4—2271no—
95C24—2272no—
96C24—2271no—
97C2×D4—4548no—
98C2×D4—2272no—
99C2×D4—2272no—
100C2×D4—2272no—
101C2×D4—2271no—
102C2×D4—2272no—
103C2×D4C2×D22712yescomplemented
104C2×D4C2×D22711yescharacteristic, complemented
105C22×C4—2271no—
106C22×C4—2272no—
107C22×C4C2×D22711yescharacteristic, complemented
108C4⋊C4—2272no—
109C4⋊C4—2271no—
110C4⋊C4C2×D22711yescharacteristic, complemented
111C22⋊C4—2272no—
112C22⋊C4—2272no—
113C22⋊C4—2272no—
114C22⋊C4—2272no—
115C22⋊C4—2272no—
116C22⋊C4C2×D22712yescomplemented
117C23—9081no—
118C23—9081no—
119C23—9082no—
120C23—9082no—
121C23—4544no—
122C23—2274no—
123C23—2272no—
124C23—2272no—
125C23C22×D22712yes—
126C23C22×D22711yescharacteristic
127D4—24no—
128D4—41no—
129D4—9081no—
130D4—9082no—
131D4—4544no—
132D4—4544no—
133D4—4544no—
134C2×C4—41no—
135C2×C4—9081no—
136C2×C4—4544no—
137C2×C4—2272no—
138C2×C4—2272no—
139C2×C4C22×D22711yescharacteristic
140C2×C4C22×D22711yescharacteristic
141C2×C4C22×D22712yes—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—22no—
146C22—22no—
147C22—9084no—
148C22—9088no—
149C22—4542no—
150C22—4542no—
151C22—4542no—
152C22—4542no—
153C22—4542no—
154C22—2274no—
155C22D4×D22712yes—
156C22C23×D22711yescharacteristic
157C4—22no—
158C4—21no—
159C4—21no—
160C4—4542no—
161C4—4542no—
162C2—22no—
163C2—42no—
164C2—9082no—
165C2—4544no—
166C2C2×D4×D22711yescharacteristic
167C2#2030033311yescharacteristic
168C2#2030033411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D227
commutator, socle, cosocleC22×C227C24
Fitting#121×C227C2
hypercenter#121D227
hypocenterC227#534
LCSG ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×D227C2semidirect
2. #20300273C2semidirect
3. #20300270C2semidirect
4. #20300262C2semidirect
5. #20300260C2semidirect
6. #20300258C2semidirect
7. #20300256C2semidirect
8. C2×(C2×D227)⋊C4C2semidirect
9. #121×C227C2semidirect
11. C23×D227C22semidirect
13. C2×D908C22semidirect
15. C2×C22⋊D227C22semidirect
16. C2×C22⋊D227C22semidirect
18. C22×Dic227C22semidirect
19. C2×C4×D227C22semidirect
20. C4⋊Dic227C22semidirect
21. Dic227⋊C4C22semidirect
22. C22⋊Dic227C22semidirect
23. (C2×D227)⋊C4C22semidirect
24. (C2×D227)⋊C4C22semidirect
25. (C2×D227)⋊C4C22semidirect
26. (C2×D227)⋊C4C22semidirect
27. C2×D4×C227C22semidirect
28. C2×D4×C227C22semidirect
29. C22×C4×C227C22semidirect
30. C4⋊C4×C227C22semidirect
31. C22⋊C4×C227C22semidirect
37. C22×D227D4semidirect
84. C227#534semidirect
91. #121D227semidirect
103. C2×D4C2×D227semidirect
104. C2×D4C2×D227semidirect
107. C22×C4C2×D227semidirect
110. C4⋊C4C2×D227semidirect
116. C22⋊C4C2×D227semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D2272
#203002732
#203002702
#203002622
#203002602
#203002582
#203002562
C2×(C2×D227)⋊C42
#121×C2272
#534227