Finite Group Databasebeta

← #28409677

Subgroups — #28409677

Order 14528.

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×#16813883C211yesmaximal, characteristic, complemented
2C2×D4×D227C211yesmaximal, characteristic, complemented
3#20300276C211yesmaximal, characteristic, complemented
4#20300272C212yesmaximal, complemented
5#20300268C211yesmaximal, characteristic, complemented
6#20300263C211yesmaximal, characteristic, complemented
7#20300262C212yesmaximal, complemented
8#20300258C212yesmaximal, complemented
9C4×D908C211yesmaximal, characteristic, complemented
10C2×(C2×D227)⋊C4C212yesmaximal, complemented
11C4×D4×C227C211yesmaximal, characteristic, complemented
12C23×D227C2212yes—
13#16813883—24no—
14C2×Dic454C2211yescharacteristic, complemented
15C2×D908C2211yescharacteristic
16C2×D908C2211yescharacteristic
17C2×D908C2212yescomplemented
18C2×C22⋊D227C2212yes—
19C2×C22⋊D227C2212yescomplemented
20D4×D227—24no—
21C22×Dic227C2212yescomplemented
22C2×C4×D227C2211yescharacteristic
23C2×C4×D227C2211yescharacteristic, complemented
24C4⋊Dic227C2211yescharacteristic, complemented
25C4⋊Dic227C2212yescomplemented
26(C2×D227)⋊C4C2211yescharacteristic
27(C2×D227)⋊C4C2212yes—
28(C2×D227)⋊C4C2212yescomplemented
29(C2×D227)⋊C4C2212yescomplemented
30(C2×D227)⋊C4C2211yescharacteristic, complemented
31(C2×D227)⋊C4C2212yescomplemented
32C2×D4×C227C2211yescharacteristic
33C22×C4×C227C2212yescomplemented
34C4⋊C4×C227C2211yescharacteristic, complemented
35C22⋊C4×C227C2212yescomplemented
36C42×C227C2211yescharacteristic, complemented
37C22×D227—24no—
38C22×D227—24no—
39C22×D227—42no—
40C22×D227C2312yes—
41C22×D227C2311yescharacteristic
42C22×D227C2311yescharacteristic
43Dic454—22no—
44D908—22no—
45D908—22no—
46D908—42no—
47C22⋊D227—24no—
48C22⋊D227—24no—
49C2×Dic227—24no—
50C2×Dic227C2311yescharacteristic
51C2×Dic227C2312yes—
52C2×Dic227C2311yescharacteristic
53C4×D227—22no—
54C4×D227—22no—
55C23×C227C2312yes—
56D4×C227D414yescomplemented
57C2×C4×C227—42no—
58C2×C4×C227C2311yescharacteristic
59C2×C4×C227C2311yescharacteristic
60C2×C4×C227C2312yes—
61C2×C4×C227C2311yescharacteristic
62C22×C227—22no—
63C22×C227—22no—
64C22×C227C2×D414yes—
65C22×C227C2411yescharacteristic
66C2×D227—21no—
67C2×D227—21no—
68C2×D227—21no—
69C2×D227—22no—
70C2×D227—21no—
71C2×D227—22no—
72C2×D227—21no—
73C2×D227—22no—
74C2×D227—48no—
75C2×D227#4112yescomplemented
76C4×C227—21no—
77C4×C227—21no—
78C4×C227—22no—
79C4×C227C2×D412yes—
80Dic227—21no—
81Dic227—21no—
82Dic227—22no—
83C2×C227—24no—
84C2×C227C22×D411yescharacteristic
85C2×C227C2×#4111yescharacteristic
86C2×C227H(5,𝔽2)11yescharacteristic
87D227—22no—
88D227—41no—
89D227—42no—
90C227#54611yescharacteristic, complemented, Sylow
91#546—2271nomaximal, Sylow
92C2×#41—2271no—
93C22×D4—2271no—
94#124—2271no—
95#123—2272no—
96C22⋊Q8—2271no—
97#121—2272no—
98#121—2271no—
99#120—2272no—
100C4×D4—2271no—
101C4×D4D22711yescharacteristic, complemented
102C2×C22⋊C4—2272no—
103C24—2272no—
104#41—4544no—
105C2×Q8—2271no—
106C2×D4—4544no—
107C2×D4—2272no—
108C2×D4—2272no—
109C2×D4—2272no—
110C2×D4—2271no—
111C2×D4—2271no—
112C2×D4C2×D22711yescharacteristic
113C22×C4—2272no—
114C22×C4—2271no—
115C22×C4—2271no—
116C22×C4C2×D22712yescomplemented
117C4⋊C4—2271no—
118C4⋊C4—2272no—
119C4⋊C4C2×D22711yescharacteristic, complemented
120C22⋊C4—2271no—
121C22⋊C4—2272no—
122C22⋊C4—2272no—
123C22⋊C4—2272no—
124C22⋊C4—2271no—
125C22⋊C4—2272no—
126C22⋊C4C2×D22712yescomplemented
127C42C2×D22711yescharacteristic, complemented
128C23—9082no—
129C23—4544no—
130C23—4544no—
131C23—2272no—
132C23—2271no—
133C23—2271no—
134C23C22×D22712yes—
135Q8—4542no—
136D4—9082no—
137D4—4542no—
138D4—4544no—
139D4—4544no—
140D4—4542no—
141D4D90814yescomplemented
142C2×C4—42no—
143C2×C4—4544no—
144C2×C4—4542no—
145C2×C4—4542no—
146C2×C4—2271no—
147C2×C4—2272no—
148C2×C4—2271no—
149C2×C4C22×D22711yescharacteristic
150C2×C4C22×D22711yescharacteristic
151C2×C4C22×D22712yes—
152C2×C4C22×D22711yescharacteristic
153C22—22no—
154C22—22no—
155C22—9088no—
156C22—4541no—
157C22—4542no—
158C22—4541no—
159C22—4542no—
160C22—4541no—
161C22—4542no—
162C22—4541no—
163C22—4541no—
164C22—2272no—
165C22C2×D90814yes—
166C22C23×D22711yescharacteristic
167C4—21no—
168C4—21no—
169C4—22no—
170C4—4541no—
171C4—4541no—
172C4—4542no—
173C4C2×D90812yes—
174C2—24no—
175C2—9082no—
176C2—9081no—
177C2—4542no—
178C2C22×D90811yescharacteristic
179C2#41×D22711yescharacteristic
180C2#2030033311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D227
commutator, socle, cosocleC22×C227C24
FittingC4×D4×C227C2
hypercenterC4×D4D227
hypocenterC227#546
LCSG ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16813883C2semidirect
2. C2×D4×D227C2semidirect
3. #20300276C2semidirect
4. #20300272C2semidirect
5. #20300268C2semidirect
6. #20300263C2semidirect
7. #20300262C2semidirect
8. #20300258C2semidirect
9. C4×D908C2semidirect
10. C2×(C2×D227)⋊C4C2semidirect
11. C4×D4×C227C2semidirect
14. C2×Dic454C22semidirect
17. C2×D908C22semidirect
19. C2×C22⋊D227C22semidirect
21. C22×Dic227C22semidirect
23. C2×C4×D227C22semidirect
24. C4⋊Dic227C22semidirect
25. C4⋊Dic227C22semidirect
28. (C2×D227)⋊C4C22semidirect
29. (C2×D227)⋊C4C22semidirect
30. (C2×D227)⋊C4C22semidirect
31. (C2×D227)⋊C4C22semidirect
33. C22×C4×C227C22semidirect
34. C4⋊C4×C227C22semidirect
35. C22⋊C4×C227C22semidirect
36. C42×C227C22semidirect
56. D4×C227D4semidirect
75. C2×D227#41semidirect
90. C227#546semidirect
101. C4×D4D227semidirect
116. C22×C4C2×D227semidirect
119. C4⋊C4C2×D227semidirect
126. C22⋊C4C2×D227semidirect
127. C42C2×D227semidirect
141. D4D908semidirect

Maximal subgroups

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

subgroupindex
C2×#168138832
C2×D4×D2272
#203002762
#203002722
#203002682
#203002632
#203002622
#203002582
C4×D9082
C2×(C2×D227)⋊C42
C4×D4×C2272
#546227