Finite Group Databasebeta

← #28409676

Subgroups — #28409676

Order 14528.

334 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 115 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
2#20300275C211yesmaximal, characteristic, complemented
3#20300269C212yesmaximal, complemented
4#20300267C211yesmaximal, characteristic, complemented
5#20300260C212yesmaximal, complemented
6#20300259C211yesmaximal, characteristic, complemented
7#20300257C212yesmaximal, complemented
8D4×Dic227C211yesmaximal, characteristic, complemented
9C4×C22⋊D227C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic227C212yesmaximal, complemented
11C22×D4×C227C211yesmaximal, characteristic, complemented
12#16813883—24no—
13C2×Dic454C2211yescharacteristic, complemented
14C2×C22⋊D227C2211yescharacteristic
15C2×C22⋊D227C2212yescomplemented
16C2×C22⋊D227C2212yescomplemented
17C22×Dic227C2212yescomplemented
18C22×Dic227C2212yescomplemented
19C2×C4×D227C2211yescharacteristic, complemented
20C4⋊Dic227C2211yescharacteristic, complemented
21Dic227⋊C4C2211yescharacteristic, complemented
22Dic227⋊C4C2212yescomplemented
23C22⋊Dic227C2211yescharacteristic
24C22⋊Dic227C2212yes—
25C22⋊Dic227C2212yescomplemented
26C22⋊Dic227C2212yescomplemented
27C22⋊Dic227C2212yescomplemented
28C22⋊Dic227C2212yescomplemented
29(C2×D227)⋊C4C2211yescharacteristic, complemented
30C4×Dic227C2211yescharacteristic, complemented
31C24×C227C2212yes—
32C2×D4×C227—24no—
33C2×D4×C227C2211yescharacteristic
34C2×D4×C227C2212yes—
35C2×D4×C227C2211yescharacteristic
36C22×C4×C227C2211yescharacteristic
37C22×D227C2311yescharacteristic
38Dic454—22no—
39C22⋊D227—24no—
40C22⋊D227—22no—
41C22⋊D227—42no—
42C2×Dic227—24no—
43C2×Dic227—42no—
44C2×Dic227C2311yescharacteristic
45C2×Dic227C2311yescharacteristic
46C2×Dic227C2312yes—
47C2×Dic227C2311yescharacteristic
48C2×Dic227C2312yes—
49C4×D227—22no—
50C23×C227—24no—
51C23×C227—24no—
52C23×C227—42no—
53C23×C227C2312yes—
54C23×C227C2312yes—
55C23×C227C2311yescharacteristic
56D4×C227—24no—
57D4×C227—22no—
58D4×C227D414yescomplemented
59C2×C4×C227—22no—
60C2×C4×C227C2311yescharacteristic
61C2×C4×C227C2311yescharacteristic
62C22×C227—21no—
63C22×C227—22no—
64C22×C227—21no—
65C22×C227—22no—
66C22×C227—22no—
67C22×C227—22no—
68C22×C227—22no—
69C22×C227—48no—
70C22×C227C2×D414yes—
71C22×C227#4112yescomplemented
72C22×C227C2411yescharacteristic
73C2×D227—21no—
74C2×D227—21no—
75C2×D227—21no—
76C4×C227—21no—
77C4×C227C2×D412yes—
78Dic227—21no—
79Dic227—22no—
80Dic227—21no—
81Dic227—21no—
82Dic227—22no—
83C2×C227—22no—
84C2×C227—24no—
85C2×C227—42no—
86C2×C227C22×D411yescharacteristic
87C2×C227C2×#4111yescharacteristic
88C2×C227H(5,𝔽2)11yescharacteristic
89D227—41no—
90C227#54611yescharacteristic, complemented, Sylow
91#546—2271nomaximal, Sylow
92C2×#41—2271no—
93C22×D4D22711yescharacteristic, complemented
94#124—2271no—
95#123—2272no—
96C22⋊Q8—2271no—
97#121—2271no—
98#121—2272no—
99#120—2272no—
100C4×D4—2271no—
101C4×D4—2271no—
102C2×C22⋊C4—2272no—
103C24C2×D22712yes—
104#41—4544no—
105C2×Q8—2271no—
106C2×D4—24no—
107C2×D4—2272no—
108C2×D4—2271no—
109C2×D4—2272no—
110C2×D4C2×D22712yes—
111C2×D4C2×D22711yescharacteristic
112C2×D4C2×D22711yescharacteristic
113C22×C4—2272no—
114C22×C4—2272no—
115C22×C4—2271no—
116C22×C4C2×D22711yescharacteristic
117C4⋊C4—2271no—
118C4⋊C4—2272no—
119C4⋊C4—2271no—
120C22⋊C4—2271no—
121C22⋊C4—2272no—
122C22⋊C4—2272no—
123C22⋊C4—2272no—
124C22⋊C4—2272no—
125C22⋊C4—2272no—
126C22⋊C4—2271no—
127C42—2271no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—2271no—
132C23C22×D22712yes—
133C23C22×D22712yes—
134C23C22×D22711yescharacteristic
135Q8—4542no—
136D4—24no—
137D4—22no—
138D4—9082no—
139D4—4544no—
140D4—4542no—
141D4C22⋊D22714yescomplemented
142C2×C4—22no—
143C2×C4—9082no—
144C2×C4—4544no—
145C2×C4—4542no—
146C2×C4—2271no—
147C2×C4—2271no—
148C2×C4—2272no—
149C2×C4—2271no—
150C2×C4—2272no—
151C2×C4C22×D22711yescharacteristic
152C2×C4C22×D22711yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—4541no—
162C22—4541no—
163C22—4541no—
164C22C2×C22⋊D22714yes—
165C22#1681388312yescomplemented
166C22C23×D22711yescharacteristic
167C4—21no—
168C4—4541no—
169C4—4542no—
170C4—4541no—
171C4—4541no—
172C4—4542no—
173C4C2×C22⋊D22712yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—9081no—
178C2C22×C22⋊D22711yescharacteristic
179C2C2×#1681388311yescharacteristic
180C2#2030033311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D227
commutator, socle, cosocleC22×C227C24
FittingC22×D4×C227C2
hypercenterC22×D4D227
hypocenterC227#546
LCSG ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16813883C2semidirect
2. #20300275C2semidirect
3. #20300269C2semidirect
4. #20300267C2semidirect
5. #20300260C2semidirect
6. #20300259C2semidirect
7. #20300257C2semidirect
8. D4×Dic227C2semidirect
9. C4×C22⋊D227C2semidirect
10. C2×C22⋊Dic227C2semidirect
11. C22×D4×C227C2semidirect
13. C2×Dic454C22semidirect
15. C2×C22⋊D227C22semidirect
16. C2×C22⋊D227C22semidirect
17. C22×Dic227C22semidirect
18. C22×Dic227C22semidirect
19. C2×C4×D227C22semidirect
20. C4⋊Dic227C22semidirect
21. Dic227⋊C4C22semidirect
22. Dic227⋊C4C22semidirect
25. C22⋊Dic227C22semidirect
26. C22⋊Dic227C22semidirect
27. C22⋊Dic227C22semidirect
28. C22⋊Dic227C22semidirect
29. (C2×D227)⋊C4C22semidirect
30. C4×Dic227C22semidirect
58. D4×C227D4semidirect
71. C22×C227#41semidirect
90. C227#546semidirect
93. C22×D4D227semidirect
141. D4C22⋊D227semidirect
165. C22#16813883semidirect

Maximal subgroups

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

subgroupindex
C2×#168138832
#203002752
#203002692
#203002672
#203002602
#203002592
#203002572
D4×Dic2272
C4×C22⋊D2272
C2×C22⋊Dic2272
C22×D4×C2272
#546227