Finite Group Databasebeta

← #28409679

Subgroups — #28409679

Order 14528.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16813884C211yesmaximal, characteristic, complemented
2C2×D4×D227C211yesmaximal, characteristic, complemented
3#20300275C211yesmaximal, characteristic, complemented
4#20300270C212yesmaximal, complemented
5#20300268C211yesmaximal, characteristic, complemented
6#20300259C212yesmaximal, complemented
7#20300256C212yesmaximal, complemented
8#20300250C211yesmaximal, characteristic, complemented
9C4×C22⋊D227C211yesmaximal, characteristic, complemented
10C22⋊C4×D227C212yesmaximal, complemented
11#121×C227C211yesmaximal, characteristic, complemented
12C23×D227C2212yes—
13#16813884—24no—
14C2×Dic454C2211yescharacteristic, complemented
15C2×D908C2211yescharacteristic
16C2×C22⋊D227C2212yes—
17C2×C22⋊D227C2211yescharacteristic
18C2×C22⋊D227C2212yescomplemented
19D4×D227—24no—
20C2×C4×D227C2211yescharacteristic
21C2×C4×D227C2212yescomplemented
22C2×C4×D227C2212yescomplemented
23C4⋊Dic227C2212yescomplemented
24Dic227⋊C4C2211yescharacteristic, complemented
25C22⋊Dic227C2212yescomplemented
26C22⋊Dic227C2211yescharacteristic, complemented
27C22⋊Dic227C2212yescomplemented
28(C2×D227)⋊C4C2211yescharacteristic
29(C2×D227)⋊C4C2212yes—
30(C2×D227)⋊C4C2212yescomplemented
31C4×Dic227C2211yescharacteristic, complemented
32C2×D4×C227C2211yescharacteristic
33C2×D4×C227C2212yescomplemented
34C22×C4×C227C2211yescharacteristic, complemented
35C4⋊C4×C227C2211yescharacteristic, complemented
36C22⋊C4×C227C2212yescomplemented
37C22×D227—24no—
38C22×D227—24no—
39C22×D227—42no—
40C22×D227C2312yes—
41C22×D227C2311yescharacteristic
42Dic454—22no—
43D908D414yescomplemented
44C22⋊D227—24no—
45C22⋊D227—24no—
46C22⋊D227—22no—
47C2×Dic227C2311yescharacteristic
48C2×Dic227C2311yescharacteristic
49C2×Dic227C2312yes—
50C2×Dic227C2311yescharacteristic
51C4×D227—24no—
52C4×D227—22no—
53C4×D227—42no—
54C23×C227C2312yes—
55C23×C227C2311yescharacteristic
56D4×C227—22no—
57D4×C227—42no—
58C2×C4×C227—22no—
59C2×C4×C227C2311yescharacteristic
60C2×C4×C227C2312yes—
61C2×C4×C227C2311yescharacteristic
62C22×C227—21no—
63C22×C227—22no—
64C22×C227—21no—
65C22×C227—22no—
66C22×C227—21no—
67C22×C227—22no—
68C22×C227C2411yescharacteristic
69C2×D227—21no—
70C2×D227—22no—
71C2×D227—22no—
72C2×D227—21no—
73C2×D227—48no—
74C2×D227C2×D414yes—
75C2×D227#4112yescomplemented
76C4×C227—21no—
77C4×C227—22no—
78C4×C227C2×D412yes—
79Dic227—21no—
80Dic227—22no—
81Dic227—21no—
82Dic227—21no—
83C2×C227—42no—
84C2×C227—41no—
85C2×C227C22×D411yescharacteristic
86C2×C227C2×#4111yescharacteristic
87C2×C227H(5,𝔽2)11yescharacteristic
88D227—22no—
89D227—24no—
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#121D22711yescharacteristic, complemented
99#120—2272no—
100C4×D4—2271no—
101C4×D4—2271no—
102C2×C22⋊C4—2272no—
103C24—2272no—
104#41—4544no—
105C2×Q8—2271no—
106C2×D4—4544no—
107C2×D4—2271no—
108C2×D4—2272no—
109C2×D4—2272no—
110C2×D4—2271no—
111C2×D4C2×D22711yescharacteristic
112C2×D4C2×D22712yescomplemented
113C22×C4—2272no—
114C22×C4—2271no—
115C22×C4—2272no—
116C22×C4C2×D22711yescharacteristic, complemented
117C4⋊C4—2271no—
118C4⋊C4—2272no—
119C4⋊C4C2×D22711yescharacteristic, complemented
120C22⋊C4—2272no—
121C22⋊C4—2271no—
122C22⋊C4—2272no—
123C22⋊C4—2271no—
124C22⋊C4—2272no—
125C22⋊C4—2272no—
126C22⋊C4C2×D22712yescomplemented
127C42—2271no—
128C23—9082no—
129C23—4544no—
130C23—4544no—
131C23—2271no—
132C23—2272no—
133C23C22×D22712yes—
134C23C22×D22711yescharacteristic
135Q8—4542no—
136D4—22no—
137D4—42no—
138D4—4544no—
139D4—4544no—
140D4—4542no—
141D4—2274no—
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×C4C22×D22712yes—
151C2×C4C22×D22711yescharacteristic
152C2×C4C22×D22711yescharacteristic
153C22—21no—
154C22—22no—
155C22—21no—
156C22—22no—
157C22—21no—
158C22—22no—
159C22—9088no—
160C22—4541no—
161C22—4542no—
162C22—4542no—
163C22—4541no—
164C22—2274no—
165C22—2272no—
166C22C23×D22711yescharacteristic
167C4—21no—
168C4—22no—
169C4—4541no—
170C4—4542no—
171C4—4541no—
172C4—4541no—
173C4D4×D22712yes—
174C2—42no—
175C2—41no—
176C2—4542no—
177C2—4544no—
178C2C2×D4×D22711yescharacteristic
179C2#41×D22711yescharacteristic
180C2#2030033311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D227
commutator, socle, cosocleC22×C227C24
Fitting#121×C227C2
hypercenter#121D227
hypocenterC227#546
LCSG ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#16813884C2semidirect
2. C2×D4×D227C2semidirect
3. #20300275C2semidirect
4. #20300270C2semidirect
5. #20300268C2semidirect
6. #20300259C2semidirect
7. #20300256C2semidirect
8. #20300250C2semidirect
9. C4×C22⋊D227C2semidirect
10. C22⋊C4×D227C2semidirect
11. #121×C227C2semidirect
14. C2×Dic454C22semidirect
18. C2×C22⋊D227C22semidirect
21. C2×C4×D227C22semidirect
22. C2×C4×D227C22semidirect
23. C4⋊Dic227C22semidirect
24. Dic227⋊C4C22semidirect
25. C22⋊Dic227C22semidirect
26. C22⋊Dic227C22semidirect
27. C22⋊Dic227C22semidirect
30. (C2×D227)⋊C4C22semidirect
31. C4×Dic227C22semidirect
33. C2×D4×C227C22semidirect
34. C22×C4×C227C22semidirect
35. C4⋊C4×C227C22semidirect
36. C22⋊C4×C227C22semidirect
43. D908D4semidirect
75. C2×D227#41semidirect
90. C227#546semidirect
98. #121D227semidirect
112. C2×D4C2×D227semidirect
116. C22×C4C2×D227semidirect
119. C4⋊C4C2×D227semidirect
126. C22⋊C4C2×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#168138842
C2×D4×D2272
#203002752
#203002702
#203002682
#203002592
#203002562
#203002502
C4×C22⋊D2272
C22⋊C4×D2272
#121×C2272
#546227