Finite Group Databasebeta

← #28409504

Subgroups — #28409504

Order 14528.

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×D227C211yesmaximal, characteristic
2C4×D908C211yesmaximal, characteristic, complemented
3D4×Dic227C211yesmaximal, characteristic, complemented
4#20300250C211yesmaximal, characteristic, complemented
5C4×C22⋊D227C212yesmaximal, complemented
6(C22⋊D227)⋊C4C212yesmaximal, complemented
7#20300246C211yesmaximal, characteristic, complemented
8#20300245C211yesmaximal, characteristic, complemented
9C2×(C2×D227)⋊C4C212yesmaximal, complemented
10C22⋊C4×D227C212yesmaximal, complemented
11C4×D4×C227C211yesmaximal, characteristic, complemented
12C23×D227C2212yes—
13C2×D908C2211yescharacteristic
14C2×C22⋊D227C2212yes—
15D4×D227C418yescomplemented
16C22×Dic227C2212yescomplemented
17C2×C4×D227C2211yescharacteristic
18C2×C4×D227C2212yescomplemented
19C2×C4×D227C2212yescomplemented
20C4⋊Dic227C2211yescharacteristic, complemented
21Dic227⋊C4C2211yescharacteristic, complemented
22Dic227⋊C4C2211yescharacteristic, complemented
23C22⋊Dic227C2212yescomplemented
24(C2×D227)⋊C4C2211yescharacteristic
25(C2×D227)⋊C4C2212yes—
26(C2×D227)⋊C4C2211yescharacteristic
27(C2×D227)⋊C4C2212yescomplemented
28(C2×D227)⋊C4C2212yescomplemented
29C4×Dic227C2211yescharacteristic, complemented
30C4×Dic227C2212yescomplemented
31C2×D4×C227C2211yescharacteristic
32C22×C4×C227C2212yescomplemented
33C4⋊C4×C227C2211yescharacteristic, complemented
34C22⋊C4×C227C2212yescomplemented
35C42×C227C2211yescharacteristic, complemented
36C22×D227—42no—
37C22×D227—42no—
38C22×D227C2×C418yescomplemented
39C22×D227C2311yescharacteristic
40C22×D227C2312yes—
41D908C2×C414yescomplemented
42C22⋊D227C2×C418yescomplemented
43C2×Dic227—42no—
44C2×Dic227C2311yescharacteristic
45C2×Dic227C2312yes—
46C2×Dic227C2311yescharacteristic
47C2×Dic227C2311yescharacteristic
48C4×D227—42no—
49C4×D227—42no—
50C4×D227C2×C414yescomplemented
51C23×C227C2312yes—
52D4×C227C2×C414yescomplemented
53C2×C4×C227—42no—
54C2×C4×C227C2311yescharacteristic
55C2×C4×C227C2311yescharacteristic
56C2×C4×C227C2312yes—
57C2×C4×C227C2311yescharacteristic
58C22×C227—22no—
59C22×C227—22no—
60C22×C227C22×C414yes—
61C22×C227C2411yescharacteristic
62C2×D227—22no—
63C2×D227—22no—
64C2×D227—21no—
65C2×D227—21no—
66C2×D227—48no—
67C2×D227C22×C412yes—
68C2×D227C22×C414yes—
69C4×C227—21no—
70C4×C227—22no—
71C4×C227—21no—
72C4×C227C22×C412yes—
73Dic227—21no—
74Dic227—22no—
75Dic227—21no—
76Dic227C22×C412yes—
77C2×C227—24no—
78C2×C227C23×C411yescharacteristic
79C2×C227H(5,𝔽2)11yescharacteristic
80C2×C227H(5,𝔽2)11yescharacteristic
81D227—22no—
82D227—24no—
83C227#51811yescharacteristic, complemented, Sylow
84#518—2271nomaximal, Sylow
85C22×D4—2271no—
86C4×D4—2271no—
87C4×D4—2272no—
88C4×D4—2271no—
89C4×D4—2272no—
90C4×D4—2271no—
91C4×D4D22711yescharacteristic, complemented
92#117—2271no—
93#117—2271no—
94C2×C22⋊C4—2272no—
95C2×C22⋊C4—2272no—
96C24—2272no—
97C2×D4—2272no—
98C2×D4—2278no—
99C2×D4—2271no—
100C2×D4C2×D22711yescharacteristic
101C22×C4—2272no—
102C22×C4—2272no—
103C22×C4—2272no—
104C22×C4—2271no—
105C22×C4C2×D22712yescomplemented
106C4⋊C4—2271no—
107C4⋊C4—2271no—
108C4⋊C4—2271no—
109C4⋊C4C2×D22711yescharacteristic, complemented
110C22⋊C4—2272no—
111C22⋊C4—2272no—
112C22⋊C4—2271no—
113C22⋊C4—2271no—
114C22⋊C4—2272no—
115C22⋊C4—2272no—
116C22⋊C4C2×D22712yescomplemented
117C42—2271no—
118C42—2272no—
119C42C2×D22711yescharacteristic, complemented
120C23—9082no—
121C23—9082no—
122C23—2278no—
123C23—2271no—
124C23—2272no—
125C23C22×D22712yes—
126D4—2278no—
127D4—2274no—
128D4C4×D22714yescomplemented
129C2×C4—42no—
130C2×C4—9082no—
131C2×C4—9082no—
132C2×C4—9082no—
133C2×C4—2271no—
134C2×C4—2272no—
135C2×C4—2271no—
136C2×C4—2271no—
137C2×C4—2274no—
138C2×C4C22×D22712yes—
139C2×C4C22×D22711yescharacteristic
140C2×C4C22×D22711yescharacteristic
141C2×C4C22×D22711yescharacteristic
142C22—22no—
143C22—22no—
144C22—9088no—
145C22—4542no—
146C22—4541no—
147C22—4542no—
148C22—4541no—
149C22—2272no—
150C22—2274no—
151C22C2×C4×D22714yes—
152C22C23×D22711yescharacteristic
153C4—21no—
154C4—22no—
155C4—21no—
156C4—4541no—
157C4—4542no—
158C4—4541no—
159C4—2272no—
160C4C2×C4×D22712yes—
161C2—24no—
162C2—4542no—
163C2—4544no—
164C2C22×C4×D22711yescharacteristic
165C2#2030033311yescharacteristic
166C2#2030033411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D227
commutatorC2×C227C23×C4
FittingC4×D4×C227C2
socle, cosocleC22×C227C24
prime radicalC2×D4×D227C2
hypercenterC4×D4D227
hypocenterC227#518
LCSG ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×D908C2semidirect
3. D4×Dic227C2semidirect
4. #20300250C2semidirect
5. C4×C22⋊D227C2semidirect
6. (C22⋊D227)⋊C4C2semidirect
7. #20300246C2semidirect
8. #20300245C2semidirect
9. C2×(C2×D227)⋊C4C2semidirect
10. C22⋊C4×D227C2semidirect
11. C4×D4×C227C2semidirect
15. D4×D227C4semidirect
16. C22×Dic227C22semidirect
18. C2×C4×D227C22semidirect
19. C2×C4×D227C22semidirect
20. C4⋊Dic227C22semidirect
21. Dic227⋊C4C22semidirect
22. Dic227⋊C4C22semidirect
23. C22⋊Dic227C22semidirect
27. (C2×D227)⋊C4C22semidirect
28. (C2×D227)⋊C4C22semidirect
29. C4×Dic227C22semidirect
30. C4×Dic227C22semidirect
32. C22×C4×C227C22semidirect
33. C4⋊C4×C227C22semidirect
34. C22⋊C4×C227C22semidirect
35. C42×C227C22semidirect
38. C22×D227C2×C4semidirect
41. D908C2×C4semidirect
42. C22⋊D227C2×C4semidirect
50. C4×D227C2×C4semidirect
52. D4×C227C2×C4semidirect
83. C227#518semidirect
91. C4×D4D227semidirect
105. C22×C4C2×D227semidirect
109. C4⋊C4C2×D227semidirect
116. C22⋊C4C2×D227semidirect
119. C42C2×D227semidirect
128. D4C4×D227semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D2272
C4×D9082
D4×Dic2272
#203002502
C4×C22⋊D2272
(C22⋊D227)⋊C42
#203002462
#203002452
C2×(C2×D227)⋊C42
C22⋊C4×D2272
C4×D4×C2272
#518227