Finite Group Databasebeta

← #31771249

Subgroups — #31771249

Order 17216.

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×#16864401C211yesmaximal, characteristic, complemented
2#22575440C211yesmaximal, characteristic, complemented
3#22575434C212yesmaximal, complemented
4#22575432C211yesmaximal, characteristic, complemented
5#22575425C212yesmaximal, complemented
6#22575424C211yesmaximal, characteristic, complemented
7#22575422C212yesmaximal, complemented
8D4×Dic269C211yesmaximal, characteristic, complemented
9C4×C22⋊D269C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic269C212yesmaximal, complemented
11C22×D4×C269C211yesmaximal, characteristic, complemented
12#16864401—24no—
13C2×Dic538C2211yescharacteristic, complemented
14C2×C22⋊D269C2211yescharacteristic
15C2×C22⋊D269C2212yescomplemented
16C2×C22⋊D269C2212yescomplemented
17C22×Dic269C2212yescomplemented
18C22×Dic269C2212yescomplemented
19C2×C4×D269C2211yescharacteristic, complemented
20C4⋊Dic269C2211yescharacteristic, complemented
21#16864381C2211yescharacteristic, complemented
22#16864381C2212yescomplemented
23C22⋊Dic269C2211yescharacteristic
24C22⋊Dic269C2212yes—
25C22⋊Dic269C2212yescomplemented
26C22⋊Dic269C2212yescomplemented
27C22⋊Dic269C2212yescomplemented
28C22⋊Dic269C2212yescomplemented
29#16864379C2211yescharacteristic, complemented
30C4×Dic269C2211yescharacteristic, complemented
31C24×C269C2212yes—
32C2×D4×C269—24no—
33C2×D4×C269C2211yescharacteristic
34C2×D4×C269C2212yes—
35C2×D4×C269C2211yescharacteristic
36C22×C4×C269C2211yescharacteristic
37C22×D269C2311yescharacteristic
38Dic538—22no—
39C22⋊D269—24no—
40C22⋊D269—22no—
41C22⋊D269—42no—
42C2×Dic269—24no—
43C2×Dic269—42no—
44C2×Dic269C2311yescharacteristic
45C2×Dic269C2311yescharacteristic
46C2×Dic269C2312yes—
47C2×Dic269C2311yescharacteristic
48C2×Dic269C2312yes—
49C4×D269—22no—
50C23×C269—24no—
51C23×C269—24no—
52C23×C269—42no—
53C23×C269C2312yes—
54C23×C269C2312yes—
55C23×C269C2311yescharacteristic
56D4×C269—24no—
57D4×C269—22no—
58D4×C269D414yescomplemented
59C2×C4×C269—22no—
60C2×C4×C269C2311yescharacteristic
61C2×C4×C269C2311yescharacteristic
62C22×C269—21no—
63C22×C269—22no—
64C22×C269—21no—
65C22×C269—22no—
66C22×C269—22no—
67C22×C269—22no—
68C22×C269—22no—
69C22×C269—48no—
70C22×C269C2×D414yes—
71C22×C269#4112yescomplemented
72C22×C269C2411yescharacteristic
73C2×D269—21no—
74C2×D269—21no—
75C2×D269—21no—
76C4×C269—21no—
77C4×C269C2×D412yes—
78Dic269—21no—
79Dic269—22no—
80Dic269—21no—
81Dic269—21no—
82Dic269—22no—
83C2×C269—22no—
84C2×C269—24no—
85C2×C269—42no—
86C2×C269C22×D411yescharacteristic
87C2×C269C2×#4111yescharacteristic
88C2×C269H(5,𝔽2)11yescharacteristic
89D269—41no—
90C269#54611yescharacteristic, complemented, Sylow
91#546—2691nomaximal, Sylow
92C2×#41—2691no—
93C22×D4D26911yescharacteristic, complemented
94#124—2691no—
95#123—2692no—
96C22⋊Q8—2691no—
97#121—2691no—
98#121—2692no—
99#120—2692no—
100C4×D4—2691no—
101C4×D4—2691no—
102C2×C22⋊C4—2692no—
103C24C2×D26912yes—
104#41—5384no—
105C2×Q8—2691no—
106C2×D4—24no—
107C2×D4—2691no—
108C2×D4—2692no—
109C2×D4—2692no—
110C2×D4C2×D26912yes—
111C2×D4C2×D26911yescharacteristic
112C2×D4C2×D26911yescharacteristic
113C22×C4—2692no—
114C22×C4—2692no—
115C22×C4—2691no—
116C22×C4C2×D26911yescharacteristic
117C4⋊C4—2691no—
118C4⋊C4—2691no—
119C4⋊C4—2692no—
120C22⋊C4—2691no—
121C22⋊C4—2692no—
122C22⋊C4—2692no—
123C22⋊C4—2692no—
124C22⋊C4—2692no—
125C22⋊C4—2692no—
126C22⋊C4—2691no—
127C42—2691no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—2691no—
132C23C22×D26912yes—
133C23C22×D26912yes—
134C23C22×D26911yescharacteristic
135Q8—5382no—
136D4—24no—
137D4—22no—
138D4—5384no—
139D4—5382no—
140D4—10762no—
141D4C22⋊D26914yescomplemented
142C2×C4—22no—
143C2×C4—2691no—
144C2×C4—2691no—
145C2×C4—2692no—
146C2×C4—2691no—
147C2×C4—2692no—
148C2×C4—5384no—
149C2×C4—5382no—
150C2×C4—10762no—
151C2×C4C22×D26911yescharacteristic
152C2×C4C22×D26911yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—5381no—
162C22—5381no—
163C22—5381no—
164C22C2×C22⋊D26914yes—
165C22#1686440112yescomplemented
166C22C23×D26911yescharacteristic
167C4—21no—
168C4—5381no—
169C4—5382no—
170C4—5381no—
171C4—5381no—
172C4—5382no—
173C4C2×C22⋊D26912yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—10761no—
178C2C22×C22⋊D26911yescharacteristic
179C2C2×#1686440111yescharacteristic
180C2#2257549811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D269
commutator, socle, cosocleC22×C269C24
FittingC22×D4×C269C2
hypercenterC22×D4D269
hypocenterC269#546
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16864401C2semidirect
2. #22575440C2semidirect
3. #22575434C2semidirect
4. #22575432C2semidirect
5. #22575425C2semidirect
6. #22575424C2semidirect
7. #22575422C2semidirect
8. D4×Dic269C2semidirect
9. C4×C22⋊D269C2semidirect
10. C2×C22⋊Dic269C2semidirect
11. C22×D4×C269C2semidirect
13. C2×Dic538C22semidirect
15. C2×C22⋊D269C22semidirect
16. C2×C22⋊D269C22semidirect
17. C22×Dic269C22semidirect
18. C22×Dic269C22semidirect
19. C2×C4×D269C22semidirect
20. C4⋊Dic269C22semidirect
21. #16864381C22semidirect
22. #16864381C22semidirect
25. C22⋊Dic269C22semidirect
26. C22⋊Dic269C22semidirect
27. C22⋊Dic269C22semidirect
28. C22⋊Dic269C22semidirect
29. #16864379C22semidirect
30. C4×Dic269C22semidirect
58. D4×C269D4semidirect
71. C22×C269#41semidirect
90. C269#546semidirect
93. C22×D4D269semidirect
141. D4C22⋊D269semidirect
165. C22#16864401semidirect

Maximal subgroups

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

subgroupindex
C2×#168644012
#225754402
#225754342
#225754322
#225754252
#225754242
#225754222
D4×Dic2692
C4×C22⋊D2692
C2×C22⋊Dic2692
C22×D4×C2692
#546269