Finite Group Databasebeta

← #27213706

Subgroups — #27213706

Order 13504.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C22×C22⋊D211C211yesmaximal, characteristic, complemented
2C2×D4×D211C211yesmaximal, characteristic, complemented
3#20269961C211yesmaximal, characteristic, complemented
4#20269959C212yesmaximal, complemented
5#20269958C211yesmaximal, characteristic, complemented
6#20269950C212yesmaximal, complemented
7#20269948C212yesmaximal, complemented
8#20269946C211yesmaximal, characteristic, complemented
9#20269945C212yesmaximal, complemented
10C22⋊C4×D211C211yesmaximal, characteristic, complemented
11#120×C211C211yesmaximal, characteristic, complemented
12C23×D211C2211yescharacteristic
13C23×D211C2211yescharacteristic, complemented
14C2×D844C2211yescharacteristic, complemented
15C2×C22⋊D211—24no—
16C2×C22⋊D211C2212yes—
17C2×C22⋊D211C2211yescharacteristic
18C2×C22⋊D211C2211yescharacteristic, complemented
19C2×C22⋊D211C2212yescomplemented
20C2×C22⋊D211C2212yescomplemented
21D4×D211—24no—
22C22×Dic211C2211yescharacteristic, complemented
23C2×C4×D211C2211yescharacteristic, complemented
24C2×C4×D211C2212yescomplemented
25C4⋊Dic211C2212yescomplemented
26Dic211⋊C4C2212yescomplemented
27C22⋊Dic211C2211yescharacteristic, complemented
28C22⋊Dic211C2212yescomplemented
29C22⋊Dic211C2212yescomplemented
30(C2×D211)⋊C4C2212yescomplemented
31(C2×D211)⋊C4C2212yescomplemented
32C24×C211C2211yescharacteristic, complemented
33C2×D4×C211C2211yescharacteristic
34C2×D4×C211C2212yescomplemented
35C22⋊C4×C211C2211yescharacteristic, complemented
36C22⋊C4×C211C2212yescomplemented
37C22×D211—22no—
38C22×D211—42no—
39C22×D211—42no—
40C22×D211D414yescomplemented
41C22×D211C2311yescharacteristic
42C22×D211C2311yescharacteristic
43C22×D211C2311yescharacteristic
44D844—22no—
45C22⋊D211—22no—
46C22⋊D211—22no—
47C22⋊D211—24no—
48C22⋊D211—24no—
49C22⋊D211—42no—
50C2×Dic211—22no—
51C2×Dic211C2312yes—
52C2×Dic211C2312yes—
53C2×Dic211C2311yescharacteristic
54C4×D211—22no—
55C4×D211—42no—
56C23×C211—22no—
57C23×C211—42no—
58C23×C211C2312yes—
59C23×C211C2311yescharacteristic
60C23×C211C2311yescharacteristic
61D4×C211—22no—
62D4×C211—42no—
63C2×C4×C211C2311yescharacteristic
64C2×C4×C211C2312yes—
65C22×C211—22no—
66C22×C211—21no—
67C22×C211—22no—
68C22×C211—22no—
69C22×C211—22no—
70C22×C211—22no—
71C22×C211—44no—
72C22×C211C2×D412yes—
73C22×C211C2411yescharacteristic
74C2×D211—22no—
75C2×D211—22no—
76C2×D211—21no—
77C2×D211—22no—
78C2×D211—44no—
79C2×D211—44no—
80C2×D211C2×D412yes—
81C2×D211C2×D412yes—
82C4×C211—21no—
83C4×C211—22no—
84Dic211—21no—
85Dic211—22no—
86Dic211—22no—
87C2×C211—22no—
88C2×C211—41no—
89C2×C211—42no—
90C2×C211C22×D411yescharacteristic
91C2×C211H(5,𝔽2)12yes—
92D211—22no—
93D211—22no—
94D211—41no—
95C211#53411yescharacteristic, complemented, Sylow
96#534—2111nomaximal, Sylow
97C22×D4—2111no—
98C22×D4—2111no—
99#123—2111no—
100#123—2112no—
101#123—2111no—
102#121—2112no—
103#121—2112no—
104#120—2111no—
105#120—2112no—
106#120D21111yescharacteristic, complemented
107C2×C22⋊C4—2111no—
108C24—2111no—
109C24—2111no—
110C24C2×D21111yescharacteristic, complemented
111C2×D4—4224no—
112C2×D4—4224no—
113C2×D4—2112no—
114C2×D4—2112no—
115C2×D4—2111no—
116C2×D4—2111no—
117C2×D4—2112no—
118C2×D4—2111no—
119C2×D4C2×D21111yescharacteristic
120C2×D4C2×D21112yescomplemented
121C22×C4—2112no—
122C22×C4—2111no—
123C22×C4—2111no—
124C4⋊C4—2112no—
125C4⋊C4—2112no—
126C22⋊C4—2111no—
127C22⋊C4—2112no—
128C22⋊C4—2112no—
129C22⋊C4—2112no—
130C22⋊C4—2112no—
131C22⋊C4C2×D21111yescharacteristic, complemented
132C22⋊C4C2×D21112yescomplemented
133C23—22no—
134C23—42no—
135C23—8442no—
136C23—8442no—
137C23—4222no—
138C23—2114no—
139C23—2111no—
140C23—2111no—
141C23—2111no—
142C23C22×D21111yescharacteristic
143C23C22×D21112yes—
144C23C22×D21111yescharacteristic
145D4—22no—
146D4—42no—
147D4—8442no—
148D4—4222no—
149D4—4222no—
150D4—4224no—
151D4—4224no—
152D4—4222no—
153C2×C4—8442no—
154C2×C4—4222no—
155C2×C4—4222no—
156C2×C4—2112no—
157C2×C4—2111no—
158C2×C4—2112no—
159C2×C4C22×D21112yes—
160C2×C4C22×D21111yescharacteristic
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—8444no—
169C22—8444no—
170C22—4221no—
171C22—4222no—
172C22—4222no—
173C22—4222no—
174C22—2112no—
175C22—2112no—
176C22D4×D21112yes—
177C22C23×D21111yescharacteristic
178C4—21no—
179C4—22no—
180C4—4221no—
181C4—4222no—
182C4—4222no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—8441no—
187C2—4222no—
188C2—4222no—
189C2C2×D4×D21111yescharacteristic
190C2#2027002212yes—
subgroupG/H
center, FrattiniC22C23×D211
commutator, socle, cosocleC22×C211C24
Fitting#120×C211C2
hypercenter#120D211
hypocenterC211#534
LCSG ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D211C2semidirect
2. C2×D4×D211C2semidirect
3. #20269961C2semidirect
4. #20269959C2semidirect
5. #20269958C2semidirect
6. #20269950C2semidirect
7. #20269948C2semidirect
8. #20269946C2semidirect
9. #20269945C2semidirect
10. C22⋊C4×D211C2semidirect
11. #120×C211C2semidirect
13. C23×D211C22semidirect
14. C2×D844C22semidirect
18. C2×C22⋊D211C22semidirect
19. C2×C22⋊D211C22semidirect
20. C2×C22⋊D211C22semidirect
22. C22×Dic211C22semidirect
23. C2×C4×D211C22semidirect
24. C2×C4×D211C22semidirect
25. C4⋊Dic211C22semidirect
26. Dic211⋊C4C22semidirect
27. C22⋊Dic211C22semidirect
28. C22⋊Dic211C22semidirect
29. C22⋊Dic211C22semidirect
30. (C2×D211)⋊C4C22semidirect
31. (C2×D211)⋊C4C22semidirect
32. C24×C211C22semidirect
34. C2×D4×C211C22semidirect
35. C22⋊C4×C211C22semidirect
36. C22⋊C4×C211C22semidirect
40. C22×D211D4semidirect
95. C211#534semidirect
106. #120D211semidirect
110. C24C2×D211semidirect
120. C2×D4C2×D211semidirect
131. C22⋊C4C2×D211semidirect
132. C22⋊C4C2×D211semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D2112
C2×D4×D2112
#202699612
#202699592
#202699582
#202699502
#202699482
#202699462
#202699452
C22⋊C4×D2112
#120×C2112
#534211