Finite Group Databasebeta

← #27213793

Subgroups — #27213793

Order 13504.

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×#16809962C211yesmaximal, characteristic, complemented
2C2×D4×D211C211yesmaximal, characteristic, complemented
3#20269964C211yesmaximal, characteristic, complemented
4#20269959C212yesmaximal, complemented
5#20269957C211yesmaximal, characteristic, complemented
6#20269948C212yesmaximal, complemented
7#20269945C212yesmaximal, complemented
8#20269939C211yesmaximal, characteristic, complemented
9C4×C22⋊D211C211yesmaximal, characteristic, complemented
10C22⋊C4×D211C212yesmaximal, complemented
11#121×C211C211yesmaximal, characteristic, complemented
12C23×D211C2212yes—
13#16809962—24no—
14C2×Dic422C2211yescharacteristic, complemented
15C2×D844C2211yescharacteristic
16C2×C22⋊D211C2211yescharacteristic
17C2×C22⋊D211C2212yes—
18C2×C22⋊D211C2212yescomplemented
19D4×D211—24no—
20C2×C4×D211C2211yescharacteristic
21C2×C4×D211C2212yescomplemented
22C2×C4×D211C2212yescomplemented
23C4⋊Dic211C2212yescomplemented
24Dic211⋊C4C2211yescharacteristic, complemented
25C22⋊Dic211C2212yescomplemented
26C22⋊Dic211C2211yescharacteristic, complemented
27C22⋊Dic211C2212yescomplemented
28(C2×D211)⋊C4C2211yescharacteristic
29(C2×D211)⋊C4C2212yes—
30(C2×D211)⋊C4C2212yescomplemented
31C4×Dic211C2211yescharacteristic, complemented
32C2×D4×C211C2211yescharacteristic
33C2×D4×C211C2212yescomplemented
34C22×C4×C211C2211yescharacteristic, complemented
35C4⋊C4×C211C2211yescharacteristic, complemented
36C22⋊C4×C211C2212yescomplemented
37C22×D211—24no—
38C22×D211—24no—
39C22×D211—42no—
40C22×D211C2312yes—
41C22×D211C2311yescharacteristic
42Dic422—22no—
43D844D414yescomplemented
44C22⋊D211—24no—
45C22⋊D211—24no—
46C22⋊D211—22no—
47C2×Dic211C2311yescharacteristic
48C2×Dic211C2311yescharacteristic
49C2×Dic211C2312yes—
50C2×Dic211C2311yescharacteristic
51C4×D211—24no—
52C4×D211—22no—
53C4×D211—42no—
54C23×C211C2312yes—
55C23×C211C2311yescharacteristic
56D4×C211—22no—
57D4×C211—42no—
58C2×C4×C211—22no—
59C2×C4×C211C2311yescharacteristic
60C2×C4×C211C2312yes—
61C2×C4×C211C2311yescharacteristic
62C22×C211—21no—
63C22×C211—22no—
64C22×C211—21no—
65C22×C211—22no—
66C22×C211—21no—
67C22×C211—22no—
68C22×C211C2411yescharacteristic
69C2×D211—21no—
70C2×D211—22no—
71C2×D211—22no—
72C2×D211—21no—
73C2×D211—48no—
74C2×D211C2×D414yes—
75C2×D211#4112yescomplemented
76C4×C211—21no—
77C4×C211—22no—
78C4×C211C2×D412yes—
79Dic211—21no—
80Dic211—22no—
81Dic211—21no—
82Dic211—21no—
83C2×C211—42no—
84C2×C211—41no—
85C2×C211C22×D411yescharacteristic
86C2×C211C2×#4111yescharacteristic
87C2×C211H(5,𝔽2)11yescharacteristic
88D211—22no—
89D211—24no—
90C211#54611yescharacteristic, complemented, Sylow
91#546—2111nomaximal, Sylow
92C2×#41—2111no—
93C22×D4—2111no—
94#124—2111no—
95#123—2112no—
96C22⋊Q8—2111no—
97#121—2112no—
98#121D21111yescharacteristic, complemented
99#120—2112no—
100C4×D4—2111no—
101C4×D4—2111no—
102C2×C22⋊C4—2112no—
103C24—2112no—
104#41—4224no—
105C2×Q8—2111no—
106C2×D4—4224no—
107C2×D4—2111no—
108C2×D4—2112no—
109C2×D4—2112no—
110C2×D4—2111no—
111C2×D4C2×D21111yescharacteristic
112C2×D4C2×D21112yescomplemented
113C22×C4—2112no—
114C22×C4—2112no—
115C22×C4—2111no—
116C22×C4C2×D21111yescharacteristic, complemented
117C4⋊C4—2111no—
118C4⋊C4—2112no—
119C4⋊C4C2×D21111yescharacteristic, complemented
120C22⋊C4—2112no—
121C22⋊C4—2111no—
122C22⋊C4—2112no—
123C22⋊C4—2111no—
124C22⋊C4—2112no—
125C22⋊C4—2112no—
126C22⋊C4C2×D21112yescomplemented
127C42—2111no—
128C23—8442no—
129C23—4224no—
130C23—4224no—
131C23—2111no—
132C23—2112no—
133C23C22×D21112yes—
134C23C22×D21111yescharacteristic
135Q8—4222no—
136D4—22no—
137D4—42no—
138D4—4224no—
139D4—4224no—
140D4—4222no—
141D4—2114no—
142C2×C4—22no—
143C2×C4—8442no—
144C2×C4—4224no—
145C2×C4—4222no—
146C2×C4—2111no—
147C2×C4—2111no—
148C2×C4—2112no—
149C2×C4—2111no—
150C2×C4C22×D21112yes—
151C2×C4C22×D21111yescharacteristic
152C2×C4C22×D21111yescharacteristic
153C22—21no—
154C22—22no—
155C22—21no—
156C22—22no—
157C22—21no—
158C22—22no—
159C22—8448no—
160C22—4221no—
161C22—4222no—
162C22—4222no—
163C22—4221no—
164C22—2114no—
165C22—2112no—
166C22C23×D21111yescharacteristic
167C4—21no—
168C4—22no—
169C4—4221no—
170C4—4222no—
171C4—4221no—
172C4—4221no—
173C4D4×D21112yes—
174C2—42no—
175C2—41no—
176C2—4222no—
177C2—4224no—
178C2C2×D4×D21111yescharacteristic
179C2#41×D21111yescharacteristic
180C2#2027002211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D211
commutator, socle, cosocleC22×C211C24
Fitting#121×C211C2
hypercenter#121D211
hypocenterC211#546
LCSG ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#16809962C2semidirect
2. C2×D4×D211C2semidirect
3. #20269964C2semidirect
4. #20269959C2semidirect
5. #20269957C2semidirect
6. #20269948C2semidirect
7. #20269945C2semidirect
8. #20269939C2semidirect
9. C4×C22⋊D211C2semidirect
10. C22⋊C4×D211C2semidirect
11. #121×C211C2semidirect
14. C2×Dic422C22semidirect
18. C2×C22⋊D211C22semidirect
21. C2×C4×D211C22semidirect
22. C2×C4×D211C22semidirect
23. C4⋊Dic211C22semidirect
24. Dic211⋊C4C22semidirect
25. C22⋊Dic211C22semidirect
26. C22⋊Dic211C22semidirect
27. C22⋊Dic211C22semidirect
30. (C2×D211)⋊C4C22semidirect
31. C4×Dic211C22semidirect
33. C2×D4×C211C22semidirect
34. C22×C4×C211C22semidirect
35. C4⋊C4×C211C22semidirect
36. C22⋊C4×C211C22semidirect
43. D844D4semidirect
75. C2×D211#41semidirect
90. C211#546semidirect
98. #121D211semidirect
112. C2×D4C2×D211semidirect
116. C22×C4C2×D211semidirect
119. C4⋊C4C2×D211semidirect
126. C22⋊C4C2×D211semidirect

Maximal subgroups

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

subgroupindex
C2×#168099622
C2×D4×D2112
#202699642
#202699592
#202699572
#202699482
#202699452
#202699392
C4×C22⋊D2112
C22⋊C4×D2112
#121×C2112
#546211