Finite Group Databasebeta

← #27213791

Subgroups — #27213791

Order 13504.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16809961C211yesmaximal, characteristic, complemented
2C2×D4×D211C211yesmaximal, characteristic, complemented
3#20269965C211yesmaximal, characteristic, complemented
4#20269961C212yesmaximal, complemented
5#20269957C211yesmaximal, characteristic, complemented
6#20269952C211yesmaximal, characteristic, complemented
7#20269951C212yesmaximal, complemented
8#20269947C212yesmaximal, complemented
9C4×D844C211yesmaximal, characteristic, complemented
10C2×(C2×D211)⋊C4C212yesmaximal, complemented
11C4×D4×C211C211yesmaximal, characteristic, complemented
12C23×D211C2212yes—
13#16809961—24no—
14C2×Dic422C2211yescharacteristic, complemented
15C2×D844C2211yescharacteristic
16C2×D844C2211yescharacteristic
17C2×D844C2212yescomplemented
18C2×C22⋊D211C2212yes—
19C2×C22⋊D211C2212yescomplemented
20D4×D211—24no—
21C22×Dic211C2212yescomplemented
22C2×C4×D211C2211yescharacteristic
23C2×C4×D211C2211yescharacteristic, complemented
24C4⋊Dic211C2211yescharacteristic, complemented
25C4⋊Dic211C2212yescomplemented
26(C2×D211)⋊C4C2211yescharacteristic
27(C2×D211)⋊C4C2212yes—
28(C2×D211)⋊C4C2212yescomplemented
29(C2×D211)⋊C4C2212yescomplemented
30(C2×D211)⋊C4C2211yescharacteristic, complemented
31(C2×D211)⋊C4C2212yescomplemented
32C2×D4×C211C2211yescharacteristic
33C22×C4×C211C2212yescomplemented
34C4⋊C4×C211C2211yescharacteristic, complemented
35C22⋊C4×C211C2212yescomplemented
36C42×C211C2211yescharacteristic, complemented
37C22×D211—24no—
38C22×D211—24no—
39C22×D211—42no—
40C22×D211C2312yes—
41C22×D211C2311yescharacteristic
42C22×D211C2311yescharacteristic
43Dic422—22no—
44D844—22no—
45D844—22no—
46D844—42no—
47C22⋊D211—24no—
48C22⋊D211—24no—
49C2×Dic211—24no—
50C2×Dic211C2311yescharacteristic
51C2×Dic211C2312yes—
52C2×Dic211C2311yescharacteristic
53C4×D211—22no—
54C4×D211—22no—
55C23×C211C2312yes—
56D4×C211D414yescomplemented
57C2×C4×C211—42no—
58C2×C4×C211C2311yescharacteristic
59C2×C4×C211C2311yescharacteristic
60C2×C4×C211C2312yes—
61C2×C4×C211C2311yescharacteristic
62C22×C211—22no—
63C22×C211—22no—
64C22×C211C2×D414yes—
65C22×C211C2411yescharacteristic
66C2×D211—21no—
67C2×D211—21no—
68C2×D211—21no—
69C2×D211—22no—
70C2×D211—21no—
71C2×D211—22no—
72C2×D211—21no—
73C2×D211—22no—
74C2×D211—48no—
75C2×D211#4112yescomplemented
76C4×C211—21no—
77C4×C211—21no—
78C4×C211—22no—
79C4×C211C2×D412yes—
80Dic211—21no—
81Dic211—21no—
82Dic211—22no—
83C2×C211—24no—
84C2×C211C22×D411yescharacteristic
85C2×C211C2×#4111yescharacteristic
86C2×C211H(5,𝔽2)11yescharacteristic
87D211—22no—
88D211—41no—
89D211—42no—
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#121—2111no—
99#120—2112no—
100C4×D4—2111no—
101C4×D4D21111yescharacteristic, complemented
102C2×C22⋊C4—2112no—
103C24—2112no—
104#41—4224no—
105C2×Q8—2111no—
106C2×D4—4224no—
107C2×D4—2112no—
108C2×D4—2112no—
109C2×D4—2112no—
110C2×D4—2111no—
111C2×D4—2111no—
112C2×D4C2×D21111yescharacteristic
113C22×C4—2112no—
114C22×C4—2111no—
115C22×C4—2111no—
116C22×C4C2×D21112yescomplemented
117C4⋊C4—2111no—
118C4⋊C4—2112no—
119C4⋊C4C2×D21111yescharacteristic, complemented
120C22⋊C4—2111no—
121C22⋊C4—2112no—
122C22⋊C4—2112no—
123C22⋊C4—2112no—
124C22⋊C4—2111no—
125C22⋊C4—2112no—
126C22⋊C4C2×D21112yescomplemented
127C42C2×D21111yescharacteristic, complemented
128C23—8442no—
129C23—4224no—
130C23—4224no—
131C23—2112no—
132C23—2111no—
133C23—2111no—
134C23C22×D21112yes—
135Q8—4222no—
136D4—8442no—
137D4—4222no—
138D4—4224no—
139D4—4224no—
140D4—4222no—
141D4D84414yescomplemented
142C2×C4—42no—
143C2×C4—4224no—
144C2×C4—4222no—
145C2×C4—4222no—
146C2×C4—2111no—
147C2×C4—2112no—
148C2×C4—2111no—
149C2×C4C22×D21111yescharacteristic
150C2×C4C22×D21111yescharacteristic
151C2×C4C22×D21112yes—
152C2×C4C22×D21111yescharacteristic
153C22—22no—
154C22—22no—
155C22—8448no—
156C22—4221no—
157C22—4222no—
158C22—4221no—
159C22—4222no—
160C22—4221no—
161C22—4222no—
162C22—4221no—
163C22—4221no—
164C22—2112no—
165C22C2×D84414yes—
166C22C23×D21111yescharacteristic
167C4—21no—
168C4—21no—
169C4—22no—
170C4—4221no—
171C4—4221no—
172C4—4222no—
173C4C2×D84412yes—
174C2—24no—
175C2—8442no—
176C2—8441no—
177C2—4222no—
178C2C22×D84411yescharacteristic
179C2#41×D21111yescharacteristic
180C2#2027002211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D211
commutator, socle, cosocleC22×C211C24
FittingC4×D4×C211C2
hypercenterC4×D4D211
hypocenterC211#546
LCSG ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16809961C2semidirect
2. C2×D4×D211C2semidirect
3. #20269965C2semidirect
4. #20269961C2semidirect
5. #20269957C2semidirect
6. #20269952C2semidirect
7. #20269951C2semidirect
8. #20269947C2semidirect
9. C4×D844C2semidirect
10. C2×(C2×D211)⋊C4C2semidirect
11. C4×D4×C211C2semidirect
14. C2×Dic422C22semidirect
17. C2×D844C22semidirect
19. C2×C22⋊D211C22semidirect
21. C22×Dic211C22semidirect
23. C2×C4×D211C22semidirect
24. C4⋊Dic211C22semidirect
25. C4⋊Dic211C22semidirect
28. (C2×D211)⋊C4C22semidirect
29. (C2×D211)⋊C4C22semidirect
30. (C2×D211)⋊C4C22semidirect
31. (C2×D211)⋊C4C22semidirect
33. C22×C4×C211C22semidirect
34. C4⋊C4×C211C22semidirect
35. C22⋊C4×C211C22semidirect
36. C42×C211C22semidirect
56. D4×C211D4semidirect
75. C2×D211#41semidirect
90. C211#546semidirect
101. C4×D4D211semidirect
116. C22×C4C2×D211semidirect
119. C4⋊C4C2×D211semidirect
126. C22⋊C4C2×D211semidirect
127. C42C2×D211semidirect
141. D4D844semidirect

Maximal subgroups

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

subgroupindex
C2×#168099612
C2×D4×D2112
#202699652
#202699612
#202699572
#202699522
#202699512
#202699472
C4×D8442
C2×(C2×D211)⋊C42
C4×D4×C2112
#546211