Finite Group Databasebeta

← #27213790

Subgroups — #27213790

Order 13504.

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×#16809961C211yesmaximal, characteristic, complemented
2#20269964C211yesmaximal, characteristic, complemented
3#20269958C212yesmaximal, complemented
4#20269956C211yesmaximal, characteristic, complemented
5#20269949C212yesmaximal, complemented
6#20269948C211yesmaximal, characteristic, complemented
7#20269946C212yesmaximal, complemented
8D4×Dic211C211yesmaximal, characteristic, complemented
9C4×C22⋊D211C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic211C212yesmaximal, complemented
11C22×D4×C211C211yesmaximal, characteristic, complemented
12#16809961—24no—
13C2×Dic422C2211yescharacteristic, complemented
14C2×C22⋊D211C2211yescharacteristic
15C2×C22⋊D211C2212yescomplemented
16C2×C22⋊D211C2212yescomplemented
17C22×Dic211C2212yescomplemented
18C22×Dic211C2212yescomplemented
19C2×C4×D211C2211yescharacteristic, complemented
20C4⋊Dic211C2211yescharacteristic, complemented
21Dic211⋊C4C2211yescharacteristic, complemented
22Dic211⋊C4C2212yescomplemented
23C22⋊Dic211C2211yescharacteristic
24C22⋊Dic211C2212yes—
25C22⋊Dic211C2212yescomplemented
26C22⋊Dic211C2212yescomplemented
27C22⋊Dic211C2212yescomplemented
28C22⋊Dic211C2212yescomplemented
29(C2×D211)⋊C4C2211yescharacteristic, complemented
30C4×Dic211C2211yescharacteristic, complemented
31C24×C211C2212yes—
32C2×D4×C211—24no—
33C2×D4×C211C2211yescharacteristic
34C2×D4×C211C2212yes—
35C2×D4×C211C2211yescharacteristic
36C22×C4×C211C2211yescharacteristic
37C22×D211C2311yescharacteristic
38Dic422—22no—
39C22⋊D211—24no—
40C22⋊D211—22no—
41C22⋊D211—42no—
42C2×Dic211—24no—
43C2×Dic211—42no—
44C2×Dic211C2311yescharacteristic
45C2×Dic211C2311yescharacteristic
46C2×Dic211C2312yes—
47C2×Dic211C2311yescharacteristic
48C2×Dic211C2312yes—
49C4×D211—22no—
50C23×C211—24no—
51C23×C211—24no—
52C23×C211—42no—
53C23×C211C2312yes—
54C23×C211C2312yes—
55C23×C211C2311yescharacteristic
56D4×C211—24no—
57D4×C211—22no—
58D4×C211D414yescomplemented
59C2×C4×C211—22no—
60C2×C4×C211C2311yescharacteristic
61C2×C4×C211C2311yescharacteristic
62C22×C211—21no—
63C22×C211—22no—
64C22×C211—21no—
65C22×C211—22no—
66C22×C211—22no—
67C22×C211—22no—
68C22×C211—22no—
69C22×C211—48no—
70C22×C211C2×D414yes—
71C22×C211#4112yescomplemented
72C22×C211C2411yescharacteristic
73C2×D211—21no—
74C2×D211—21no—
75C2×D211—21no—
76C4×C211—21no—
77C4×C211C2×D412yes—
78Dic211—21no—
79Dic211—22no—
80Dic211—21no—
81Dic211—21no—
82Dic211—22no—
83C2×C211—22no—
84C2×C211—24no—
85C2×C211—42no—
86C2×C211C22×D411yescharacteristic
87C2×C211C2×#4111yescharacteristic
88C2×C211H(5,𝔽2)11yescharacteristic
89D211—41no—
90C211#54611yescharacteristic, complemented, Sylow
91#546—2111nomaximal, Sylow
92C2×#41—2111no—
93C22×D4D21111yescharacteristic, complemented
94#124—2111no—
95#123—2112no—
96C22⋊Q8—2111no—
97#121—2111no—
98#121—2112no—
99#120—2112no—
100C4×D4—2111no—
101C4×D4—2111no—
102C2×C22⋊C4—2112no—
103C24C2×D21112yes—
104#41—4224no—
105C2×Q8—2111no—
106C2×D4—24no—
107C2×D4—2112no—
108C2×D4—2111no—
109C2×D4—2112no—
110C2×D4C2×D21112yes—
111C2×D4C2×D21111yescharacteristic
112C2×D4C2×D21111yescharacteristic
113C22×C4—2112no—
114C22×C4—2112no—
115C22×C4—2111no—
116C22×C4C2×D21111yescharacteristic
117C4⋊C4—2111no—
118C4⋊C4—2111no—
119C4⋊C4—2112no—
120C22⋊C4—2111no—
121C22⋊C4—2112no—
122C22⋊C4—2112no—
123C22⋊C4—2112no—
124C22⋊C4—2112no—
125C22⋊C4—2112no—
126C22⋊C4—2111no—
127C42—2111no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—2111no—
132C23C22×D21112yes—
133C23C22×D21112yes—
134C23C22×D21111yescharacteristic
135Q8—4222no—
136D4—24no—
137D4—22no—
138D4—8442no—
139D4—4224no—
140D4—4222no—
141D4C22⋊D21114yescomplemented
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×C4—2112no—
151C2×C4C22×D21111yescharacteristic
152C2×C4C22×D21111yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—4221no—
162C22—4221no—
163C22—4221no—
164C22C2×C22⋊D21114yes—
165C22#1680996112yescomplemented
166C22C23×D21111yescharacteristic
167C4—21no—
168C4—4221no—
169C4—4222no—
170C4—4221no—
171C4—4221no—
172C4—4222no—
173C4C2×C22⋊D21112yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—8441no—
178C2C22×C22⋊D21111yescharacteristic
179C2C2×#1680996111yescharacteristic
180C2#2027002211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D211
commutator, socle, cosocleC22×C211C24
FittingC22×D4×C211C2
hypercenterC22×D4D211
hypocenterC211#546
LCSG ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16809961C2semidirect
2. #20269964C2semidirect
3. #20269958C2semidirect
4. #20269956C2semidirect
5. #20269949C2semidirect
6. #20269948C2semidirect
7. #20269946C2semidirect
8. D4×Dic211C2semidirect
9. C4×C22⋊D211C2semidirect
10. C2×C22⋊Dic211C2semidirect
11. C22×D4×C211C2semidirect
13. C2×Dic422C22semidirect
15. C2×C22⋊D211C22semidirect
16. C2×C22⋊D211C22semidirect
17. C22×Dic211C22semidirect
18. C22×Dic211C22semidirect
19. C2×C4×D211C22semidirect
20. C4⋊Dic211C22semidirect
21. Dic211⋊C4C22semidirect
22. Dic211⋊C4C22semidirect
25. C22⋊Dic211C22semidirect
26. C22⋊Dic211C22semidirect
27. C22⋊Dic211C22semidirect
28. C22⋊Dic211C22semidirect
29. (C2×D211)⋊C4C22semidirect
30. C4×Dic211C22semidirect
58. D4×C211D4semidirect
71. C22×C211#41semidirect
90. C211#546semidirect
93. C22×D4D211semidirect
141. D4C22⋊D211semidirect
165. C22#16809961semidirect

Maximal subgroups

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

subgroupindex
C2×#168099612
#202699642
#202699582
#202699562
#202699492
#202699482
#202699462
D4×Dic2112
C4×C22⋊D2112
C2×C22⋊Dic2112
C22×D4×C2112
#546211