Finite Group Databasebeta

← #27213945

Subgroups — #27213945

Order 13504.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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×#16809963C211yesmaximal, characteristic, complemented
2#41×D211C211yesmaximal, characteristic, complemented
3#20270009C212yesmaximal, complemented
4#20270008C211yesmaximal, characteristic, complemented
5#20270005C211yesmaximal, characteristic, complemented
6#20270004C212yesmaximal, complemented
7#20270002C211yesmaximal, characteristic, complemented
8#20270000C212yesmaximal, complemented
9#20269998C212yesmaximal, complemented
10M16×D211C211yesmaximal, characteristic, complemented
11#137×C211C211yesmaximal, characteristic, complemented
12#16809963—21no—
13#16809963—21no—
14#16809963C2212yes—
15#16809963C2212yescomplemented
16#16809963C2211yescharacteristic, complemented
17#16809962—21no—
18#16809962C2211yescharacteristic
19#16809961—21no—
20#16809961C2211yescharacteristic, complemented
21Q8×D211C2211yescharacteristic, complemented
22C2×D844—21no—
23C2×D844C2211yescharacteristic, complemented
24D4×D211—21no—
25D4×D211C2211yescharacteristic, complemented
26C2×C4×D211—21no—
27C2×C4×D211—21no—
28C2×C4×D211C2211yescharacteristic
29#16809952C2212yescomplemented
30#16809951C2212yescomplemented
31#16809949C2212yescomplemented
32#16809949C2212yescomplemented
33D1688C2212yescomplemented
34#16809947C2212yescomplemented
35#16809946C2211yescharacteristic, complemented
36#16809945C2212yescomplemented
37C8×D211C2212yescomplemented
38#41×C211C2211yescharacteristic
39C2×Q8×C211C2211yescharacteristic, complemented
40Q16×C211C2212yescomplemented
41SD16×C211C2212yescomplemented
42M16×C211C2211yescharacteristic, complemented
43C22×D211—21no—
44C22×D211—21no—
45C22×D211D411yescharacteristic
46Dic422—21no—
47Dic422C2311yescharacteristic
48D844—21no—
49D844—22no—
50D844—22no—
51D844—21no—
52D844C2312yes—
53D844C2311yescharacteristic
54C22⋊D211—21no—
55C22⋊D211—21no—
56C22⋊D211—21no—
57C2×Dic211—21no—
58C2×Dic211D411yescharacteristic
59C4×D211—21no—
60C4×D211—21no—
61C4×D211—21no—
62C4×D211—22no—
63C4×D211—22no—
64C4×D211D411yescharacteristic
65C4×D211D411yescharacteristic
66C4×D211C2311yescharacteristic
67C4×D211C2311yescharacteristic
68C211⋊C8C2312yes—
69Q8×C211—21no—
70Q8×C211C2312yes—
71Q8×C211C2311yescharacteristic
72D4×C211—21no—
73D4×C211C2311yescharacteristic
74C2×C4×C211—21no—
75C2×C4×C211—21no—
76C2×C4×C211C2311yescharacteristic
77C8×C211C2312yes—
78C22×C211—21no—
79C22×C211C2×D411yescharacteristic
80C2×D211—21no—
81C2×D211—22no—
82C2×D211—41no—
83C2×D211—42no—
84C2×D211—42no—
85C2×D211C2×D411yescharacteristic
86C2×D211C2×D411yescharacteristic
87C4×C211—21no—
88C4×C211—22no—
89C4×C211C2×D411yescharacteristic
90C4×C211C2411yescharacteristic
91Dic211—21no—
92Dic211C2×D411yescharacteristic
93Dic211C2×D411yescharacteristic
94C2×C211—21no—
95C2×C211—41no—
96C2×C211C22×D411yescharacteristic
97D211—21no—
98D211—21no—
99D211—41no—
100D211—42no—
101C211#57511yescharacteristic, complemented, Sylow
102#575—2111nomaximal, Sylow
103C2×#41—2111no—
104C2×#41—2111no—
105#137—2111no—
106#137—2112no—
107#137D21111yescharacteristic, complemented
108AGL(1,ℤ/8)—2111no—
109AGL(1,ℤ/8)—2112no—
110AGL(1,ℤ/8)—2111no—
111#135—2112no—
112#135—2112no—
113C2×M16—2111no—
114#41—4221no—
115#41—4221no—
116#41—4221no—
117#41—4221no—
118#41—2111no—
119#41—2112no—
120#41—2112no—
121#41—2111no—
122#41—2111no—
123#41C2×D21111yescharacteristic
124C2×Q8—2111no—
125C2×Q8C2×D21111yescharacteristic, complemented
126C2×D4—4221no—
127C2×D4—4221no—
128C2×D4—2111no—
129C2×D4—2111no—
130C22×C4—4221no—
131C22×C4—4221no—
132C22×C4—2111no—
133Q16—2112no—
134Q16C2×D21112yescomplemented
135SD16—2112no—
136SD16—2112no—
137SD16—2112no—
138SD16C2×D21112yescomplemented
139D8—2112no—
140D8—2112no—
141M16—2111no—
142M16—2112no—
143M16C2×D21111yescharacteristic, complemented
144C2×C8—2112no—
145C23—4221no—
146C23—4221no—
147C23—2111no—
148Q8—21no—
149Q8—4221no—
150Q8—2111no—
151Q8C22×D21111yescharacteristic
152Q8C22×D21112yes—
153D4—21no—
154D4—4221no—
155D4—4222no—
156D4—4222no—
157D4—4221no—
158D4—4221no—
159D4—4221no—
160D4—4221no—
161D4—2112no—
162D4—2111no—
163D4C22×D21111yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—4221no—
167C2×C4—4221no—
168C2×C4—4222no—
169C2×C4—4222no—
170C2×C4—4221no—
171C2×C4—4221no—
172C2×C4—2111no—
173C2×C4—2111no—
174C2×C4—2111no—
175C2×C4—2111no—
176C2×C4—2111no—
177C2×C4C22×D21111yescharacteristic
178C8—2112no—
179C8C22×D21112yes—
180C22—21no—
181C22—8442no—
182C22—8441no—
183C22—8442no—
184C22—4222no—
185C22—4221no—
186C22—2111no—
187C22—2111no—
188C22D4×D21111yescharacteristic
189C4—21no—
190C4—22no—
191C4—4221no—
192C4—2111no—
193C4—2111no—
194C4D4×D21111yescharacteristic
195C4C23×D21111yescharacteristic
196C2—21no—
197C2—41no—
198C2—8442no—
199C2—8441no—
200C2—4221no—
201C2—4221no—
202C2C2×D4×D21111yescharacteristic
subgroupG/H
centerC2C2×D4×D211
commutator, cosocleC4×C211C24
FrattiniC4C23×D211
Fitting#137×C211C2
socleC2×C211C22×D4
hypercenter#137D211
hypocenterC211#575
LCSG ▷ C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#16809963C2semidirect
2. #41×D211C2semidirect
3. #20270009C2semidirect
4. #20270008C2semidirect
5. #20270005C2semidirect
6. #20270004C2semidirect
7. #20270002C2semidirect
8. #20270000C2semidirect
9. #20269998C2semidirect
10. M16×D211C2semidirect
11. #137×C211C2semidirect
15. #16809963C22semidirect
16. #16809963C22semidirect
20. #16809961C22semidirect
21. Q8×D211C22semidirect
23. C2×D844C22semidirect
25. D4×D211C22semidirect
29. #16809952C22semidirect
30. #16809951C22semidirect
31. #16809949C22semidirect
32. #16809949C22semidirect
33. D1688C22semidirect
34. #16809947C22semidirect
35. #16809946C22semidirect
36. #16809945C22semidirect
37. C8×D211C22semidirect
39. C2×Q8×C211C22semidirect
40. Q16×C211C22semidirect
41. SD16×C211C22semidirect
42. M16×C211C22semidirect
101. C211#575semidirect
107. #137D211semidirect
125. C2×Q8C2×D211semidirect
134. Q16C2×D211semidirect
138. SD16C2×D211semidirect
143. M16C2×D211semidirect

Maximal subgroups

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

subgroupindex
C2×#168099632
#41×D2112
#202700092
#202700082
#202700052
#202700042
#202700022
#202700002
#202699982
M16×D2112
#137×C2112
#575211