Finite Group Databasebeta

← #27184265

Subgroups — #27184265

Order 12928.

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

normal subgroups only

This group has 364 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
1#135×D101C212yesmaximal, complemented
2#20265619C212yesmaximal, complemented
3#20265616C212yesmaximal, complemented
4#20265614C212yesmaximal, complemented
5#20265611C212yesmaximal, complemented
6#20265609C211yesmaximal, characteristic, complemented
7#20265607C212yesmaximal, complemented
8C2×#15692194C211yesmaximal, characteristic, complemented
9#508×C101C211yesmaximal, characteristic, complemented
10#41×D101—22no—
11#15692278C2212yes—
12#15692277C2212yescomplemented
13#15692276—22no—
14#15692275—22no—
15#15692274C2212yes—
16Q16×D101C2212yescomplemented
17SD16×D101—22no—
18D8×D101C2212yescomplemented
19C2×C8×D101C2211yescharacteristic
20Dic808C2211yescharacteristic, complemented
21#15692201C2212yescomplemented
22#15692200C2212yescomplemented
23#15692199C2212yescomplemented
24#15692198C2212yescomplemented
25D1616C2211yescharacteristic, complemented
26#15692196C2212yescomplemented
27#15692194C2212yescomplemented
28#15692194C2212yescomplemented
29C2×#13108049C2211yescharacteristic, complemented
30#135×C101C2212yescomplemented
31Q32×C101C2211yescharacteristic, complemented
32SD32×C101C2212yescomplemented
33D16×C101C2211yescharacteristic, complemented
34C2×C16×C101C2211yescharacteristic, complemented
35#13108075—22no—
36#13108074—22no—
37#13108074—42no—
38#13108073—22no—
39#13108073—42no—
40Q8×D101—22no—
41D4×D101—22no—
42D4×D101—42no—
43C2×C4×D101—42no—
44C2×C4×D101D411yescharacteristic
45Dic404C2312yes—
46#13108064—22no—
47#13108063—22no—
48#13108062—22no—
49#13108061—22no—
50D808C2312yes—
51#13108059—22no—
52C2×#11717579D411yescharacteristic
53C8×D101D412yes—
54C8×D101C2312yes—
55#13108049C2312yes—
56#41×C101—22no—
57Q16×C101C2312yes—
58SD16×C101—22no—
59D8×C101C2312yes—
60C2×C8×C101C2311yescharacteristic
61C16×C101C2311yescharacteristic
62C16×C101C2311yescharacteristic
63C22×D101—42no—
64C22×D101D811yescharacteristic
65Q8×C101—22no—
66D4×C101—22no—
67D4×C101—42no—
68C2×C4×C101—42no—
69C2×C4×C101C2×D411yescharacteristic
70C22⋊D101—42no—
71C22⋊D101—42no—
72C22⋊D101—42no—
73C2×Dic101—42no—
74C2×Dic101D811yescharacteristic
75D404—22no—
76D404—42no—
77C4×D101—42no—
78C4×D101—42no—
79C4×D101—42no—
80C4×D101D812yes—
81C4×D101C2×D412yes—
82Dic202—22no—
83Dic202—42no—
84C8×C101C2×D411yescharacteristic
85C8×C101C2411yescharacteristic
86#11717579C2×D412yes—
87C22×C101—42no—
88C22×C101C2×D811yescharacteristic
89C2×D101—41no—
90C2×D101—42no—
91C2×D101—84no—
92C2×D101C2×D812yes—
93C4×C101—42no—
94C4×C101C2×D811yescharacteristic
95C4×C101C22×D411yescharacteristic
96Dic101—42no—
97Dic101C2×D812yes—
98C2×C101—21no—
99C2×C101—82no—
100C2×C101C22×D811yescharacteristic
101D101—22no—
102D101—82no—
103#3414—1011nomaximal, Sylow
104C101#341411yescharacteristic, complemented, Sylow
105C2×#135—1012no—
106#510—1012no—
107#510—1012no—
108#509—1012no—
109#509—1012no—
110#508—1012no—
111#508—1011no—
112#508D10111yescharacteristic, complemented
113C2×M32—1011no—
114C2×#41—2022no—
115#135—1012no—
116#135—1012no—
117#135—1012no—
118#135—2022no—
119#135—2022no—
120#135C2×D10112yescomplemented
121C2×Q16—1012no—
122C2×SD16—2022no—
123C2×D8—1012no—
124C22×C8—1011no—
125Q32—1012no—
126Q32—1011no—
127Q32C2×D10111yescharacteristic, complemented
128SD32—1012no—
129SD32—1012no—
130SD32—1012no—
131SD32C2×D10112yescomplemented
132D16—1012no—
133D16—1011no—
134D16C2×D10111yescharacteristic, complemented
135M32—1012no—
136M32—1012no—
137C2×C16—1011no—
138C2×C16C2×D10111yescharacteristic, complemented
139#41—22no—
140#41—4042no—
141#41—4042no—
142#41—2022no—
143#41—2022no—
144#41—2022no—
145C2×Q8—2022no—
146C2×D4—4042no—
147C2×D4—2022no—
148C22×C4—1011no—
149C22×C4—4042no—
150Q16—1012no—
151Q16—2022no—
152Q16C22×D10112yes—
153SD16—22no—
154SD16—2022no—
155SD16—2022no—
156SD16—2022no—
157D8—1012no—
158D8—2022no—
159D8C22×D10112yes—
160C2×C8—1011no—
161C2×C8—1012no—
162C2×C8—1012no—
163C2×C8C22×D10111yescharacteristic
164C16—1012no—
165C16C22×D10111yescharacteristic
166C16C22×D10111yescharacteristic
167C23—1011no—
168C23—4042no—
169Q8—22no—
170Q8—4042no—
171Q8—2022no—
172D4—22no—
173D4—42no—
174D4—4042no—
175D4—4042no—
176D4—4042no—
177D4—4042no—
178D4—2022no—
179C2×C4—42no—
180C2×C4—1011no—
181C2×C4—1012no—
182C2×C4—1012no—
183C2×C4—4042no—
184C2×C4—4042no—
185C2×C4—4042no—
186C2×C4—4042no—
187C2×C4D4×D10111yescharacteristic
188C8—1012no—
189C8D4×D10111yescharacteristic
190C8C23×D10111yescharacteristic
191C22—42no—
192C22—8084no—
193C22—1012no—
194C22—4041no—
195C22—4042no—
196C22D8×D10111yescharacteristic
197C4—42no—
198C4—1012no—
199C4—4042no—
200C4D8×D10111yescharacteristic
201C4C2×D4×D10111yescharacteristic
202C2—21no—
203C2—82no—
204C2—8082no—
205C2—2022no—
206C2C2×D8×D10111yescharacteristic
subgroupG/H
centerC4D8×D101
commutator, cosocleC8×C101C24
FrattiniC8C23×D101
Fitting#508×C101C2
socleC2×C101C22×D8
hypercenter#508D101
hypocenterC101#3414
LCSG ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D101C2semidirect
2. #20265619C2semidirect
3. #20265616C2semidirect
4. #20265614C2semidirect
5. #20265611C2semidirect
6. #20265609C2semidirect
7. #20265607C2semidirect
8. C2×#15692194C2semidirect
9. #508×C101C2semidirect
12. #15692277C22semidirect
16. Q16×D101C22semidirect
18. D8×D101C22semidirect
20. Dic808C22semidirect
21. #15692201C22semidirect
22. #15692200C22semidirect
23. #15692199C22semidirect
24. #15692198C22semidirect
25. D1616C22semidirect
26. #15692196C22semidirect
27. #15692194C22semidirect
28. #15692194C22semidirect
29. C2×#13108049C22semidirect
30. #135×C101C22semidirect
31. Q32×C101C22semidirect
32. SD32×C101C22semidirect
33. D16×C101C22semidirect
34. C2×C16×C101C22semidirect
104. C101#3414semidirect
112. #508D101semidirect
120. #135C2×D101semidirect
127. Q32C2×D101semidirect
131. SD32C2×D101semidirect
134. D16C2×D101semidirect
138. C2×C16C2×D101semidirect

Maximal subgroups

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

subgroupindex
#135×D1012
#202656192
#202656162
#202656142
#202656112
#202656092
#202656072
C2×#156921942
#508×C1012
#3414101