Finite Group Databasebeta

← #27152202

Subgroups — #27152202

Order 12416.

572 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15690843C212yesmaximal, complemented
2C2×#15690838C212yesmaximal, complemented
3#20261285C211yesmaximal, characteristic, complemented
4C2×#15690798C211yesmaximal, characteristic, complemented
5#20261129C212yesmaximal, complemented
6#20261124C212yesmaximal, complemented
7#20261119C214yesmaximal, complemented
8C4×M16×C97C211yesmaximal, characteristic, complemented
9C2×#13095511C2212yes—
10C22×Dic194C2211yescharacteristic
11C22×D388C2211yescharacteristic
12#15690843—28no—
13#15690838—28no—
14C2×Dic388C2214yescomplemented
15C2×#13095500C2214yes—
16C2×#13095500C2214yescomplemented
17C2×D776C2214yes—
18#15690810C2211yescharacteristic, complemented
19#15690807C2211yescharacteristic, complemented
20#15690798—22no—
21#15690798C2212yes—
22#15690798C2212yes—
23#15690798C2212yescomplemented
24#15690784—22no—
25C4×D388—22no—
26C2×#13095488—22no—
27C2×M16×C97C2212yes—
28C2×C42×C97C2211yescharacteristic
29#97×C97C2212yescomplemented
30C4×C8×C97C2212yescomplemented
31C23×D97—21no—
32#13095511—24no—
33#13095511—24no—
34C2×Dic194—24no—
35C2×Dic194—21no—
36C2×Dic194C2312yes—
37C2×Dic194C2312yes—
38C2×D388—24no—
39C2×D388—22no—
40C2×D388—21no—
41C2×D388C2312yes—
42C2×D388C2312yes—
43C2×C22⋊D97—22no—
44C22×Dic97—21no—
45C2×C4×D97—22no—
46Dic388—28no—
47#13095500—28no—
48#13095500—28no—
49D776—28no—
50C4⋊Dic97—22no—
51#13095488—24no—
52#13095488—24no—
53#13095488—24no—
54C22×C4×C97D412yes—
55C22×C4×C97C2311yescharacteristic
56M16×C97D418yescomplemented
57C2×C8×C97C2314yes—
58C42×C97D412yes—
59C42×C97C2311yescharacteristic
60C42×C97C2311yescharacteristic
61C23×C97C2×D411yescharacteristic
62C22×D97—22no—
63C22×D97—22no—
64C22×D97—44no—
65C22×D97—42no—
66C2×C4×C97—42no—
67C2×C4×C97C2×D412yes—
68C2×C4×C97C2×D412yes—
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×D414yes—
71C2×C4×C97C2411yescharacteristic
72C22⋊D97—44no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—42no—
76D388—22no—
77D388—24no—
78D388—22no—
79D388—44no—
80D388—42no—
81C4×D97—44no—
82Dic194—22no—
83Dic194—24no—
84Dic194—22no—
85Dic194—42no—
86C8×C97C2×D418yes—
87C22×C97—22no—
88C22×C97#12712yes—
89C22×C97C22×D411yescharacteristic
90C2×D97—42no—
91C2×D97—42no—
92C2×D97—44no—
93C2×D97—44no—
94C2×D97—84no—
95C4×C97—22no—
96C4×C97—22no—
97C4×C97#12712yes—
98C4×C97C22×D412yes—
99Dic97—42no—
100Dic97—42no—
101C2×C97—22no—
102C2×C97C2×#12711yescharacteristic
103C2×C97#57512yes—
104D97—82no—
105D97—82no—
106#3153—971nomaximal, Sylow
107C97#315311yescharacteristic, complemented, Sylow
108C2×#137—972no—
109C2×AGL(1,ℤ/8)—972no—
110#532—971no—
111C2×#124—971no—
112#497—972no—
113#496—972no—
114#495—974no—
115C4×M16D9711yescharacteristic, complemented
116C2×#41—972no—
117C22×Q8—971no—
118C22×D4—971no—
119#137—1948no—
120AGL(1,ℤ/8)—1948no—
121C2×Q16—974no—
122C2×SD16—974no—
123C2×SD16—974no—
124C2×D8—974no—
125C2×M16C2×D9712yes—
126C4⋊Q8—971no—
127#127—971no—
128#124—972no—
129#124—972no—
130#124—972no—
131#124—1942no—
132#121—1942no—
133C4×D4—1942no—
134C2×C22⋊C4—1942no—
135C2×C42C2×D9711yescharacteristic
136#97C2×D9712yescomplemented
137C4×C8C2×D9712yescomplemented
138C24—1941no—
139#41—1944no—
140#41—1944no—
141C2×Q8—972no—
142C2×Q8—972no—
143C2×Q8—1944no—
144C2×Q8—1941no—
145C2×D4—972no—
146C2×D4—972no—
147C2×D4—1942no—
148C2×D4—1942no—
149C2×D4—1941no—
150C2×D4—1944no—
151C22×C4—1941no—
152C22×C4—1942no—
153C22×C4D38812yes—
154C22×C4C22×D9711yescharacteristic
155Q16—1948no—
156SD16—1948no—
157SD16—1948no—
158D8—1948no—
159M16D38818yescomplemented
160C2×C8C22×D9714yes—
161C4⋊C4—1942no—
162C22⋊C4—1944no—
163C22⋊C4—1944no—
164C22⋊C4—1944no—
165C42D38812yes—
166C42C22×D9711yescharacteristic
167C42C22×D9711yescharacteristic
168C23—3884no—
169C23—3882no—
170C23—1942no—
171C23—1942no—
172C23C2×D38811yescharacteristic
173Q8—3882no—
174Q8—1942no—
175Q8—1944no—
176Q8—1942no—
177D4—3884no—
178D4—3884no—
179D4—3882no—
180D4—1942no—
181D4—1944no—
182D4—1942no—
183C2×C4—42no—
184C2×C4—3882no—
185C2×C4—3884no—
186C2×C4—1942no—
187C2×C4—1942no—
188C2×C4C2×D38812yes—
189C2×C4C2×D38812yes—
190C2×C4C2×D38811yescharacteristic
191C2×C4C2×D38814yes—
192C2×C4C23×D9711yescharacteristic
193C8C2×D38818yes—
194C22—22no—
195C22—7764no—
196C22—3882no—
197C22—3882no—
198C22—3884no—
199C22—3884no—
200C22#1569080712yes—
201C22C22×D38811yescharacteristic
202C4—22no—
203C4—22no—
204C4—3882no—
205C4—3882no—
206C4#1569080712yes—
207C4C22×D38812yes—
208C2—22no—
209C2—7762no—
210C2—7762no—
211C2C2×#1569080711yescharacteristic
212C2#2026153512yes—
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×M16×C97C2
socleC22×C97C22×D4
hypercenterC4×M16D97
hypocenterC97#3153
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#15690843C2semidirect
2. C2×#15690838C2semidirect
3. #20261285C2semidirect
4. C2×#15690798C2semidirect
5. #20261129C2semidirect
6. #20261124C2semidirect
7. #20261119C2semidirect
8. C4×M16×C97C2semidirect
14. C2×Dic388C22semidirect
16. C2×#13095500C22semidirect
18. #15690810C22semidirect
19. #15690807C22semidirect
23. #15690798C22semidirect
29. #97×C97C22semidirect
30. C4×C8×C97C22semidirect
56. M16×C97D4semidirect
107. C97#3153semidirect
115. C4×M16D97semidirect
136. #97C2×D97semidirect
137. C4×C8C2×D97semidirect
159. M16D388semidirect

Maximal subgroups

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

subgroupindex
C2×#156908432
C2×#156908382
#202612852
C2×#156907982
#202611292
#202611242
#202611192
C4×M16×C972
#315397