Finite Group Databasebeta

← #27152204

Subgroups — #27152204

Order 12416.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15690840C212yesmaximal, complemented
2C2×#15690835C212yesmaximal, complemented
3C2×#15690798C211yesmaximal, characteristic, complemented
4#20261127C212yesmaximal, complemented
5#20261122C212yesmaximal, complemented
6#20261117C214yesmaximal, complemented
7C4×#13095495C211yesmaximal, characteristic, complemented
8#532×C97C211yesmaximal, characteristic, complemented
9C22×Dic194C2211yescharacteristic
10C22×D388C2211yescharacteristic
11#15690840—28no—
12#15690835—28no—
13C2×#13095501C2214yescomplemented
14C2×#13095499C2214yescomplemented
15C2×#13095498C2214yes—
16C2×#13095496C2214yes—
17C2×#13095495C2212yes—
18#15690798—22no—
19#15690798C2212yes—
20#15690798C2212yes—
21C2×#13095488—22no—
22#15690721C2212yescomplemented
23C4×#11715895C2212yescomplemented
24C2×#41×C97C2212yes—
25C4⋊Q8×C97C2211yescharacteristic, complemented
26#127×C97C2211yescharacteristic, complemented
27#124×C97C2212yescomplemented
28#121×C97—22no—
29C4×D4×C97—22no—
30C2×C42×C97C2211yescharacteristic
31C23×D97—21no—
32C2×Dic194—24no—
33C2×Dic194—21no—
34C2×Dic194C2312yes—
35C2×D388—24no—
36C2×D388—21no—
37C2×D388C2312yes—
38C22×Dic97—21no—
39#13095501—28no—
40#13095499—28no—
41#13095498—28no—
42#13095496—28no—
43#13095495D418yescomplemented
44C2×#11715895C2314yes—
45#13095488—24no—
46#13095488—24no—
47#41×C97—24no—
48#41×C97—24no—
49C2×Q8×C97C2312yes—
50C2×D4×C97—22no—
51C2×D4×C97—22no—
52C2×D4×C97C2312yes—
53C22×C4×C97—22no—
54C22×C4×C97D412yes—
55C22×C4×C97C2311yescharacteristic
56C4⋊C4×C97—22no—
57C22⋊C4×C97—24no—
58C42×C97D412yes—
59C42×C97C2311yescharacteristic
60C42×C97C2311yescharacteristic
61C23×C97—22no—
62C23×C97C2×D411yescharacteristic
63C22×D97—22no—
64C22×D97—44no—
65C22×D97—42no—
66Q8×C97—22no—
67Q8×C97—22no—
68D4×C97—22no—
69D4×C97—22no—
70D4×C97—44no—
71D4×C97—44no—
72C2×C4×C97—22no—
73C2×C4×C97—42no—
74C2×C4×C97—44no—
75C2×C4×C97C2×D414yes—
76C2×C4×C97C2×D412yes—
77C2×C4×C97C2×D412yes—
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2411yescharacteristic
80C2×Dic97—22no—
81C2×Dic97—42no—
82D388—24no—
83D388—42no—
84Dic194—24no—
85Dic194—42no—
86#11715895C2×D418yes—
87C22×C97—22no—
88C22×C97—42no—
89C22×C97—44no—
90C22×C97#12712yes—
91C22×C97C22×D411yescharacteristic
92C2×D97—42no—
93C2×D97—44no—
94C2×D97—84no—
95C4×C97—22no—
96C4×C97—22no—
97C4×C97—42no—
98C4×C97#12712yes—
99C4×C97C22×D412yes—
100Dic97—42no—
101C2×C97—22no—
102C2×C97—82no—
103C2×C97C2×#12711yescharacteristic
104C2×C97#57512yes—
105D97—82no—
106#3153—971nomaximal, Sylow
107C97#315311yescharacteristic, complemented, Sylow
108C2×#137—972no—
109C2×AGL(1,ℤ/8)—972no—
110#532D9711yescharacteristic, complemented
111C2×#124—971no—
112#497—972no—
113#496—972no—
114#495—974no—
115C4×M16—971no—
116C2×#41C2×D9712yes—
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×M16—972no—
126C4⋊Q8C2×D9711yescharacteristic, complemented
127#127C2×D9711yescharacteristic, complemented
128#124—972no—
129#124—972no—
130#124—1942no—
131#124C2×D9712yescomplemented
132#121—22no—
133C4×D4—22no—
134C2×C22⋊C4—1942no—
135C2×C42C2×D9711yescharacteristic
136#97—972no—
137C4×C8—972no—
138C24—1941no—
139#41—24no—
140#41—24no—
141C2×Q8—972no—
142C2×Q8—1944no—
143C2×Q8—1941no—
144C2×Q8C22×D9712yes—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—972no—
148C2×D4—1944no—
149C2×D4—1941no—
150C2×D4C22×D9712yes—
151C22×C4—22no—
152C22×C4—1941no—
153C22×C4C22⋊D9712yes—
154C22×C4C22×D9711yescharacteristic
155Q16—1948no—
156SD16—1948no—
157SD16—1948no—
158D8—1948no—
159M16—978no—
160C2×C8—974no—
161C4⋊C4—22no—
162C22⋊C4—24no—
163C22⋊C4—1944no—
164C22⋊C4—1944no—
165C42C22⋊D9712yes—
166C42C22×D9711yescharacteristic
167C42C22×D9711yescharacteristic
168C23—22no—
169C23—3884no—
170C23—3882no—
171C23—1942no—
172C23C2×C22⋊D9711yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—3882no—
176Q8—1944no—
177D4—22no—
178D4—22no—
179D4—44no—
180D4—44no—
181D4—3882no—
182D4—1944no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4—3882no—
187C2×C4—1942no—
188C2×C4D4×D9714yes—
189C2×C4C2×C22⋊D9712yes—
190C2×C4C2×C22⋊D9712yes—
191C2×C4C2×C22⋊D9711yescharacteristic
192C2×C4C23×D9711yescharacteristic
193C8—978no—
194C22—22no—
195C22—42no—
196C22—44no—
197C22—7764no—
198C22—3882no—
199C22—3884no—
200C22#1569080612yes—
201C22C22×C22⋊D9711yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—3882no—
206C4#1569080612yes—
207C4C2×D4×D9712yes—
208C2—22no—
209C2—82no—
210C2—7762no—
211C2C2×#1569080611yescharacteristic
212C2#2026153312yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#532×C97C2
socleC22×C97C22×D4
hypercenter#532D97
hypocenterC97#3153
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#15690840C2semidirect
2. C2×#15690835C2semidirect
3. C2×#15690798C2semidirect
4. #20261127C2semidirect
5. #20261122C2semidirect
6. #20261117C2semidirect
7. C4×#13095495C2semidirect
8. #532×C97C2semidirect
13. C2×#13095501C22semidirect
14. C2×#13095499C22semidirect
22. #15690721C22semidirect
23. C4×#11715895C22semidirect
25. C4⋊Q8×C97C22semidirect
26. #127×C97C22semidirect
27. #124×C97C22semidirect
43. #13095495D4semidirect
107. C97#3153semidirect
110. #532D97semidirect
126. C4⋊Q8C2×D97semidirect
127. #127C2×D97semidirect
131. #124C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908402
C2×#156908352
C2×#156907982
#202611272
#202611222
#202611172
C4×#130954952
#532×C972
#315397