Finite Group Databasebeta

← #27152200

Subgroups — #27152200

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×#15690841C212yesmaximal, complemented
2C2×#15690836C212yesmaximal, complemented
3#20261285C211yesmaximal, characteristic, complemented
4#20261130C212yesmaximal, complemented
5#20261125C212yesmaximal, complemented
6#20261117C214yesmaximal, complemented
7C4×#13095495C211yesmaximal, characteristic, complemented
8C2×#124×C97C211yesmaximal, characteristic, complemented
9C2×#13095511C2212yes—
10#15690841—28no—
11#15690836—28no—
12C2×#13095501C2214yescomplemented
13C2×#13095499C2214yes—
14C2×#13095498C2214yescomplemented
15C2×#13095496C2214yes—
16C2×#13095495C2212yes—
17#15690810C2211yescharacteristic, complemented
18#15690807C2211yescharacteristic, complemented
19#15690798C2212yescomplemented
20#15690784—22no—
21C4×D388—22no—
22#15690721C2212yescomplemented
23C4×#11715895C2212yescomplemented
24C22×Q8×C97C2211yescharacteristic
25C22×D4×C97C2211yescharacteristic
26#124×C97—22no—
27#124×C97C2212yes—
28#124×C97C2212yes—
29C2×C22⋊C4×C97—22no—
30C2×C42×C97C2211yescharacteristic
31#13095511—24no—
32#13095511—24no—
33C2×Dic194C2312yes—
34C2×D388—22no—
35C2×D388C2312yes—
36C2×C22⋊D97—22no—
37C2×C4×D97—22no—
38#13095501—28no—
39#13095499—28no—
40#13095498—28no—
41#13095496—28no—
42#13095495D418yescomplemented
43C2×#11715895C2314yes—
44C4⋊Dic97—22no—
45#13095488—24no—
46C24×C97—21no—
47C2×Q8×C97—21no—
48C2×Q8×C97—24no—
49C2×Q8×C97C2312yes—
50C2×D4×C97—24no—
51C2×D4×C97—21no—
52C2×D4×C97C2312yes—
53C22×C4×C97—21no—
54C22×C4×C97D412yes—
55C22×C4×C97C2311yescharacteristic
56C22⋊C4×C97—24no—
57C22⋊C4×C97—24no—
58C42×C97D412yes—
59C42×C97C2311yescharacteristic
60C42×C97C2311yescharacteristic
61C23×C97—22no—
62C23×C97—44no—
63C23×C97—42no—
64C23×C97C2×D411yescharacteristic
65C22×D97—22no—
66Q8×C97—24no—
67Q8×C97—42no—
68D4×C97—24no—
69D4×C97—42no—
70C2×C4×C97—22no—
71C2×C4×C97—42no—
72C2×C4×C97—42no—
73C2×C4×C97C2×D414yes—
74C2×C4×C97C2×D412yes—
75C2×C4×C97C2×D412yes—
76C2×C4×C97C2×D411yescharacteristic
77C2×C4×C97C2411yescharacteristic
78C22⋊D97—44no—
79C2×Dic97—22no—
80D388—22no—
81D388—22no—
82D388—44no—
83C4×D97—44no—
84Dic194—22no—
85Dic194—22no—
86#11715895C2×D418yes—
87C22×C97—22no—
88C22×C97—42no—
89C22×C97—44no—
90C22×C97—84no—
91C22×C97#12712yes—
92C22×C97C22×D411yescharacteristic
93C2×D97—42no—
94C2×D97—44no—
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#532—971no—
111C2×#124D9711yescharacteristic, complemented
112#497—972no—
113#496—972no—
114#495—974no—
115C4×M16—971no—
116C2×#41—972no—
117C22×Q8C2×D9711yescharacteristic
118C22×D4C2×D9711yescharacteristic
119#137—1948no—
120AGL(1,ℤ/8)—1948no—
121C2×Q16—974no—
122C2×SD16—974no—
123C2×SD16—974no—
124C2×D8—974no—
125C2×M16—972no—
126C4⋊Q8—971no—
127#127—971no—
128#124—22no—
129#124—972no—
130#124C2×D9712yes—
131#124C2×D9712yes—
132#121—1942no—
133C4×D4—1942no—
134C2×C22⋊C4—22no—
135C2×C42C2×D9711yescharacteristic
136#97—972no—
137C4×C8—972no—
138C24—21no—
139#41—1944no—
140#41—1944no—
141C2×Q8—21no—
142C2×Q8—24no—
143C2×Q8—972no—
144C2×Q8C22×D9712yes—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—972no—
148C2×D4—1942no—
149C2×D4—1942no—
150C2×D4C22×D9712yes—
151C22×C4—21no—
152C22×C4—1942no—
153C22×C4C22⋊D9712yes—
154C22×C4C22×D9711yescharacteristic
155Q16—1948no—
156SD16—1948no—
157SD16—1948no—
158D8—1948no—
159M16—978no—
160C2×C8—974no—
161C4⋊C4—1942no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—1944no—
165C42C22⋊D9712yes—
166C42C22×D9711yescharacteristic
167C42C22×D9711yescharacteristic
168C23—22no—
169C23—44no—
170C23—42no—
171C23—1942no—
172C23C2×C22⋊D9711yescharacteristic
173Q8—24no—
174Q8—42no—
175Q8—1942no—
176Q8—1942no—
177D4—24no—
178D4—42no—
179D4—3884no—
180D4—3884no—
181D4—1942no—
182D4—1942no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—3884no—
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—84no—
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
FittingC2×#124×C97C2
socleC22×C97C22×D4
hypercenterC2×#124D97
hypocenterC97#3153
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#15690841C2semidirect
2. C2×#15690836C2semidirect
3. #20261285C2semidirect
4. #20261130C2semidirect
5. #20261125C2semidirect
6. #20261117C2semidirect
7. C4×#13095495C2semidirect
8. C2×#124×C97C2semidirect
12. C2×#13095501C22semidirect
14. C2×#13095498C22semidirect
17. #15690810C22semidirect
18. #15690807C22semidirect
19. #15690798C22semidirect
22. #15690721C22semidirect
23. C4×#11715895C22semidirect
42. #13095495D4semidirect
107. C97#3153semidirect
111. C2×#124D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
C2×#156908362
#202612852
#202611302
#202611252
#202611172
C4×#130954952
C2×#124×C972
#315397