Finite Group Databasebeta

← #27152199

Subgroups — #27152199

Order 12416.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15690842C212yesmaximal, complemented
2C2×#15690837C212yesmaximal, complemented
3#20261286C211yesmaximal, characteristic, complemented
4#124×D97C211yesmaximal, characteristic, complemented
5#20261127C211yesmaximal, characteristic, complemented
6#20261125C211yesmaximal, characteristic, complemented
7#20261124C211yesmaximal, characteristic, complemented
8#20261120C212yesmaximal, complemented
9#20261116C212yesmaximal, complemented
10#20260687C211yesmaximal, characteristic, complemented
11#497×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2212yes—
13C2×Q8×D97C2211yescharacteristic
14C2×D4×D97C2211yescharacteristic
15#15690842—28no—
16#15690837—28no—
17C2×#13095501C2212yescomplemented
18C2×#13095500C2212yescomplemented
19C2×#13095498C2212yes—
20C2×#13095498C2212yescomplemented
21C2×D776C2212yes—
22C2×#13095496C2212yes—
23C2×#13095494C2212yes—
24#15690807C2211yescharacteristic, complemented
25#15690799C2211yescharacteristic
26#15690799C2212yescomplemented
27#15690798C2211yescharacteristic
28#15690797C2211yescharacteristic
29#15690796—22no—
30#15690785—22no—
31#15690772—22no—
32C22⋊C4×D97—22no—
33C42×D97C2211yescharacteristic
34#15690721C2211yescharacteristic, complemented
35C8×Dic97C2212yescomplemented
36C2×Q16×C97C2212yescomplemented
37C2×SD16×C97C2212yes—
38C4⋊Q8×C97C2211yescharacteristic, complemented
39#124×C97C2211yescharacteristic
40#97×C97C2211yescharacteristic, complemented
41C23×D97—21no—
42#13095512—24no—
43#13095512—24no—
44C2×Dic194—21no—
45C2×Dic194C2311yescharacteristic
46Q8×D97—24no—
47C2×D388—22no—
48C2×D388—22no—
49C2×D388C2311yescharacteristic
50C2×D388C2312yes—
51C2×C22⋊D97—21no—
52D4×D97—24no—
53C2×C4×D97—21no—
54C2×C4×D97—22no—
55C2×C4×D97D412yes—
56C2×C4×D97C2311yescharacteristic
57#13095501—24no—
58#13095500—24no—
59#13095498—24no—
60#13095498—24no—
61D776—24no—
62#13095496—24no—
63#13095494D418yescomplemented
64C2×#11715895C2312yes—
65C22⋊Dic97—22no—
66#13095488—22no—
67#13095488—22no—
68#13095488—24no—
69C4×Dic97D412yes—
70C4×Dic97C2311yescharacteristic
71C2×Q8×C97C2311yescharacteristic
72C2×Q8×C97C2312yes—
73C2×D4×C97C2311yescharacteristic
74Q16×C97—24no—
75SD16×C97—24no—
76C2×C8×C97C2312yes—
77C4⋊C4×C97—22no—
78C22⋊C4×C97—22no—
79C42×C97C2311yescharacteristic
80C23×C97—21no—
81C22×D97—21no—
82C22×D97—22no—
83C22×D97—44no—
84C22×D97—42no—
85C22×D97C2×D411yescharacteristic
86Q8×C97—22no—
87Q8×C97—22no—
88Q8×C97—22no—
89D4×C97—22no—
90C2×C4×C97—21no—
91C2×C4×C97—22no—
92C2×C4×C97C2×D412yes—
93C2×C4×C97C2411yescharacteristic
94C22⋊D97—42no—
95C2×Dic97—21no—
96C2×Dic97C2×D412yes—
97C2×Dic97C2×D411yescharacteristic
98D388—22no—
99D388—22no—
100D388—22no—
101D388—44no—
102D388—44no—
103C4×D97—42no—
104C4×D97—42no—
105C4×D97—44no—
106C4×D97C2×D414yes—
107Dic194—22no—
108Dic194—42no—
109C8×C97C2×D414yes—
110#11715895C2×D414yes—
111C22×C97—41no—
112C22×C97—42no—
113C22×C97C22×D411yescharacteristic
114C2×D97—22no—
115C2×D97—42no—
116C2×D97—41no—
117C2×D97—44no—
118C2×D97—42no—
119C2×D97—84no—
120C2×D97#12712yes—
121C4×C97—22no—
122C4×C97—41no—
123C4×C97—42no—
124C4×C97C22×D412yes—
125Dic97—22no—
126Dic97—41no—
127Dic97#12712yes—
128C2×C97—81no—
129C2×C97C2×#12711yescharacteristic
130C2×C97#57512yes—
131D97—22no—
132D97—82no—
133D97—81no—
134#3153—971nomaximal, Sylow
135C97#315311yescharacteristic, complemented, Sylow
136C2×#137—972no—
137C2×AGL(1,ℤ/8)—972no—
138#532—971no—
139C2×#124—971no—
140#497—971no—
141#497D9711yescharacteristic, complemented
142#496—971no—
143#496—971no—
144#495—972no—
145#495—972no—
146C4×M16—971no—
147C2×#41—972no—
148C22×Q8—971no—
149C22×D4—971no—
150#137—1948no—
151AGL(1,ℤ/8)—1948no—
152C2×Q16—972no—
153C2×Q16C2×D9712yescomplemented
154C2×SD16—972no—
155C2×SD16—972no—
156C2×SD16—972no—
157C2×SD16C2×D9712yes—
158C2×D8—972no—
159C2×D8—972no—
160C2×M16—972no—
161C4⋊Q8C2×D9711yescharacteristic, complemented
162#127—971no—
163#124—971no—
164#124—972no—
165#124—971no—
166#124—971no—
167#124—1942no—
168#124C2×D9711yescharacteristic
169#121—1942no—
170C4×D4—1942no—
171C2×C22⋊C4—1942no—
172C2×C42—971no—
173#97—971no—
174#97C2×D9711yescharacteristic, complemented
175C4×C8—972no—
176C24—1941no—
177#41—1944no—
178#41—1944no—
179C2×Q8—971no—
180C2×Q8—1941no—
181C2×Q8—1944no—
182C2×Q8C22×D9711yescharacteristic
183C2×Q8C22×D9712yes—
184C2×D4—971no—
185C2×D4—972no—
186C2×D4—1942no—
187C2×D4—1944no—
188C2×D4—1941no—
189C2×D4—1942no—
190C2×D4C22×D9711yescharacteristic
191C22×C4—971no—
192C22×C4—972no—
193C22×C4—1942no—
194C22×C4—1941no—
195Q16—24no—
196Q16—1944no—
197SD16—24no—
198SD16—1944no—
199SD16—1944no—
200SD16—1944no—
201D8—1944no—
202D8—1944no—
203M16—978no—
204C2×C8—972no—
205C2×C8C22×D9712yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—1942no—
209C22⋊C4—1944no—
210C22⋊C4—1942no—
211C22⋊C4—1942no—
212C42—972no—
213C42—971no—
214C42C22×D9711yescharacteristic
215C23—21no—
216C23—971no—
217C23—3882no—
218C23—3884no—
219C23—1941no—
220C23—1942no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—3882no—
225Q8—1942no—
226D4—22no—
227D4—3884no—
228D4—3884no—
229D4—3882no—
230D4—1942no—
231D4—1942no—
232D4—1942no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—971no—
236C2×C4—974no—
237C2×C4—972no—
238C2×C4—3882no—
239C2×C4—3884no—
240C2×C4—3882no—
241C2×C4—1941no—
242C2×C4D4×D9712yes—
243C2×C4C23×D9711yescharacteristic
244C8—974no—
245C8D4×D9714yes—
246C22—41no—
247C22—42no—
248C22—7764no—
249C22—972no—
250C22—3882no—
251C22—3881no—
252C22—3884no—
253C22—3882no—
254C22—1942no—
255C22C2×D4×D9711yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—972no—
260C4—3881no—
261C4—1942no—
262C4C2×D4×D9712yes—
263C2—81no—
264C2—7761no—
265C2—7762no—
266C2—1942no—
267C2#127×D9711yescharacteristic
268C2#2026153412yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#497×C97C2
socleC22×C97C22×D4
hypercenter#497D97
hypocenterC97#3153
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#15690842C2semidirect
2. C2×#15690837C2semidirect
3. #20261286C2semidirect
4. #124×D97C2semidirect
5. #20261127C2semidirect
6. #20261125C2semidirect
7. #20261124C2semidirect
8. #20261120C2semidirect
9. #20261116C2semidirect
10. #20260687C2semidirect
11. #497×C97C2semidirect
17. C2×#13095501C22semidirect
18. C2×#13095500C22semidirect
20. C2×#13095498C22semidirect
24. #15690807C22semidirect
26. #15690799C22semidirect
34. #15690721C22semidirect
35. C8×Dic97C22semidirect
36. C2×Q16×C97C22semidirect
38. C4⋊Q8×C97C22semidirect
40. #97×C97C22semidirect
63. #13095494D4semidirect
135. C97#3153semidirect
141. #497D97semidirect
153. C2×Q16C2×D97semidirect
161. C4⋊Q8C2×D97semidirect
174. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908422
C2×#156908372
#202612862
#124×D972
#202611272
#202611252
#202611242
#202611202
#202611162
#202606872
#497×C972
#315397