Finite Group Databasebeta

← #27153346

Subgroups — #27153346

Order 12416.

520 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 129 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×#15690834C212yesmaximal, complemented
2#20261380C212yesmaximal, complemented
3#20261119C211yesmaximal, characteristic, complemented
4#20260982C212yesmaximal, complemented
5#20260950C212yesmaximal, complemented
6#20260898C212yesmaximal, complemented
7#20260870C212yesmaximal, complemented
8#20260814C211yesmaximal, characteristic, complemented
9C4×D8×C97C211yesmaximal, characteristic, complemented
10C2×#13095510C2212yes—
11C2×D4×D97C2212yescomplemented
12#15690834—28no—
13C2×Dic388C2211yescharacteristic, complemented
14C2×#13095500C2212yescomplemented
15C2×#13095499C2212yescomplemented
16C2×D776C2211yescharacteristic
17C2×#13095496C2212yes—
18C2×#13095494C2212yescomplemented
19#15690798C2212yescomplemented
20#15690794—22no—
21#15690790C2212yescomplemented
22#15690785C2212yes—
23#15690784—22no—
24#15690780—22no—
25C4×D388C2211yescharacteristic
26C2×#13095488—22no—
27C4⋊#11715895C2211yescharacteristic, complemented
28#15690733C2212yescomplemented
29#15690730C2212yescomplemented
30#15690722C2212yescomplemented
31C2×D8×C97C2211yescharacteristic
32C4×D4×C97C2212yescomplemented
33#107×C97C2211yescharacteristic, complemented
34#102×C97C2212yescomplemented
35C4×C8×C97C2211yescharacteristic, complemented
36C23×D97—22no—
37#13095510—24no—
38#13095510—42no—
39C2×Dic194C2312yes—
40C2×D388—22no—
41C2×D388D411yescharacteristic
42C2×D388C2312yes—
43C2×C22⋊D97—22no—
44C2×C22⋊D97—22no—
45D4×D97—24no—
46D4×D97—42no—
47C22×Dic97—22no—
48C2×C4×D97C2312yes—
49Dic388—41no—
50#13095500—24no—
51#13095499—24no—
52D776—41no—
53#13095496—24no—
54#13095494—24no—
55C2×#11715895C2312yes—
56C4⋊Dic97—22no—
57C4⋊Dic97D411yescharacteristic
58#13095488—22no—
59#13095488—22no—
60#13095488—22no—
61#13095488—22no—
62#13095488D412yes—
63C2×D4×C97C2312yes—
64C22×C4×C97—22no—
65D8×C97D414yescomplemented
66C2×C8×C97C2311yescharacteristic
67C2×C8×C97C2311yescharacteristic
68C4⋊C4×C97C2312yes—
69C22⋊C4×C97—22no—
70C42×C97C2311yescharacteristic
71C23×C97—22no—
72C22×D97—22no—
73C22×D97—44no—
74C22×D97—44no—
75C22×D97—82no—
76C22×D97C2×D412yes—
77D4×C97—22no—
78D4×C97C2×D414yes—
79C2×C4×C97—22no—
80C2×C4×C97—82no—
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97C2411yescharacteristic
84C22⋊D97—44no—
85C22⋊D97—44no—
86C2×Dic97—22no—
87C2×Dic97—44no—
88C2×Dic97C2×D412yes—
89D388—22no—
90D388—22no—
91D388—41no—
92D388—82no—
93C4×D97—22no—
94C4×D97—22no—
95Dic194—22no—
96Dic194—22no—
97C8×C97—21no—
98C8×C97C2×D412yes—
99#11715895—22no—
100C22×C97—24no—
101C22×C97—42no—
102C22×C97—42no—
103C22×C97C22×D411yescharacteristic
104C2×D97—22no—
105C2×D97—22no—
106C2×D97—22no—
107C2×D97—42no—
108C2×D97—42no—
109C2×D97—42no—
110C2×D97—88no—
111C4×C97—21no—
112C4×C97—21no—
113C4×C97—42no—
114C4×C97C22×D411yescharacteristic
115C4×C97H(5,𝔽2)11yescharacteristic
116Dic97—22no—
117Dic97—42no—
118C2×C97—44no—
119C2×C97D4211yescharacteristic
120C2×C97#57511yescharacteristic
121C2×C97#57611yescharacteristic
122D97—42no—
123D97—82no—
124#3273—971nomaximal, Sylow
125C97#327311yescharacteristic, complemented, Sylow
126C2×AGL(1,ℤ/8)—972no—
127#546—972no—
128#495—971no—
129#468—972no—
130#463—972no—
131#452—972no—
132#447—972no—
133C4×D8D9711yescharacteristic, complemented
134#436—971no—
135C2×#41—972no—
136C22×D4—972no—
137AGL(1,ℤ/8)—1948no—
138C2×Q16—971no—
139C2×SD16—972no—
140C2×SD16—972no—
141C2×D8—972no—
142C2×D8—971no—
143C2×D8C2×D9711yescharacteristic
144C2×M16—972no—
145#124—972no—
146#123—1942no—
147C22⋊Q8—972no—
148#121—972no—
149#121—1942no—
150#120—1942no—
151C4×D4—971no—
152C4×D4C2×D9712yescomplemented
153C2×C22⋊C4—1942no—
154#107C2×D9711yescharacteristic, complemented
155C4⋊C8—971no—
156#103—972no—
157#102—972no—
158#102C2×D9712yescomplemented
159C22⋊C8—972no—
160C4×C8C2×D9711yescharacteristic, complemented
161C24—1942no—
162#41—3882no—
163#41—1944no—
164C2×Q8—972no—
165C2×D4—971no—
166C2×D4—972no—
167C2×D4—3882no—
168C2×D4—1942no—
169C2×D4—1942no—
170C2×D4—1942no—
171C2×D4—1944no—
172C2×D4C22×D9712yes—
173C22×C4—22no—
174C22×C4—972no—
175C22×C4—1942no—
176Q16—3881no—
177SD16—1944no—
178SD16—1944no—
179D8—3881no—
180D8—1944no—
181D8D38814yescomplemented
182M16—1944no—
183C2×C8—972no—
184C2×C8C22×D9711yescharacteristic
185C2×C8C22×D9711yescharacteristic
186C4⋊C4—971no—
187C4⋊C4—1942no—
188C4⋊C4C22×D9712yes—
189C22⋊C4—22no—
190C22⋊C4—972no—
191C22⋊C4—1942no—
192C22⋊C4—1942no—
193C22⋊C4—1942no—
194C22⋊C4—1942no—
195C42C22×D9711yescharacteristic
196C23—22no—
197C23—7762no—
198C23—972no—
199C23—3884no—
200C23—3884no—
201C23—1942no—
202Q8—1942no—
203Q8—1942no—
204D4—22no—
205D4—7762no—
206D4—3881no—
207D4—3884no—
208D4—3884no—
209D4—1942no—
210D4—1942no—
211D4C2×D38814yes—
212C2×C4—22no—
213C2×C4—82no—
214C2×C4—972no—
215C2×C4—3884no—
216C2×C4—1942no—
217C2×C4—1942no—
218C2×C4—1942no—
219C2×C4D4×D9711yescharacteristic
220C2×C4D4×D9711yescharacteristic
221C2×C4C23×D9711yescharacteristic
222C8—21no—
223C8—1942no—
224C8C2×D38812yes—
225C22—24no—
226C22—42no—
227C22—42no—
228C22—7768no—
229C22—3882no—
230C22—3882no—
231C22—3882no—
232C22—1942no—
233C22—1942no—
234C22—1942no—
235C22C2×D4×D9711yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—3882no—
240C4—1942no—
241C4C22×D38811yescharacteristic
242C4#1569085611yescharacteristic
243C2—44no—
244C2—7762no—
245C2—3882no—
246C2D4×D38811yescharacteristic
247C2#2026153211yescharacteristic
248C2#2026153611yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×D8×C97C2
socleC22×C97C22×D4
hypercenterC4×D8D97
hypocenterC97#3273
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#15690834C2semidirect
2. #20261380C2semidirect
3. #20261119C2semidirect
4. #20260982C2semidirect
5. #20260950C2semidirect
6. #20260898C2semidirect
7. #20260870C2semidirect
8. #20260814C2semidirect
9. C4×D8×C97C2semidirect
11. C2×D4×D97C22semidirect
13. C2×Dic388C22semidirect
14. C2×#13095500C22semidirect
15. C2×#13095499C22semidirect
18. C2×#13095494C22semidirect
19. #15690798C22semidirect
21. #15690790C22semidirect
27. C4⋊#11715895C22semidirect
28. #15690733C22semidirect
29. #15690730C22semidirect
30. #15690722C22semidirect
32. C4×D4×C97C22semidirect
33. #107×C97C22semidirect
34. #102×C97C22semidirect
35. C4×C8×C97C22semidirect
65. D8×C97D4semidirect
125. C97#3273semidirect
133. C4×D8D97semidirect
152. C4×D4C2×D97semidirect
154. #107C2×D97semidirect
158. #102C2×D97semidirect
160. C4×C8C2×D97semidirect
181. D8D388semidirect

Maximal subgroups

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

subgroupindex
C2×#156908342
#202613802
#202611192
#202609822
#202609502
#202608982
#202608702
#202608142
C4×D8×C972
#327397