Finite Group Databasebeta

← #27153348

Subgroups — #27153348

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×#15690838C212yesmaximal, complemented
2#20261381C212yesmaximal, complemented
3#20261119C211yesmaximal, characteristic, complemented
4#20260981C212yesmaximal, complemented
5#20260951C212yesmaximal, complemented
6#20260899C212yesmaximal, complemented
7#20260871C212yesmaximal, complemented
8C4×D776C211yesmaximal, characteristic, complemented
9#436×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2212yes—
11C22×D388C2212yescomplemented
12#15690838—28no—
13C2×Dic388C2211yescharacteristic, complemented
14C2×#13095500C2212yescomplemented
15C2×#13095500C2212yescomplemented
16C2×D776C2212yes—
17C2×D776C2211yescharacteristic
18C2×D776C2211yescharacteristic
19#15690798C2212yescomplemented
20#15690795—22no—
21#15690789C2212yescomplemented
22#15690784C2212yes—
23#15690783—22no—
24#15690778—22no—
25C4×D388C2212yescomplemented
26C22⋊C4×D97—22no—
27#15690744C2211yescharacteristic, complemented
28#15690735C2212yescomplemented
29#15690732C2212yescomplemented
30#15690732C2212yescomplemented
31C2×M16×C97C2212yescomplemented
32C4×D4×C97C2211yescharacteristic
33C4⋊C8×C97C2211yescharacteristic, complemented
34C22⋊C8×C97C2212yescomplemented
35C4×C8×C97C2211yescharacteristic, complemented
36C23×D97—22no—
37#13095511—24no—
38#13095511—42no—
39C2×Dic194C2312yes—
40C2×D388—24no—
41C2×D388—22no—
42C2×D388—42no—
43C2×D388C2312yes—
44C2×D388C2312yes—
45C2×C22⋊D97—22no—
46C2×C22⋊D97—22no—
47C2×C4×D97—22no—
48C2×C4×D97—22no—
49Dic388—41no—
50#13095500—24no—
51#13095500—24no—
52D776—24no—
53D776—41no—
54D776D414yescomplemented
55C4⋊Dic97C2312yes—
56#13095490—22no—
57C22⋊Dic97—22no—
58#13095488—22no—
59#13095488—22no—
60#13095488—22no—
61#13095488—22no—
62C2×D4×C97D411yescharacteristic
63C22×C4×C97C2312yes—
64M16×C97—24no—
65C2×C8×C97C2312yes—
66C2×C8×C97C2311yescharacteristic
67C2×C8×C97C2311yescharacteristic
68C4⋊C4×C97D411yescharacteristic
69C22⋊C4×C97D412yes—
70C42×C97C2311yescharacteristic
71C23×C97C2×D412yes—
72C22×D97—22no—
73C22×D97—22no—
74C22×D97—44no—
75C22×D97—44no—
76C22×D97—82no—
77D4×C97—41no—
78C2×C4×C97—22no—
79C2×C4×C97—22no—
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D412yes—
83C2×C4×C97C2411yescharacteristic
84C22⋊D97—44no—
85C22⋊D97—82no—
86C2×Dic97—22no—
87C2×Dic97—22no—
88D388—22no—
89D388—22no—
90D388—22no—
91D388—44no—
92D388C2×D414yes—
93C4×D97—44no—
94C4×D97—82no—
95Dic194—22no—
96Dic194—22no—
97C8×C97—22no—
98C8×C97—21no—
99C8×C97C2×D412yes—
100C22×C97—22no—
101C22×C97—22no—
102C22×C97—22no—
103C22×C97C22×D411yescharacteristic
104C2×D97—24no—
105C2×D97—42no—
106C2×D97—42no—
107C2×D97—42no—
108C2×D97—42no—
109C2×D97—42no—
110C2×D97—88no—
111C4×C97—22no—
112C4×C97—21no—
113C4×C97—21no—
114C4×C97C22×D411yescharacteristic
115C4×C97H(5,𝔽2)11yescharacteristic
116Dic97—42no—
117Dic97—42no—
118C2×C97—42no—
119C2×C97D4211yescharacteristic
120C2×C97#57511yescharacteristic
121C2×C97#57611yescharacteristic
122D97—44no—
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×D8—971no—
134#436D9711yescharacteristic, complemented
135C2×#41—972no—
136C22×D4—972no—
137AGL(1,ℤ/8)—1948no—
138C2×Q16—971no—
139C2×SD16—972no—
140C2×SD16—972no—
141C2×D8—971no—
142C2×D8—971no—
143C2×D8—972no—
144C2×M16C2×D9712yescomplemented
145#124—972no—
146#123—1942no—
147C22⋊Q8—972no—
148#121—972no—
149#121—1942no—
150#120—1942no—
151C4×D4—972no—
152C4×D4C2×D9711yescharacteristic
153C2×C22⋊C4—1942no—
154#107—971no—
155C4⋊C8C2×D9711yescharacteristic, complemented
156#103—972no—
157#102—972no—
158#102—972no—
159C22⋊C8C2×D9712yescomplemented
160C4×C8C2×D9711yescharacteristic, complemented
161C24—1942no—
162#41—3882no—
163#41—1944no—
164C2×Q8—972no—
165C2×D4—972no—
166C2×D4—972no—
167C2×D4—3882no—
168C2×D4—1944no—
169C2×D4—1942no—
170C2×D4—1942no—
171C2×D4—1942no—
172C2×D4D38811yescharacteristic
173C22×C4—1942no—
174C22×C4—1942no—
175C22×C4C22×D9712yes—
176Q16—3881no—
177SD16—1944no—
178SD16—1944no—
179D8—974no—
180D8—3881no—
181D8—1944no—
182M16—24no—
183C2×C8C22×D9711yescharacteristic
184C2×C8C22×D9712yes—
185C2×C8C22×D9711yescharacteristic
186C4⋊C4—972no—
187C4⋊C4—1942no—
188C4⋊C4D38811yescharacteristic
189C22⋊C4—1942no—
190C22⋊C4—1942no—
191C22⋊C4—1942no—
192C22⋊C4—1942no—
193C22⋊C4—1942no—
194C22⋊C4D38812yes—
195C42C22×D9711yescharacteristic
196C23—7762no—
197C23—3884no—
198C23—3884no—
199C23—1942no—
200C23—1942no—
201C23C2×D38812yes—
202Q8—1942no—
203Q8—1942no—
204D4—41no—
205D4—7762no—
206D4—974no—
207D4—3884no—
208D4—3884no—
209D4—1942no—
210D4—1942no—
211D4—1942no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—7762no—
215C2×C4—3884no—
216C2×C4—1942no—
217C2×C4—1942no—
218C2×C4C2×D38811yescharacteristic
219C2×C4C2×D38811yescharacteristic
220C2×C4C2×D38812yes—
221C2×C4C23×D9711yescharacteristic
222C8—22no—
223C8—21no—
224C8D4×D9712yes—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—7768no—
229C22—3882no—
230C22—3882no—
231C22—3882no—
232C22—3882no—
233C22—3882no—
234C22—1944no—
235C22C22×D38811yescharacteristic
236C4—21no—
237C4—21no—
238C4—22no—
239C4—3882no—
240C4—3882no—
241C4C2×D4×D9711yescharacteristic
242C4#1569085611yescharacteristic
243C2—42no—
244C2—7762no—
245C2—3884no—
246C2D4×D38811yescharacteristic
247C2#2026153511yescharacteristic
248C2#2026154011yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#436×C97C2
socleC22×C97C22×D4
hypercenter#436D97
hypocenterC97#3273
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#15690838C2semidirect
2. #20261381C2semidirect
3. #20261119C2semidirect
4. #20260981C2semidirect
5. #20260951C2semidirect
6. #20260899C2semidirect
7. #20260871C2semidirect
8. C4×D776C2semidirect
9. #436×C97C2semidirect
11. C22×D388C22semidirect
13. C2×Dic388C22semidirect
14. C2×#13095500C22semidirect
15. C2×#13095500C22semidirect
19. #15690798C22semidirect
21. #15690789C22semidirect
25. C4×D388C22semidirect
27. #15690744C22semidirect
28. #15690735C22semidirect
29. #15690732C22semidirect
30. #15690732C22semidirect
31. C2×M16×C97C22semidirect
33. C4⋊C8×C97C22semidirect
34. C22⋊C8×C97C22semidirect
35. C4×C8×C97C22semidirect
54. D776D4semidirect
125. C97#3273semidirect
134. #436D97semidirect
144. C2×M16C2×D97semidirect
155. C4⋊C8C2×D97semidirect
159. C22⋊C8C2×D97semidirect
160. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908382
#202613812
#202611192
#202609812
#202609512
#202608992
#202608712
C4×D7762
#436×C972
#327397