Finite Group Databasebeta

← #27153345

Subgroups — #27153345

Order 12416.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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×#15690837C212yesmaximal, complemented
2#20261378C212yesmaximal, complemented
3#20260978C212yesmaximal, complemented
4#20260947C212yesmaximal, complemented
5#20260896C212yesmaximal, complemented
6#20260867C212yesmaximal, complemented
7C4×D776C211yesmaximal, characteristic, complemented
8#20260814C211yesmaximal, characteristic, complemented
9#495×C97C211yesmaximal, characteristic, complemented
10C2×#13095512C2212yes—
11C2×D4×D97C2212yescomplemented
12#15690837—28no—
13C2×#13095498C2212yescomplemented
14C2×D776C2211yescharacteristic
15C2×#13095496C2212yes—
16C2×#13095494C2212yescomplemented
17#15690792—22no—
18#15690786C2212yescomplemented
19#15690783—22no—
20#15690781C2212yes—
21#15690780—22no—
22C4×D388C2211yescharacteristic
23C4×D388C2212yescomplemented
24C22⋊C4×D97—22no—
25#15690744C2211yescharacteristic, complemented
26C4⋊#11715895C2211yescharacteristic, complemented
27#15690734C2212yescomplemented
28#15690732C2212yescomplemented
29#15690731C2212yescomplemented
30#15690722C2212yescomplemented
31C2×Q16×C97C2211yescharacteristic, complemented
32C2×SD16×C97C2212yescomplemented
33C2×D8×C97C2211yescharacteristic
34#124×C97C2212yescomplemented
35C4×C8×C97C2211yescharacteristic, complemented
36C23×D97—22no—
37#13095512—24no—
38#13095512—42no—
39C2×D388—22no—
40C2×D388D411yescharacteristic
41C2×D388C2312yes—
42C2×C22⋊D97—22no—
43C2×C22⋊D97—22no—
44D4×D97—24no—
45D4×D97—42no—
46C2×C4×D97—22no—
47C2×C4×D97—22no—
48C2×C4×D97C2312yes—
49#13095498—24no—
50D776D414yescomplemented
51#13095496—24no—
52#13095494—24no—
53C2×#11715895C2312yes—
54C4⋊Dic97D411yescharacteristic
55C4⋊Dic97C2312yes—
56#13095490—22no—
57C22⋊Dic97—22no—
58#13095488—22no—
59#13095488—22no—
60#13095488D412yes—
61C2×Q8×C97C2312yes—
62C2×D4×C97C2312yes—
63Q16×C97—41no—
64SD16×C97—24no—
65D8×C97—41no—
66C2×C8×C97C2311yescharacteristic
67C2×C8×C97C2311yescharacteristic
68C22⋊C4×C97—22no—
69C22⋊C4×C97—22no—
70C42×C97C2311yescharacteristic
71C23×C97—22no—
72C22×D97—22no—
73C22×D97—44no—
74C22×D97—44no—
75C22×D97—82no—
76C22×D97C2×D412yes—
77Q8×C97—22no—
78Q8×C97—22no—
79D4×C97—22no—
80D4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97C2×D411yescharacteristic
84C2×C4×C97C2411yescharacteristic
85C22⋊D97—44no—
86C22⋊D97—82no—
87C2×Dic97—22no—
88C2×Dic97C2×D412yes—
89D388—22no—
90D388—41no—
91D388—44no—
92D388C2×D414yes—
93C4×D97—22no—
94C4×D97—22no—
95C4×D97—44no—
96C4×D97—82no—
97C8×C97—21no—
98C8×C97C2×D412yes—
99#11715895—22no—
100C22×C97—42no—
101C22×C97—42no—
102C22×C97—42no—
103C22×C97C22×D411yescharacteristic
104C2×D97—22no—
105C2×D97—24no—
106C2×D97—22no—
107C2×D97—22no—
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—82no—
119C2×C97D4211yescharacteristic
120C2×C97#57511yescharacteristic
121C2×C97#57611yescharacteristic
122D97—42no—
123D97—44no—
124#3273—971nomaximal, Sylow
125C97#327311yescharacteristic, complemented, Sylow
126C2×AGL(1,ℤ/8)—972no—
127#546—972no—
128#495D9711yescharacteristic, complemented
129#468—972no—
130#463—972no—
131#452—972no—
132#447—972no—
133C4×D8—971no—
134#436—971no—
135C2×#41—972no—
136C22×D4—972no—
137AGL(1,ℤ/8)—1948no—
138C2×Q16C2×D9711yescharacteristic, complemented
139C2×SD16—972no—
140C2×SD16C2×D9712yescomplemented
141C2×D8—972no—
142C2×D8—971no—
143C2×D8C2×D9711yescharacteristic
144C2×M16—972no—
145#124C2×D9712yescomplemented
146#123—1942no—
147C22⋊Q8—972no—
148#121—972no—
149#121—1942no—
150#120—1942no—
151C4×D4—971no—
152C4×D4—972no—
153C2×C22⋊C4—1942no—
154#107—971no—
155C4⋊C8—971no—
156#103—972no—
157#102—972no—
158#102—972no—
159C22⋊C8—972no—
160C4×C8C2×D9711yescharacteristic, complemented
161C24—1942no—
162#41—3882no—
163#41—1944no—
164C2×Q8C22×D9712yes—
165C2×D4—971no—
166C2×D4—972no—
167C2×D4—3882no—
168C2×D4—1942no—
169C2×D4—1942no—
170C2×D4—1944no—
171C2×D4—1942no—
172C2×D4C22×D9712yes—
173C22×C4—972no—
174C22×C4—1942no—
175C22×C4—1942no—
176Q16—41no—
177SD16—24no—
178SD16—1944no—
179D8—41no—
180D8—974no—
181D8—1944no—
182M16—1944no—
183C2×C8—972no—
184C2×C8C22×D9711yescharacteristic
185C2×C8C22×D9711yescharacteristic
186C4⋊C4—971no—
187C4⋊C4—972no—
188C4⋊C4—1942no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—972no—
192C22⋊C4—1942no—
193C22⋊C4—1942no—
194C22⋊C4—1942no—
195C42C22×D9711yescharacteristic
196C23—22no—
197C23—7762no—
198C23—972no—
199C23—3884no—
200C23—3884no—
201C23—1942no—
202Q8—22no—
203Q8—22no—
204D4—22no—
205D4—22no—
206D4—7762no—
207D4—974no—
208D4—3881no—
209D4—3884no—
210D4—3884no—
211D4—1942no—
212C2×C4—22no—
213C2×C4—7762no—
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—
224C8D4×D9712yes—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—7768no—
229C22—3882no—
230C22—3882no—
231C22—1942no—
232C22—1942no—
233C22—1944no—
234C22—1942no—
235C22C2×D4×D9711yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—3882no—
240C4—1942no—
241C4C2×D4×D9711yescharacteristic
242C4#1569085511yescharacteristic
243C2—82no—
244C2—3882no—
245C2—3884no—
246C2#2026137011yescharacteristic
247C2#2026153211yescharacteristic
248C2#2026153711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#495×C97C2
socleC22×C97C22×D4
hypercenter#495D97
hypocenterC97#3273
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. #20261378C2semidirect
3. #20260978C2semidirect
4. #20260947C2semidirect
5. #20260896C2semidirect
6. #20260867C2semidirect
7. C4×D776C2semidirect
8. #20260814C2semidirect
9. #495×C97C2semidirect
11. C2×D4×D97C22semidirect
13. C2×#13095498C22semidirect
16. C2×#13095494C22semidirect
18. #15690786C22semidirect
23. C4×D388C22semidirect
25. #15690744C22semidirect
26. C4⋊#11715895C22semidirect
27. #15690734C22semidirect
28. #15690732C22semidirect
29. #15690731C22semidirect
30. #15690722C22semidirect
31. C2×Q16×C97C22semidirect
32. C2×SD16×C97C22semidirect
34. #124×C97C22semidirect
35. C4×C8×C97C22semidirect
50. D776D4semidirect
125. C97#3273semidirect
128. #495D97semidirect
138. C2×Q16C2×D97semidirect
140. C2×SD16C2×D97semidirect
145. #124C2×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×#156908372
#202613782
#202609782
#202609472
#202608962
#202608672
C4×D7762
#202608142
#495×C972
#327397