Finite Group Databasebeta

← #496×D269 #46821315

Subgroups — #496×D269 #46821315

Order 34432.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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×SD16×D269C212yesmaximal, complemented
2C2×D8×D269C212yesmaximal, complemented
3#127×D269C211yesmaximal, characteristic, complemented
4#124×D269C211yesmaximal, characteristic, complemented
5#31770995C211yesmaximal, characteristic, complemented
6#31770994C211yesmaximal, characteristic, complemented
7#31770993C212yesmaximal, complemented
8#31770992C211yesmaximal, characteristic, complemented
9#31770991C212yesmaximal, complemented
10#97×D269C211yesmaximal, characteristic, complemented
11#496×C269C211yesmaximal, characteristic, complemented
12C2×Q8×D269C2211yescharacteristic
13C2×D4×D269—22no—
14C2×D4×D269C2212yes—
15C2×D4×D269C2211yescharacteristic
16C2×#16864391C2212yescomplemented
17C2×#16864390C2212yescomplemented
18C2×#16864389C2212yescomplemented
19SD16×D269—28no—
20C2×D2152C2212yescomplemented
21C2×#16864387C2212yescomplemented
22C2×#16864387C2212yescomplemented
23D8×D269—28no—
24C2×C8×D269C2212yes—
25#22575450C2211yescharacteristic, complemented
26#22575449—22no—
27#22575449C2212yescomplemented
28#22575442C2211yescharacteristic, complemented
29#22575441C2211yescharacteristic, complemented
30#22575440C2211yescharacteristic, complemented
31#22575439—22no—
32C22⋊C4×D269—22no—
33C42×D269C2211yescharacteristic
34#22575364C2211yescharacteristic, complemented
35#22575363C2212yescomplemented
36C2×SD16×C269C2212yescomplemented
37C2×D8×C269C2212yescomplemented
38#127×C269C2211yescharacteristic, complemented
39#124×C269C2211yescharacteristic, complemented
40#97×C269C2211yescharacteristic, complemented
41C23×D269—21no—
42C23×D269—22no—
43C2×Dic538—21no—
44C2×Dic538C2311yescharacteristic
45Q8×D269—24no—
46C2×D1076—22no—
47C2×D1076C2311yescharacteristic
48C2×D1076C2312yes—
49C2×C22⋊D269—21no—
50C2×C22⋊D269—22no—
51C2×C22⋊D269—24no—
52D4×D269—24no—
53D4×D269—24no—
54D4×D269—24no—
55D4×D269—48no—
56C2×C4×D269—21no—
57C2×C4×D269D412yes—
58C2×C4×D269C2311yescharacteristic
59#16864391—24no—
60#16864390—24no—
61#16864389—24no—
62D2152—24no—
63#16864387—24no—
64#16864387—24no—
65C2×#14524130C2312yes—
66C8×D269D418yescomplemented
67C22⋊Dic269—22no—
68#16864379—22no—
69#16864379—22no—
70C4×Dic269D412yes—
71C4×Dic269C2311yescharacteristic
72C2×Q8×C269C2311yescharacteristic
73C2×D4×C269—22no—
74C2×D4×C269C2311yescharacteristic
75C2×D4×C269C2312yes—
76SD16×C269—24no—
77D8×C269—24no—
78C2×C8×C269C2312yes—
79C22⋊C4×C269—22no—
80C42×C269C2311yescharacteristic
81C22×D269—21no—
82C22×D269—22no—
83C22×D269—42no—
84C22×D269—44no—
85C22×D269—48no—
86C22×D269—44no—
87C22×D269C2×D411yescharacteristic
88Dic538—22no—
89Dic538—42no—
90D1076—22no—
91D1076—22no—
92D1076—22no—
93D1076—44no—
94C22⋊D269—48no—
95C22⋊D269—42no—
96C22⋊D269—44no—
97C2×Dic269—21no—
98C2×Dic269C2×D412yes—
99C2×Dic269C2×D411yescharacteristic
100C4×D269—24no—
101C4×D269—42no—
102C4×D269C2×D414yes—
103#14524130C2×D414yescomplemented
104C23×C269—21no—
105C23×C269—22no—
106Q8×C269—22no—
107D4×C269—22no—
108D4×C269—22no—
109D4×C269—22no—
110D4×C269—44no—
111C2×C4×C269—21no—
112C2×C4×C269C2×D412yes—
113C2×C4×C269C2411yescharacteristic
114C8×C269C2×D414yescomplemented
115C22×C269—41no—
116C22×C269—42no—
117C22×C269—44no—
118C22×C269—42no—
119C22×C269C22×D411yescharacteristic
120C2×D269—41no—
121C2×D269—42no—
122C2×D269—44no—
123C2×D269—42no—
124C2×D269—88no—
125C2×D269—84no—
126C2×D269#12712yes—
127C2×D269AGL(1,ℤ/8)14yes—
128C4×C269—22no—
129C4×C269—41no—
130C4×C269C22×D412yes—
131Dic269—22no—
132Dic269—41no—
133Dic269#12712yes—
134C2×C269—82no—
135C2×C269—81no—
136C2×C269C2×#12711yescharacteristic
137C2×C269C2×AGL(1,ℤ/8)12yes—
138D269—82no—
139D269—81no—
140D269#49614yesdirect product factor
141C269C2×#49611yescharacteristic, complemented, Sylow
142C2×#496—2691nomaximal, Sylow
143C22×SD16—2692no—
144C22×D8—2692no—
145C2×#127—2691no—
146C2×#124—2691no—
147#496—2692no—
148#496—2692no—
149#496—2691no—
150#496—2691no—
151#496—2691no—
152#496D26911yescharacteristic, direct product factor
153C2×#97—2691no—
154C22×Q8—2691no—
155C22×D4—2691no—
156C22×D4—2692no—
157C22×D4—5382no—
158C2×SD16—2692no—
159C2×SD16—2692no—
160C2×SD16—2692no—
161C2×SD16—5388no—
162C2×SD16C2×D26912yescomplemented
163C2×D8—2692no—
164C2×D8—2692no—
165C2×D8—2692no—
166C2×D8—5388no—
167C2×D8C2×D26912yescomplemented
168C22×C8—2692no—
169#127—2692no—
170#127—2691no—
171#127—5382no—
172#127C2×D26911yescharacteristic, complemented
173#124—2691no—
174#124—2691no—
175#124—2691no—
176#124—5382no—
177#124C2×D26911yescharacteristic, complemented
178C2×C22⋊C4—5382no—
179C2×C42—2691no—
180#97—2692no—
181#97—2691no—
182#97C2×D26911yescharacteristic, complemented
183C24—5381no—
184C24—5382no—
185C2×Q8—2691no—
186C2×Q8—5381no—
187C2×Q8—5384no—
188C2×Q8C22×D26911yescharacteristic
189C2×D4—22no—
190C2×D4—2691no—
191C2×D4—2692no—
192C2×D4—5384no—
193C2×D4—5384no—
194C2×D4—5384no—
195C2×D4—5381no—
196C2×D4—5384no—
197C2×D4—5382no—
198C2×D4—5382no—
199C2×D4—10768no—
200C2×D4C22×D26911yescharacteristic
201C2×D4C22×D26912yes—
202C22×C4—2691no—
203C22×C4—2692no—
204C22×C4—5381no—
205SD16—24no—
206SD16—5384no—
207SD16—5384no—
208SD16—5384no—
209D8—24no—
210D8—5384no—
211D8—5384no—
212D8—5384no—
213C2×C8—2692no—
214C2×C8—2698no—
215C2×C8C22×D26912yes—
216C22⋊C4—22no—
217C22⋊C4—5382no—
218C22⋊C4—5382no—
219C22⋊C4—5382no—
220C42—2692no—
221C42—2691no—
222C42C22×D26911yescharacteristic
223C23—21no—
224C23—22no—
225C23—2691no—
226C23—5381no—
227C23—5382no—
228C23—10762no—
229C23—10764no—
230C23—10768no—
231C23—10764no—
232Q8—22no—
233Q8—5382no—
234Q8—10762no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—5382no—
240D4—5382no—
241D4—5382no—
242D4—10764no—
243D4—10762no—
244D4—10764no—
245D4—10768no—
246C2×C4—21no—
247C2×C4—2691no—
248C2×C4—2694no—
249C2×C4—2692no—
250C2×C4—5384no—
251C2×C4—5381no—
252C2×C4—10762no—
253C2×C4D4×D26912yes—
254C2×C4C23×D26911yescharacteristic
255C8—2694no—
256C8D4×D26914yescomplemented
257C22—41no—
258C22—42no—
259C22—44no—
260C22—42no—
261C22—2692no—
262C22—2694no—
263C22—10761no—
264C22—10762no—
265C22—10764no—
266C22—10762no—
267C22—21528no—
268C22—21524no—
269C22C2×D4×D26911yescharacteristic
270C4—22no—
271C4—41no—
272C4—2692no—
273C4—5382no—
274C4—10761no—
275C4C2×D4×D26912yes—
276C2—82no—
277C2—81no—
278C2—2694no—
279C2—21522no—
280C2—21521no—
281C2#127×D26911yescharacteristic
282C2AGL(1,ℤ/8)×D26912yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#496×C269C2
socleC22×C269C22×D4
hypercenter#496D269
hypocenterC269C2×#496
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D269C2semidirect
2. C2×D8×D269C2semidirect
3. #127×D269C2semidirect
4. #124×D269C2semidirect
5. #31770995C2semidirect
6. #31770994C2semidirect
7. #31770993C2semidirect
8. #31770992C2semidirect
9. #31770991C2semidirect
10. #97×D269C2semidirect
11. #496×C269C2semidirect
16. C2×#16864391C22semidirect
17. C2×#16864390C22semidirect
18. C2×#16864389C22semidirect
20. C2×D2152C22semidirect
21. C2×#16864387C22semidirect
22. C2×#16864387C22semidirect
25. #22575450C22semidirect
27. #22575449C22semidirect
28. #22575442C22semidirect
29. #22575441C22semidirect
30. #22575440C22semidirect
34. #22575364C22semidirect
35. #22575363C22semidirect
36. C2×SD16×C269C22semidirect
37. C2×D8×C269C22semidirect
38. #127×C269C22semidirect
39. #124×C269C22semidirect
40. #97×C269C22semidirect
66. C8×D269D4semidirect
103. #14524130C2×D4semidirect
114. C8×C269C2×D4semidirect
140. D269#496direct (N × H)
141. C269C2×#496semidirect
152. #496D269direct (N × H)
162. C2×SD16C2×D269semidirect
167. C2×D8C2×D269semidirect
172. #127C2×D269semidirect
177. #124C2×D269semidirect
182. #97C2×D269semidirect
256. C8D4×D269semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D2692
C2×D8×D2692
#127×D2692
#124×D2692
#317709952
#317709942
#317709932
#317709922
#317709912
#97×D2692
#496×C2692
C2×#496269