Finite Group Databasebeta

← #46821024

Subgroups — #46821024

Order 34432.

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

normal subgroups only

This group has 450 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
1#31771141C211yesmaximal, characteristic, complemented
2C2×D4×Dic269C211yesmaximal, characteristic, complemented
3#31770933C211yesmaximal, characteristic, complemented
4#31770919C211yesmaximal, characteristic, complemented
5#31770903C212yesmaximal, complemented
6#31770822C212yesmaximal, complemented
7#31770739C211yesmaximal, characteristic
8#31770714C212yesmaximal, complemented
9#31770706C212yesmaximal, complemented
10#31770659C211yesmaximal, characteristic, complemented
11#450×C269C211yesmaximal, characteristic, complemented
12C23×Dic269—21no—
13C2×#16864387C2212yes—
14#22575448—21no—
15#22575444C2211yescharacteristic, complemented
16#22575442C2211yescharacteristic, complemented
17#22575437—21no—
18#22575427C2211yescharacteristic
19Q8×Dic269C2211yescharacteristic, complemented
20D4×Dic269—21no—
21D4×Dic269—21no—
22D4×Dic269C2212yes—
23D4×Dic269C2212yes—
24#22575415C2211yescharacteristic, complemented
25#22575413—21no—
26#22575411—21no—
27C2×C4⋊Dic269C2211yescharacteristic
28C2×C22⋊Dic269—21no—
29C2×C4×Dic269C2211yescharacteristic
30#22575387C2212yescomplemented
31#22575384C2212yescomplemented
32#22575382C2212yescomplemented
33#22575377C2212yescomplemented
34#22575375C2212yescomplemented
35#22575374C2212yes—
36C22⋊#14524130C2211yescharacteristic
37#22575363C2212yescomplemented
38C22×D4×C269C2211yescharacteristic
39C2×SD16×C269C2212yes—
40C22⋊Q8×C269C2211yescharacteristic
41#102×C269C2212yes—
42C22⋊C8×C269C2211yescharacteristic
43C2×D1076C2311yescharacteristic
44C2×C22⋊D269—21no—
45C22×Dic269—22no—
46C22×Dic269—22no—
47C22×Dic269—21no—
48C22×Dic269—44no—
49C22×Dic269D411yescharacteristic
50C22×Dic269D411yescharacteristic
51C2×C4×D269—21no—
52#16864387—24no—
53C2×#14524130C2312yes—
54C4⋊Dic269—21no—
55C4⋊Dic269—21no—
56C4⋊Dic269—21no—
57C4⋊Dic269C2311yescharacteristic
58C4⋊Dic269C2312yes—
59#16864381—21no—
60C22⋊Dic269—22no—
61C22⋊Dic269—22no—
62#16864379—21no—
63#16864379—21no—
64#16864379—21no—
65#16864379—21no—
66C4×Dic269—21no—
67C4×Dic269D411yescharacteristic
68C4×Dic269D411yescharacteristic
69C4×Dic269C2311yescharacteristic
70C4×Dic269C2311yescharacteristic
71C24×C269—21no—
72C2×Q8×C269C2311yescharacteristic
73C2×D4×C269—24no—
74C2×D4×C269—21no—
75C2×D4×C269C2312yes—
76C22×C4×C269C2311yescharacteristic
77SD16×C269—24no—
78C2×C8×C269C2312yes—
79C4⋊C4×C269—21no—
80C4⋊C4×C269C2311yescharacteristic
81C22⋊C4×C269—21no—
82C22×D269—21no—
83D1076—22no—
84C22⋊D269—42no—
85C2×Dic269—22no—
86C2×Dic269—21no—
87C2×Dic269—22no—
88C2×Dic269—41no—
89C2×Dic269—44no—
90C2×Dic269—44no—
91C2×Dic269—42no—
92C2×Dic269C2×D411yescharacteristic
93C2×Dic269C2×D411yescharacteristic
94C2×Dic269C2×D411yescharacteristic
95C2×Dic269C2×D411yescharacteristic
96C4×D269—42no—
97#14524130—22no—
98C23×C269—24no—
99C23×C269—22no—
100C23×C269—42no—
101C23×C269—42no—
102C23×C269C2×D411yescharacteristic
103Q8×C269—22no—
104D4×C269—24no—
105D4×C269—22no—
106D4×C269#4114yes—
107C2×C4×C269—22no—
108C2×C4×C269—21no—
109C2×C4×C269—21no—
110C2×C4×C269C2×D411yescharacteristic
111C2×C4×C269C2411yescharacteristic
112C8×C269—22no—
113C22×C269—21no—
114C22×C269—21no—
115C22×C269—24no—
116C22×C269—48no—
117C22×C269—42no—
118C22×C269—42no—
119C22×C269AGL(1,ℤ/8)12yes—
120C22×C269C22×D411yescharacteristic
121C2×D269—41no—
122C2×D269—41no—
123C2×D269—41no—
124C4×C269—21no—
125C4×C269—41no—
126C4×C269—41no—
127C4×C269C2×#4112yes—
128Dic269—21no—
129Dic269—21no—
130Dic269—21no—
131Dic269—21no—
132Dic269—42no—
133Dic269—41no—
134C2×C269—22no—
135C2×C269—44no—
136C2×C269#52511yescharacteristic
137C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C269#57511yescharacteristic
139D269—81no—
140C269#311411yescharacteristic, complemented, Sylow
141#3114—2691nomaximal, Sylow
142#529—2691no—
143C2×C4×D4—2691no—
144#483—2691no—
145#480—2691no—
146#478—2692no—
147#463—2692no—
148#450D26911yescharacteristic, complemented
149#447—2691no—
150#442—2692no—
151#440—2692no—
152#432—2691no—
153C22×D4C2×D26911yescharacteristic
154C23×C4—5381no—
155C2×SD16C2×D26912yes—
156C2×D8—2692no—
157#126—5381no—
158#125—2691no—
159#124—2691no—
160#123—5381no—
161C22⋊Q8C2×D26911yescharacteristic
162#121—2691no—
163C4×Q8—2691no—
164C4×D4—2692no—
165C4×D4—2692no—
166C4×D4—2691no—
167C4×D4—5381no—
168C4×D4—5381no—
169C4×D4—5381no—
170#117—5381no—
171C2×C4⋊C4—2691no—
172C2×C22⋊C4—5381no—
173C2×C42—2691no—
174#107—2692no—
175#106—2692no—
176C4⋊C8—2692no—
177#103—2692no—
178#102—2692no—
179#102—2692no—
180#102C2×D26912yes—
181C22⋊C8—2691no—
182C22⋊C8C2×D26911yescharacteristic
183#97—2692no—
184C24—21no—
185C2×Q8C22×D26911yescharacteristic
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—2691no—
189C2×D4—5381no—
190C2×D4C22×D26912yes—
191C22×C4—2691no—
192C22×C4—2691no—
193C22×C4—5381no—
194C22×C4—5382no—
195C22×C4—5382no—
196C22×C4—5381no—
197C22×C4—10764no—
198C22×C4C22×D26911yescharacteristic
199SD16—24no—
200D8—5384no—
201C2×C8—2692no—
202C2×C8C22×D26912yes—
203C4⋊C4—21no—
204C4⋊C4—2691no—
205C4⋊C4—2692no—
206C4⋊C4—5381no—
207C4⋊C4—5381no—
208C4⋊C4—5381no—
209C4⋊C4—5381no—
210C4⋊C4C22×D26911yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—5381no—
213C22⋊C4—5381no—
214C22⋊C4—5381no—
215C22⋊C4—5382no—
216C22⋊C4—5382no—
217C22⋊C4—5381no—
218C42—2691no—
219C42—2691no—
220C42—2691no—
221C42—2691no—
222C42—5381no—
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—5381no—
228C23D4×D26911yescharacteristic
229Q8—22no—
230D4—24no—
231D4—22no—
232D4—5382no—
233D4—10762no—
234D4#1686440114yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—2691no—
239C2×C4—2691no—
240C2×C4—2691no—
241C2×C4—2691no—
242C2×C4—5382no—
243C2×C4—5381no—
244C2×C4—5382no—
245C2×C4—10762no—
246C2×C4—10761no—
247C2×C4—10764no—
248C2×C4—10764no—
249C2×C4—10762no—
250C2×C4D4×D26911yescharacteristic
251C2×C4C23×D26911yescharacteristic
252C8—22no—
253C8—5382no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—10761no—
261C22—10761no—
262C22—10761no—
263C22#2257548012yes—
264C22C2×D4×D26911yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—5381no—
269C4—5381no—
270C4—5381no—
271C4—5381no—
272C4—10762no—
273C4—10761no—
274C4C2×#1686440112yes—
275C2—22no—
276C2—44no—
277C2—21521no—
278C2#3177111111yescharacteristic
279C2C2×#2257548011yescharacteristic
280C2#3177140111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#450×C269C2
socleC22×C269C22×D4
prime radical#31770739C2
hypercenter#450D269
hypocenterC269#3114
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #31771141C2semidirect
2. C2×D4×Dic269C2semidirect
3. #31770933C2semidirect
4. #31770919C2semidirect
5. #31770903C2semidirect
6. #31770822C2semidirect
8. #31770714C2semidirect
9. #31770706C2semidirect
10. #31770659C2semidirect
11. #450×C269C2semidirect
15. #22575444C22semidirect
16. #22575442C22semidirect
19. Q8×Dic269C22semidirect
24. #22575415C22semidirect
30. #22575387C22semidirect
31. #22575384C22semidirect
32. #22575382C22semidirect
33. #22575377C22semidirect
34. #22575375C22semidirect
37. #22575363C22semidirect
140. C269#3114semidirect
148. #450D269semidirect

Maximal subgroups

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

subgroupindex
#317711412
C2×D4×Dic2692
#317709332
#317709192
#317709032
#317708222
#317707392
#317707142
#317707062
#317706592
#450×C2692
#3114269