Finite Group Databasebeta

← #46821023

Subgroups — #46821023

Order 34432.

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

normal subgroups only

This group has 450 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
1#31771139C211yesmaximal, characteristic, complemented
2#31770935C211yesmaximal, characteristic, complemented
3#31770918C211yesmaximal, characteristic, complemented
4#31770905C212yesmaximal, complemented
5#31770820C212yesmaximal, complemented
6#31770756C211yesmaximal, characteristic, complemented
7#31770739C211yesmaximal, characteristic
8#31770715C212yesmaximal, complemented
9#31770705C212yesmaximal, complemented
10#31770662C211yesmaximal, characteristic, complemented
11C2×C4×D4×C269C211yesmaximal, characteristic, complemented
12C2×#16864390C2212yes—
13C2×#16864387C2212yes—
14#22575447—21no—
15#22575445C2211yescharacteristic, complemented
16#22575441C2211yescharacteristic, complemented
17#22575436—21no—
18#22575432C2211yescharacteristic
19#22575427C2211yescharacteristic
20C4×Dic538C2211yescharacteristic, complemented
21C4×D1076C2211yescharacteristic, complemented
22C4×C22⋊D269—21no—
23#22575410—21no—
24#22575386C2212yescomplemented
25#22575384C2212yescomplemented
26C4⋊#14524130C2212yescomplemented
27#22575376C2212yescomplemented
28#22575374C2212yes—
29#22575374C2212yes—
30#22575373C2212yescomplemented
31C22⋊#14524130C2211yescharacteristic
32C22⋊#14524130C2211yescharacteristic
33#22575364C2212yescomplemented
34C22×D4×C269C2211yescharacteristic
35C23×C4×C269—21no—
36C4×D4×C269—21no—
37C4×D4×C269—21no—
38C4×D4×C269C2212yes—
39C4×D4×C269C2212yes—
40C2×C4⋊C4×C269C2211yescharacteristic
41C2×C22⋊C4×C269—21no—
42C2×C42×C269C2211yescharacteristic
43C2×Dic538C2311yescharacteristic
44C2×D1076C2311yescharacteristic
45C2×C22⋊D269—21no—
46C2×C4×D269—21no—
47#16864390—24no—
48#16864387—24no—
49C2×#14524130C2312yes—
50C2×#14524130C2312yes—
51C4⋊Dic269C2311yescharacteristic
52C4⋊Dic269C2311yescharacteristic
53#16864381—21no—
54#16864381—21no—
55#16864381—21no—
56#16864381—21no—
57C22⋊Dic269—21no—
58#16864379—21no—
59#16864379—21no—
60#16864379—21no—
61#16864379—21no—
62C4×Dic269—21no—
63C24×C269—21no—
64C2×D4×C269—24no—
65C2×D4×C269—21no—
66C2×D4×C269C2312yes—
67C22×C4×C269—21no—
68C22×C4×C269—22no—
69C22×C4×C269—22no—
70C22×C4×C269—44no—
71C22×C4×C269D411yescharacteristic
72C22×C4×C269D411yescharacteristic
73C22×C4×C269C2311yescharacteristic
74C4⋊C4×C269—21no—
75C4⋊C4×C269C2312yes—
76C22⋊C4×C269—22no—
77C22⋊C4×C269—22no—
78C42×C269D411yescharacteristic
79C42×C269D411yescharacteristic
80C42×C269C2311yescharacteristic
81C42×C269C2311yescharacteristic
82C22×D269—21no—
83Dic538—22no—
84D1076—22no—
85C22⋊D269—42no—
86C2×Dic269—21no—
87C2×Dic269—21no—
88C2×Dic269—21no—
89C4×D269—42no—
90#14524130—22no—
91#14524130—22no—
92C23×C269—24no—
93C23×C269—22no—
94C23×C269—42no—
95C23×C269—42no—
96C23×C269C2×D411yescharacteristic
97D4×C269—24no—
98D4×C269—22no—
99D4×C269#4114yes—
100C2×C4×C269—22no—
101C2×C4×C269—22no—
102C2×C4×C269—22no—
103C2×C4×C269—41no—
104C2×C4×C269—44no—
105C2×C4×C269—44no—
106C2×C4×C269—42no—
107C2×C4×C269C2×D411yescharacteristic
108C2×C4×C269C2×D411yescharacteristic
109C2×C4×C269C2×D411yescharacteristic
110C2×C4×C269C2×D411yescharacteristic
111C2×C4×C269C2×D411yescharacteristic
112C2×C4×C269C2411yescharacteristic
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—21no—
126C4×C269—21no—
127C4×C269—21no—
128C4×C269—21no—
129C4×C269—42no—
130C4×C269C2×#4112yes—
131Dic269—41no—
132Dic269—41no—
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×D4D26911yescharacteristic, complemented
144#483—2691no—
145#480—2691no—
146#478—2692no—
147#463—2692no—
148#450—2691no—
149#447—2691no—
150#442—2692no—
151#440—2692no—
152#432—2691no—
153C22×D4C2×D26911yescharacteristic
154C23×C4—21no—
155C2×SD16—2692no—
156C2×D8—2692no—
157#126—5381no—
158#125—2691no—
159#124—2691no—
160#123—5381no—
161C22⋊Q8—2691no—
162#121—2691no—
163C4×Q8—2691no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—2691no—
167C4×D4—5381no—
168C4×D4C2×D26912yes—
169C4×D4C2×D26912yes—
170#117—5381no—
171C2×C4⋊C4C2×D26911yescharacteristic
172C2×C22⋊C4—21no—
173C2×C42C2×D26911yescharacteristic
174#107—2692no—
175#106—2692no—
176C4⋊C8—2692no—
177#103—2692no—
178#102—2692no—
179#102—2692no—
180#102—2692no—
181C22⋊C8—2691no—
182C22⋊C8—2691no—
183#97—2692no—
184C24—21no—
185C2×Q8—2691no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—2691no—
189C2×D4—5381no—
190C2×D4C22×D26912yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—5381no—
196C22×C4C22⋊D26911yescharacteristic
197C22×C4C22⋊D26911yescharacteristic
198C22×C4C22×D26911yescharacteristic
199SD16—5384no—
200D8—5384no—
201C2×C8—2692no—
202C2×C8—2692no—
203C4⋊C4—21no—
204C4⋊C4—2691no—
205C4⋊C4—2691no—
206C4⋊C4—5381no—
207C4⋊C4—5381no—
208C4⋊C4—5381no—
209C4⋊C4—5381no—
210C4⋊C4C22×D26912yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—5381no—
214C22⋊C4—5381no—
215C22⋊C4—5381no—
216C22⋊C4—5381no—
217C22⋊C4—5381no—
218C42—5381no—
219C42C22⋊D26911yescharacteristic
220C42C22⋊D26911yescharacteristic
221C42C22×D26911yescharacteristic
222C42C22×D26911yescharacteristic
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—5381no—
228C23C2×C22⋊D26911yescharacteristic
229Q8—5382no—
230D4—24no—
231D4—22no—
232D4—5382no—
233D4—10762no—
234D4#1686440214yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—5381no—
243C2×C4—5381no—
244C2×C4—5381no—
245C2×C4—10762no—
246C2×C4C2×C22⋊D26911yescharacteristic
247C2×C4C2×C22⋊D26911yescharacteristic
248C2×C4C2×C22⋊D26911yescharacteristic
249C2×C4C2×C22⋊D26911yescharacteristic
250C2×C4C2×C22⋊D26911yescharacteristic
251C2×C4C23×D26911yescharacteristic
252C8—5382no—
253C8—5382no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—10761no—
261C22—10761no—
262C22—10761no—
263C22#2257547912yes—
264C22C22×C22⋊D26911yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—10761no—
272C4—10761no—
273C4—10761no—
274C4C2×#1686440212yes—
275C2—22no—
276C2—44no—
277C2—21521no—
278C2#3177111611yescharacteristic
279C2C2×#2257547911yescharacteristic
280C2#3177140311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×C4×D4×C269C2
socleC22×C269C22×D4
prime radical#31770739C2
hypercenterC2×C4×D4D269
hypocenterC269#3114
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #31771139C2semidirect
2. #31770935C2semidirect
3. #31770918C2semidirect
4. #31770905C2semidirect
5. #31770820C2semidirect
6. #31770756C2semidirect
8. #31770715C2semidirect
9. #31770705C2semidirect
10. #31770662C2semidirect
11. C2×C4×D4×C269C2semidirect
15. #22575445C22semidirect
16. #22575441C22semidirect
20. C4×Dic538C22semidirect
21. C4×D1076C22semidirect
24. #22575386C22semidirect
25. #22575384C22semidirect
26. C4⋊#14524130C22semidirect
27. #22575376C22semidirect
30. #22575373C22semidirect
33. #22575364C22semidirect
140. C269#3114semidirect
143. C2×C4×D4D269semidirect

Maximal subgroups

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

subgroupindex
#317711392
#317709352
#317709182
#317709052
#317708202
#317707562
#317707392
#317707152
#317707052
#317706622
C2×C4×D4×C2692
#3114269