Finite Group Databasebeta

← #46820160

Subgroups — #46820160

Order 34432.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22575499C211yesmaximal, characteristic
2C2×#22575484C211yesmaximal, characteristic, complemented
3C22×#16864389C211yesmaximal, characteristic, complemented
4#31770765C214yesmaximal, complemented
5#31770755C214yesmaximal, complemented
6#31770609C211yesmaximal, characteristic, complemented
7#31770599C211yesmaximal, characteristic, complemented
8#31770572C211yesmaximal, characteristic, complemented
9#536×C269C211yesmaximal, characteristic, complemented
10#22575499—24no—
11#22575499—24no—
12C2×#16864403—21no—
13C2×#16864402C2212yes—
14C2×#16864402C2211yescharacteristic
15C22×D1076C2212yes—
16C22×D1076C2211yescharacteristic
17C2×D4×D269—22no—
18C2×D4×D269—21no—
19#22575484—24no—
20C2×#16864392C2212yescomplemented
21C2×#16864389—24no—
22C2×#16864389C2212yescomplemented
23C2×#16864389C2214yescomplemented
24C2×#16864386C2211yescharacteristic, complemented
25C22×#14524130C2211yescharacteristic, complemented
26#22575376C2212yescomplemented
27#22575373C2212yescomplemented
28#22575373C2214yescomplemented
29C22⋊#14524130C2214yescomplemented
30C2×#41×C269C2211yescharacteristic
31C22×Q8×C269C2211yescharacteristic, complemented
32C4⋊Q8×C269—21no—
33#124×C269—21no—
34#123×C269—22no—
35C22⋊Q8×C269C2214yescomplemented
36#117×C269C2211yescharacteristic, complemented
37C23×D269—22no—
38C23×D269—21no—
39#16864403—28no—
40#16864402—24no—
41#16864402—24no—
42#16864402—22no—
43#16864402D414yes—
44C2×Dic538C2311yescharacteristic
45C2×D1076—24no—
46C2×D1076—24no—
47C2×D1076—22no—
48C2×D1076—21no—
49C2×D1076—22no—
50C2×D1076D414yes—
51C2×D1076C2311yescharacteristic
52C2×D1076C2312yes—
53C2×C22⋊D269—22no—
54C2×C22⋊D269—21no—
55D4×D269—216no—
56D4×D269—28no—
57C2×C4×D269—22no—
58C2×C4×D269—21no—
59#16864392—24no—
60#16864389—28no—
61#16864389—24no—
62#16864386—22no—
63C2×#14524130—22no—
64C2×#14524130C2312yes—
65C2×#14524130C2312yes—
66#41×C269—22no—
67#41×C269—22no—
68C2×Q8×C269—21no—
69C2×Q8×C269—22no—
70C2×Q8×C269—22no—
71C2×Q8×C269D411yescharacteristic
72C2×Q8×C269C2312yes—
73C2×D4×C269D412yes—
74C2×D4×C269D411yescharacteristic
75C22×C4×C269—21no—
76C22×C4×C269C2312yes—
77C22×C4×C269C2311yescharacteristic
78C4⋊C4×C269—24no—
79C4⋊C4×C269C2312yes—
80C22⋊C4×C269—21no—
81C22⋊C4×C269—24no—
82C42×C269—21no—
83C22×D269—22no—
84C22×D269—21no—
85C22×D269—28no—
86C22×D269—24no—
87C22×D269—82no—
88C22×D269—84no—
89Dic538—21no—
90Dic538C2×D412yes—
91D1076—28no—
92D1076—24no—
93D1076—22no—
94D1076—21no—
95D1076C2×D412yes—
96D1076C2×D414yes—
97C22⋊D269—28no—
98C22⋊D269—24no—
99C2×Dic269—21no—
100C4×D269—28no—
101C4×D269—24no—
102#14524130—22no—
103#14524130—22no—
104C23×C269C2×D412yes—
105C23×C269C2×D411yescharacteristic
106Q8×C269—22no—
107Q8×C269—22no—
108Q8×C269—22no—
109Q8×C269—42no—
110D4×C269—24no—
111D4×C269—22no—
112C2×C4×C269—22no—
113C2×C4×C269—22no—
114C2×C4×C269—22no—
115C2×C4×C269—22no—
116C2×C4×C269—42no—
117C2×C4×C269C2×D412yes—
118C2×C4×C269C2×D412yes—
119C2×C4×C269C2×D412yes—
120C2×C4×C269C2×D411yescharacteristic
121C2×C4×C269C2411yescharacteristic
122C22×C269—22no—
123C22×C269—22no—
124C22×C269—24no—
125C22×C269#12012yes—
126C22×C269C22×D411yescharacteristic
127C2×D269—24no—
128C2×D269—22no—
129C2×D269—44no—
130C2×D269—42no—
131C2×D269—84no—
132C2×D269—88no—
133C4×C269—22no—
134C4×C269—42no—
135C4×C269—42no—
136C4×C269#12012yes—
137C4×C269C22×D411yescharacteristic
138C4×C269C22×D411yescharacteristic
139Dic269—22no—
140C2×C269—22no—
141C2×C269—42no—
142C2×C269C2×#12011yescharacteristic
143C2×C269#57712yes—
144D269—44no—
145D269—42no—
146C269#300911yescharacteristic, complemented, Sylow
147#3009—2691nomaximal, Sylow
148C2×H(5,𝔽2)—2691no—
149C2×#137—2691no—
150C22×SD16—2691no—
151#536D26911yescharacteristic, complemented
152#452—2694no—
153#450—2694no—
154#418—2691no—
155#416—2691no—
156#408—2691no—
157H(5,𝔽2)—5384no—
158H(5,𝔽2)—5384no—
159C2×#41—2691no—
160C2×#41—2692no—
161C2×#41—5381no—
162C2×#41C2×D26911yescharacteristic
163C22×Q8C2×D26911yescharacteristic, complemented
164C22×D4—2691no—
165C22×D4—2692no—
166C22×D4—5382no—
167C22×D4—5381no—
168#137—5384no—
169C2×Q16—2692no—
170C2×SD16—2694no—
171C2×SD16—2692no—
172C2×SD16—5384no—
173C2×M16—2691no—
174C22×C8—2691no—
175C4⋊Q8—21no—
176#124—21no—
177#123—22no—
178C22⋊Q8C2×D26914yescomplemented
179#117C2×D26911yescharacteristic, complemented
180#103—2692no—
181#102—2692no—
182#102—2694no—
183C22⋊C8—2694no—
184C24—5382no—
185C24—5381no—
186#41—22no—
187#41—22no—
188#41—2694no—
189#41—5384no—
190#41—5384no—
191#41—5388no—
192#41—5382no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—2691no—
197C2×Q8C22⋊D26911yescharacteristic
198C2×Q8C22×D26912yes—
199C2×D4—2694no—
200C2×D4—2692no—
201C2×D4—2691no—
202C2×D4—5384no—
203C2×D4—5384no—
204C2×D4—53816no—
205C2×D4—5388no—
206C2×D4—5382no—
207C2×D4—5381no—
208C2×D4—5382no—
209C2×D4—5382no—
210C2×D4—5381no—
211C2×D4C22⋊D26912yes—
212C2×D4C22⋊D26911yescharacteristic
213C22×C4—21no—
214C22×C4—5382no—
215C22×C4—5381no—
216C22×C4C22×D26912yes—
217C22×C4C22×D26911yescharacteristic
218Q16—5384no—
219SD16—5388no—
220SD16—5384no—
221M16—5382no—
222C2×C8—2692no—
223C2×C8—2692no—
224C2×C8—5382no—
225C4⋊C4—24no—
226C4⋊C4C22×D26912yes—
227C22⋊C4—21no—
228C22⋊C4—24no—
229C42—21no—
230C23—5388no—
231C23—5382no—
232C23—5381no—
233C23—5384no—
234C23—21522no—
235C23—21524no—
236C23C2×C22⋊D26912yes—
237C23C2×C22⋊D26911yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—2692no—
243Q8—5381no—
244D4—24no—
245D4—22no—
246D4—2694no—
247D4—2692no—
248D4—5388no—
249D4—5388no—
250D4—5384no—
251D4—5384no—
252D4—5381no—
253D4—5382no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—5388no—
260C2×C4—5381no—
261C2×C4—5384no—
262C2×C4D4×D26912yes—
263C2×C4D4×D26912yes—
264C2×C4C2×C22⋊D26912yes—
265C2×C4C2×C22⋊D26911yescharacteristic
266C2×C4C23×D26911yescharacteristic
267C8—5382no—
268C8—5382no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—5384no—
273C22—5382no—
274C22—10764no—
275C22—10762no—
276C22—21524no—
277C22—21528no—
278C22#2257542112yes—
279C22C22×C22⋊D26911yescharacteristic
280C4—22no—
281C4—42no—
282C4—42no—
283C4—5382no—
284C4#2257542112yes—
285C4C2×D4×D26911yescharacteristic
286C4C2×D4×D26911yescharacteristic
287C2—22no—
288C2—42no—
289C2—10764no—
290C2—10762no—
291C2C2×#2257542111yescharacteristic
292C2#3177141312yes—
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#536×C269C2
socleC22×C269C22×D4
prime radicalC2×#22575499C2
hypercenter#536D269
hypocenterC269#3009
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#22575484C2semidirect
3. C22×#16864389C2semidirect
4. #31770765C2semidirect
5. #31770755C2semidirect
6. #31770609C2semidirect
7. #31770599C2semidirect
8. #31770572C2semidirect
9. #536×C269C2semidirect
20. C2×#16864392C22semidirect
22. C2×#16864389C22semidirect
23. C2×#16864389C22semidirect
24. C2×#16864386C22semidirect
25. C22×#14524130C22semidirect
26. #22575376C22semidirect
27. #22575373C22semidirect
28. #22575373C22semidirect
29. C22⋊#14524130C22semidirect
31. C22×Q8×C269C22semidirect
35. C22⋊Q8×C269C22semidirect
36. #117×C269C22semidirect
146. C269#3009semidirect
151. #536D269semidirect
163. C22×Q8C2×D269semidirect
178. C22⋊Q8C2×D269semidirect
179. #117C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754992
C2×#225754842
C22×#168643892
#317707652
#317707552
#317706092
#317705992
#317705722
#536×C2692
#3009269