Finite Group Databasebeta

← #46820153

Subgroups — #46820153

Order 34432.

784 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 165 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×#22575478C211yesmaximal, characteristic, complemented
2C22×#16864387C211yesmaximal, characteristic, complemented
3#31771169C211yesmaximal, characteristic, complemented
4#31770750C214yesmaximal, complemented
5#31770739C214yesmaximal, complemented
6#31770610C211yesmaximal, characteristic, complemented
7#31770601C211yesmaximal, characteristic, complemented
8#31770572C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C269C211yesmaximal, characteristic, complemented
10C22×D1076C2211yescharacteristic, complemented
11#22575478—24no—
12C2×#16864389C2212yescomplemented
13C2×#16864387—24no—
14C2×#16864387C2212yescomplemented
15C2×#16864387C2214yescomplemented
16C2×#16864386C2211yescharacteristic, complemented
17C22×#14524130C2211yescharacteristic, complemented
18#22575449—21no—
19#22575442—21no—
20#22575427C2214yescomplemented
21#22575421—22no—
22#22575412C2211yescharacteristic, complemented
23#22575377C2212yescomplemented
24#22575374C2212yescomplemented
25#22575374C2214yescomplemented
26C22⋊#14524130C2214yescomplemented
27H(5,𝔽2)×C269—24no—
28H(5,𝔽2)×C269—24no—
29C2×#41×C269—21no—
30C2×#41×C269C2211yescharacteristic
31C2×#41×C269C2211yescharacteristic
32C2×#41×C269C2212yescomplemented
33C22×D4×C269—21no—
34C22×D4×C269—22no—
35C22×D4×C269C2211yescharacteristic
36C22×D4×C269C2212yescomplemented
37C23×D269—21no—
38C2×D1076—22no—
39C2×D1076—22no—
40C2×D1076—21no—
41C2×D1076C2312yes—
42C2×C22⋊D269—24no—
43#16864389—24no—
44#16864387—28no—
45#16864387—24no—
46#16864386—22no—
47C2×#14524130—22no—
48C2×#14524130C2312yes—
49C2×#14524130C2312yes—
50C4⋊Dic269C2312yes—
51C22⋊Dic269—21no—
52#16864379—24no—
53C4×Dic269—21no—
54C24×C269—21no—
55C24×C269—22no—
56#41×C269—28no—
57#41×C269—24no—
58#41×C269—24no—
59#41×C269—22no—
60#41×C269—22no—
61#41×C269—22no—
62#41×C269D414yescomplemented
63C2×Q8×C269D411yescharacteristic
64C2×Q8×C269C2311yescharacteristic
65C2×D4×C269—216no—
66C2×D4×C269—28no—
67C2×D4×C269—21no—
68C2×D4×C269—22no—
69C2×D4×C269—24no—
70C2×D4×C269—24no—
71C2×D4×C269—21no—
72C2×D4×C269—22no—
73C2×D4×C269—22no—
74C2×D4×C269D412yes—
75C2×D4×C269D411yescharacteristic
76C2×D4×C269D414yescomplemented
77C2×D4×C269C2312yes—
78C2×D4×C269C2311yescharacteristic
79C22×C4×C269—21no—
80C22×C4×C269—22no—
81C22×C4×C269C2312yes—
82C22×C4×C269C2311yescharacteristic
83C22×D269—22no—
84C22×D269—42no—
85C22×D269—82no—
86D1076—22no—
87D1076—22no—
88D1076—42no—
89C22⋊D269—84no—
90C2×Dic269—22no—
91#14524130—22no—
92#14524130—22no—
93C23×C269—28no—
94C23×C269—21no—
95C23×C269—22no—
96C23×C269—24no—
97C23×C269—82no—
98C23×C269—84no—
99C23×C269C2×D412yes—
100C23×C269C2×D411yescharacteristic
101Q8×C269—22no—
102Q8×C269—21no—
103Q8×C269C2×D412yes—
104D4×C269—28no—
105D4×C269—28no—
106D4×C269—24no—
107D4×C269—24no—
108D4×C269—22no—
109D4×C269—24no—
110D4×C269—22no—
111D4×C269—21no—
112D4×C269C2×D412yes—
113D4×C269C2×D414yes—
114C2×C4×C269—28no—
115C2×C4×C269—21no—
116C2×C4×C269—24no—
117C2×C4×C269—22no—
118C2×C4×C269—22no—
119C2×C4×C269C2×D412yes—
120C2×C4×C269C2×D411yescharacteristic
121C2×C4×C269C2×D412yes—
122C2×C4×C269C2×D412yes—
123C2×C4×C269C2411yescharacteristic
124C22×C269—22no—
125C22×C269—22no—
126C22×C269—24no—
127C22×C269—22no—
128C22×C269—24no—
129C22×C269—44no—
130C22×C269—42no—
131C22×C269—84no—
132C22×C269—88no—
133C22×C269#12012yes—
134C22×C269C22×D411yescharacteristic
135C2×D269—42no—
136C2×D269—44no—
137C2×D269—84no—
138C4×C269—22no—
139C4×C269—22no—
140C4×C269#12012yes—
141C4×C269C22×D411yescharacteristic
142C4×C269C22×D411yescharacteristic
143Dic269—42no—
144C2×C269—22no—
145C2×C269—42no—
146C2×C269—44no—
147C2×C269—42no—
148C2×C269C2×#12011yescharacteristic
149C2×C269#57612yes—
150D269—82no—
151C269#300811yescharacteristic, complemented, Sylow
152#3008—2691nomaximal, Sylow
153C2×H(5,𝔽2)D26911yescharacteristic, complemented
154C2×AGL(1,ℤ/8)—2691no—
155C22×D8—2691no—
156#535—2691no—
157#449—2694no—
158#447—2694no—
159#418—2691no—
160#416—2691no—
161#408—2691no—
162H(5,𝔽2)—24no—
163H(5,𝔽2)—24no—
164C2×#41—21no—
165C2×#41C2×D26911yescharacteristic
166C2×#41C2×D26911yescharacteristic
167C2×#41C2×D26912yescomplemented
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—2691no—
171C22×D4C2×D26911yescharacteristic
172C22×D4C2×D26912yescomplemented
173AGL(1,ℤ/8)—5384no—
174C2×SD16—2692no—
175C2×D8—2694no—
176C2×D8—2692no—
177C2×D8—5384no—
178C2×M16—2691no—
179C22×C8—2691no—
180#127—5381no—
181#124—5381no—
182#121—2694no—
183#120—5382no—
184#117—2691no—
185#103—2692no—
186#102—2692no—
187#102—2694no—
188C22⋊C8—2694no—
189C24—21no—
190C24—22no—
191C24—5381no—
192#41—28no—
193#41—24no—
194#41—24no—
195#41—22no—
196#41—22no—
197#41—22no—
198#41C22⋊D26914yescomplemented
199C2×Q8C22⋊D26911yescharacteristic
200C2×Q8C22×D26911yescharacteristic
201C2×D4—216no—
202C2×D4—28no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—2692no—
211C2×D4—5384no—
212C2×D4—5382no—
213C2×D4—5382no—
214C2×D4—5381no—
215C2×D4C22⋊D26912yes—
216C2×D4C22⋊D26911yescharacteristic
217C2×D4C22⋊D26914yescomplemented
218C2×D4C22×D26912yes—
219C2×D4C22×D26911yescharacteristic
220C22×C4—21no—
221C22×C4—22no—
222C22×C4C22×D26912yes—
223C22×C4C22×D26911yescharacteristic
224SD16—5384no—
225D8—5388no—
226D8—5384no—
227M16—5382no—
228C2×C8—2692no—
229C2×C8—2692no—
230C2×C8—5382no—
231C4⋊C4—2692no—
232C22⋊C4—5381no—
233C22⋊C4—5384no—
234C42—5381no—
235C23—28no—
236C23—21no—
237C23—22no—
238C23—24no—
239C23—82no—
240C23—84no—
241C23—5382no—
242C23—10762no—
243C23—21522no—
244C23C2×C22⋊D26912yes—
245C23C2×C22⋊D26911yescharacteristic
246Q8—22no—
247Q8—21no—
248Q8C2×C22⋊D26912yes—
249D4—28no—
250D4—28no—
251D4—24no—
252D4—24no—
253D4—22no—
254D4—24no—
255D4—22no—
256D4—21no—
257D4—5382no—
258D4—5382no—
259D4—10762no—
260D4—21524no—
261D4C2×C22⋊D26912yes—
262D4C2×C22⋊D26914yes—
263C2×C4—28no—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—5382no—
269C2×C4C2×C22⋊D26912yes—
270C2×C4C2×C22⋊D26911yescharacteristic
271C2×C4C2×C22⋊D26912yes—
272C2×C4C2×C22⋊D26912yes—
273C2×C4C23×D26911yescharacteristic
274C8—5382no—
275C8—5382no—
276C22—22no—
277C22—24no—
278C22—22no—
279C22—24no—
280C22—22no—
281C22—44no—
282C22—42no—
283C22—84no—
284C22—88no—
285C22—10764no—
286C22—10762no—
287C22—21524no—
288C22#2257542212yes—
289C22C22×C22⋊D26911yescharacteristic
290C4—22no—
291C4—22no—
292C4—10762no—
293C4#2257542212yes—
294C4C22×C22⋊D26911yescharacteristic
295C4C22×C22⋊D26911yescharacteristic
296C2—22no—
297C2—42no—
298C2—44no—
299C2—42no—
300C2—21522no—
301C2C2×#2257542211yescharacteristic
302C2#3177140812yes—
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×H(5,𝔽2)×C269C2
socleC22×C269C22×D4
hypercenterC2×H(5,𝔽2)D269
hypocenterC269#3008
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#22575478C2semidirect
2. C22×#16864387C2semidirect
3. #31771169C2semidirect
4. #31770750C2semidirect
5. #31770739C2semidirect
6. #31770610C2semidirect
7. #31770601C2semidirect
8. #31770572C2semidirect
9. C2×H(5,𝔽2)×C269C2semidirect
10. C22×D1076C22semidirect
12. C2×#16864389C22semidirect
14. C2×#16864387C22semidirect
15. C2×#16864387C22semidirect
16. C2×#16864386C22semidirect
17. C22×#14524130C22semidirect
20. #22575427C22semidirect
22. #22575412C22semidirect
23. #22575377C22semidirect
24. #22575374C22semidirect
25. #22575374C22semidirect
26. C22⋊#14524130C22semidirect
32. C2×#41×C269C22semidirect
36. C22×D4×C269C22semidirect
62. #41×C269D4semidirect
76. C2×D4×C269D4semidirect
151. C269#3008semidirect
153. C2×H(5,𝔽2)D269semidirect
167. C2×#41C2×D269semidirect
172. C22×D4C2×D269semidirect
198. #41C22⋊D269semidirect
217. C2×D4C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754782
C22×#168643872
#317711692
#317707502
#317707392
#317706102
#317706012
#317705722
C2×H(5,𝔽2)×C2692
#3008269