Finite Group Databasebeta

← #46820164

Subgroups — #46820164

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×#22575486C211yesmaximal, characteristic, complemented
3C22×#16864391C211yesmaximal, characteristic, complemented
4#31771176C211yesmaximal, characteristic, complemented
5#31770769C214yesmaximal, complemented
6#31770758C214yesmaximal, complemented
7#31770611C211yesmaximal, characteristic, complemented
8#31770602C211yesmaximal, characteristic, complemented
9#408×C269C211yesmaximal, characteristic, complemented
10#22575499—24no—
11#22575499—24no—
12C2×#16864403—21no—
13C2×#16864402C2212yes—
14C2×#16864402C2211yescharacteristic
15C22×Dic538C2211yescharacteristic, complemented
16C22×D1076C2212yes—
17C22×D1076C2211yescharacteristic
18C2×D4×D269—22no—
19C2×D4×D269—21no—
20#22575486—24no—
21C2×Dic1076C2212yescomplemented
22C2×#16864391—24no—
23C2×#16864391C2212yescomplemented
24C2×#16864391C2214yescomplemented
25#22575451—21no—
26#22575440—21no—
27#22575434—22no—
28#22575432C2214yescomplemented
29#22575412C2211yescharacteristic, complemented
30#22575378C2212yescomplemented
31#22575375C2212yescomplemented
32#22575375C2214yescomplemented
33C2×#41×C269C2211yescharacteristic
34C2×M16×C269C2211yescharacteristic, complemented
35C22×C8×C269C2211yescharacteristic, complemented
36C22⋊C8×C269C2214yescomplemented
37C23×D269—22no—
38C23×D269—21no—
39#16864403—28no—
40#16864402—24no—
41#16864402—24no—
42#16864402—22no—
43#16864402D414yes—
44C2×Dic538—21no—
45C2×Dic538—22no—
46C2×Dic538—22no—
47C2×Dic538C2312yes—
48C2×Dic538C2311yescharacteristic
49C2×D1076—24no—
50C2×D1076—24no—
51C2×D1076—22no—
52C2×D1076—21no—
53C2×D1076—22no—
54C2×D1076D414yes—
55C2×D1076C2311yescharacteristic
56C2×D1076C2312yes—
57C2×C22⋊D269—22no—
58C2×C22⋊D269—21no—
59D4×D269—216no—
60D4×D269—28no—
61C22×Dic269—21no—
62C2×C4×D269—22no—
63C2×C4×D269—21no—
64Dic1076—24no—
65#16864391—28no—
66#16864391—24no—
67C4⋊Dic269C2312yes—
68#16864381—24no—
69C22⋊Dic269—21no—
70C22⋊Dic269—24no—
71C4×Dic269—21no—
72#41×C269—22no—
73#41×C269—22no—
74C2×Q8×C269D411yescharacteristic
75C2×D4×C269D412yes—
76C2×D4×C269D411yescharacteristic
77C22×C4×C269C2312yes—
78C22×C4×C269C2311yescharacteristic
79M16×C269—22no—
80C2×C8×C269—22no—
81C2×C8×C269C2312yes—
82C2×C8×C269C2312yes—
83C22×D269—22no—
84C22×D269—21no—
85C22×D269—28no—
86C22×D269—24no—
87C22×D269—82no—
88C22×D269—84no—
89Dic538—21no—
90Dic538—22no—
91Dic538—22no—
92Dic538—42no—
93Dic538C2×D412yes—
94D1076—28no—
95D1076—24no—
96D1076—22no—
97D1076—21no—
98D1076C2×D412yes—
99D1076C2×D414yes—
100C22⋊D269—28no—
101C22⋊D269—24no—
102C2×Dic269—21no—
103C2×Dic269—22no—
104C2×Dic269—22no—
105C2×Dic269—42no—
106C4×D269—28no—
107C4×D269—24no—
108C23×C269C2×D412yes—
109C23×C269C2×D411yescharacteristic
110Q8×C269—22no—
111D4×C269—24no—
112D4×C269—22no—
113C2×C4×C269—22no—
114C2×C4×C269—22no—
115C2×C4×C269C2×D412yes—
116C2×C4×C269C2×D412yes—
117C2×C4×C269C2×D412yes—
118C2×C4×C269C2×D411yescharacteristic
119C2×C4×C269C2411yescharacteristic
120C8×C269—22no—
121C8×C269—22no—
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#12012yes—
135C4×C269C22×D411yescharacteristic
136C4×C269C22×D411yescharacteristic
137Dic269—22no—
138Dic269—42no—
139Dic269—42no—
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#536—2691no—
152#452—2694no—
153#450—2694no—
154#418—2691no—
155#416—2691no—
156#408D26911yescharacteristic, complemented
157H(5,𝔽2)—5384no—
158H(5,𝔽2)—5384no—
159C2×#41—2691no—
160C2×#41—2692no—
161C2×#41—5381no—
162C2×#41C2×D26911yescharacteristic
163C22×Q8—2691no—
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×M16C2×D26911yescharacteristic, complemented
174C22×C8C2×D26911yescharacteristic, complemented
175C4⋊Q8—5381no—
176#124—5381no—
177#123—5382no—
178C22⋊Q8—2694no—
179#117—2691no—
180#103—2692no—
181#102—2692no—
182#102—2694no—
183C22⋊C8C2×D26914yescomplemented
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—2691no—
194C2×Q8—2692no—
195C2×Q8—5381no—
196C2×Q8—5382no—
197C2×Q8—5382no—
198C2×Q8D107611yescharacteristic
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×D4D107612yes—
212C2×D4D107611yescharacteristic
213C22×C4—5382no—
214C22×C4—5381no—
215C22×C4—5381no—
216C22×C4C22×D26912yes—
217C22×C4C22×D26911yescharacteristic
218Q16—5384no—
219SD16—5388no—
220SD16—5384no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D26912yes—
224C2×C8C22×D26912yes—
225C4⋊C4—2692no—
226C4⋊C4—5384no—
227C22⋊C4—5381no—
228C22⋊C4—5384no—
229C42—5381no—
230C23—5388no—
231C23—5382no—
232C23—5381no—
233C23—5384no—
234C23—21522no—
235C23—21524no—
236C23C2×D107612yes—
237C23C2×D107611yescharacteristic
238Q8—22no—
239Q8—2692no—
240Q8—5381no—
241Q8—5382no—
242Q8—5382no—
243Q8—10762no—
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—5388no—
257C2×C4—5381no—
258C2×C4—5384no—
259C2×C4—5382no—
260C2×C4—5382no—
261C2×C4—10762no—
262C2×C4D4×D26912yes—
263C2×C4D4×D26912yes—
264C2×C4C2×D107612yes—
265C2×C4C2×D107611yescharacteristic
266C2×C4C23×D26911yescharacteristic
267C8—22no—
268C8—22no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—5384no—
273C22—5382no—
274C22—10764no—
275C22—10762no—
276C22—21524no—
277C22—21528no—
278C22#2257542312yes—
279C22C22×D107611yescharacteristic
280C4—22no—
281C4—5382no—
282C4—10762no—
283C4—10762no—
284C4#2257542312yes—
285C4C2×D4×D26911yescharacteristic
286C4C2×D4×D26911yescharacteristic
287C2—22no—
288C2—42no—
289C2—10764no—
290C2—10762no—
291C2C2×#2257542311yescharacteristic
292C2#3177141712yes—
subgroupG/H
centerC22C22×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#408×C269C2
socleC22×C269C22×D4
prime radicalC2×#22575499C2
hypercenter#408D269
hypocenterC269#3009
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#22575486C2semidirect
3. C22×#16864391C2semidirect
4. #31771176C2semidirect
5. #31770769C2semidirect
6. #31770758C2semidirect
7. #31770611C2semidirect
8. #31770602C2semidirect
9. #408×C269C2semidirect
15. C22×Dic538C22semidirect
21. C2×Dic1076C22semidirect
23. C2×#16864391C22semidirect
24. C2×#16864391C22semidirect
28. #22575432C22semidirect
29. #22575412C22semidirect
30. #22575378C22semidirect
31. #22575375C22semidirect
32. #22575375C22semidirect
34. C2×M16×C269C22semidirect
35. C22×C8×C269C22semidirect
36. C22⋊C8×C269C22semidirect
146. C269#3009semidirect
156. #408D269semidirect
173. C2×M16C2×D269semidirect
174. C22×C8C2×D269semidirect
183. C22⋊C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754992
C2×#225754862
C22×#168643912
#317711762
#317707692
#317707582
#317706112
#317706022
#408×C2692
#3009269