Finite Group Databasebeta

← #46820173

Subgroups — #46820173

Order 34432.

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

normal subgroups only

This group has 732 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×#22575500C211yesmaximal, characteristic, complemented
2C2×#22575481C211yesmaximal, characteristic, complemented
3C22×#16864391C211yesmaximal, characteristic, complemented
4#31771169C211yesmaximal, characteristic, complemented
5#31770752C214yesmaximal, complemented
6#31770746C214yesmaximal, complemented
7#31770615C211yesmaximal, characteristic, complemented
8#31770602C211yesmaximal, characteristic, complemented
9#408×C269C211yesmaximal, characteristic, complemented
10#22575500—24no—
11#22575500—24no—
12C2×#16864402C2211yescharacteristic
13C2×#16864402C2212yescomplemented
14C2×#16864401—22no—
15C2×#16864401—21no—
16C22×Dic538C2211yescharacteristic
17C22×Dic538C2212yescomplemented
18C2×Q8×D269—21no—
19C22×D1076C2211yescharacteristic
20#22575481—24no—
21C2×#16864391—24no—
22C2×#16864391C2212yes—
23C2×#16864391C2214yes—
24C2×D2152C2212yes—
25#22575449—21no—
26#22575442—21no—
27#22575427C2214yes—
28#22575421—22no—
29#22575412C2211yescharacteristic
30#22575378C2212yescomplemented
31#22575378C2214yescomplemented
32#22575375C2212yescomplemented
33C2×#41×C269C2211yescharacteristic
34C2×M16×C269C2211yescharacteristic
35C22×C8×C269C2211yescharacteristic
36C22⋊C8×C269C2214yescomplemented
37C23×D269—21no—
38#16864402—24no—
39#16864402—24no—
40#16864402—22no—
41#16864402D414yescomplemented
42#16864401—216no—
43#16864401—28no—
44C2×Dic538—24no—
45C2×Dic538—24no—
46C2×Dic538—21no—
47C2×Dic538—22no—
48C2×Dic538—22no—
49C2×Dic538D414yescomplemented
50C2×Dic538C2312yes—
51C2×Dic538C2311yescharacteristic
52Q8×D269—28no—
53C2×D1076—22no—
54C2×D1076—22no—
55C2×D1076—21no—
56C2×D1076C2312yes—
57C2×D1076C2311yescharacteristic
58C2×C22⋊D269—21no—
59C2×C22⋊D269—22no—
60C2×C22⋊D269—24no—
61C22×Dic269—21no—
62C22×Dic269—22no—
63C2×C4×D269—21no—
64C2×C4×D269—22no—
65#16864391—28no—
66#16864391—24no—
67D2152—24no—
68C4⋊Dic269C2312yes—
69C22⋊Dic269—21no—
70#16864379—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—21no—
84C22×D269—22no—
85C22×D269—42no—
86C22×D269—82no—
87Dic538—28no—
88Dic538—24no—
89Dic538—21no—
90Dic538—22no—
91Dic538C2×D412yes—
92Dic538C2×D414yes—
93D1076—21no—
94D1076—22no—
95D1076—22no—
96D1076—42no—
97D1076C2×D412yes—
98C22⋊D269—28no—
99C22⋊D269—24no—
100C22⋊D269—84no—
101C2×Dic269—28no—
102C2×Dic269—24no—
103C2×Dic269—21no—
104C2×Dic269—22no—
105C2×Dic269—22no—
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—22no—
128C2×D269—42no—
129C2×D269—42no—
130C2×D269—44no—
131C2×D269—84no—
132C4×C269—22no—
133C4×C269#12012yes—
134C4×C269C22×D411yescharacteristic
135C4×C269C22×D411yescharacteristic
136Dic269—24no—
137Dic269—22no—
138Dic269—42no—
139C2×C269—22no—
140C2×C269—42no—
141C2×C269C2×#12011yescharacteristic
142C2×C269#57712yes—
143D269—42no—
144D269—82no—
145C269#301011yescharacteristic, complemented, Sylow
146#3010—2691nomaximal, Sylow
147C2×2−1+4—2691no—
148C2×AGL(1,ℤ/8)—2691no—
149C22×SD16—2691no—
150#535—2691no—
151#449—2694no—
152#448—2694no—
153#419—2691no—
154#416—2691no—
155#408D26911yescharacteristic, complemented
1562−1+4—5384no—
1572−1+4—5384no—
158C2×#41—2691no—
159C2×#41—2692no—
160C2×#41—5382no—
161C2×#41—5381no—
162C2×#41C2×D26911yescharacteristic
163C22×Q8—2691no—
164C22×Q8—2692no—
165C22×Q8—5381no—
166C22×D4—2691no—
167AGL(1,ℤ/8)—5384no—
168C2×SD16—2694no—
169C2×SD16—2692no—
170C2×SD16—5384no—
171C2×D8—2692no—
172C2×M16C2×D26911yescharacteristic
173C22×C8C2×D26911yescharacteristic
174#127—5381no—
175#124—5381no—
176#121—2694no—
177#120—5382no—
178#117—2691no—
179#103—2692no—
180#103—2694no—
181#102—2692no—
182C22⋊C8C2×D26914yescomplemented
183C24—5381no—
184#41—22no—
185#41—22no—
186#41—2694no—
187#41—5384no—
188#41—5384no—
189#41—53816no—
190#41—5388no—
191#41—5382no—
192C2×Q8—2694no—
193C2×Q8—2692no—
194C2×Q8—2691no—
195C2×Q8—5384no—
196C2×Q8—5384no—
197C2×Q8—5388no—
198C2×Q8—5382no—
199C2×Q8—5382no—
200C2×Q8—5381no—
201C2×Q8D107611yescharacteristic
202C2×D4—2691no—
203C2×D4—2692no—
204C2×D4—5382no—
205C2×D4—5381no—
206C2×D4—5384no—
207C2×D4—5382no—
208C2×D4—5382no—
209C2×D4—5381no—
210C2×D4D107612yes—
211C2×D4D107611yescharacteristic
212C22×C4—5382no—
213C22×C4—5381no—
214C22×C4—5382no—
215C22×C4—5381no—
216C22×C4C22×D26912yes—
217C22×C4C22×D26911yescharacteristic
218SD16—5388no—
219SD16—5384no—
220D8—5384no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D26912yes—
224C2×C8C22×D26912yes—
225C4⋊C4—2692no—
226C22⋊C4—5381no—
227C22⋊C4—5384no—
228C42—5381no—
229C23—5381no—
230C23—5382no—
231C23—10762no—
232C23—21522no—
233C23C2×D107612yes—
234C23C2×D107611yescharacteristic
235Q8—22no—
236Q8—2694no—
237Q8—2692no—
238Q8—5388no—
239Q8—5384no—
240Q8—5382no—
241Q8—5381no—
242D4—24no—
243D4—22no—
244D4—2692no—
245D4—5388no—
246D4—5384no—
247D4—5381no—
248D4—5382no—
249D4—5382no—
250D4—10762no—
251D4—21524no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—5388no—
255C2×C4—5388no—
256C2×C4—5384no—
257C2×C4—5382no—
258C2×C4—5381no—
259C2×C4—5384no—
260C2×C4—5382no—
261C2×C4D4×D26912yes—
262C2×C4D4×D26912yes—
263C2×C4C2×D107612yes—
264C2×C4C2×D107611yescharacteristic
265C2×C4C23×D26911yescharacteristic
266C8—22no—
267C8—22no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—5382no—
272C22—10762no—
273C22—10764no—
274C22—10762no—
275C22—21524no—
276C22#2257542312yes—
277C22C22×D107611yescharacteristic
278C4—22no—
279C4—5384no—
280C4—5382no—
281C4—10762no—
282C4#2257542312yes—
283C4C2×D4×D26911yescharacteristic
284C4C2×D4×D26911yescharacteristic
285C2—22no—
286C2—42no—
287C2—10762no—
288C2—21522no—
289C2C2×#2257542311yescharacteristic
290C2#3177141712yes—
subgroupG/H
centerC22C22×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#408×C269C2
socleC22×C269C22×D4
hypercenter#408D269
hypocenterC269#3010
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
1. C2×#22575500C2semidirect
2. C2×#22575481C2semidirect
3. C22×#16864391C2semidirect
4. #31771169C2semidirect
5. #31770752C2semidirect
6. #31770746C2semidirect
7. #31770615C2semidirect
8. #31770602C2semidirect
9. #408×C269C2semidirect
13. C2×#16864402C22semidirect
17. C22×Dic538C22semidirect
30. #22575378C22semidirect
31. #22575378C22semidirect
32. #22575375C22semidirect
36. C22⋊C8×C269C22semidirect
41. #16864402D4semidirect
49. C2×Dic538D4semidirect
145. C269#3010semidirect
155. #408D269semidirect
182. C22⋊C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225755002
C2×#225754812
C22×#168643912
#317711692
#317707522
#317707462
#317706152
#317706022
#408×C2692
#3010269