Finite Group Databasebeta

← #46820121

Subgroups — #46820121

Order 34432.

738 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×Q8×D269C211yesmaximal, characteristic, complemented
2C2×#22575486C211yesmaximal, characteristic, complemented
3C2×#22575484C211yesmaximal, characteristic, complemented
4#31771117C211yesmaximal, characteristic
5#31770762C214yesmaximal, complemented
6#31770745C214yesmaximal, complemented
7#31770613C211yesmaximal, characteristic, complemented
8#31770568C211yesmaximal, characteristic, complemented
9#419×C269C211yesmaximal, characteristic, complemented
10C2×#16864402C2211yescharacteristic
11C22×Dic538—21no—
12C22×Dic538C2211yescharacteristic
13C2×Q8×D269—24no—
14C2×Q8×D269—22no—
15C2×Q8×D269—24no—
16C2×Q8×D269C2214yes—
17C22×C4×D269—21no—
18C22×C4×D269C2211yescharacteristic
19#22575486—24no—
20#22575484—24no—
21C2×Dic1076C2212yescomplemented
22C2×#16864392C2212yescomplemented
23C2×#16864391C2212yescomplemented
24C2×#16864389C2212yescomplemented
25C2×#16864386C2211yescharacteristic
26#22575437—21no—
27#22575433C2212yes—
28#22575428C2212yes—
29#22575423—21no—
30C4×D1076—22no—
31#22575376C2214yescomplemented
32#22575365C2214yes—
33C22×Q8×C269C2211yescharacteristic
34C2×M16×C269C2211yescharacteristic
35#117×C269C2211yescharacteristic
36#103×C269C2214yescomplemented
37C23×D269D411yescharacteristic
38#16864402—22no—
39#16864402—22no—
40C2×Dic538—28no—
41C2×Dic538—24no—
42C2×Dic538—21no—
43C2×Dic538—22no—
44C2×Dic538D414yescomplemented
45C2×Dic538C2311yescharacteristic
46C2×Dic538C2312yes—
47Q8×D269—216no—
48Q8×D269—28no—
49Q8×D269—28no—
50C2×D1076—22no—
51C2×D1076C2311yescharacteristic
52C2×C22⋊D269—21no—
53C22×Dic269—21no—
54C22×Dic269D411yescharacteristic
55C2×C4×D269—28no—
56C2×C4×D269—24no—
57C2×C4×D269—22no—
58C2×C4×D269—22no—
59C2×C4×D269—21no—
60C2×C4×D269D412yes—
61C2×C4×D269C2312yes—
62Dic1076—24no—
63#16864392—24no—
64#16864391—24no—
65#16864389—24no—
66#16864386—22no—
67C2×#14524130C2312yes—
68C4⋊Dic269—22no—
69#16864379—22no—
70#16864379—22no—
71C2×Q8×C269—22no—
72C2×Q8×C269—21no—
73C2×Q8×C269D414yescomplemented
74C2×Q8×C269C2312yes—
75C22×C4×C269—21no—
76C22×C4×C269C2311yescharacteristic
77M16×C269—22no—
78C2×C8×C269C2312yes—
79C4⋊C4×C269C2312yes—
80C22⋊C4×C269—21no—
81C42×C269—21no—
82C22×D269—22no—
83C22×D269—24no—
84C22×D269—21no—
85C22×D269C2×D412yes—
86Dic538—216no—
87Dic538—24no—
88Dic538—22no—
89Dic538—21no—
90Dic538—21no—
91Dic538C2×D414yes—
92D1076—21no—
93D1076—21no—
94D1076—44no—
95C22⋊D269—42no—
96C2×Dic269—24no—
97C2×Dic269—22no—
98C2×Dic269—22no—
99C2×Dic269—21no—
100C2×Dic269C2×D412yes—
101C4×D269—216no—
102C4×D269—24no—
103C4×D269—22no—
104C4×D269—22no—
105C4×D269—42no—
106#14524130—22no—
107C23×C269C2×D411yescharacteristic
108Q8×C269—24no—
109Q8×C269—22no—
110Q8×C269C2×D414yes—
111C2×C4×C269—24no—
112C2×C4×C269—22no—
113C2×C4×C269—22no—
114C2×C4×C269C2×D411yescharacteristic
115C2×C4×C269C2×D412yes—
116C2×C4×C269C2×D412yes—
117C2×C4×C269C2411yescharacteristic
118C8×C269—22no—
119C22×C269—22no—
120C22×C269#12012yes—
121C22×C269C22×D411yescharacteristic
122C2×D269—28no—
123C2×D269—22no—
124C2×D269—22no—
125C2×D269—41no—
126C2×D269—42no—
127C2×D269#13714yescomplemented
128C4×C269—24no—
129C4×C269—42no—
130C4×C269#12012yes—
131C4×C269C22×D411yescharacteristic
132C4×C269C22×D411yescharacteristic
133Dic269—24no—
134Dic269—22no—
135Dic269—41no—
136C2×C269—22no—
137C2×C269C2×#12011yescharacteristic
138C2×C269C2×#13712yes—
139D269—24no—
140D269—81no—
141C269#300411yescharacteristic, complemented, Sylow
142#3004—2691nomaximal, Sylow
143C23×Q8—2691no—
144C2×#137—2691no—
145C2×#137—2691no—
146#525—2691no—
147#451—2694no—
148#448—2694no—
149#419—2691no—
150#419D26911yescharacteristic, complemented
151#407—2691no—
152C2×#41—2691no—
153C22×Q8—2694no—
154C22×Q8—2691no—
155C22×Q8—5381no—
156C22×Q8—5382no—
157C22×Q8—5384no—
158C22×Q8—5384no—
159C22×Q8C2×D26911yescharacteristic
160C23×C4—2691no—
161C23×C4—5381no—
162#137—5384no—
163#137—5384no—
164C2×Q16—2692no—
165C2×Q16—2692no—
166C2×SD16—2692no—
167C2×SD16—2692no—
168C2×M16—2691no—
169C2×M16C2×D26911yescharacteristic
170#123—5381no—
171C22⋊Q8—2692no—
172#121—2692no—
173#120—5381no—
174C4×D4—5382no—
175#117C2×D26911yescharacteristic
176#103—2694no—
177#103C2×D26914yescomplemented
178C22⋊C8—2694no—
179C24—2691no—
180#41—5382no—
181#41—5382no—
182C2×Q8—22no—
183C2×Q8—21no—
184C2×Q8—2694no—
185C2×Q8—2692no—
186C2×Q8—2691no—
187C2×Q8—5384no—
188C2×Q8—5388no—
189C2×Q8—5388no—
190C2×Q8—53816no—
191C2×Q8—5388no—
192C2×Q8—5381no—
193C2×Q8—5382no—
194C2×Q8D107614yescomplemented
195C2×Q8C22×D26912yes—
196C2×D4—2691no—
197C2×D4—5382no—
198C2×D4—5381no—
199C22×C4—21no—
200C22×C4—2691no—
201C22×C4—2692no—
202C22×C4—2692no—
203C22×C4—5384no—
204C22×C4—5388no—
205C22×C4—5381no—
206C22×C4—5382no—
207C22×C4—5382no—
208C22×C4—5381no—
209C22×C4C22×D26911yescharacteristic
210Q16—5384no—
211Q16—5384no—
212SD16—5384no—
213SD16—5384no—
214M16—22no—
215M16—5382no—
216C2×C8—2692no—
217C2×C8C22×D26912yes—
218C4⋊C4—5382no—
219C4⋊C4C22×D26912yes—
220C22⋊C4—21no—
221C22⋊C4—5382no—
222C22⋊C4—5382no—
223C42—21no—
224C23—2692no—
225C23—5384no—
226C23—5382no—
227C23—5381no—
228C23D4×D26911yescharacteristic
229Q8—24no—
230Q8—22no—
231Q8—2694no—
232Q8—53816no—
233Q8—5384no—
234Q8—5382no—
235Q8—5381no—
236Q8—5381no—
237Q8C2×D107614yes—
238D4—5381no—
239D4—5381no—
240D4—10764no—
241D4—10762no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—2692no—
246C2×C4—53816no—
247C2×C4—5382no—
248C2×C4—5384no—
249C2×C4—5382no—
250C2×C4—5384no—
251C2×C4—5382no—
252C2×C4—5382no—
253C2×C4—5381no—
254C2×C4—10762no—
255C2×C4D4×D26911yescharacteristic
256C2×C4D4×D26912yes—
257C2×C4C2×D107612yes—
258C2×C4C23×D26911yescharacteristic
259C8—22no—
260C8—5382no—
261C22—22no—
262C22—2694no—
263C22—5388no—
264C22—5382no—
265C22—5382no—
266C22—10762no—
267C22—10761no—
268C22#2257542312yes—
269C22C2×D4×D26911yescharacteristic
270C4—24no—
271C4—42no—
272C4—5382no—
273C4—5384no—
274C4—10761no—
275C4#2257542312yes—
276C4C2×D4×D26911yescharacteristic
277C4C22×D107611yescharacteristic
278C2—22no—
279C2—5384no—
280C2—21521no—
281C2C2×#2257542311yescharacteristic
282C2#137×D26912yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#419×C269C2
socleC22×C269C22×D4
prime radical#31771117C2
hypercenter#419D269
hypocenterC269#3004
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D269C2semidirect
2. C2×#22575486C2semidirect
3. C2×#22575484C2semidirect
5. #31770762C2semidirect
6. #31770745C2semidirect
7. #31770613C2semidirect
8. #31770568C2semidirect
9. #419×C269C2semidirect
21. C2×Dic1076C22semidirect
22. C2×#16864392C22semidirect
23. C2×#16864391C22semidirect
24. C2×#16864389C22semidirect
31. #22575376C22semidirect
36. #103×C269C22semidirect
44. C2×Dic538D4semidirect
73. C2×Q8×C269D4semidirect
127. C2×D269#137semidirect
141. C269#3004semidirect
150. #419D269semidirect
177. #103C2×D269semidirect
194. C2×Q8D1076semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D2692
C2×#225754862
C2×#225754842
#317711172
#317707622
#317707452
#317706132
#317705682
#419×C2692
#3004269