Finite Group Databasebeta

← #46820117

Subgroups — #46820117

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×#22575483C211yesmaximal, characteristic, complemented
3#31771112C211yesmaximal, characteristic
4#31770759C214yesmaximal, complemented
5#31770742C214yesmaximal, complemented
6#31770615C211yesmaximal, characteristic, complemented
7#31770614C211yesmaximal, characteristic, complemented
8#31770568C211yesmaximal, characteristic, complemented
9C2×#137×C269C211yesmaximal, characteristic, complemented
10C22×Dic538—21no—
11C22×Dic538C2211yescharacteristic
12C2×Q8×D269—24no—
13C2×Q8×D269—22no—
14C2×Q8×D269—24no—
15C2×Q8×D269C2214yes—
16C22×C4×D269—21no—
17C22×C4×D269C2211yescharacteristic
18#22575483—24no—
19C2×#16864392C2212yescomplemented
20C2×#16864390C2212yescomplemented
21C2×#16864386C2211yescharacteristic
22#22575434—21no—
23#22575429C2212yes—
24#22575424C2212yes—
25#22575421—21no—
26C4×C22⋊D269—22no—
27#22575412C2211yescharacteristic
28#22575378C2214yescomplemented
29#22575377C2214yescomplemented
30#22575365C2214yes—
31C2×#41×C269C2211yescharacteristic
32C22×Q8×C269C2211yescharacteristic
33#137×C269—24no—
34C2×Q16×C269C2212yescomplemented
35C2×SD16×C269C2212yescomplemented
36C2×M16×C269C2211yescharacteristic
37C23×D269D411yescharacteristic
38C2×Dic538—28no—
39C2×Dic538—24no—
40C2×Dic538—21no—
41C2×Dic538—22no—
42C2×Dic538D414yescomplemented
43C2×Dic538C2312yes—
44Q8×D269—216no—
45Q8×D269—28no—
46Q8×D269—28no—
47C2×C22⋊D269—22no—
48C22×Dic269—21no—
49C22×Dic269D411yescharacteristic
50C2×C4×D269—28no—
51C2×C4×D269—24no—
52C2×C4×D269—22no—
53C2×C4×D269—22no—
54C2×C4×D269D412yes—
55C2×C4×D269C2312yes—
56#16864392—24no—
57#16864390—24no—
58#16864386—22no—
59C2×#14524130C2312yes—
60C4⋊Dic269C2312yes—
61#16864381—22no—
62C22⋊Dic269—22no—
63C22⋊Dic269—21no—
64#16864379—22no—
65C4×Dic269—21no—
66#41×C269—22no—
67#41×C269—22no—
68C2×Q8×C269—22no—
69C2×Q8×C269—21no—
70C2×Q8×C269D414yescomplemented
71C2×Q8×C269C2311yescharacteristic
72C2×Q8×C269C2312yes—
73C2×D4×C269—21no—
74C2×D4×C269C2311yescharacteristic
75C22×C4×C269—21no—
76C22×C4×C269—21no—
77C22×C4×C269C2311yescharacteristic
78Q16×C269—24no—
79SD16×C269—24no—
80M16×C269—22no—
81C2×C8×C269C2312yes—
82C22×D269—22no—
83C22×D269—24no—
84C22×D269C2×D412yes—
85Dic538—216no—
86Dic538—24no—
87Dic538—22no—
88Dic538C2×D414yes—
89C22⋊D269—44no—
90C2×Dic269—24no—
91C2×Dic269—22no—
92C2×Dic269—22no—
93C2×Dic269—22no—
94C2×Dic269C2×D412yes—
95C4×D269—216no—
96C4×D269—24no—
97C4×D269—22no—
98C4×D269—22no—
99#14524130—22no—
100C23×C269—21no—
101C23×C269C2×D411yescharacteristic
102Q8×C269—24no—
103Q8×C269—22no—
104Q8×C269—21no—
105Q8×C269—21no—
106Q8×C269C2×D414yes—
107D4×C269—21no—
108D4×C269—21no—
109D4×C269—42no—
110C2×C4×C269—24no—
111C2×C4×C269—22no—
112C2×C4×C269—21no—
113C2×C4×C269—42no—
114C2×C4×C269C2×D411yescharacteristic
115C2×C4×C269C2×D412yes—
116C2×C4×C269C2×D412yes—
117C2×C4×C269C2411yescharacteristic
118C8×C269—22no—
119C22×C269—22no—
120C22×C269—41no—
121C22×C269—42no—
122C22×C269#12012yes—
123C22×C269C22×D411yescharacteristic
124C2×D269—28no—
125C2×D269—22no—
126C2×D269—22no—
127C2×D269#13714yescomplemented
128C4×C269—24no—
129C4×C269—41no—
130C4×C269#12012yes—
131C4×C269C22×D411yescharacteristic
132C4×C269C22×D411yescharacteristic
133Dic269—24no—
134Dic269—22no—
135Dic269—42no—
136C2×C269—22no—
137C2×C269—81no—
138C2×C269C2×#12011yescharacteristic
139C2×C269C2×#13712yes—
140D269—24no—
141C269#300411yescharacteristic, complemented, Sylow
142#3004—2691nomaximal, Sylow
143C23×Q8—2691no—
144C2×#137—2691no—
145C2×#137D26911yescharacteristic, complemented
146#525—2691no—
147#451—2694no—
148#448—2694no—
149#419—2691no—
150#419—2691no—
151#407—2691no—
152C2×#41C2×D26911yescharacteristic
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—24no—
163#137—5384no—
164C2×Q16—2692no—
165C2×Q16C2×D26912yescomplemented
166C2×SD16—2692no—
167C2×SD16C2×D26912yescomplemented
168C2×M16—2691no—
169C2×M16C2×D26911yescharacteristic
170#123—5381no—
171C22⋊Q8—2692no—
172#121—2692no—
173#120—5381no—
174C4×D4—5382no—
175#117—2691no—
176#103—2694no—
177#103—2694no—
178C22⋊C8—2694no—
179C24—2691no—
180#41—22no—
181#41—22no—
182C2×Q8—22no—
183C2×Q8—21no—
184C2×Q8—2694no—
185C2×Q8—2692no—
186C2×Q8—5384no—
187C2×Q8—5388no—
188C2×Q8—5388no—
189C2×Q8—53816no—
190C2×Q8—5388no—
191C2×Q8—5381no—
192C2×Q8—5382no—
193C2×Q8C22⋊D26914yescomplemented
194C2×Q8C22×D26911yescharacteristic
195C2×Q8C22×D26912yes—
196C2×D4—21no—
197C2×D4—5382no—
198C2×D4C22×D26911yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—2691no—
202C22×C4—2692no—
203C22×C4—2692no—
204C22×C4—5384no—
205C22×C4—5388no—
206C22×C4—5381no—
207C22×C4—5382no—
208C22×C4—5382no—
209C22×C4C22×D26911yescharacteristic
210Q16—24no—
211Q16—5384no—
212SD16—24no—
213SD16—5384no—
214M16—22no—
215M16—5382no—
216C2×C8—2692no—
217C2×C8C22×D26912yes—
218C4⋊C4—2692no—
219C4⋊C4—5382no—
220C22⋊C4—5382no—
221C22⋊C4—5382no—
222C22⋊C4—5381no—
223C42—5381no—
224C23—21no—
225C23—2692no—
226C23—5384no—
227C23—5382no—
228C23D4×D26911yescharacteristic
229Q8—24no—
230Q8—22no—
231Q8—21no—
232Q8—21no—
233Q8—2694no—
234Q8—53816no—
235Q8—5384no—
236Q8—5382no—
237Q8C2×C22⋊D26914yes—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—10764no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—42no—
246C2×C4—2692no—
247C2×C4—53816no—
248C2×C4—5382no—
249C2×C4—5384no—
250C2×C4—5382no—
251C2×C4—5384no—
252C2×C4—5382no—
253C2×C4—5382no—
254C2×C4—5382no—
255C2×C4D4×D26911yescharacteristic
256C2×C4D4×D26912yes—
257C2×C4C2×C22⋊D26912yes—
258C2×C4C23×D26911yescharacteristic
259C8—22no—
260C8—5382no—
261C22—22no—
262C22—42no—
263C22—41no—
264C22—2694no—
265C22—5388no—
266C22—5382no—
267C22—5382no—
268C22#2257542112yes—
269C22C2×D4×D26911yescharacteristic
270C4—24no—
271C4—41no—
272C4—5382no—
273C4—5384no—
274C4—10762no—
275C4#2257542112yes—
276C4C2×D4×D26911yescharacteristic
277C4C22×C22⋊D26911yescharacteristic
278C2—22no—
279C2—81no—
280C2—5384no—
281C2C2×#2257542111yescharacteristic
282C2#137×D26912yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#137×C269C2
socleC22×C269C22×D4
prime radical#31771112C2
hypercenterC2×#137D269
hypocenterC269#3004
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D269C2semidirect
2. C2×#22575483C2semidirect
4. #31770759C2semidirect
5. #31770742C2semidirect
6. #31770615C2semidirect
7. #31770614C2semidirect
8. #31770568C2semidirect
9. C2×#137×C269C2semidirect
19. C2×#16864392C22semidirect
20. C2×#16864390C22semidirect
28. #22575378C22semidirect
29. #22575377C22semidirect
34. C2×Q16×C269C22semidirect
35. C2×SD16×C269C22semidirect
42. C2×Dic538D4semidirect
70. C2×Q8×C269D4semidirect
127. C2×D269#137semidirect
141. C269#3004semidirect
145. C2×#137D269semidirect
165. C2×Q16C2×D269semidirect
167. C2×SD16C2×D269semidirect
193. C2×Q8C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D2692
C2×#225754832
#317711122
#317707592
#317707422
#317706152
#317706142
#317705682
C2×#137×C2692
#3004269