Finite Group Databasebeta

← #46821333

Subgroups — #46821333

Order 34432.

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

normal subgroups only

This group has 656 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
1C2×#22575481C211yesmaximal, characteristic, complemented
2C2×#22575479C211yesmaximal, characteristic, complemented
3C2×#22575478C211yesmaximal, characteristic, complemented
4#31771151C211yesmaximal, characteristic, complemented
5C2×#22575449C211yesmaximal, characteristic, complemented
6#31770993C212yesmaximal, complemented
7#31770991C212yesmaximal, complemented
8#31770980C212yesmaximal, complemented
9#31770976C212yesmaximal, complemented
10M16×Dic269C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C269C211yesmaximal, characteristic, complemented
12C2×#16864402C2211yescharacteristic
13C22×D1076C2211yescharacteristic
14C22×C22⋊D269—22no—
15#22575481—24no—
16#22575479—24no—
17#22575478—24no—
18C2×#16864391C2212yescomplemented
19C2×#16864390C2212yescomplemented
20C2×#16864389C2212yescomplemented
21C2×D2152C2212yescomplemented
22C2×#16864387C2212yescomplemented
23C2×#16864387C2212yescomplemented
24C2×#16864386C2211yescharacteristic
25#22575451C2211yescharacteristic, complemented
26#22575449—22no—
27#22575449C2212yes—
28#22575449C2212yes—
29#22575449C2211yescharacteristic, complemented
30#22575442C2211yescharacteristic, complemented
31#22575440C2211yescharacteristic, complemented
32#22575425—22no—
33C4×C22⋊D269—22no—
34C2×C4×Dic269C2211yescharacteristic
35#22575363C2212yescomplemented
36C8×Dic269C2212yescomplemented
37C2×#41×C269C2211yescharacteristic
38C22×D4×C269C2211yescharacteristic
39AGL(1,ℤ/8)×C269—24no—
40C2×SD16×C269C2212yescomplemented
41C2×D8×C269C2212yescomplemented
42C2×M16×C269C2211yescharacteristic
43C23×D269—21no—
44#16864402—22no—
45#16864402—22no—
46C2×Dic538C2311yescharacteristic
47C2×D1076—22no—
48C2×D1076—22no—
49C2×D1076—21no—
50C2×D1076C2311yescharacteristic
51C2×D1076C2312yes—
52C2×C22⋊D269—22no—
53C2×C22⋊D269—24no—
54C2×C22⋊D269—24no—
55C2×C22⋊D269—21no—
56C2×C22⋊D269—48no—
57C22×Dic269D412yes—
58C2×C4×D269—21no—
59#16864391—24no—
60#16864390—24no—
61#16864389—24no—
62D2152—24no—
63#16864387—24no—
64#16864387—24no—
65#16864386D414yescomplemented
66C2×#14524130C2312yes—
67#16864381—22no—
68C22⋊Dic269—22no—
69#16864379—22no—
70C4×Dic269D412yes—
71C4×Dic269C2311yescharacteristic
72C4×Dic269C2311yescharacteristic
73C24×C269—21no—
74#41×C269—22no—
75#41×C269—22no—
76C2×Q8×C269C2311yescharacteristic
77C2×D4×C269—21no—
78C2×D4×C269—22no—
79C2×D4×C269—22no—
80C2×D4×C269—21no—
81C2×D4×C269C2311yescharacteristic
82C2×D4×C269C2312yes—
83C22×C4×C269—21no—
84C22×C4×C269C2311yescharacteristic
85SD16×C269—24no—
86D8×C269—24no—
87M16×C269D414yescomplemented
88C2×C8×C269C2312yes—
89C22×D269—21no—
90C22×D269—22no—
91C22×D269—44no—
92C22×D269—42no—
93Dic538—21no—
94Dic538—21no—
95D1076—21no—
96D1076—22no—
97D1076—21no—
98D1076—22no—
99D1076—42no—
100C22⋊D269—48no—
101C22⋊D269—42no—
102C22⋊D269—44no—
103C22⋊D269—48no—
104C2×Dic269—24no—
105C2×Dic269—21no—
106C2×Dic269C2×D412yes—
107C2×Dic269C2×D412yes—
108C4×D269—42no—
109#14524130C2×D414yescomplemented
110C23×C269—21no—
111C23×C269—22no—
112C23×C269—44no—
113C23×C269—42no—
114C23×C269C2×D411yescharacteristic
115Q8×C269—21no—
116Q8×C269—21no—
117D4×C269—21no—
118D4×C269—22no—
119D4×C269—22no—
120D4×C269—21no—
121D4×C269—42no—
122D4×C269—42no—
123C2×C4×C269—21no—
124C2×C4×C269—42no—
125C2×C4×C269C2×D411yescharacteristic
126C2×C4×C269C2×D412yes—
127C2×C4×C269C2×D412yes—
128C2×C4×C269C2411yescharacteristic
129C8×C269C2×D414yescomplemented
130C22×C269—22no—
131C22×C269—41no—
132C22×C269—42no—
133C22×C269—44no—
134C22×C269—42no—
135C22×C269—84no—
136C22×C269#12712yes—
137C22×C269C22×D411yescharacteristic
138C2×D269—41no—
139C2×D269—42no—
140C2×D269—44no—
141C2×D269—42no—
142C2×D269—84no—
143C4×C269—41no—
144C4×C269#12712yes—
145C4×C269C22×D411yescharacteristic
146C4×C269C22×D411yescharacteristic
147Dic269—22no—
148Dic269—41no—
149Dic269AGL(1,ℤ/8)14yescomplemented
150C2×C269—22no—
151C2×C269—81no—
152C2×C269—82no—
153C2×C269C2×#12711yescharacteristic
154C2×C269C2×AGL(1,ℤ/8)12yes—
155D269—81no—
156D269—82no—
157C269#315111yescharacteristic, complemented, Sylow
158#3151—2691nomaximal, Sylow
159C2×AGL(1,ℤ/8)—2691no—
160C2×AGL(1,ℤ/8)—2691no—
161C2×AGL(1,ℤ/8)—2691no—
162C2×AGL(1,ℤ/8)D26911yescharacteristic, complemented
163#532—2691no—
164C2×#127—2691no—
165#496—2692no—
166#496—2692no—
167#493—2692no—
168#492—2692no—
169C4×M16—2691no—
170C2×#41—2691no—
171C2×#41C2×D26911yescharacteristic
172C22×D4—2691no—
173C22×D4—5382no—
174C22×D4C2×D26911yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—5384no—
177AGL(1,ℤ/8)—5384no—
178AGL(1,ℤ/8)—5384no—
179C2×SD16—2692no—
180C2×SD16—2692no—
181C2×SD16—2692no—
182C2×SD16C2×D26912yescomplemented
183C2×D8—2692no—
184C2×D8—2692no—
185C2×D8—2692no—
186C2×D8C2×D26912yescomplemented
187C2×M16—2691no—
188C2×M16C2×D26911yescharacteristic
189C4⋊Q8—2691no—
190#127—2692no—
191#127—2692no—
192#127—2691no—
193#127—5382no—
194#124—2691no—
195#124—2691no—
196#121—5382no—
197C4×D4—5382no—
198C2×C42—2691no—
199#97—2692no—
200C4×C8—2692no—
201C24—21no—
202C24—5381no—
203#41—22no—
204#41—22no—
205#41—5382no—
206#41—5382no—
207C2×Q8—2691no—
208C2×Q8C22×D26911yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—2691no—
214C2×D4—2692no—
215C2×D4—5382no—
216C2×D4—5384no—
217C2×D4—5384no—
218C2×D4—5381no—
219C2×D4—5382no—
220C2×D4—5381no—
221C2×D4—5382no—
222C2×D4—10768no—
223C2×D4C22×D26911yescharacteristic
224C2×D4C22×D26912yes—
225C22×C4—21no—
226C22×C4—2692no—
227C22×C4—5381no—
228C22×C4C22×D26911yescharacteristic
229SD16—24no—
230SD16—5384no—
231SD16—5384no—
232SD16—5384no—
233D8—24no—
234D8—5384no—
235D8—5384no—
236D8—5384no—
237M16—2694no—
238M16C22⋊D26914yescomplemented
239C2×C8—2692no—
240C2×C8C22×D26912yes—
241C4⋊C4—5382no—
242C22⋊C4—5382no—
243C22⋊C4—5382no—
244C42—2692no—
245C42—2691no—
246C42—2691no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—5381no—
252C23—5382no—
253C23—10764no—
254C23—10762no—
255C23D4×D26911yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—5381no—
259Q8—5381no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—5381no—
267D4—5382no—
268D4—5381no—
269D4—5382no—
270D4—10768no—
271D4—10764no—
272D4—10768no—
273D4—10762no—
274D4—10762no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—2692no—
278C2×C4—2692no—
279C2×C4—5384no—
280C2×C4—5381no—
281C2×C4—10762no—
282C2×C4D4×D26911yescharacteristic
283C2×C4D4×D26912yes—
284C2×C4C2×C22⋊D26912yes—
285C2×C4C23×D26911yescharacteristic
286C8—2694no—
287C8C2×C22⋊D26914yescomplemented
288C22—22no—
289C22—41no—
290C22—42no—
291C22—44no—
292C22—42no—
293C22—84no—
294C22—10761no—
295C22—10762no—
296C22—10764no—
297C22—10762no—
298C22—21524no—
299C22#2257544912yes—
300C22C2×D4×D26911yescharacteristic
301C4—41no—
302C4—2694no—
303C4—5382no—
304C4—10761no—
305C4#2257544912yes—
306C4C2×D4×D26911yescharacteristic
307C4C22×C22⋊D26911yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—21522no—
312C2—21521no—
313C2C2×#2257544911yescharacteristic
314C2AGL(1,ℤ/8)×D26912yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×AGL(1,ℤ/8)×C269C2
socleC22×C269C22×D4
hypercenterC2×AGL(1,ℤ/8)D269
hypocenterC269#3151
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22575481C2semidirect
2. C2×#22575479C2semidirect
3. C2×#22575478C2semidirect
4. #31771151C2semidirect
5. C2×#22575449C2semidirect
6. #31770993C2semidirect
7. #31770991C2semidirect
8. #31770980C2semidirect
9. #31770976C2semidirect
10. M16×Dic269C2semidirect
11. C2×AGL(1,ℤ/8)×C269C2semidirect
18. C2×#16864391C22semidirect
19. C2×#16864390C22semidirect
20. C2×#16864389C22semidirect
21. C2×D2152C22semidirect
22. C2×#16864387C22semidirect
23. C2×#16864387C22semidirect
25. #22575451C22semidirect
29. #22575449C22semidirect
30. #22575442C22semidirect
31. #22575440C22semidirect
35. #22575363C22semidirect
36. C8×Dic269C22semidirect
40. C2×SD16×C269C22semidirect
41. C2×D8×C269C22semidirect
65. #16864386D4semidirect
87. M16×C269D4semidirect
109. #14524130C2×D4semidirect
129. C8×C269C2×D4semidirect
149. Dic269AGL(1,ℤ/8)semidirect
157. C269#3151semidirect
162. C2×AGL(1,ℤ/8)D269semidirect
182. C2×SD16C2×D269semidirect
186. C2×D8C2×D269semidirect
238. M16C22⋊D269semidirect
287. C8C2×C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754812
C2×#225754792
C2×#225754782
#317711512
C2×#225754492
#317709932
#317709912
#317709802
#317709762
M16×Dic2692
C2×AGL(1,ℤ/8)×C2692
#3151269