Finite Group Databasebeta

← #46821370

Subgroups — #46821370

Order 34432.

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

normal subgroups only

This group has 596 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×#22575481C211yesmaximal, characteristic, complemented
2C2×#22575473C211yesmaximal, characteristic, complemented
3C22×#16864387C211yesmaximal, characteristic, complemented
4#31771172C211yesmaximal, characteristic, complemented
5#31771169C211yesmaximal, characteristic, complemented
6#31770993C212yesmaximal, complemented
7#31770991C212yesmaximal, complemented
8#31770986C212yesmaximal, complemented
9#31770980C212yesmaximal, complemented
10#31770563C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C269C211yesmaximal, characteristic, complemented
12C2×#16864402C2211yescharacteristic
13C22×D1076C2211yescharacteristic
14#22575481—24no—
15#22575473—24no—
16C2×#16864392C2211yescharacteristic, complemented
17C2×#16864391C2212yescomplemented
18C2×#16864390C2211yescharacteristic, complemented
19C2×#16864389C2211yescharacteristic, complemented
20C2×D2152C2212yescomplemented
21C2×#16864387—24no—
22C2×#16864387C2212yes—
23C2×#16864387C2212yes—
24C2×#16864387C2211yescharacteristic, complemented
25C22×#14524130C2211yescharacteristic
26#22575449C2212yescomplemented
27#22575449C2212yescomplemented
28#22575442C2212yescomplemented
29#22575440C2212yescomplemented
30#22575427—22no—
31#22575424—22no—
32#22575422—22no—
33#22575421—22no—
34#22575412C2211yescharacteristic
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×D1076C2312yes—
51C2×D1076C2311yescharacteristic
52C2×C22⋊D269—24no—
53C2×C22⋊D269—24no—
54C2×C22⋊D269—21no—
55C2×C4×D269—21no—
56#16864392—22no—
57#16864391—24no—
58#16864390—22no—
59#16864389—22no—
60D2152—24no—
61#16864387—24no—
62#16864387—24no—
63#16864387—22no—
64C2×#14524130D414yescomplemented
65C2×#14524130C2311yescharacteristic
66C2×#14524130C2311yescharacteristic
67C4⋊Dic269D412yes—
68C22⋊Dic269—24no—
69C22⋊Dic269D412yes—
70#16864379—24no—
71C4×Dic269C2312yes—
72C24×C269—21no—
73#41×C269—22no—
74#41×C269—22no—
75C2×Q8×C269C2311yescharacteristic
76C2×D4×C269—21no—
77C2×D4×C269—22no—
78C2×D4×C269—22no—
79C2×D4×C269—21no—
80C2×D4×C269C2312yes—
81C2×D4×C269C2311yescharacteristic
82C22×C4×C269—21no—
83C22×C4×C269C2311yescharacteristic
84SD16×C269—24no—
85D8×C269—24no—
86M16×C269D414yescomplemented
87C2×C8×C269C2312yes—
88C22×D269—22no—
89C22×D269—21no—
90C22×D269—42no—
91C22×D269—82no—
92Dic538—21no—
93Dic538—21no—
94D1076—22no—
95D1076—21no—
96D1076—21no—
97D1076—22no—
98D1076—42no—
99C22⋊D269—42no—
100C22⋊D269—84no—
101C22⋊D269—84no—
102C2×Dic269—21no—
103C2×Dic269C2×D414yes—
104C4×D269—42no—
105#14524130C2×D412yes—
106#14524130C2×D412yes—
107C23×C269—21no—
108C23×C269—22no—
109C23×C269—42no—
110C23×C269—82no—
111C23×C269C2×D411yescharacteristic
112Q8×C269—21no—
113Q8×C269—21no—
114D4×C269—21no—
115D4×C269—22no—
116D4×C269—22no—
117D4×C269—21no—
118D4×C269—42no—
119D4×C269—42no—
120C2×C4×C269—21no—
121C2×C4×C269—42no—
122C2×C4×C269C2×D411yescharacteristic
123C2×C4×C269C2×D412yes—
124C2×C4×C269C2×D412yes—
125C2×C4×C269C2411yescharacteristic
126C8×C269C2×D414yes—
127C22×C269—22no—
128C22×C269—41no—
129C22×C269—42no—
130C22×C269—44no—
131C22×C269—42no—
132C22×C269—84no—
133C22×C269#12712yes—
134C22×C269C22×D411yescharacteristic
135C2×D269—41no—
136C2×D269—42no—
137C2×D269—42no—
138C2×D269—44no—
139C2×D269—84no—
140C4×C269—41no—
141C4×C269#12712yes—
142C4×C269C22×D411yescharacteristic
143C4×C269C22×D411yescharacteristic
144Dic269—24no—
145Dic269—41no—
146C2×C269—22no—
147C2×C269—81no—
148C2×C269—82no—
149C2×C269C2×#12711yescharacteristic
150C2×C269#57612yes—
151D269—81no—
152D269—82no—
153C269#315511yescharacteristic, complemented, Sylow
154#3155—2691nomaximal, Sylow
155C2×AGL(1,ℤ/8)—2691no—
156C2×AGL(1,ℤ/8)D26911yescharacteristic, complemented
157C2×#135—2691no—
158C22×D8—2691no—
159#535—2691no—
160#535—2691no—
161#496—2692no—
162#496—2692no—
163#495—2692no—
164#493—2692no—
165#405—2691no—
166C2×#41—2691no—
167C2×#41C2×D26911yescharacteristic
168C22×D4—2691no—
169C22×D4C2×D26911yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—5384no—
172#135—5384no—
173C2×Q16—2691no—
174C2×SD16—2691no—
175C2×SD16—2691no—
176C2×SD16—2692no—
177C2×SD16C2×D26912yescomplemented
178C2×D8—2692no—
179C2×D8—2692no—
180C2×D8—2691no—
181C2×D8—2692no—
182C2×D8—5384no—
183C2×D8C2×D26912yescomplemented
184C2×M16C2×D26911yescharacteristic
185C22×C8—2691no—
186#127—2692no—
187#127—2692no—
188#124—2692no—
189#124—2692no—
190#121—5382no—
191#121—5382no—
192#120—5382no—
193#120—5382no—
194#117—2691no—
195#97—2692no—
196C4×C8—2692no—
197C24—21no—
198C24—5381no—
199#41—22no—
200#41—22no—
201#41—5382no—
202#41—5382no—
203C2×Q8—2691no—
204C2×Q8C22×D26911yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—2692no—
210C2×D4—2691no—
211C2×D4—5384no—
212C2×D4—5384no—
213C2×D4—5381no—
214C2×D4—5382no—
215C2×D4—5381no—
216C2×D4—5382no—
217C2×D4C22×D26912yes—
218C2×D4C22×D26911yescharacteristic
219C22×C4—21no—
220C22×C4—5381no—
221C22×C4C22×D26911yescharacteristic
222Q16—5382no—
223SD16—24no—
224SD16—5382no—
225SD16—5382no—
226SD16—5384no—
227D8—24no—
228D8—5384no—
229D8—5384no—
230D8—5382no—
231D8—5384no—
232M16C22⋊D26914yescomplemented
233C2×C8—2694no—
234C2×C8—2691no—
235C2×C8—2691no—
236C2×C8C22×D26912yes—
237C4⋊C4—2692no—
238C22⋊C4—2692no—
239C22⋊C4—5384no—
240C22⋊C4—5384no—
241C42—2692no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—5382no—
247C23—5381no—
248C23—10762no—
249C23—21522no—
250C23D4×D26911yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—5381no—
254Q8—5381no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—5382no—
262D4—5381no—
263D4—5381no—
264D4—5382no—
265D4—10762no—
266D4—10762no—
267D4—21524no—
268D4—21524no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—2694no—
272C2×C4—5381no—
273C2×C4—10762no—
274C2×C4D4×D26911yescharacteristic
275C2×C4D4×D26912yes—
276C2×C4C2×C22⋊D26912yes—
277C2×C4C23×D26911yescharacteristic
278C8—2692no—
279C8—2692no—
280C8C2×C22⋊D26914yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—10761no—
288C22—10762no—
289C22—10762no—
290C22—10764no—
291C22—21524no—
292C22#2257544912yes—
293C22C2×D4×D26911yescharacteristic
294C4—41no—
295C4—5384no—
296C4—10761no—
297C4#2257544912yes—
298C4C2×D4×D26911yescharacteristic
299C4C22×C22⋊D26911yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—21522no—
304C2—21521no—
305C2C2×#2257544911yescharacteristic
306C2#3177140912yes—
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#3155
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×#22575473C2semidirect
3. C22×#16864387C2semidirect
4. #31771172C2semidirect
5. #31771169C2semidirect
6. #31770993C2semidirect
7. #31770991C2semidirect
8. #31770986C2semidirect
9. #31770980C2semidirect
10. #31770563C2semidirect
11. C2×AGL(1,ℤ/8)×C269C2semidirect
16. C2×#16864392C22semidirect
17. C2×#16864391C22semidirect
18. C2×#16864390C22semidirect
19. C2×#16864389C22semidirect
20. C2×D2152C22semidirect
24. C2×#16864387C22semidirect
26. #22575449C22semidirect
27. #22575449C22semidirect
28. #22575442C22semidirect
29. #22575440C22semidirect
35. #22575363C22semidirect
36. C8×Dic269C22semidirect
40. C2×SD16×C269C22semidirect
41. C2×D8×C269C22semidirect
64. C2×#14524130D4semidirect
86. M16×C269D4semidirect
153. C269#3155semidirect
156. C2×AGL(1,ℤ/8)D269semidirect
177. C2×SD16C2×D269semidirect
183. C2×D8C2×D269semidirect
232. M16C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754812
C2×#225754732
C22×#168643872
#317711722
#317711692
#317709932
#317709912
#317709862
#317709802
#317705632
C2×AGL(1,ℤ/8)×C2692
#3155269