Finite Group Databasebeta

← #46821368

Subgroups — #46821368

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×#22575479C211yesmaximal, characteristic, complemented
2C2×#22575478C211yesmaximal, characteristic, complemented
3C2×#22575473C211yesmaximal, characteristic, complemented
4C22×#16864387C211yesmaximal, characteristic, complemented
5#31771174C211yesmaximal, characteristic, complemented
6#31770995C212yesmaximal, complemented
7#31770992C212yesmaximal, complemented
8#31770987C212yesmaximal, complemented
9#31770981C212yesmaximal, complemented
10#31770561C211yesmaximal, characteristic, complemented
11#535×C269C211yesmaximal, characteristic, complemented
12C2×#16864402C2211yescharacteristic
13C22×D1076C2211yescharacteristic
14#22575479—24no—
15#22575478—24no—
16#22575473—24no—
17C2×#16864392C2211yescharacteristic, complemented
18C2×#16864390C2212yescomplemented
19C2×#16864390C2211yescharacteristic, complemented
20C2×#16864389C2212yescomplemented
21C2×#16864389C2211yescharacteristic, complemented
22C2×#16864387—24no—
23C2×#16864387C2212yes—
24C2×#16864387C2212yes—
25C2×#16864387C2212yescomplemented
26C2×#16864387C2211yescharacteristic, complemented
27C2×#16864387C2212yescomplemented
28C2×#16864386C2211yescharacteristic
29C22×#14524130C2211yescharacteristic
30#22575450C2212yescomplemented
31#22575441C2212yescomplemented
32#22575428—22no—
33#22575423—22no—
34#22575364C2212yescomplemented
35C4×#14524130C2212yescomplemented
36C2×#41×C269C2211yescharacteristic
37C22×D4×C269C2211yescharacteristic
38#127×C269C2212yescomplemented
39#124×C269C2212yescomplemented
40#121×C269—22no—
41#120×C269—22no—
42#117×C269C2211yescharacteristic
43C23×D269—21no—
44#16864402—22no—
45#16864402—22no—
46C2×Dic538C2311yescharacteristic
47C2×D1076—22no—
48C2×D1076—22no—
49C2×D1076—21no—
50C2×D1076—24no—
51C2×D1076C2312yes—
52C2×D1076C2311yescharacteristic
53C2×C22⋊D269—21no—
54C2×C4×D269—21no—
55#16864392—22no—
56#16864390—22no—
57#16864390—24no—
58#16864389—22no—
59#16864389—24no—
60#16864387—24no—
61#16864387—24no—
62#16864387—22no—
63#16864387—24no—
64#16864387—24no—
65#16864386D414yescomplemented
66C2×#14524130D414yescomplemented
67C2×#14524130C2311yescharacteristic
68C2×#14524130C2312yes—
69C2×#14524130C2311yescharacteristic
70#16864379—24no—
71C24×C269—21no—
72#41×C269—22no—
73#41×C269—22no—
74C2×Q8×C269C2311yescharacteristic
75C2×D4×C269—21no—
76C2×D4×C269—22no—
77C2×D4×C269—22no—
78C2×D4×C269—21no—
79C2×D4×C269—24no—
80C2×D4×C269C2312yes—
81C2×D4×C269C2311yescharacteristic
82C22×C4×C269—21no—
83C22×C4×C269C2311yescharacteristic
84C4⋊C4×C269D412yes—
85C22⋊C4×C269—24no—
86C22⋊C4×C269D412yes—
87C42×C269C2312yes—
88C22×D269—22no—
89C22×D269—21no—
90C22×D269—42no—
91C22×D269—82no—
92Dic538—21no—
93Dic538—21no—
94D1076—22no—
95D1076—21no—
96D1076—22no—
97D1076—21no—
98D1076—42no—
99D1076—84no—
100C22⋊D269—42no—
101C2×Dic269—21no—
102C4×D269—42no—
103#14524130C2×D412yes—
104#14524130C2×D412yes—
105#14524130C2×D414yes—
106C23×C269—22no—
107C23×C269—21no—
108C23×C269—42no—
109C23×C269—82no—
110C23×C269C2×D411yescharacteristic
111Q8×C269—21no—
112Q8×C269—21no—
113D4×C269—22no—
114D4×C269—22no—
115D4×C269—21no—
116D4×C269—21no—
117D4×C269—42no—
118D4×C269—42no—
119D4×C269—84no—
120C2×C4×C269—21no—
121C2×C4×C269—42no—
122C2×C4×C269C2×D412yes—
123C2×C4×C269C2×D412yes—
124C2×C4×C269C2×D414yes—
125C2×C4×C269C2×D411yescharacteristic
126C2×C4×C269C2411yescharacteristic
127C22×C269—22no—
128C22×C269—41no—
129C22×C269—42no—
130C22×C269—42no—
131C22×C269—44no—
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—24no—
141C4×C269—41no—
142C4×C269#12712yes—
143C4×C269C22×D411yescharacteristic
144C4×C269C22×D411yescharacteristic
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)—2691no—
157C2×#135—2691no—
158C22×D8—2691no—
159#535—2691no—
160#535D26911yescharacteristic, complemented
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)—5384no—
171AGL(1,ℤ/8)—5384no—
172#135—5384no—
173C2×Q16—2691no—
174C2×SD16—2691no—
175C2×SD16—2692no—
176C2×SD16—2691no—
177C2×SD16—2692no—
178C2×D8—2692no—
179C2×D8—2692no—
180C2×D8—2691no—
181C2×D8—2692no—
182C2×D8—2692no—
183C2×D8—5384no—
184C2×M16—2691no—
185C22×C8—2691no—
186#127—2692no—
187#127C2×D26912yescomplemented
188#124—2692no—
189#124C2×D26912yescomplemented
190#121—22no—
191#121—5382no—
192#120—22no—
193#120—5382no—
194#117C2×D26911yescharacteristic
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—22no—
208C2×D4—21no—
209C2×D4—24no—
210C2×D4—2692no—
211C2×D4—2691no—
212C2×D4—5381no—
213C2×D4—5382no—
214C2×D4—5382no—
215C2×D4—5381no—
216C2×D4—5384no—
217C2×D4C22×D26912yes—
218C2×D4C22×D26911yescharacteristic
219C22×C4—21no—
220C22×C4—5381no—
221C22×C4C22×D26911yescharacteristic
222Q16—5382no—
223SD16—5382no—
224SD16—5384no—
225SD16—5382no—
226SD16—5384no—
227D8—5384no—
228D8—5384no—
229D8—5382no—
230D8—5384no—
231D8—5384no—
232M16—2694no—
233C2×C8—2694no—
234C2×C8—2691no—
235C2×C8—2692no—
236C2×C8—2691no—
237C4⋊C4C22⋊D26912yes—
238C22⋊C4—24no—
239C22⋊C4—5384no—
240C22⋊C4C22⋊D26912yes—
241C42C22×D26912yes—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—5382no—
247C23—5381no—
248C23—10762no—
249C23—21522no—
250C23C2×C22⋊D26911yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—5381no—
254Q8—5381no—
255D4—22no—
256D4—21no—
257D4—22no—
258D4—21no—
259D4—42no—
260D4—42no—
261D4—84no—
262D4—5382no—
263D4—5381no—
264D4—5382no—
265D4—5381no—
266D4—10762no—
267D4—10762no—
268D4—21524no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—5381no—
272C2×C4—10762no—
273C2×C4D4×D26912yes—
274C2×C4D4×D26912yes—
275C2×C4C2×C22⋊D26914yes—
276C2×C4C2×C22⋊D26911yescharacteristic
277C2×C4C23×D26911yescharacteristic
278C8—2692no—
279C8—2692no—
280C8—2694no—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—10761no—
288C22—10762no—
289C22—10762no—
290C22—10764no—
291C22—21524no—
292C22#2257544912yes—
293C22C22×C22⋊D26911yescharacteristic
294C4—24no—
295C4—41no—
296C4—10761no—
297C4#2257544912yes—
298C4C2×D4×D26911yescharacteristic
299C4C2×D4×D26911yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—21522no—
304C2—21521no—
305C2C2×#2257544911yescharacteristic
306C2#3177140812yes—
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#535×C269C2
socleC22×C269C22×D4
hypercenter#535D269
hypocenterC269#3155
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#22575479C2semidirect
2. C2×#22575478C2semidirect
3. C2×#22575473C2semidirect
4. C22×#16864387C2semidirect
5. #31771174C2semidirect
6. #31770995C2semidirect
7. #31770992C2semidirect
8. #31770987C2semidirect
9. #31770981C2semidirect
10. #31770561C2semidirect
11. #535×C269C2semidirect
17. C2×#16864392C22semidirect
18. C2×#16864390C22semidirect
19. C2×#16864390C22semidirect
20. C2×#16864389C22semidirect
21. C2×#16864389C22semidirect
25. C2×#16864387C22semidirect
26. C2×#16864387C22semidirect
27. C2×#16864387C22semidirect
30. #22575450C22semidirect
31. #22575441C22semidirect
34. #22575364C22semidirect
35. C4×#14524130C22semidirect
38. #127×C269C22semidirect
39. #124×C269C22semidirect
65. #16864386D4semidirect
66. C2×#14524130D4semidirect
153. C269#3155semidirect
160. #535D269semidirect
187. #127C2×D269semidirect
189. #124C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754792
C2×#225754782
C2×#225754732
C22×#168643872
#317711742
#317709952
#317709922
#317709872
#317709812
#317705612
#535×C2692
#3155269