Finite Group Databasebeta

← #46821365

Subgroups — #46821365

Order 34432.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#22575477C212yesmaximal, complemented
2C2×#22575472C211yesmaximal, characteristic, complemented
3C2×D8×D269C211yesmaximal, characteristic, complemented
4#31771168C212yesmaximal, complemented
5#31770993C212yesmaximal, complemented
6#31770992C212yesmaximal, complemented
7#31770989C211yesmaximal, characteristic, complemented
8#31770988C211yesmaximal, characteristic, complemented
9#31770980C211yesmaximal, characteristic, complemented
10#31770559C211yesmaximal, characteristic, complemented
11#493×C269C211yesmaximal, characteristic, complemented
12C2×#16864401C2212yes—
13C2×D4×D269C2212yes—
14#22575477—28no—
15#22575472—24no—
16C2×Dic1076C2211yescharacteristic, complemented
17C2×#16864391C2212yescomplemented
18C2×#16864390C2212yescomplemented
19C2×#16864390C2212yescomplemented
20C2×D2152C2211yescharacteristic
21C2×#16864387C2212yes—
22C2×#16864387C2212yescomplemented
23D8×D269—24no—
24C2×#16864385C2211yescharacteristic
25C2×C8×D269C2211yescharacteristic
26#22575449C2212yescomplemented
27#22575441C2212yescomplemented
28#22575440C2212yescomplemented
29#22575425—24no—
30#22575421—24no—
31#22575410C2211yescharacteristic
32#22575364C2211yescharacteristic, complemented
33#22575363C2211yescharacteristic, complemented
34C8×Dic269C2211yescharacteristic, complemented
35C2×D8×C269C2211yescharacteristic
36C2×D8×C269C2211yescharacteristic, complemented
37C2×D8×C269C2212yescomplemented
38#127×C269C2212yescomplemented
39C4×C8×C269C2211yescharacteristic, complemented
40C23×D269—22no—
41#16864401—24no—
42#16864401—24no—
43C2×Dic538C2312yes—
44C2×D1076C2312yes—
45C2×C22⋊D269—22no—
46C2×C22⋊D269—22no—
47C2×C22⋊D269—24no—
48D4×D269—24no—
49D4×D269—24no—
50C22×Dic269—22no—
51C2×C4×D269C2311yescharacteristic
52Dic1076—22no—
53#16864391—24no—
54#16864390—24no—
55#16864390—24no—
56D2152—22no—
57#16864387—24no—
58#16864387—24no—
59#16864385D414yescomplemented
60C2×#14524130C2311yescharacteristic
61C2×#14524130C2311yescharacteristic
62C8×D269D414yescomplemented
63#16864381D412yes—
64C22⋊Dic269—24no—
65#16864379—24no—
66#16864379D412yes—
67C4×Dic269C2311yescharacteristic
68C2×D4×C269—24no—
69C2×D4×C269C2312yes—
70C2×D4×C269C2312yes—
71D8×C269—22no—
72D8×C269—22no—
73D8×C269—24no—
74C2×C8×C269C2311yescharacteristic
75C2×C8×C269C2311yescharacteristic
76C42×C269C2311yescharacteristic
77C22×D269—22no—
78C22×D269—44no—
79C22×D269—84no—
80C22×D269C2×D411yescharacteristic
81Dic538—22no—
82Dic538—22no—
83D1076—22no—
84D1076—22no—
85C22⋊D269—44no—
86C22⋊D269—44no—
87C22⋊D269—84no—
88C2×Dic269—22no—
89C2×Dic269—44no—
90C2×Dic269C2×D412yes—
91C2×Dic269C2×D411yescharacteristic
92C4×D269C2×D412yes—
93C4×D269C2×D412yes—
94#14524130C2×D412yes—
95#14524130C2×D412yes—
96C23×C269—22no—
97C23×C269—22no—
98D4×C269—22no—
99D4×C269—22no—
100D4×C269—22no—
101D4×C269—22no—
102D4×C269—84no—
103C2×C4×C269C2×D412yes—
104C2×C4×C269C2411yescharacteristic
105C8×C269C2×D412yes—
106C8×C269C2×D412yes—
107C22×C269—42no—
108C22×C269—42no—
109C22×C269—44no—
110C22×C269—44no—
111C22×C269C22×D411yescharacteristic
112C2×D269—22no—
113C2×D269—42no—
114C2×D269—44no—
115C2×D269—88no—
116C2×D269#12712yes—
117C4×C269—22no—
118C4×C269C22×D411yescharacteristic
119C4×C269C22×D411yescharacteristic
120Dic269—22no—
121Dic269—42no—
122Dic269#12712yes—
123C2×C269—82no—
124C2×C269—82no—
125C2×C269C2×#12711yescharacteristic
126C2×C269#57612yes—
127D269—22no—
128D269—82no—
129C269#315511yescharacteristic, complemented, Sylow
130#3155—2691nomaximal, Sylow
131C2×AGL(1,ℤ/8)—2692no—
132C2×#135—2691no—
133C22×D8—2691no—
134#535—2692no—
135#496—2692no—
136#496—2692no—
137#495—2691no—
138#495—2691no—
139#493—2691no—
140#493D26911yescharacteristic, complemented
141#405—2691no—
142C2×#41—2692no—
143C22×D4—2692no—
144AGL(1,ℤ/8)—5388no—
145#135—5384no—
146C2×Q16—2691no—
147C2×SD16—2692no—
148C2×SD16—2692no—
149C2×SD16—2692no—
150C2×D8—2692no—
151C2×D8—2692no—
152C2×D8—2691no—
153C2×D8—5384no—
154C2×D8C2×D26911yescharacteristic
155C2×D8C2×D26911yescharacteristic, complemented
156C2×D8C2×D26912yescomplemented
157C2×M16—2691no—
158C22×C8—2691no—
159#127—2692no—
160#127C2×D26912yescomplemented
161#124—2692no—
162#124—2692no—
163#121—5384no—
164#120—5384no—
165#117—2691no—
166#97—2691no—
167#97—2691no—
168C4×C8—2691no—
169C4×C8C2×D26911yescharacteristic, complemented
170C24—5382no—
171#41—5384no—
172#41—5384no—
173C2×Q8—2692no—
174C2×D4—24no—
175C2×D4—2692no—
176C2×D4—5384no—
177C2×D4—5384no—
178C2×D4—5382no—
179C2×D4—5384no—
180C2×D4—5382no—
181C2×D4C22×D26912yes—
182C2×D4C22×D26912yes—
183C22×C4—2691no—
184C22×C4—5382no—
185Q16—5382no—
186SD16—5384no—
187SD16—5384no—
188SD16—5384no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—5384no—
193D8—5384no—
194D8—5382no—
195M16—2694no—
196C2×C8—2691no—
197C2×C8—2691no—
198C2×C8—2694no—
199C2×C8C22×D26911yescharacteristic
200C2×C8C22×D26911yescharacteristic
201C4⋊C4—2692no—
202C22⋊C4—2692no—
203C22⋊C4—5384no—
204C22⋊C4—5384no—
205C42—2691no—
206C42C22×D26911yescharacteristic
207C23—22no—
208C23—22no—
209C23—2691no—
210C23—5382no—
211C23—10764no—
212C23—21524no—
213Q8—5382no—
214Q8—5382no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—84no—
220D4—5382no—
221D4—5382no—
222D4—10764no—
223D4—10764no—
224D4—21524no—
225C2×C4—2691no—
226C2×C4—2692no—
227C2×C4—2692no—
228C2×C4—2692no—
229C2×C4—5382no—
230C2×C4—10764no—
231C2×C4D4×D26912yes—
232C2×C4C23×D26911yescharacteristic
233C8—2692no—
234C8—2692no—
235C8D4×D26912yes—
236C8D4×D26912yes—
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—2692no—
242C22—5382no—
243C22—10762no—
244C22—10764no—
245C22—21528no—
246C22C2×D4×D26911yescharacteristic
247C4—22no—
248C4—2692no—
249C4—5382no—
250C4—10762no—
251C4C2×D4×D26911yescharacteristic
252C4C2×D4×D26911yescharacteristic
253C2—82no—
254C2—82no—
255C2—5382no—
256C2—21522no—
257C2#127×D26911yescharacteristic
258C2#3177140612yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#493×C269C2
socleC22×C269C22×D4
hypercenter#493D269
hypocenterC269#3155
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#22575477C2semidirect
2. C2×#22575472C2semidirect
3. C2×D8×D269C2semidirect
4. #31771168C2semidirect
5. #31770993C2semidirect
6. #31770992C2semidirect
7. #31770989C2semidirect
8. #31770988C2semidirect
9. #31770980C2semidirect
10. #31770559C2semidirect
11. #493×C269C2semidirect
16. C2×Dic1076C22semidirect
17. C2×#16864391C22semidirect
18. C2×#16864390C22semidirect
19. C2×#16864390C22semidirect
22. C2×#16864387C22semidirect
26. #22575449C22semidirect
27. #22575441C22semidirect
28. #22575440C22semidirect
32. #22575364C22semidirect
33. #22575363C22semidirect
34. C8×Dic269C22semidirect
36. C2×D8×C269C22semidirect
37. C2×D8×C269C22semidirect
38. #127×C269C22semidirect
39. C4×C8×C269C22semidirect
59. #16864385D4semidirect
62. C8×D269D4semidirect
129. C269#3155semidirect
140. #493D269semidirect
155. C2×D8C2×D269semidirect
156. C2×D8C2×D269semidirect
160. #127C2×D269semidirect
169. C4×C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754772
C2×#225754722
C2×D8×D2692
#317711682
#317709932
#317709922
#317709892
#317709882
#317709802
#317705592
#493×C2692
#3155269