Finite Group Databasebeta

← #46821367

Subgroups — #46821367

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×#22575480C212yesmaximal, complemented
2C2×#22575476C211yesmaximal, characteristic, complemented
3C2×D8×D269C211yesmaximal, characteristic, complemented
4#31771173C212yesmaximal, complemented
5#31770995C212yesmaximal, complemented
6#31770991C212yesmaximal, complemented
7#31770990C211yesmaximal, characteristic, complemented
8#31770982C211yesmaximal, characteristic, complemented
9#31770980C211yesmaximal, characteristic, complemented
10#31770559C211yesmaximal, characteristic, complemented
11#495×C269C211yesmaximal, characteristic, complemented
12C2×#16864403C2212yes—
13C2×D4×D269C2212yes—
14#22575480—28no—
15#22575476—24no—
16C2×#16864389C2212yescomplemented
17C2×#16864389C2212yescomplemented
18C2×D2152C2211yescharacteristic
19C2×D2152C2212yescomplemented
20C2×D2152C2211yescharacteristic, complemented
21C2×#16864387C2212yes—
22C2×#16864387C2212yescomplemented
23D8×D269—24no—
24C2×#16864385C2211yescharacteristic
25C2×C8×D269C2211yescharacteristic
26#22575450C2212yescomplemented
27#22575449C2212yescomplemented
28#22575442C2212yescomplemented
29#22575426—24no—
30#22575423—24no—
31#22575410C2211yescharacteristic
32#22575364C2211yescharacteristic, complemented
33#22575363C2211yescharacteristic, complemented
34C8×Dic269C2211yescharacteristic, complemented
35C2×Q16×C269C2211yescharacteristic, complemented
36C2×SD16×C269C2212yescomplemented
37C2×D8×C269C2211yescharacteristic
38#124×C269C2212yescomplemented
39C4×C8×C269C2211yescharacteristic, complemented
40C23×D269—22no—
41#16864403—24no—
42#16864403—24no—
43C2×D1076—22no—
44C2×D1076—24no—
45C2×D1076C2312yes—
46C2×D1076C2312yes—
47C2×C22⋊D269—22no—
48C2×C22⋊D269—24no—
49D4×D269—24no—
50D4×D269—24no—
51C2×C4×D269—22no—
52C2×C4×D269C2311yescharacteristic
53#16864389—24no—
54#16864389—24no—
55D2152—22no—
56D2152—22no—
57D2152—24no—
58#16864387—24no—
59#16864387—24no—
60#16864385D414yescomplemented
61C2×#14524130C2311yescharacteristic
62C2×#14524130C2311yescharacteristic
63C8×D269D414yescomplemented
64#16864381D412yes—
65#16864379—24no—
66#16864379D412yes—
67C4×Dic269C2311yescharacteristic
68C2×Q8×C269C2312yes—
69C2×D4×C269C2312yes—
70Q16×C269—22no—
71SD16×C269—24no—
72D8×C269—22no—
73C2×C8×C269C2311yescharacteristic
74C2×C8×C269C2311yescharacteristic
75C22⋊C4×C269—24no—
76C42×C269C2311yescharacteristic
77C22×D269—22no—
78C22×D269—22no—
79C22×D269—44no—
80C22×D269—84no—
81C22×D269C2×D411yescharacteristic
82D1076—22no—
83D1076—22no—
84D1076—22no—
85D1076—22no—
86D1076—44no—
87D1076—84no—
88C22⋊D269—44no—
89C22⋊D269—84no—
90C2×Dic269C2×D412yes—
91C2×Dic269C2×D411yescharacteristic
92C4×D269—44no—
93C4×D269C2×D412yes—
94C4×D269C2×D412yes—
95#14524130C2×D412yes—
96#14524130C2×D412yes—
97C23×C269—22no—
98Q8×C269—22no—
99Q8×C269—22no—
100D4×C269—22no—
101D4×C269—22no—
102C2×C4×C269—22no—
103C2×C4×C269C2×D412yes—
104C2×C4×C269C2411yescharacteristic
105C8×C269C2×D412yes—
106C8×C269C2×D412yes—
107C22×C269—42no—
108C22×C269—44no—
109C22×C269C22×D411yescharacteristic
110C2×D269—22no—
111C2×D269—42no—
112C2×D269—42no—
113C2×D269—44no—
114C2×D269—44no—
115C2×D269—88no—
116C2×D269#12712yes—
117C4×C269—22no—
118C4×C269—42no—
119C4×C269C22×D411yescharacteristic
120C4×C269C22×D411yescharacteristic
121Dic269—22no—
122Dic269#12712yes—
123C2×C269—82no—
124C2×C269C2×#12711yescharacteristic
125C2×C269#57612yes—
126D269—22no—
127D269—82no—
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#495D26911yescharacteristic, complemented
139#493—2691no—
140#493—2691no—
141#405—2691no—
142C2×#41—2692no—
143C22×D4—2692no—
144AGL(1,ℤ/8)—5388no—
145#135—5384no—
146C2×Q16C2×D26911yescharacteristic, complemented
147C2×SD16—2692no—
148C2×SD16—2692no—
149C2×SD16C2×D26912yescomplemented
150C2×D8—2692no—
151C2×D8—2692no—
152C2×D8—2691no—
153C2×D8—2691no—
154C2×D8—2692no—
155C2×D8—5384no—
156C2×D8C2×D26911yescharacteristic
157C2×M16—2691no—
158C22×C8—2691no—
159#127—2692no—
160#127—2692no—
161#124—2692no—
162#124C2×D26912yescomplemented
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×Q8C22×D26912yes—
174C2×D4—2692no—
175C2×D4—2692no—
176C2×D4—5382no—
177C2×D4—5384no—
178C2×D4—5384no—
179C2×D4—5382no—
180C2×D4—5384no—
181C2×D4—5384no—
182C2×D4C22×D26912yes—
183C22×C4—2691no—
184C22×C4—5382no—
185Q16—22no—
186SD16—24no—
187SD16—5384no—
188SD16—5384no—
189D8—22no—
190D8—5384no—
191D8—5384no—
192D8—5382no—
193D8—5382no—
194D8—5384no—
195M16—2694no—
196C2×C8—2691no—
197C2×C8—2691no—
198C2×C8—2694no—
199C2×C8C22×D26911yescharacteristic
200C2×C8C22×D26911yescharacteristic
201C4⋊C4—2692no—
202C22⋊C4—24no—
203C22⋊C4—2692no—
204C22⋊C4—5384no—
205C42—2691no—
206C42C22×D26911yescharacteristic
207C23—22no—
208C23—2691no—
209C23—5382no—
210C23—5382no—
211C23—10764no—
212C23—21524no—
213Q8—22no—
214Q8—22no—
215D4—22no—
216D4—22no—
217D4—5382no—
218D4—5382no—
219D4—5382no—
220D4—5382no—
221D4—10764no—
222D4—10764no—
223D4—21524no—
224D4—21524no—
225C2×C4—22no—
226C2×C4—2691no—
227C2×C4—2692no—
228C2×C4—2692no—
229C2×C4—2692no—
230C2×C4—10764no—
231C2×C4D4×D26912yes—
232C2×C4C23×D26911yescharacteristic
233C8—2692no—
234C8—2692no—
235C8D4×D26912yes—
236C8D4×D26912yes—
237C22—42no—
238C22—44no—
239C22—2692no—
240C22—5382no—
241C22—10762no—
242C22—10762no—
243C22—10764no—
244C22—10764no—
245C22—21528no—
246C22C2×D4×D26911yescharacteristic
247C4—22no—
248C4—42no—
249C4—2692no—
250C4—5382no—
251C4C2×D4×D26911yescharacteristic
252C4C2×D4×D26911yescharacteristic
253C2—82no—
254C2—5382no—
255C2—21522no—
256C2—21522no—
257C2#127×D26911yescharacteristic
258C2#3177140712yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#495×C269C2
socleC22×C269C22×D4
hypercenter#495D269
hypocenterC269#3155
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#22575480C2semidirect
2. C2×#22575476C2semidirect
3. C2×D8×D269C2semidirect
4. #31771173C2semidirect
5. #31770995C2semidirect
6. #31770991C2semidirect
7. #31770990C2semidirect
8. #31770982C2semidirect
9. #31770980C2semidirect
10. #31770559C2semidirect
11. #495×C269C2semidirect
16. C2×#16864389C22semidirect
17. C2×#16864389C22semidirect
19. C2×D2152C22semidirect
20. C2×D2152C22semidirect
22. C2×#16864387C22semidirect
26. #22575450C22semidirect
27. #22575449C22semidirect
28. #22575442C22semidirect
32. #22575364C22semidirect
33. #22575363C22semidirect
34. C8×Dic269C22semidirect
35. C2×Q16×C269C22semidirect
36. C2×SD16×C269C22semidirect
38. #124×C269C22semidirect
39. C4×C8×C269C22semidirect
60. #16864385D4semidirect
63. C8×D269D4semidirect
129. C269#3155semidirect
138. #495D269semidirect
146. C2×Q16C2×D269semidirect
149. C2×SD16C2×D269semidirect
162. #124C2×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×#225754802
C2×#225754762
C2×D8×D2692
#317711732
#317709952
#317709912
#317709902
#317709822
#317709802
#317705592
#495×C2692
#3155269