Finite Group Databasebeta

← #46821334

Subgroups — #46821334

Order 34432.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22575477C212yesmaximal, complemented
3#31771154C211yesmaximal, characteristic, complemented
4#127×D269C211yesmaximal, characteristic, complemented
5#31770995C211yesmaximal, characteristic, complemented
6#31770994C211yesmaximal, characteristic, complemented
7#31770992C211yesmaximal, characteristic, complemented
8#31770980C212yesmaximal, complemented
9#31770976C212yesmaximal, complemented
10#31770557C211yesmaximal, characteristic, complemented
11#496×C269C211yesmaximal, characteristic, complemented
12C2×#16864403C2211yescharacteristic
13C2×#16864401C2211yescharacteristic
14C2×D4×D269—22no—
15C2×D4×D269C2212yes—
16#22575480—28no—
17#22575477—28no—
18C2×#16864391C2212yescomplemented
19C2×#16864390C2212yescomplemented
20C2×#16864389C2212yescomplemented
21C2×D2152C2212yescomplemented
22C2×#16864387C2212yescomplemented
23C2×#16864387C2212yescomplemented
24C2×#16864385C2212yes—
25#22575451C2211yescharacteristic, complemented
26#22575450C2211yescharacteristic
27#22575449—22no—
28#22575449C2212yes—
29#22575449C2211yescharacteristic, complemented
30#22575441C2211yescharacteristic, complemented
31#22575426—22no—
32#22575413—22no—
33C42×D269C2211yescharacteristic
34#22575364C2211yescharacteristic, complemented
35C8×Dic269C2212yescomplemented
36C2×SD16×C269C2212yescomplemented
37C2×D8×C269C2212yescomplemented
38#127×C269C2211yescharacteristic
39#124×C269C2211yescharacteristic, complemented
40#97×C269C2211yescharacteristic, complemented
41C23×D269—22no—
42#16864403—24no—
43#16864401—24no—
44C2×Dic538C2311yescharacteristic
45C2×D1076—21no—
46C2×D1076—22no—
47C2×D1076C2311yescharacteristic
48C2×D1076C2312yes—
49C2×C22⋊D269—21no—
50C2×C22⋊D269—22no—
51C2×C22⋊D269—22no—
52C2×C22⋊D269—24no—
53D4×D269—24no—
54D4×D269—24no—
55D4×D269—48no—
56C22×Dic269—21no—
57C2×C4×D269—21no—
58C2×C4×D269D412yes—
59C2×C4×D269C2311yescharacteristic
60#16864391—24no—
61#16864390—24no—
62#16864389—24no—
63D2152—24no—
64#16864387—24no—
65#16864387—24no—
66#16864385D418yescomplemented
67C2×#14524130C2312yes—
68#16864381—22no—
69#16864379—22no—
70C4×Dic269D412yes—
71C4×Dic269C2311yescharacteristic
72C2×Q8×C269C2311yescharacteristic
73C2×D4×C269—22no—
74C2×D4×C269C2312yes—
75C2×D4×C269C2311yescharacteristic
76SD16×C269—24no—
77D8×C269—24no—
78C2×C8×C269C2312yes—
79C22⋊C4×C269—22no—
80C42×C269C2311yescharacteristic
81C22×D269—21no—
82C22×D269—22no—
83C22×D269—48no—
84C22×D269—44no—
85C22×D269C2×D411yescharacteristic
86Dic538—22no—
87D1076—22no—
88D1076—22no—
89D1076—22no—
90D1076—44no—
91D1076—42no—
92C22⋊D269—42no—
93C22⋊D269—44no—
94C22⋊D269—48no—
95C22⋊D269—44no—
96C2×Dic269—21no—
97C2×Dic269—42no—
98C2×Dic269C2×D411yescharacteristic
99C2×Dic269C2×D412yes—
100C4×D269—24no—
101C4×D269—42no—
102C4×D269C2×D414yes—
103#14524130C2×D414yescomplemented
104C23×C269—21no—
105C23×C269—22no—
106Q8×C269—22no—
107D4×C269—22no—
108D4×C269—22no—
109D4×C269—22no—
110D4×C269—44no—
111C2×C4×C269—21no—
112C2×C4×C269C2×D412yes—
113C2×C4×C269C2411yescharacteristic
114C8×C269C2×D414yescomplemented
115C22×C269—41no—
116C22×C269—42no—
117C22×C269—44no—
118C22×C269—42no—
119C22×C269C22×D411yescharacteristic
120C2×D269—22no—
121C2×D269—41no—
122C2×D269—42no—
123C2×D269—44no—
124C2×D269—42no—
125C2×D269—88no—
126C2×D269#12712yes—
127C4×C269—22no—
128C4×C269—41no—
129C4×C269C22×D412yes—
130Dic269—41no—
131Dic269#12712yes—
132Dic269AGL(1,ℤ/8)14yescomplemented
133C2×C269—82no—
134C2×C269—81no—
135C2×C269C2×#12711yescharacteristic
136C2×C269C2×AGL(1,ℤ/8)12yes—
137D269—22no—
138D269—81no—
139D269—82no—
140C269#315111yescharacteristic, complemented, Sylow
141#3151—2691nomaximal, Sylow
142C2×AGL(1,ℤ/8)—2692no—
143C2×AGL(1,ℤ/8)—2692no—
144#532—2691no—
145C2×#127—2691no—
146#496—2691no—
147#496—2691no—
148#496—2691no—
149#496D26911yescharacteristic, complemented
150#493—2692no—
151#492—2692no—
152C4×M16—2691no—
153C2×#41—2691no—
154C2×#41—2691no—
155C22×D4—2692no—
156C22×D4—5382no—
157AGL(1,ℤ/8)—5388no—
158AGL(1,ℤ/8)—5388no—
159C2×SD16—2692no—
160C2×SD16—2692no—
161C2×SD16—2692no—
162C2×SD16C2×D26912yescomplemented
163C2×D8—2692no—
164C2×D8—2692no—
165C2×D8—2692no—
166C2×D8C2×D26912yescomplemented
167C2×M16—2692no—
168C4⋊Q8—2691no—
169#127—2692no—
170#127—2691no—
171#127—2691no—
172#127—5382no—
173#127C2×D26911yescharacteristic
174#124—2691no—
175#124C2×D26911yescharacteristic, complemented
176#121—5382no—
177C4×D4—5382no—
178C2×C42—2691no—
179#97—2691no—
180#97C2×D26911yescharacteristic, complemented
181C4×C8—2692no—
182C24—5382no—
183#41—5384no—
184#41—5384no—
185C2×Q8—2691no—
186C2×Q8C22×D26911yescharacteristic
187C2×D4—22no—
188C2×D4—2691no—
189C2×D4—2692no—
190C2×D4—5381no—
191C2×D4—5384no—
192C2×D4—5384no—
193C2×D4—5382no—
194C2×D4—5384no—
195C2×D4—5382no—
196C2×D4—5381no—
197C2×D4—5382no—
198C2×D4—10768no—
199C2×D4C22×D26911yescharacteristic
200C2×D4C22×D26912yes—
201C22×C4—2691no—
202C22×C4—2692no—
203C22×C4—5381no—
204C22×C4—5381no—
205SD16—24no—
206SD16—5384no—
207SD16—5384no—
208SD16—5384no—
209D8—24no—
210D8—5384no—
211D8—5384no—
212D8—5384no—
213M16—2698no—
214C2×C8—2692no—
215C2×C8C22×D26912yes—
216C4⋊C4—5382no—
217C22⋊C4—22no—
218C22⋊C4—5382no—
219C42—2692no—
220C42—2691no—
221C42C22×D26911yescharacteristic
222C23—21no—
223C23—22no—
224C23—2691no—
225C23—5381no—
226C23—5382no—
227C23—10764no—
228C23—10768no—
229Q8—22no—
230Q8—5382no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—5382no—
236D4—5382no—
237D4—5382no—
238D4—10764no—
239D4—10762no—
240D4—10764no—
241D4—10768no—
242D4—10764no—
243D4—10762no—
244C2×C4—21no—
245C2×C4—2692no—
246C2×C4—2691no—
247C2×C4—2694no—
248C2×C4—5384no—
249C2×C4—5381no—
250C2×C4—10762no—
251C2×C4—10762no—
252C2×C4D4×D26912yes—
253C2×C4C23×D26911yescharacteristic
254C8—2694no—
255C8D4×D26914yescomplemented
256C22—41no—
257C22—42no—
258C22—44no—
259C22—42no—
260C22—2692no—
261C22—5382no—
262C22—10761no—
263C22—10762no—
264C22—10764no—
265C22—10762no—
266C22—21528no—
267C22C2×D4×D26911yescharacteristic
268C4—22no—
269C4—41no—
270C4—2692no—
271C4—2694no—
272C4—10761no—
273C4C2×D4×D26912yes—
274C2—82no—
275C2—81no—
276C2—5382no—
277C2—21522no—
278C2—21521no—
279C2#127×D26911yescharacteristic
280C2AGL(1,ℤ/8)×D26912yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#496×C269C2
socleC22×C269C22×D4
hypercenter#496D269
hypocenterC269#3151
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22575480C2semidirect
2. C2×#22575477C2semidirect
3. #31771154C2semidirect
4. #127×D269C2semidirect
5. #31770995C2semidirect
6. #31770994C2semidirect
7. #31770992C2semidirect
8. #31770980C2semidirect
9. #31770976C2semidirect
10. #31770557C2semidirect
11. #496×C269C2semidirect
18. C2×#16864391C22semidirect
19. C2×#16864390C22semidirect
20. C2×#16864389C22semidirect
21. C2×D2152C22semidirect
22. C2×#16864387C22semidirect
23. C2×#16864387C22semidirect
25. #22575451C22semidirect
29. #22575449C22semidirect
30. #22575441C22semidirect
34. #22575364C22semidirect
35. C8×Dic269C22semidirect
36. C2×SD16×C269C22semidirect
37. C2×D8×C269C22semidirect
39. #124×C269C22semidirect
40. #97×C269C22semidirect
66. #16864385D4semidirect
103. #14524130C2×D4semidirect
114. C8×C269C2×D4semidirect
132. Dic269AGL(1,ℤ/8)semidirect
140. C269#3151semidirect
149. #496D269semidirect
162. C2×SD16C2×D269semidirect
166. C2×D8C2×D269semidirect
175. #124C2×D269semidirect
180. #97C2×D269semidirect
255. C8D4×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754802
C2×#225754772
#317711542
#127×D2692
#317709952
#317709942
#317709922
#317709802
#317709762
#317705572
#496×C2692
#3151269