Finite Group Databasebeta

← #46822653

Subgroups — #46822653

Order 34432.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#31771413C212yesmaximal, complemented
2#31771409C212yesmaximal, complemented
3#31770995C211yesmaximal, characteristic, complemented
4#31770873C212yesmaximal, complemented
5#31770871C212yesmaximal, complemented
6#31770782C211yesmaximal, characteristic, complemented
7#31770774C212yesmaximal, complemented
8#31770724C212yesmaximal, complemented
9#454×C269C211yesmaximal, characteristic, complemented
10#22575499C2212yescomplemented
11#22575498C2211yescharacteristic
12#22575484—22no—
13#22575484C2212yescomplemented
14#22575481C2212yes—
15#22575480—22no—
16#22575478—22no—
17#22575478C2212yescomplemented
18#22575477—22no—
19#22575474—22no—
20#22575473—22no—
21#22575473C2212yes—
22C2×#16864389—22no—
23C2×#16864389C2212yescomplemented
24C2×#16864387C2212yescomplemented
25D8×D269—22no—
26#22575461C2212yescomplemented
27#22575460C2212yescomplemented
28#22575450C2211yescharacteristic, complemented
29#22575388C2212yescomplemented
30#22575381C2212yescomplemented
31#22575370C2211yescharacteristic, complemented
32#22575369C2212yescomplemented
33#22575364C2211yescharacteristic, complemented
342−1+4×C269C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C269C2212yescomplemented
36#124×C269C2211yescharacteristic, complemented
37#104×C269C2212yescomplemented
38#101×C269C2211yescharacteristic, complemented
39#16864403—22no—
40#16864402—22no—
41#16864402C2312yes—
42#16864401—22no—
43C2×D1076—22no—
44C2×D1076—41no—
45C2×D1076C2312yes—
46C2×C22⋊D269—41no—
47D4×D269—22no—
48D4×D269—22no—
49D4×D269—42no—
50#16864392—22no—
51#16864392D412yescomplemented
52#16864391—22no—
53#16864390D412yescomplemented
54#16864389—22no—
55#16864389—22no—
56#16864389—22no—
57#16864389—22no—
58#16864389D412yescomplemented
59D2152—22no—
60#16864387—22no—
61#16864387—22no—
62#16864387—22no—
63#16864387D412yescomplemented
64#16864386—22no—
65#16864386C2312yes—
66#16864385—22no—
67C2×#14524130—22no—
68C2×#14524130C2312yes—
69C8×D269—22no—
70#41×C269—22no—
71#41×C269—41no—
72#41×C269C2312yes—
73C2×Q8×C269—22no—
74C2×Q8×C269C2311yescharacteristic
75C2×D4×C269C2311yescharacteristic
76SD16×C269—22no—
77D8×C269—22no—
78M16×C269C2312yes—
79C22⋊C4×C269—41no—
80C42×C269C2311yescharacteristic
81C22×D269—22no—
82C22×D269—41no—
83C22×D269—42no—
84Dic538C2×D412yes—
85D1076—22no—
86D1076—22no—
87D1076—42no—
88D1076—82no—
89D1076C2×D412yes—
90C22⋊D269—22no—
91C22⋊D269—42no—
92C22⋊D269—42no—
93C2×Dic269—41no—
94C4×D269—22no—
95C4×D269—42no—
96#14524130—22no—
97#14524130C2×D412yes—
98#14524130C2×D412yes—
99C23×C269—21no—
100Q8×C269—22no—
101Q8×C269—41no—
102Q8×C269C2×D412yes—
103D4×C269—22no—
104D4×C269—22no—
105D4×C269—41no—
106D4×C269C2×D412yes—
107C2×C4×C269—22no—
108C2×C4×C269—21no—
109C2×C4×C269—21no—
110C2×C4×C269—42no—
111C2×C4×C269C2411yescharacteristic
112C8×C269—22no—
113C22×C269—22no—
114C22×C269—41no—
115C22×C269—81no—
116C22×C269H(5,𝔽2)11yescharacteristic
117C2×D269—22no—
118C2×D269—42no—
119C2×D269—82no—
120C2×D269—161no—
121C2×D269—162no—
122C4×C269—22no—
123C4×C269—41no—
124C4×C269—41no—
125C4×C269C22×D412yes—
126Dic269—22no—
127C2×C269—21no—
128C2×C269—42no—
129C2×C269—81no—
130C2×C269D4211yescharacteristic
131D269—42no—
132D269—82no—
133C269#328811yescharacteristic, complemented, Sylow
134#3288—2691nomaximal, Sylow
135#577—2692no—
136#576—2692no—
137#496—2691no—
138#472—2692no—
139#471—2692no—
140#455—2691no—
141#454D26911yescharacteristic, complemented
142#453—2692no—
143#444—2692no—
1442−1+4C2×D26911yescharacteristic, complemented
145H(5,𝔽2)—2692no—
146H(5,𝔽2)—2691no—
147#137—2692no—
148#137—5382no—
149AGL(1,ℤ/8)—2692no—
150AGL(1,ℤ/8)—2692no—
151AGL(1,ℤ/8)—5382no—
152AGL(1,ℤ/8)—5382no—
153AGL(1,ℤ/8)—5382no—
154AGL(1,ℤ/8)C2×D26912yescomplemented
155#135—2692no—
156#135—5382no—
157#135—5382no—
158C2×SD16—2692no—
159C2×SD16—5382no—
160C2×D8—2692no—
161C2×D8—5382no—
162#131—2692no—
163#131—2692no—
164#127—2691no—
165#124C2×D26911yescharacteristic, complemented
166#108—2692no—
167#104—2692no—
168#104C2×D26912yescomplemented
169#101C2×D26911yescharacteristic, complemented
170#100—2691no—
171#100—2692no—
172#97—2691no—
173#41—22no—
174#41—41no—
175#41—2692no—
176#41—5382no—
177#41—5382no—
178#41—5382no—
179#41C22×D26912yes—
180C2×Q8—22no—
181C2×Q8C22×D26911yescharacteristic
182C2×D4—2692no—
183C2×D4—5382no—
184C2×D4—5382no—
185C2×D4—5382no—
186C2×D4—10761no—
187C2×D4—10762no—
188C2×D4—10761no—
189C2×D4C22×D26911yescharacteristic
190Q16—2692no—
191Q16—5382no—
192SD16—22no—
193SD16—2692no—
194SD16—2692no—
195SD16—5382no—
196SD16—5382no—
197SD16—5382no—
198SD16—5382no—
199SD16—5382no—
200D8—22no—
201D8—2692no—
202D8—5382no—
203D8—5382no—
204D8—5382no—
205D8—5382no—
206M16—2692no—
207M16—5382no—
208M16—5382no—
209M16C22×D26912yes—
210C2×C8—2692no—
211C2×C8—5382no—
212C2×C8—5382no—
213C22⋊C4—41no—
214C42C22×D26911yescharacteristic
215C23—21no—
216C23—5382no—
217C23—10762no—
218C23—10761no—
219Q8—22no—
220Q8—41no—
221Q8—2692no—
222Q8D4×D26912yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—2692no—
227D4—5382no—
228D4—5382no—
229D4—5382no—
230D4—10762no—
231D4—10762no—
232D4—10762no—
233D4—21522no—
234D4D4×D26912yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—5382no—
240C2×C4—10762no—
241C2×C4—10761no—
242C2×C4C23×D26911yescharacteristic
243C8—22no—
244C8—2692no—
245C8—2692no—
246C8—5382no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—5382no—
251C22—10762no—
252C22—21522no—
253C22—43042no—
254C22—43041no—
255C22#2257549811yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—5382no—
260C4C2×D4×D26912yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—10762no—
265C2—21522no—
266C2#3177123711yescharacteristic
subgroupG/H
centerC2#31771237
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#454×C269C2
socleC2×C269D42
hypercenter#454D269
hypocenterC269#3288
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #31771413C2semidirect
2. #31771409C2semidirect
3. #31770995C2semidirect
4. #31770873C2semidirect
5. #31770871C2semidirect
6. #31770782C2semidirect
7. #31770774C2semidirect
8. #31770724C2semidirect
9. #454×C269C2semidirect
10. #22575499C22semidirect
13. #22575484C22semidirect
17. #22575478C22semidirect
23. C2×#16864389C22semidirect
24. C2×#16864387C22semidirect
26. #22575461C22semidirect
27. #22575460C22semidirect
28. #22575450C22semidirect
29. #22575388C22semidirect
30. #22575381C22semidirect
31. #22575370C22semidirect
32. #22575369C22semidirect
33. #22575364C22semidirect
34. 2−1+4×C269C22semidirect
35. AGL(1,ℤ/8)×C269C22semidirect
36. #124×C269C22semidirect
37. #104×C269C22semidirect
38. #101×C269C22semidirect
51. #16864392D4semidirect
53. #16864390D4semidirect
58. #16864389D4semidirect
63. #16864387D4semidirect
133. C269#3288semidirect
141. #454D269semidirect
144. 2−1+4C2×D269semidirect
154. AGL(1,ℤ/8)C2×D269semidirect
165. #124C2×D269semidirect
168. #104C2×D269semidirect
169. #101C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714132
#317714092
#317709952
#317708732
#317708712
#317707822
#317707742
#317707242
#454×C2692
#3288269