Finite Group Databasebeta

← #46822675

Subgroups — #46822675

Order 34432.

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

normal subgroups only

This group has 460 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#31771414C212yesmaximal, complemented
2#31771413C212yesmaximal, complemented
3#31770979C211yesmaximal, characteristic, complemented
4#31770871C212yesmaximal, complemented
5#31770870C212yesmaximal, complemented
6#31770787C211yesmaximal, characteristic, complemented
7#31770774C212yesmaximal, complemented
8#31770719C212yesmaximal, complemented
9#456×C269C211yesmaximal, characteristic, complemented
10#22575501C2211yescharacteristic
11#22575499C2212yescomplemented
12#22575485—22no—
13#22575484—22no—
14#22575484C2212yescomplemented
15#22575481C2212yes—
16#22575480—22no—
17#22575478—22no—
18#22575478C2212yes—
19#22575476—22no—
20#22575473—22no—
21#22575473C2212yes—
22C2×#16864389—22no—
23C2×#16864389C2212yescomplemented
24C2×#16864389C2212yescomplemented
25SD16×D269—22no—
26#22575461C2212yescomplemented
27#22575460C2212yescomplemented
28#22575450C2211yescharacteristic
29#22575388C2212yescomplemented
30#22575381C2212yescomplemented
31#22575372C2211yescharacteristic, complemented
32#22575369C2212yescomplemented
33C4×#14524130C2211yescharacteristic, complemented
342−1+4×C269C2211yescharacteristic
35#137×C269C2212yescomplemented
36C4⋊Q8×C269C2211yescharacteristic, complemented
37#104×C269C2212yescomplemented
38#101×C269C2211yescharacteristic, complemented
39#16864403—22no—
40#16864403—22no—
41#16864402—22no—
42#16864402—41no—
43#16864402C2312yes—
44Q8×D269—22no—
45C2×D1076—22no—
46C2×D1076—41no—
47C2×D1076C2312yes—
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#16864389—22no—
59#16864389—22no—
60#16864389D412yescomplemented
61D2152—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×Q8×C269C2311yescharacteristic
76Q16×C269—22no—
77SD16×C269—22no—
78M16×C269C2312yes—
79C4⋊C4×C269—41no—
80C42×C269C2311yescharacteristic
81C22×D269—22no—
82C22×D269—42no—
83Dic538—41no—
84Dic538C2×D412yes—
85D1076—22no—
86D1076—22no—
87D1076—42no—
88D1076—41no—
89D1076—82no—
90D1076C2×D412yes—
91C22⋊D269—22no—
92C22⋊D269—42no—
93C4×D269—22no—
94C4×D269—42no—
95C4×D269—42no—
96#14524130—22no—
97#14524130C2×D412yes—
98#14524130C2×D412yes—
99Q8×C269—22no—
100Q8×C269—22no—
101Q8×C269—41no—
102Q8×C269C2×D412yes—
103D4×C269—22no—
104D4×C269—41no—
105D4×C269C2×D412yes—
106C2×C4×C269—21no—
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×C269H(5,𝔽2)11yescharacteristic
115C2×D269—22no—
116C2×D269—42no—
117C2×D269—82no—
118C2×D269—162no—
119C4×C269—22no—
120C4×C269—22no—
121C4×C269—41no—
122C4×C269—41no—
123C4×C269C22×D412yes—
124Dic269—22no—
125C2×C269—21no—
126C2×C269—42no—
127C2×C269D4211yescharacteristic
128D269—42no—
129D269—82no—
130C269#329111yescharacteristic, complemented, Sylow
131#3291—2691nomaximal, Sylow
132#577—2692no—
133#577—2692no—
134#492—2691no—
135#471—2692no—
136#471—2692no—
137#456—2691no—
138#456D26911yescharacteristic, complemented
139#453—2692no—
140#443—2692no—
1412−1+4—2691no—
1422−1+4C2×D26911yescharacteristic
143H(5,𝔽2)—2692no—
144#137—2692no—
145#137—5382no—
146#137—5382no—
147#137C2×D26912yescomplemented
148AGL(1,ℤ/8)—2692no—
149AGL(1,ℤ/8)—2692no—
150AGL(1,ℤ/8)—5382no—
151AGL(1,ℤ/8)—5382no—
152#135—2692no—
153#135—5382no—
154#135—5382no—
155C2×SD16—2692no—
156C2×SD16—2692no—
157C2×SD16—5382no—
158C2×SD16—5382no—
159#131—2692no—
160#131—2692no—
161C4⋊Q8C2×D26911yescharacteristic, complemented
162#127—2691no—
163#108—2692no—
164#104—2692no—
165#104C2×D26912yescomplemented
166#101—2691no—
167#101C2×D26911yescharacteristic, complemented
168#100—2692no—
169C4×C8—2691no—
170#41—22no—
171#41—41no—
172#41—2692no—
173#41—5382no—
174#41—5382no—
175#41—5382no—
176#41—10761no—
177#41C22×D26912yes—
178C2×Q8—22no—
179C2×Q8—5382no—
180C2×Q8C22×D26911yescharacteristic
181C2×Q8C22×D26911yescharacteristic
182C2×D4—2692no—
183C2×D4—5382no—
184C2×D4—5382no—
185C2×D4—10761no—
186C2×D4—10762no—
187Q16—22no—
188Q16—2692no—
189Q16—5382no—
190SD16—22no—
191SD16—2692no—
192SD16—2692no—
193SD16—5382no—
194SD16—5382no—
195SD16—5382no—
196SD16—5382no—
197SD16—5382no—
198SD16—5382no—
199SD16—5382no—
200D8—2692no—
201D8—5382no—
202D8—5382no—
203M16—2692no—
204M16—5382no—
205M16—5382no—
206M16C22×D26912yes—
207C2×C8—2692no—
208C2×C8—5382no—
209C2×C8—5382no—
210C4⋊C4—41no—
211C42C22×D26911yescharacteristic
212C23—5382no—
213C23—10762no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—2692no—
218Q8—10761no—
219Q8D4×D26912yes—
220D4—22no—
221D4—41no—
222D4—2692no—
223D4—5382no—
224D4—5382no—
225D4—5382no—
226D4—10762no—
227D4—10761no—
228D4—10762no—
229D4—21522no—
230D4D4×D26912yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—5382no—
237C2×C4—10762no—
238C2×C4—10762no—
239C2×C4C23×D26911yescharacteristic
240C8—22no—
241C8—2692no—
242C8—2692no—
243C8—5382no—
244C22—22no—
245C22—5382no—
246C22—10762no—
247C22—21522no—
248C22—43042no—
249C22#2257549811yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—5382no—
255C4C2×D4×D26912yes—
256C2—21no—
257C2—42no—
258C2—10762no—
259C2—21522no—
260C2#3177123711yescharacteristic
subgroupG/H
centerC2#31771237
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#456×C269C2
socleC2×C269D42
hypercenter#456D269
hypocenterC269#3291
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #31771414C2semidirect
2. #31771413C2semidirect
3. #31770979C2semidirect
4. #31770871C2semidirect
5. #31770870C2semidirect
6. #31770787C2semidirect
7. #31770774C2semidirect
8. #31770719C2semidirect
9. #456×C269C2semidirect
11. #22575499C22semidirect
14. #22575484C22semidirect
23. C2×#16864389C22semidirect
24. C2×#16864389C22semidirect
26. #22575461C22semidirect
27. #22575460C22semidirect
29. #22575388C22semidirect
30. #22575381C22semidirect
31. #22575372C22semidirect
32. #22575369C22semidirect
33. C4×#14524130C22semidirect
35. #137×C269C22semidirect
36. C4⋊Q8×C269C22semidirect
37. #104×C269C22semidirect
38. #101×C269C22semidirect
51. #16864392D4semidirect
53. #16864390D4semidirect
60. #16864389D4semidirect
63. #16864387D4semidirect
130. C269#3291semidirect
138. #456D269semidirect
147. #137C2×D269semidirect
161. C4⋊Q8C2×D269semidirect
165. #104C2×D269semidirect
167. #101C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714142
#317714132
#317709792
#317708712
#317708702
#317707872
#317707742
#317707192
#456×C2692
#3291269