Finite Group Databasebeta

← #46822679

Subgroups — #46822679

Order 34432.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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#31771417C212yesmaximal, complemented
2#31771412C212yesmaximal, complemented
3#31770978C211yesmaximal, characteristic, complemented
4#31770869C212yesmaximal, complemented
5#31770868C212yesmaximal, complemented
6#31770789C212yesmaximal, complemented
7#31770774C212yesmaximal, complemented
8#31770723C211yesmaximal, characteristic, complemented
9#443×C269C211yesmaximal, characteristic, complemented
10#22575500C2212yes—
11#22575499C2212yescomplemented
12#22575486—22no—
13#22575486C2212yescomplemented
14#22575485—22no—
15#22575483C2212yescomplemented
16#22575481—22no—
17#22575481C2212yes—
18#22575478C2212yes—
19#22575477—22no—
20#22575475—22no—
21#22575475C2211yescharacteristic
22#22575474—22no—
23C2×#16864391—22no—
24C2×#16864391C2212yescomplemented
25C2×#16864391C2212yescomplemented
26SD16×D269—22no—
27#22575459C2212yescomplemented
28#22575453C2211yescharacteristic, complemented
29#22575450C2211yescharacteristic
30#22575389C2211yescharacteristic, complemented
31#22575381C2212yescomplemented
32#22575371C2212yescomplemented
33#22575369C2212yescomplemented
34#135×C269C2211yescharacteristic
35#131×C269C2212yescomplemented
36#108×C269C2211yescharacteristic, complemented
37#104×C269C2212yescomplemented
38C4×C8×C269C2211yescharacteristic, complemented
39#16864403—22no—
40#16864402—22no—
41#16864402—22no—
42#16864402C2312yes—
43#16864401—22no—
44#16864401—42no—
45C2×Dic538—22no—
46C2×Dic538C2312yes—
47Q8×D269—22no—
48C2×D1076—22no—
49C2×D1076—41no—
50C2×D1076C2312yes—
51D4×D269—22no—
52D4×D269—42no—
53Dic1076—22no—
54Dic1076D411yescharacteristic, complemented
55#16864392—22no—
56#16864391—22no—
57#16864391—22no—
58#16864391—22no—
59#16864391—22no—
60#16864391—22no—
61#16864391—22no—
62#16864391D412yescomplemented
63#16864390—22no—
64#16864389—22no—
65D2152—22no—
66D2152D411yescharacteristic, complemented
67#16864387—22no—
68#16864386C2312yes—
69#16864385—22no—
70C8×D269—22no—
71C4⋊Dic269—41no—
72#41×C269C2312yes—
73Q16×C269D411yescharacteristic, complemented
74SD16×C269D412yescomplemented
75D8×C269D411yescharacteristic, complemented
76M16×C269—22no—
77M16×C269C2312yes—
78C2×C8×C269—22no—
79C2×C8×C269C2311yescharacteristic
80C2×C8×C269C2311yescharacteristic
81C42×C269C2311yescharacteristic
82C22×D269—22no—
83C22×D269—42no—
84Dic538—22no—
85Dic538—22no—
86Dic538—42no—
87Dic538C2×D412yes—
88D1076—22no—
89D1076—22no—
90D1076—42no—
91D1076—82no—
92D1076C2×D412yes—
93C22⋊D269—22no—
94C22⋊D269—42no—
95C22⋊D269—42no—
96C2×Dic269—22no—
97C2×Dic269—42no—
98C4×D269—22no—
99C4×D269—42no—
100C4×D269—42no—
101#14524130—22no—
102Q8×C269C2×D412yes—
103D4×C269—22no—
104D4×C269C2×D412yes—
105C2×C4×C269—21no—
106C2×C4×C269—22no—
107C2×C4×C269C2411yescharacteristic
108C8×C269—22no—
109C8×C269C2×D411yescharacteristic
110C8×C269C2×D411yescharacteristic
111C8×C269C2×D411yescharacteristic
112C8×C269C2×D411yescharacteristic
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×C269C22×D411yescharacteristic
122C4×C269C22×D411yescharacteristic
123Dic269—22no—
124Dic269—42no—
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—2692no—
138#453—2692no—
139#443—2691no—
140#443D26911yescharacteristic, complemented
1412−1+4—2692no—
142H(5,𝔽2)—2692no—
143#137—2692no—
144#137—2692no—
145#137—5382no—
146#137—5382no—
147AGL(1,ℤ/8)—2692no—
148AGL(1,ℤ/8)—2692no—
149AGL(1,ℤ/8)—5382no—
150AGL(1,ℤ/8)—5382no—
151#135—2691no—
152#135—5382no—
153#135—5382no—
154#135C2×D26911yescharacteristic
155C2×SD16—2692no—
156C2×SD16—2692no—
157C2×SD16—5382no—
158C2×SD16—5382no—
159#131—2692no—
160#131C2×D26912yescomplemented
161C4⋊Q8—2691no—
162#127—2691no—
163#108—2691no—
164#108C2×D26911yescharacteristic, complemented
165#104—2692no—
166#104C2×D26912yescomplemented
167#101—2692no—
168#100—2692no—
169C4×C8C2×D26911yescharacteristic, complemented
170#41—2692no—
171#41—5382no—
172#41—5382no—
173#41—5382no—
174#41—5382no—
175#41—10762no—
176#41C22×D26912yes—
177C2×Q8—2692no—
178C2×Q8—5382no—
179C2×Q8—5382no—
180C2×D4—2692no—
181C2×D4—5382no—
182C2×D4—5382no—
183C2×D4—10761no—
184C2×D4—10762no—
185Q16—2691no—
186Q16—5382no—
187Q16—5382no—
188Q16D107611yescharacteristic, complemented
189SD16—2692no—
190SD16—5382no—
191SD16—5382no—
192SD16—5382no—
193SD16—5382no—
194SD16—5382no—
195SD16—5382no—
196SD16—5382no—
197SD16—5382no—
198SD16D107612yescomplemented
199D8—2691no—
200D8—5382no—
201D8—5382no—
202D8D107611yescharacteristic, complemented
203M16—22no—
204M16—2692no—
205M16—5382no—
206M16C22×D26912yes—
207C2×C8—22no—
208C2×C8—5382no—
209C2×C8C22×D26911yescharacteristic
210C2×C8C22×D26911yescharacteristic
211C4⋊C4—10761no—
212C42C22×D26911yescharacteristic
213C23—5382no—
214C23—10762no—
215Q8—2692no—
216Q8—5382no—
217Q8—5382no—
218Q8—10762no—
219Q8C2×D107612yes—
220D4—22no—
221D4—2692no—
222D4—5382no—
223D4—5382no—
224D4—5382no—
225D4—10762no—
226D4—10762no—
227D4—10762no—
228D4—21522no—
229D4C2×D107612yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—5382no—
233C2×C4—5382no—
234C2×C4—10762no—
235C2×C4—10762no—
236C2×C4—10762no—
237C2×C4C23×D26911yescharacteristic
238C8—22no—
239C8—5382no—
240C8D4×D26911yescharacteristic
241C8D4×D26911yescharacteristic
242C8C2×D107611yescharacteristic
243C8C2×D107611yescharacteristic
244C22—22no—
245C22—5382no—
246C22—10762no—
247C22—21522no—
248C22—43042no—
249C22#2257549911yescharacteristic
250C4—22no—
251C4—22no—
252C4—5382no—
253C4—10762no—
254C4C2×D4×D26911yescharacteristic
255C4C22×D107611yescharacteristic
256C2—21no—
257C2—42no—
258C2—10762no—
259C2—21522no—
260C2D4×D107611yescharacteristic
subgroupG/H
centerC2D4×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#443×C269C2
socleC2×C269D42
hypercenter#443D269
hypocenterC269#3291
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #31771417C2semidirect
2. #31771412C2semidirect
3. #31770978C2semidirect
4. #31770869C2semidirect
5. #31770868C2semidirect
6. #31770789C2semidirect
7. #31770774C2semidirect
8. #31770723C2semidirect
9. #443×C269C2semidirect
11. #22575499C22semidirect
13. #22575486C22semidirect
15. #22575483C22semidirect
24. C2×#16864391C22semidirect
25. C2×#16864391C22semidirect
27. #22575459C22semidirect
28. #22575453C22semidirect
30. #22575389C22semidirect
31. #22575381C22semidirect
32. #22575371C22semidirect
33. #22575369C22semidirect
35. #131×C269C22semidirect
36. #108×C269C22semidirect
37. #104×C269C22semidirect
38. C4×C8×C269C22semidirect
54. Dic1076D4semidirect
62. #16864391D4semidirect
66. D2152D4semidirect
73. Q16×C269D4semidirect
74. SD16×C269D4semidirect
75. D8×C269D4semidirect
130. C269#3291semidirect
140. #443D269semidirect
160. #131C2×D269semidirect
164. #108C2×D269semidirect
166. #104C2×D269semidirect
169. C4×C8C2×D269semidirect
188. Q16D1076semidirect
198. SD16D1076semidirect
202. D8D1076semidirect

Maximal subgroups

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

subgroupindex
#317714172
#317714122
#317709782
#317708692
#317708682
#317707892
#317707742
#317707232
#443×C2692
#3291269