Finite Group Databasebeta

← #46822651

Subgroups — #46822651

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#31771411C212yesmaximal, complemented
2#31771406C212yesmaximal, complemented
3#31770872C212yesmaximal, complemented
4#31770865C212yesmaximal, complemented
5#31770782C211yesmaximal, characteristic, complemented
6#31770777C211yesmaximal, characteristic, complemented
7#31770772C212yesmaximal, complemented
8#31770728C212yesmaximal, complemented
9#496×C269C211yesmaximal, characteristic, complemented
10#22575501C2211yescharacteristic, complemented
11#22575498C2211yescharacteristic
12#22575498C2212yescomplemented
13#22575484C2212yescomplemented
14#22575482—22no—
15#22575480—22no—
16#22575479C2212yes—
17#22575479C2212yescomplemented
18#22575479C2212yescomplemented
19#22575477—22no—
20#22575477—22no—
21#22575475C2212yes—
22#22575474—22no—
23#22575472—22no—
24SD16×D269—22no—
25D8×D269—22no—
26#22575459C2212yescomplemented
27#22575459C2212yescomplemented
28#22575389C2212yescomplemented
29#22575381C2212yescomplemented
30#22575381C2212yescomplemented
31#22575372C2211yescharacteristic, complemented
32#22575370C2212yescomplemented
33#22575370C2211yescharacteristic, complemented
34C2×SD16×C269C2212yescomplemented
35C2×D8×C269C2212yescomplemented
36#127×C269C2211yescharacteristic, complemented
37#124×C269C2211yescharacteristic, complemented
38#97×C269C2211yescharacteristic, complemented
39#16864403—22no—
40#16864402—41no—
41#16864402C2312yes—
42#16864402C2312yes—
43#16864401—22no—
44#16864401—22no—
45#16864401—22no—
46Q8×D269—22no—
47C2×C22⋊D269—41no—
48C2×C22⋊D269—42no—
49D4×D269—22no—
50D4×D269—22no—
51D4×D269—22no—
52Dic1076D412yescomplemented
53#16864392—22no—
54#16864391D412yescomplemented
55#16864391D412yescomplemented
56#16864390—22no—
57#16864390—22no—
58#16864390—22no—
59#16864389—22no—
60D2152D412yescomplemented
61#16864387—22no—
62#16864387—22no—
63#16864387—22no—
64#16864386C2312yes—
65#16864386C2312yes—
66#16864385—22no—
67#16864385—22no—
68C8×D269—22no—
69C8×D269—22no—
70C2×Q8×C269C2311yescharacteristic
71C2×D4×C269—41no—
72C2×D4×C269C2312yes—
73C2×D4×C269C2311yescharacteristic
74SD16×C269—22no—
75SD16×C269—22no—
76D8×C269—22no—
77D8×C269—22no—
78C2×C8×C269C2312yes—
79C22⋊C4×C269—41no—
80C42×C269C2311yescharacteristic
81C22×D269—42no—
82C22×D269—41no—
83Dic538—41no—
84Dic538C2×D412yes—
85Dic538C2×D412yes—
86D1076—41no—
87D1076C2×D412yes—
88D1076C2×D412yes—
89C22⋊D269—22no—
90C22⋊D269—22no—
91C22⋊D269—42no—
92C22⋊D269—42no—
93C22⋊D269—42no—
94C2×Dic269—42no—
95C2×Dic269—41no—
96C4×D269—22no—
97C4×D269—22no—
98C4×D269—42no—
99#14524130—22no—
100#14524130—22no—
101C23×C269—21no—
102C23×C269—22no—
103Q8×C269—22no—
104D4×C269—22no—
105D4×C269—22no—
106D4×C269—22no—
107D4×C269—82no—
108C2×C4×C269—21no—
109C2×C4×C269—21no—
110C2×C4×C269C2411yescharacteristic
111C8×C269C2×D412yes—
112C8×C269C2×D412yes—
113C22×C269—41no—
114C22×C269—42no—
115C22×C269—82no—
116C22×C269—81no—
117C22×C269H(5,𝔽2)11yescharacteristic
118C2×D269—22no—
119C2×D269—22no—
120C2×D269—162no—
121C2×D269—161no—
122C4×C269—41no—
123C4×C269—41no—
124C4×C269C22×D412yes—
125Dic269—22no—
126Dic269—22no—
127C2×C269—21no—
128C2×C269—82no—
129C2×C269—81no—
130C2×C269D4211yescharacteristic
131D269—42no—
132D269—42no—
133C269#328811yescharacteristic, complemented, Sylow
134#3288—2691nomaximal, Sylow
135#577—2692no—
136#576—2692no—
137#496D26911yescharacteristic, complemented
138#472—2692no—
139#471—2692no—
140#455—2691no—
141#454—2691no—
142#453—2692no—
143#444—2692no—
1442−1+4—2691no—
145H(5,𝔽2)—2691no—
146H(5,𝔽2)—2692no—
147#137—2692no—
148#137—5382no—
149AGL(1,ℤ/8)—2692no—
150AGL(1,ℤ/8)—2692no—
151AGL(1,ℤ/8)—2692no—
152AGL(1,ℤ/8)—5382no—
153AGL(1,ℤ/8)—5382no—
154AGL(1,ℤ/8)—5382no—
155#135—2692no—
156#135—5382no—
157#135—5382no—
158C2×SD16—5382no—
159C2×SD16C2×D26912yescomplemented
160C2×D8—5382no—
161C2×D8C2×D26912yescomplemented
162#131—2692no—
163#131—2692no—
164#127C2×D26911yescharacteristic, complemented
165#124C2×D26911yescharacteristic, complemented
166#108—2692no—
167#104—2692no—
168#104—2692no—
169#101—2691no—
170#100—2692no—
171#100—2691no—
172#97C2×D26911yescharacteristic, complemented
173#41—2692no—
174#41—2692no—
175#41—5382no—
176#41—5382no—
177#41—5382no—
178#41—5382no—
179#41—10761no—
180C2×Q8—5382no—
181C2×Q8C22×D26911yescharacteristic
182C2×D4—41no—
183C2×D4—5382no—
184C2×D4—5382no—
185C2×D4—5382no—
186C2×D4—10761no—
187C2×D4—10762no—
188C2×D4C22×D26911yescharacteristic
189C2×D4C22×D26912yes—
190Q16—2692no—
191Q16—5382no—
192SD16—22no—
193SD16—22no—
194SD16—2692no—
195SD16—2692no—
196SD16—5382no—
197SD16—5382no—
198SD16—5382no—
199SD16—5382no—
200D8—22no—
201D8—22no—
202D8—2692no—
203D8—5382no—
204D8—5382no—
205D8—5382no—
206M16—2692no—
207M16—2692no—
208M16—5382no—
209M16—5382no—
210C2×C8—5382no—
211C2×C8—5382no—
212C2×C8C22×D26912yes—
213C22⋊C4—41no—
214C42C22×D26911yescharacteristic
215C23—21no—
216C23—22no—
217C23—10762no—
218C23—10761no—
219Q8—22no—
220Q8—2692no—
221Q8—2692no—
222Q8—10761no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—2692no—
228D4—2692no—
229D4—5382no—
230D4—5382no—
231D4—10761no—
232D4—10762no—
233D4—10762no—
234D4—10762no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—5382no—
238C2×C4—5382no—
239C2×C4—10762no—
240C2×C4—10761no—
241C2×C4—10762no—
242C2×C4C23×D26911yescharacteristic
243C8—5382no—
244C8—5382no—
245C8D4×D26912yes—
246C8D4×D26912yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—5382no—
252C22—5382no—
253C22—43041no—
254C22—43042no—
255C22#2257549811yescharacteristic
256C4—41no—
257C4—41no—
258C4—5382no—
259C4—5382no—
260C4C2×D4×D26912yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—10762no—
265C2—10762no—
266C2#3177124011yescharacteristic
subgroupG/H
centerC2#31771240
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#496×C269C2
socleC2×C269D42
hypercenter#496D269
hypocenterC269#3288
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #31771411C2semidirect
2. #31771406C2semidirect
3. #31770872C2semidirect
4. #31770865C2semidirect
5. #31770782C2semidirect
6. #31770777C2semidirect
7. #31770772C2semidirect
8. #31770728C2semidirect
9. #496×C269C2semidirect
10. #22575501C22semidirect
12. #22575498C22semidirect
13. #22575484C22semidirect
17. #22575479C22semidirect
18. #22575479C22semidirect
26. #22575459C22semidirect
27. #22575459C22semidirect
28. #22575389C22semidirect
29. #22575381C22semidirect
30. #22575381C22semidirect
31. #22575372C22semidirect
32. #22575370C22semidirect
33. #22575370C22semidirect
34. C2×SD16×C269C22semidirect
35. C2×D8×C269C22semidirect
36. #127×C269C22semidirect
37. #124×C269C22semidirect
38. #97×C269C22semidirect
52. Dic1076D4semidirect
54. #16864391D4semidirect
55. #16864391D4semidirect
60. D2152D4semidirect
133. C269#3288semidirect
137. #496D269semidirect
159. C2×SD16C2×D269semidirect
161. C2×D8C2×D269semidirect
164. #127C2×D269semidirect
165. #124C2×D269semidirect
172. #97C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714112
#317714062
#317708722
#317708652
#317707822
#317707772
#317707722
#317707282
#496×C2692
#3288269