Finite Group Databasebeta

← #46824025

Subgroups — #46824025

Order 34432.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#31771407C211yesmaximal, characteristic, complemented
2#31771406C211yesmaximal, characteristic, complemented
3#31771043C211yesmaximal, characteristic, complemented
4#31771042C212yesmaximal, complemented
5#31771039C212yesmaximal, complemented
6#31771036C212yesmaximal, complemented
7#31771035C211yesmaximal, characteristic, complemented
8C2×#22575394C211yesmaximal, characteristic, complemented
9D16×D269C212yesmaximal, complemented
10#31771023C211yesmaximal, characteristic, complemented
11#509×C269C211yesmaximal, characteristic, complemented
12#22575499—21no—
13#22575498—21no—
14#22575480—22no—
15#22575479—21no—
16#22575478—21no—
17#22575477—22no—
18#22575476C2212yescomplemented
19#22575475C2211yescharacteristic, complemented
20#22575472C2212yescomplemented
21C2×D2152C2211yescharacteristic, complemented
22D8×D269C2212yescomplemented
23D8×D269C2212yescomplemented
24#22575459C2211yescharacteristic
25#22575399C2211yescharacteristic, complemented
26#22575398C2212yescomplemented
27#22575397C2211yescharacteristic, complemented
28#22575396C2211yescharacteristic, complemented
29D4304C2212yescomplemented
30#22575394C2211yescharacteristic, complemented
31#22575394C2212yescomplemented
32#22575394C2212yescomplemented
33#22575392C2212yescomplemented
34C2×#16864377C2211yescharacteristic, complemented
35C16×D269C2212yescomplemented
36#135×C269C2211yescharacteristic, complemented
37C2×D8×C269C2211yescharacteristic, complemented
38SD32×C269C2212yescomplemented
39D16×C269C2212yescomplemented
40M32×C269C2211yescharacteristic, complemented
41#16864403—22no—
42#16864402—21no—
43#16864402—41no—
44#16864402D411yescharacteristic
45#16864401—22no—
46#16864401—41no—
47C2×D1076—21no—
48C2×D1076—41no—
49C2×C22⋊D269—42no—
50D4×D269—22no—
51D4×D269—22no—
52D4×D269—41no—
53D4×D269—42no—
54Dic1076C2311yescharacteristic
55#16864391—21no—
56#16864390—22no—
57#16864389—22no—
58D2152—21no—
59D2152C2312yes—
60D2152C2311yescharacteristic
61#16864387—22no—
62#16864387—22no—
63#16864386D411yescharacteristic
64#16864385D412yes—
65C8×D269C2312yes—
66#16864377C2311yescharacteristic
67#16864377C2311yescharacteristic
68#41×C269—21no—
69C2×D4×C269—21no—
70Q16×C269C2311yescharacteristic
71SD16×C269—21no—
72D8×C269—21no—
73D8×C269C2312yes—
74D8×C269C2311yescharacteristic
75C2×C8×C269C2311yescharacteristic
76C16×C269C2312yes—
77C22×D269—41no—
78C22×D269—42no—
79C22×D269—42no—
80Dic538—21no—
81Dic538D811yescharacteristic
82D1076—21no—
83D1076—22no—
84D1076—42no—
85D1076—41no—
86D1076D811yescharacteristic
87C22⋊D269—41no—
88C22⋊D269—42no—
89C22⋊D269—42no—
90C22⋊D269—42no—
91C22⋊D269D812yes—
92C2×Dic269—42no—
93C4×D269—41no—
94C4×D269—42no—
95C4×D269C2×D412yes—
96#14524130C2×D412yes—
97C23×C269—41no—
98Q8×C269—21no—
99D4×C269—22no—
100D4×C269—21no—
101D4×C269—41no—
102D4×C269—41no—
103C2×C4×C269—41no—
104C2×C4×C269C2×D411yescharacteristic
105C8×C269C2×D411yescharacteristic
106C8×C269C2411yescharacteristic
107C22×C269—41no—
108C22×C269—42no—
109C22×C269—161no—
110C22×C269C2×D811yescharacteristic
111C2×D269—41no—
112C2×D269—42no—
113C2×D269—162no—
114C2×D269—162no—
115C2×D269—161no—
116C2×D269C2×D812yes—
117C4×C269—41no—
118C4×C269C2×D811yescharacteristic
119C4×C269C22×D411yescharacteristic
120Dic269—41no—
121Dic269C2×D812yes—
122C2×C269—21no—
123C2×C269—81no—
124C2×C269—82no—
125C2×C269C22×D811yescharacteristic
126D269—22no—
127D269—81no—
128D269—82no—
129C269#341511yescharacteristic, complemented, Sylow
130#3415—2691nomaximal, Sylow
131#576—2691no—
132#576—2691no—
133#509—2692no—
134#509—2692no—
135#509—2691no—
136#509D26911yescharacteristic, complemented
137#508—2691no—
138#508—2692no—
139C2×D16—2691no—
140C2×D16—2692no—
141#504—2691no—
142H(5,𝔽2)—5381no—
143H(5,𝔽2)—5381no—
144AGL(1,ℤ/8)—5382no—
145AGL(1,ℤ/8)—5382no—
146AGL(1,ℤ/8)—5381no—
147AGL(1,ℤ/8)—5381no—
148#135—2692no—
149#135—2692no—
150#135—2691no—
151#135C2×D26911yescharacteristic, complemented
152C2×D8—2692no—
153C2×D8—2692no—
154C2×D8—2691no—
155C2×D8C2×D26911yescharacteristic, complemented
156#131—2691no—
157Q32—2691no—
158SD32—2691no—
159SD32—2691no—
160SD32—2692no—
161SD32C2×D26912yescomplemented
162D16—2692no—
163D16—2691no—
164D16—2692no—
165D16—2692no—
166D16C2×D26912yescomplemented
167M32—2692no—
168M32C2×D26911yescharacteristic, complemented
169C2×C16—2691no—
170C2×C16—2692no—
171#41—21no—
172#41—2691no—
173#41—5382no—
174#41—5382no—
175#41—5381no—
176#41—10761no—
177#41—10761no—
178C2×D4—21no—
179C2×D4—5382no—
180C2×D4—5382no—
181C2×D4—5381no—
182C2×D4—10761no—
183C2×D4—10762no—
184C2×D4—10761no—
185C2×D4—10762no—
186Q16—2691no—
187Q16C22×D26911yescharacteristic
188SD16—21no—
189SD16—5382no—
190SD16—5382no—
191SD16—5381no—
192D8—21no—
193D8—2692no—
194D8—2691no—
195D8—5382no—
196D8—5382no—
197D8—5381no—
198D8C22×D26912yes—
199D8C22×D26911yescharacteristic
200M16—2691no—
201M16—2692no—
202C2×C8—2692no—
203C2×C8C22×D26911yescharacteristic
204C16—2691no—
205C16—2691no—
206C16C22×D26912yes—
207C23—41no—
208C23—10762no—
209C23—10762no—
210C23—10761no—
211Q8—21no—
212Q8—2691no—
213Q8—5381no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—2692no—
219D4—2691no—
220D4—5381no—
221D4—5382no—
222D4—10762no—
223D4—10762no—
224D4—10762no—
225D4—10762no—
226D4—10761no—
227D4—10761no—
228C2×C4—41no—
229C2×C4—2692no—
230C2×C4—10762no—
231C2×C4—10762no—
232C2×C4—10761no—
233C2×C4D4×D26911yescharacteristic
234C8—2692no—
235C8D4×D26911yescharacteristic
236C8C23×D26911yescharacteristic
237C22—41no—
238C22—42no—
239C22—2692no—
240C22—161no—
241C22—10762no—
242C22—10761no—
243C22—43041no—
244C22—43042no—
245C22—43042no—
246C22D8×D26911yescharacteristic
247C4—41no—
248C4—2692no—
249C4—10761no—
250C4D8×D26911yescharacteristic
251C4C2×D4×D26911yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—5382no—
256C2—21522no—
257C2—21521no—
258C2C2×D8×D26911yescharacteristic
subgroupG/H
centerC2C2×D8×D269
commutator, cosocleC8×C269C24
FrattiniC8C23×D269
Fitting#509×C269C2
socleC2×C269C22×D8
hypercenter#509D269
hypocenterC269#3415
LCSG ▷ C8×C269 ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #31771407C2semidirect
2. #31771406C2semidirect
3. #31771043C2semidirect
4. #31771042C2semidirect
5. #31771039C2semidirect
6. #31771036C2semidirect
7. #31771035C2semidirect
8. C2×#22575394C2semidirect
9. D16×D269C2semidirect
10. #31771023C2semidirect
11. #509×C269C2semidirect
18. #22575476C22semidirect
19. #22575475C22semidirect
20. #22575472C22semidirect
21. C2×D2152C22semidirect
22. D8×D269C22semidirect
23. D8×D269C22semidirect
25. #22575399C22semidirect
26. #22575398C22semidirect
27. #22575397C22semidirect
28. #22575396C22semidirect
29. D4304C22semidirect
30. #22575394C22semidirect
31. #22575394C22semidirect
32. #22575394C22semidirect
33. #22575392C22semidirect
34. C2×#16864377C22semidirect
35. C16×D269C22semidirect
36. #135×C269C22semidirect
37. C2×D8×C269C22semidirect
38. SD32×C269C22semidirect
39. D16×C269C22semidirect
40. M32×C269C22semidirect
129. C269#3415semidirect
136. #509D269semidirect
151. #135C2×D269semidirect
155. C2×D8C2×D269semidirect
161. SD32C2×D269semidirect
166. D16C2×D269semidirect
168. M32C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714072
#317714062
#317710432
#317710422
#317710392
#317710362
#317710352
C2×#225753942
D16×D2692
#317710232
#509×C2692
#3415269