Finite Group Databasebeta

← #46824027

Subgroups — #46824027

Order 34432.

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

normal subgroups only

This group has 360 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#31771418C211yesmaximal, characteristic, complemented
2#31771406C211yesmaximal, characteristic, complemented
3#31771046C211yesmaximal, characteristic, complemented
4#31771044C212yesmaximal, complemented
5#31771042C212yesmaximal, complemented
6#31771041C211yesmaximal, characteristic, complemented
7#31771037C211yesmaximal, characteristic, complemented
8#31771036C212yesmaximal, complemented
9SD32×D269C212yesmaximal, complemented
10#31771021C211yesmaximal, characteristic, complemented
11C2×SD32×C269C211yesmaximal, characteristic, complemented
12#22575501—21no—
13#22575498—21no—
14#22575485—22no—
15#22575484—21no—
16#22575479—21no—
17#22575477—22no—
18#22575476C2212yescomplemented
19#22575475C2211yescharacteristic
20#22575475C2211yescharacteristic, complemented
21#22575472C2212yes—
22Q16×D269C2212yescomplemented
23D8×D269C2212yes—
24#22575459C2211yescharacteristic
25Dic2152C2211yescharacteristic, complemented
26#22575399C2212yescomplemented
27#22575398C2211yescharacteristic, complemented
28#22575398C2211yescharacteristic, complemented
29#22575397C2212yescomplemented
30#22575396C2212yescomplemented
31D4304C2211yescharacteristic, complemented
32#22575394C2212yescomplemented
33#22575393C2211yescharacteristic, complemented
34#22575392C2212yescomplemented
35C16×D269C2212yescomplemented
36C2×Q16×C269C2211yescharacteristic, complemented
37C2×D8×C269C2211yescharacteristic
38SD32×C269C2212yescomplemented
39SD32×C269C2212yescomplemented
40C2×C16×C269C2211yescharacteristic, complemented
41#16864403—22no—
42#16864403—41no—
43#16864402—21no—
44#16864402—21no—
45#16864402—42no—
46#16864402D411yescharacteristic
47#16864401—22no—
48#16864401—41no—
49Q8×D269—22no—
50Q8×D269—41no—
51C2×C22⋊D269—42no—
52D4×D269—22no—
53D4×D269—41no—
54Dic1076C2311yescharacteristic
55Dic1076C2311yescharacteristic
56#16864392—22no—
57#16864391—21no—
58#16864391—21no—
59#16864390—22no—
60#16864389—22no—
61D2152C2311yescharacteristic
62D2152C2311yescharacteristic
63#16864387—22no—
64#16864386D411yescharacteristic
65#16864385D412yes—
66C8×D269C2312yes—
67#16864377C2312yes—
68C2×Q8×C269—21no—
69C2×D4×C269—21no—
70Q16×C269—21no—
71Q16×C269C2312yes—
72D8×C269—21no—
73D8×C269C2312yes—
74C2×C8×C269C2311yescharacteristic
75C16×C269C2311yescharacteristic
76C16×C269C2311yescharacteristic
77C22×D269—42no—
78Dic538—21no—
79Dic538—21no—
80Dic538—42no—
81Dic538D811yescharacteristic
82D1076—21no—
83D1076—21no—
84D1076—42no—
85D1076D811yescharacteristic
86C22⋊D269—41no—
87C22⋊D269—41no—
88C22⋊D269—42no—
89C22⋊D269—42no—
90C22⋊D269D812yes—
91C2×Dic269—42no—
92C4×D269—42no—
93C4×D269—41no—
94C4×D269—41no—
95C4×D269—42no—
96C4×D269C2×D412yes—
97#14524130C2×D412yes—
98C23×C269—41no—
99Q8×C269—22no—
100Q8×C269—41no—
101D4×C269—22no—
102D4×C269—41no—
103C2×C4×C269—41no—
104C2×C4×C269C2×D411yescharacteristic
105C8×C269C2×D411yescharacteristic
106C8×C269C2411yescharacteristic
107C22×C269—42no—
108C22×C269—161no—
109C22×C269C2×D811yescharacteristic
110C2×D269—41no—
111C2×D269—41no—
112C2×D269—162no—
113C2×D269C2×D812yes—
114C4×C269—42no—
115C4×C269C2×D811yescharacteristic
116C4×C269C22×D411yescharacteristic
117Dic269—41no—
118Dic269—41no—
119Dic269C2×D812yes—
120C2×C269—21no—
121C2×C269—82no—
122C2×C269C22×D811yescharacteristic
123D269—22no—
124D269—81no—
125D269—81no—
126C269#341611yescharacteristic, complemented, Sylow
127#3416—2691nomaximal, Sylow
128#578—2691no—
129#576—2691no—
130#510—2691no—
131#510—2692no—
132#509—2691no—
133#509—2692no—
134#508—2692no—
135#508—2691no—
136C2×SD32—2692no—
137C2×SD32D26911yescharacteristic, complemented
138#504—2691no—
1392−1+4—5381no—
140H(5,𝔽2)—5381no—
141#137—5382no—
142#137—5381no—
143AGL(1,ℤ/8)—5382no—
144AGL(1,ℤ/8)—5381no—
145#135—2692no—
146#135—2692no—
147#135—2691no—
148#135—2691no—
149C2×Q16—2692no—
150C2×Q16C2×D26911yescharacteristic, complemented
151C2×D8—2692no—
152C2×D8C2×D26911yescharacteristic
153#131—2691no—
154Q32—2692no—
155Q32—2691no—
156SD32—2692no—
157SD32—2692no—
158SD32—2691no—
159SD32—2691no—
160SD32C2×D26912yescomplemented
161SD32C2×D26912yescomplemented
162D16—2692no—
163D16—2691no—
164M32—2691no—
165M32—2692no—
166C2×C16—2692no—
167C2×C16C2×D26911yescharacteristic, complemented
168#41—2691no—
169#41—5382no—
170#41—5382no—
171#41—5381no—
172#41—5381no—
173#41—10762no—
174#41—10761no—
175#41—10761no—
176C2×Q8—21no—
177C2×Q8—5382no—
178C2×Q8—10761no—
179C2×D4—21no—
180C2×D4—5382no—
181C2×D4—10761no—
182C2×D4—10762no—
183Q16—21no—
184Q16—2691no—
185Q16—2691no—
186Q16—5382no—
187Q16C22×D26912yes—
188SD16—5382no—
189SD16—5382no—
190SD16—5381no—
191SD16—5381no—
192D8—21no—
193D8—2691no—
194D8—2691no—
195D8—5382no—
196D8C22×D26912yes—
197M16—2691no—
198M16—2692no—
199C2×C8—2692no—
200C2×C8C22×D26911yescharacteristic
201C16—2692no—
202C16C22×D26911yescharacteristic
203C16C22×D26911yescharacteristic
204C23—41no—
205C23—10762no—
206Q8—22no—
207Q8—41no—
208Q8—2691no—
209Q8—5381no—
210Q8—5381no—
211Q8—10762no—
212D4—22no—
213D4—41no—
214D4—2692no—
215D4—2691no—
216D4—5381no—
217D4—5381no—
218D4—10762no—
219D4—10762no—
220D4—10762no—
221D4—10761no—
222D4—10761no—
223C2×C4—41no—
224C2×C4—2692no—
225C2×C4—10762no—
226C2×C4—10762no—
227C2×C4—10762no—
228C2×C4—10761no—
229C2×C4—10761no—
230C2×C4D4×D26911yescharacteristic
231C8—2692no—
232C8D4×D26911yescharacteristic
233C8C23×D26911yescharacteristic
234C22—42no—
235C22—2692no—
236C22—161no—
237C22—10761no—
238C22—10761no—
239C22—43042no—
240C22D8×D26911yescharacteristic
241C4—42no—
242C4—2692no—
243C4—10761no—
244C4—10761no—
245C4D8×D26911yescharacteristic
246C4C2×D4×D26911yescharacteristic
247C2—21no—
248C2—82no—
249C2—5382no—
250C2—21521no—
251C2—21521no—
252C2C2×D8×D26911yescharacteristic
subgroupG/H
centerC2C2×D8×D269
commutator, cosocleC8×C269C24
FrattiniC8C23×D269
FittingC2×SD32×C269C2
socleC2×C269C22×D8
hypercenterC2×SD32D269
hypocenterC269#3416
LCSG ▷ C8×C269 ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
1. #31771418C2semidirect
2. #31771406C2semidirect
3. #31771046C2semidirect
4. #31771044C2semidirect
5. #31771042C2semidirect
6. #31771041C2semidirect
7. #31771037C2semidirect
8. #31771036C2semidirect
9. SD32×D269C2semidirect
10. #31771021C2semidirect
11. C2×SD32×C269C2semidirect
18. #22575476C22semidirect
20. #22575475C22semidirect
22. Q16×D269C22semidirect
25. Dic2152C22semidirect
26. #22575399C22semidirect
27. #22575398C22semidirect
28. #22575398C22semidirect
29. #22575397C22semidirect
30. #22575396C22semidirect
31. D4304C22semidirect
32. #22575394C22semidirect
33. #22575393C22semidirect
34. #22575392C22semidirect
35. C16×D269C22semidirect
36. C2×Q16×C269C22semidirect
38. SD32×C269C22semidirect
39. SD32×C269C22semidirect
40. C2×C16×C269C22semidirect
126. C269#3416semidirect
137. C2×SD32D269semidirect
150. C2×Q16C2×D269semidirect
160. SD32C2×D269semidirect
161. SD32C2×D269semidirect
167. C2×C16C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714182
#317714062
#317710462
#317710442
#317710422
#317710412
#317710372
#317710362
SD32×D2692
#317710212
C2×SD32×C2692
#3416269