Finite Group Databasebeta

← #46824028

Subgroups — #46824028

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#31771419C211yesmaximal, characteristic, complemented
2#31771407C211yesmaximal, characteristic, complemented
3#31771047C212yesmaximal, complemented
4#31771045C211yesmaximal, characteristic, complemented
5#31771040C211yesmaximal, characteristic, complemented
6#31771039C212yesmaximal, complemented
7#31771036C212yesmaximal, complemented
8C2×#22575398C211yesmaximal, characteristic, complemented
9SD32×D269C212yesmaximal, complemented
10#31771021C211yesmaximal, characteristic, complemented
11#508×C269C211yesmaximal, characteristic, complemented
12#22575500—21no—
13#22575499—21no—
14#22575483—21no—
15#22575482—22no—
16#22575480—22no—
17#22575478—21no—
18#22575476C2212yes—
19#22575472C2212yescomplemented
20C2×Dic1076C2211yescharacteristic, complemented
21Q16×D269C2212yescomplemented
22C2×D2152C2211yescharacteristic
23D8×D269C2212yes—
24#22575459C2211yescharacteristic
25#22575399C2212yescomplemented
26#22575398C2212yescomplemented
27#22575398C2212yescomplemented
28#22575397C2212yescomplemented
29#22575396C2212yescomplemented
30#22575394C2212yescomplemented
31#22575393C2211yescharacteristic, complemented
32#22575392C2212yescomplemented
33C16×D269C2212yescomplemented
34#135×C269C2211yescharacteristic
35#135×C269C2211yescharacteristic, complemented
36Q32×C269C2211yescharacteristic, complemented
37SD32×C269C2211yescharacteristic, complemented
38SD32×C269C2211yescharacteristic, complemented
39D16×C269C2211yescharacteristic, complemented
40C2×C16×C269C2211yescharacteristic, complemented
41#16864403—22no—
42#16864402—41no—
43#16864402—41no—
44#16864402D411yescharacteristic
45#16864401—22no—
46#16864401—42no—
47C2×Dic538—21no—
48C2×Dic538—41no—
49Q8×D269—22no—
50C2×D1076—21no—
51C2×D1076—41no—
52D4×D269—22no—
53D4×D269—42no—
54Dic1076—21no—
55Dic1076C2312yes—
56#16864392—22no—
57#16864390—22no—
58#16864389—22no—
59D2152—21no—
60D2152C2312yes—
61#16864387—22no—
62#16864386D411yescharacteristic
63#16864385D412yes—
64C8×D269C2312yes—
65#16864377C2312yes—
66#41×C269—21no—
67#41×C269—21no—
68Q16×C269C2311yescharacteristic
69Q16×C269C2311yescharacteristic
70SD16×C269—21no—
71SD16×C269—21no—
72D8×C269C2311yescharacteristic
73D8×C269C2311yescharacteristic
74C2×C8×C269C2311yescharacteristic
75C16×C269C2311yescharacteristic
76C16×C269C2311yescharacteristic
77C22×D269—41no—
78C22×D269—42no—
79Dic538—22no—
80Dic538—41no—
81Dic538—42no—
82Dic538D811yescharacteristic
83D1076—22no—
84D1076—42no—
85D1076—41no—
86D1076D811yescharacteristic
87C22⋊D269—42no—
88C22⋊D269—42no—
89C22⋊D269D812yes—
90C2×Dic269—41no—
91C2×Dic269—42no—
92C4×D269—42no—
93C4×D269—42no—
94C4×D269C2×D412yes—
95#14524130C2×D412yes—
96Q8×C269—21no—
97Q8×C269—21no—
98D4×C269—21no—
99D4×C269—21no—
100D4×C269—41no—
101D4×C269—41no—
102C2×C4×C269—41no—
103C2×C4×C269—41no—
104C2×C4×C269C2×D411yescharacteristic
105C8×C269C2×D411yescharacteristic
106C8×C269C2411yescharacteristic
107C22×C269—41no—
108C22×C269—41no—
109C22×C269C2×D811yescharacteristic
110C2×D269—42no—
111C2×D269—162no—
112C2×D269—161no—
113C2×D269C2×D812yes—
114C4×C269—41no—
115C4×C269—41no—
116C4×C269C2×D811yescharacteristic
117C4×C269C22×D411yescharacteristic
118Dic269—42no—
119Dic269C2×D812yes—
120C2×C269—21no—
121C2×C269—81no—
122C2×C269—81no—
123C2×C269C22×D811yescharacteristic
124D269—22no—
125D269—82no—
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#508D26911yescharacteristic, complemented
136C2×SD32—2692no—
137C2×SD32—2691no—
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#135C2×D26911yescharacteristic
148#135C2×D26911yescharacteristic, complemented
149C2×Q16—2692no—
150C2×Q16—2691no—
151C2×D8—2692no—
152C2×D8—2691no—
153#131—2691no—
154Q32—2692no—
155Q32C2×D26911yescharacteristic, complemented
156SD32—2692no—
157SD32—2692no—
158SD32—2692no—
159SD32—2692no—
160SD32C2×D26911yescharacteristic, complemented
161SD32C2×D26911yescharacteristic, complemented
162D16—2692no—
163D16C2×D26911yescharacteristic, complemented
164M32—2691no—
165M32—2692no—
166C2×C16—2692no—
167C2×C16C2×D26911yescharacteristic, complemented
168#41—21no—
169#41—21no—
170#41—2691no—
171#41—5382no—
172#41—5382no—
173#41—10761no—
174#41—10761no—
175#41—10762no—
176C2×Q8—5382no—
177C2×Q8—5381no—
178C2×Q8—10761no—
179C2×D4—5382no—
180C2×D4—5381no—
181C2×D4—10761no—
182C2×D4—10762no—
183Q16—2692no—
184Q16—5382no—
185Q16—5381no—
186Q16C22×D26911yescharacteristic
187Q16C22×D26911yescharacteristic
188SD16—21no—
189SD16—21no—
190SD16—5382no—
191SD16—5382no—
192D8—2692no—
193D8—5382no—
194D8—5381no—
195D8C22×D26911yescharacteristic
196D8C22×D26911yescharacteristic
197M16—2691no—
198M16—2692no—
199C2×C8—2692no—
200C2×C8C22×D26911yescharacteristic
201C16—2692no—
202C16C22×D26911yescharacteristic
203C16C22×D26911yescharacteristic
204C23—10762no—
205C23—10761no—
206Q8—21no—
207Q8—21no—
208Q8—2691no—
209Q8—5382no—
210Q8—10762no—
211Q8—10761no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—2692no—
217D4—2691no—
218D4—5382no—
219D4—10762no—
220D4—10762no—
221D4—10762no—
222D4—10761no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—2692no—
226C2×C4—10762no—
227C2×C4—10762no—
228C2×C4—10762no—
229C2×C4—10761no—
230C2×C4D4×D26911yescharacteristic
231C8—2692no—
232C8D4×D26911yescharacteristic
233C8C23×D26911yescharacteristic
234C22—41no—
235C22—41no—
236C22—2692no—
237C22—10762no—
238C22—43041no—
239C22—43042no—
240C22D8×D26911yescharacteristic
241C4—41no—
242C4—41no—
243C4—2692no—
244C4—10762no—
245C4D8×D26911yescharacteristic
246C4C2×D4×D26911yescharacteristic
247C2—21no—
248C2—81no—
249C2—81no—
250C2—5382no—
251C2—21522no—
252C2C2×D8×D26911yescharacteristic
subgroupG/H
centerC2C2×D8×D269
commutator, cosocleC8×C269C24
FrattiniC8C23×D269
Fitting#508×C269C2
socleC2×C269C22×D8
hypercenter#508D269
hypocenterC269#3416
LCSG ▷ C8×C269 ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #31771419C2semidirect
2. #31771407C2semidirect
3. #31771047C2semidirect
4. #31771045C2semidirect
5. #31771040C2semidirect
6. #31771039C2semidirect
7. #31771036C2semidirect
8. C2×#22575398C2semidirect
9. SD32×D269C2semidirect
10. #31771021C2semidirect
11. #508×C269C2semidirect
19. #22575472C22semidirect
20. C2×Dic1076C22semidirect
21. Q16×D269C22semidirect
25. #22575399C22semidirect
26. #22575398C22semidirect
27. #22575398C22semidirect
28. #22575397C22semidirect
29. #22575396C22semidirect
30. #22575394C22semidirect
31. #22575393C22semidirect
32. #22575392C22semidirect
33. C16×D269C22semidirect
35. #135×C269C22semidirect
36. Q32×C269C22semidirect
37. SD32×C269C22semidirect
38. SD32×C269C22semidirect
39. D16×C269C22semidirect
40. C2×C16×C269C22semidirect
126. C269#3416semidirect
135. #508D269semidirect
148. #135C2×D269semidirect
155. Q32C2×D269semidirect
160. SD32C2×D269semidirect
161. SD32C2×D269semidirect
163. D16C2×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
#317714192
#317714072
#317710472
#317710452
#317710402
#317710392
#317710362
C2×#225753982
SD32×D2692
#317710212
#508×C2692
#3416269