Finite Group Databasebeta

← #46820259

Subgroups — #46820259

Order 34432.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 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#31771441C211yesmaximal, characteristic, complemented
2C2×#22575481C212yesmaximal, complemented
3#31771169C211yesmaximal, characteristic, complemented
4#31770780C214yesmaximal, complemented
5#31770775C214yesmaximal, complemented
6#31770621C212yesmaximal, complemented
7#413×C269C211yesmaximal, characteristic, complemented
8#22575501C2212yescomplemented
9#22575500—22no—
10#22575499—22no—
11#22575498—22no—
12#22575498C2212yescomplemented
13C2×#16864402—22no—
14C2×#16864402C2212yes—
15#41×D269—24no—
16C22×D1076C2211yescharacteristic, complemented
17#22575481—24no—
18#22575481C2218yescomplemented
19C2×#16864391—22no—
20C2×D2152—22no—
21#22575449C2212yescomplemented
22#22575442C2212yescomplemented
23#22575427—22no—
24#22575421—22no—
25#22575412C2211yescharacteristic, complemented
26#22575379C2218yescomplemented
27C2×#41×C269C2211yescharacteristic
28C2×M16×C269C2212yescomplemented
29#101×C269C2212yescomplemented
30#100×C269C2212yescomplemented
31C23×D269—21no—
32#16864403—24no—
33#16864402—24no—
34#16864402—24no—
35#16864402—22no—
36#16864402—24no—
37#16864402—22no—
38#16864402D414yescomplemented
39#16864402C2314yes—
40#16864401—28no—
41#16864401—24no—
42C2×Dic538—22no—
43C2×Dic538D412yescomplemented
44Q8×D269—24no—
45C2×D1076—22no—
46C2×D1076—22no—
47C2×D1076—22no—
48C2×D1076—21no—
49C2×D1076D412yescomplemented
50C2×D1076C2312yes—
51C2×C22⋊D269—24no—
52C2×C22⋊D269—22no—
53C2×C22⋊D269—24no—
54D4×D269—28no—
55D4×D269—24no—
56C2×C4×D269—24no—
57C2×C4×D269—22no—
58#16864391—28no—
59D2152—28no—
60C4⋊Dic269—21no—
61C22⋊Dic269—21no—
62#16864379—24no—
63C4×Dic269C2312yes—
64#41×C269—22no—
65#41×C269—22no—
66C2×Q8×C269C2311yescharacteristic
67C2×D4×C269D412yes—
68C2×D4×C269C2311yescharacteristic
69C22×C4×C269D412yes—
70C22×C4×C269C2311yescharacteristic
71M16×C269—22no—
72M16×C269C2314yes—
73C2×C8×C269—22no—
74C22×D269—24no—
75C22×D269—22no—
76C22×D269—22no—
77C22×D269—42no—
78C22×D269—82no—
79Dic538—24no—
80Dic538—22no—
81Dic538C2×D414yes—
82D1076—24no—
83D1076—22no—
84D1076—24no—
85D1076—41no—
86D1076—41no—
87D1076C2×D414yes—
88C22⋊D269—28no—
89C22⋊D269—24no—
90C22⋊D269—24no—
91C22⋊D269—84no—
92C2×Dic269—24no—
93C2×Dic269—22no—
94C2×Dic269—22no—
95C4×D269—28no—
96C4×D269—24no—
97C4×D269—24no—
98C23×C269C2×D412yes—
99C23×C269C2×D411yescharacteristic
100Q8×C269—22no—
101D4×C269—24no—
102D4×C269—22no—
103C2×C4×C269—22no—
104C2×C4×C269—22no—
105C2×C4×C269C2×D412yes—
106C2×C4×C269C2×D412yes—
107C2×C4×C269C2×D412yes—
108C2×C4×C269C2×D411yescharacteristic
109C2×C4×C269C2411yescharacteristic
110C8×C269—24no—
111C22×C269—22no—
112C22×C269—41no—
113C22×C269—42no—
114C22×C269#12012yes—
115C22×C269C22×D411yescharacteristic
116C2×D269—24no—
117C2×D269—44no—
118C2×D269—42no—
119C2×D269—82no—
120C2×D269—84no—
121C2×D269—84no—
122C4×C269—22no—
123C4×C269#12011yescharacteristic
124C4×C269#12011yescharacteristic
125C4×C269C22×D412yes—
126Dic269—24no—
127Dic269—42no—
128C2×C269—21no—
129C2×C269—22no—
130C2×C269—42no—
131C2×C269C2×#12011yescharacteristic
132D269—44no—
133D269—82no—
134C269#302111yescharacteristic, complemented, Sylow
135#3021—2691nomaximal, Sylow
136#585—2691no—
137C2×AGL(1,ℤ/8)—2692no—
138#535—2691no—
139#454—2694no—
140#453—2694no—
141#421—2692no—
142#413D26911yescharacteristic, complemented
1432−1+4—2692no—
1442−1+4—5382no—
145H(5,𝔽2)—2692no—
146H(5,𝔽2)—5382no—
147H(5,𝔽2)—5382no—
148C2×#41—2692no—
149C2×#41—5384no—
150C2×#41—5382no—
151C2×#41C2×D26911yescharacteristic
152C22×D4—2691no—
153AGL(1,ℤ/8)—2698no—
154AGL(1,ℤ/8)—5384no—
155C2×SD16—5382no—
156C2×D8—5382no—
157C2×M16C2×D26912yescomplemented
158#127—2692no—
159#124—2692no—
160#121—5382no—
161#120—5382no—
162#117—2691no—
163#104—2698no—
164#101C2×D26912yescomplemented
165#100C2×D26912yescomplemented
166C24—5381no—
167#41—22no—
168#41—22no—
169#41—2694no—
170#41—2694no—
171#41—5384no—
172#41—5384no—
173#41—5384no—
174#41—5388no—
175#41—5384no—
176#41—5382no—
177#41—5382no—
178#41—5384no—
179C2×Q8—2692no—
180C2×Q8—5382no—
181C2×Q8—5384no—
182C2×Q8C22×D26911yescharacteristic
183C2×D4—2692no—
184C2×D4—2692no—
185C2×D4—5382no—
186C2×D4—5384no—
187C2×D4—5388no—
188C2×D4—5382no—
189C2×D4—5384no—
190C2×D4—5384no—
191C2×D4—5382no—
192C2×D4—5382no—
193C2×D4—5381no—
194C2×D4D107612yes—
195C2×D4C22×D26911yescharacteristic
196C22×C4—5384no—
197C22×C4—5382no—
198C22×C4D107612yes—
199C22×C4C22×D26911yescharacteristic
200SD16—5388no—
201D8—5388no—
202M16—22no—
203M16C22×D26914yes—
204C2×C8—22no—
205C4⋊C4—5381no—
206C22⋊C4—5381no—
207C22⋊C4—5384no—
208C42—2692no—
209C23—5384no—
210C23—5382no—
211C23—5382no—
212C23—10762no—
213C23—21522no—
214C23C2×D107612yes—
215C23C2×D107611yescharacteristic
216Q8—22no—
217Q8—2694no—
218Q8—5384no—
219Q8—5382no—
220D4—24no—
221D4—22no—
222D4—2694no—
223D4—5384no—
224D4—5388no—
225D4—5384no—
226D4—5384no—
227D4—5382no—
228D4—5384no—
229D4—10761no—
230D4—10761no—
231D4—21524no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—5388no—
235C2×C4—5384no—
236C2×C4—5382no—
237C2×C4—5384no—
238C2×C4—5384no—
239C2×C4—5382no—
240C2×C4D4×D26912yes—
241C2×C4D4×D26912yes—
242C2×C4C2×D107612yes—
243C2×C4C2×D107611yescharacteristic
244C2×C4C23×D26911yescharacteristic
245C8—24no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—5384no—
250C22—10764no—
251C22—10762no—
252C22—21522no—
253C22—21524no—
254C22—21524no—
255C22#2257542312yes—
256C22C22×D107611yescharacteristic
257C4—22no—
258C4—5384no—
259C4—10762no—
260C4#2257542311yescharacteristic
261C4#2257542311yescharacteristic
262C4C2×D4×D26912yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—10764no—
267C2—21522no—
268C2C2×#2257542311yescharacteristic
subgroupG/H
centerC2C2×#22575423
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#413×C269C2
socleC2×C269C2×#120
hypercenter#413D269
hypocenterC269#3021
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #31771441C2semidirect
2. C2×#22575481C2semidirect
3. #31771169C2semidirect
4. #31770780C2semidirect
5. #31770775C2semidirect
6. #31770621C2semidirect
7. #413×C269C2semidirect
8. #22575501C22semidirect
12. #22575498C22semidirect
16. C22×D1076C22semidirect
18. #22575481C22semidirect
21. #22575449C22semidirect
22. #22575442C22semidirect
25. #22575412C22semidirect
26. #22575379C22semidirect
28. C2×M16×C269C22semidirect
29. #101×C269C22semidirect
30. #100×C269C22semidirect
38. #16864402D4semidirect
43. C2×Dic538D4semidirect
49. C2×D1076D4semidirect
134. C269#3021semidirect
142. #413D269semidirect
157. C2×M16C2×D269semidirect
164. #101C2×D269semidirect
165. #100C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714412
C2×#225754812
#317711692
#317707802
#317707752
#317706212
#413×C2692
#3021269