Finite Group Databasebeta

← #46820288

Subgroups — #46820288

Order 34432.

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

normal subgroups only

This group has 772 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
2#31771174C212yesmaximal, complemented
3#31771169C211yesmaximal, characteristic, complemented
4#31770793C218yesmaximal, complemented
5#31770578C212yesmaximal, complemented
6#410×C269C211yesmaximal, characteristic, complemented
7#22575501—21no—
8#22575500—22no—
9#22575499—22no—
10#22575498—21no—
11#22575498C2214yes—
12C2×#16864402—22no—
13C2×#16864402C2212yes—
14#41×D269—24no—
15C22×D1076C2211yescharacteristic, complemented
16#22575450—22no—
17#22575449—21no—
18#22575442—21no—
19#22575441—22no—
20#22575428—24no—
21#22575427—22no—
22#22575423C2218yescomplemented
23#22575421C2214yescomplemented
24#22575412C2211yescharacteristic, complemented
25#22575367C2218yescomplemented
26C2×#41×C269C2211yescharacteristic
27#117×C269C2212yescomplemented
28#99×C269C2214yescomplemented
29C23×D269C2312yes—
30#16864403—24no—
31#16864402—24no—
32#16864402—24no—
33#16864402—22no—
34#16864402—22no—
35#16864402—24no—
36#16864402—22no—
37#16864402—22no—
38#16864401—24no—
39#16864401—28no—
40C2×Dic538—22no—
41C2×Dic538D412yescomplemented
42Q8×D269—24no—
43C2×D1076—22no—
44C2×D1076—28no—
45C2×D1076—21no—
46C2×D1076—22no—
47C2×D1076—22no—
48C2×D1076—21no—
49C2×D1076D412yescomplemented
50C2×C22⋊D269—24no—
51C2×C22⋊D269—24no—
52C2×C22⋊D269C2314yes—
53D4×D269—24no—
54D4×D269—28no—
55C2×C4×D269—24no—
56C2×C4×D269D414yescomplemented
57C4⋊Dic269—21no—
58C22⋊Dic269C2312yes—
59#16864379—28no—
60#16864379—24no—
61C4×Dic269—21no—
62#41×C269—22no—
63#41×C269—22no—
64C2×Q8×C269D411yescharacteristic, complemented
65C2×D4×C269D411yescharacteristic, complemented
66C2×D4×C269C2312yes—
67C22×C4×C269D412yescomplemented
68C22×C4×C269C2311yescharacteristic
69C4⋊C4×C269—22no—
70C22⋊C4×C269C2314yes—
71C42×C269—22no—
72C22×D269—24no—
73C22×D269—22no—
74C22×D269—24no—
75C22×D269—82no—
76C22×D269C2×D414yes—
77Dic538—24no—
78Dic538—22no—
79Dic538—22no—
80D1076—24no—
81D1076—22no—
82D1076—22no—
83D1076—41no—
84D1076—41no—
85D1076—42no—
86D1076—88no—
87C22⋊D269—28no—
88C22⋊D269—24no—
89C22⋊D269—24no—
90C22⋊D269—84no—
91C2×Dic269—24no—
92C2×Dic269—22no—
93C2×Dic269C2×D414yes—
94C4×D269—28no—
95C4×D269—24no—
96C4×D269—24no—
97C23×C269C2×D412yes—
98C23×C269C2411yescharacteristic
99Q8×C269—22no—
100D4×C269—24no—
101D4×C269—22no—
102C2×C4×C269—22no—
103C2×C4×C269—22no—
104C2×C4×C269—24no—
105C2×C4×C269C2×D412yes—
106C2×C4×C269C2×D412yes—
107C2×C4×C269C2×D412yes—
108C2×C4×C269C2×D411yescharacteristic
109C2×C4×C269C2×D411yescharacteristic
110C22×C269—22no—
111C22×C269—41no—
112C22×C269—42no—
113C22×C269C22×D412yes—
114C22×C269C22×D411yescharacteristic
115C2×D269—24no—
116C2×D269—44no—
117C2×D269—44no—
118C2×D269—48no—
119C2×D269—42no—
120C2×D269—84no—
121C4×C269—22no—
122C4×C269—44no—
123C4×C269#12011yescharacteristic
124C4×C269#12011yescharacteristic
125C4×C269#12012yes—
126Dic269—24no—
127Dic269—42no—
128C2×C269—21no—
129C2×C269—22no—
130C2×C269—42no—
131C2×C269C2×#12011yescharacteristic
132D269—44no—
133D269—82no—
134C269#302711yescharacteristic, complemented, Sylow
135#3027—2691nomaximal, Sylow
136#585—2691no—
137#535—2691no—
138#535—2692no—
139B(4,𝔽2)—2698no—
140#410—2692no—
141#410D26911yescharacteristic, complemented
1422−1+4—5382no—
1432−1+4—5381no—
144H(5,𝔽2)—2694no—
145H(5,𝔽2)—5382no—
146H(5,𝔽2)—5381no—
147C2×#41—2692no—
148C2×#41—5384no—
149C2×#41—5382no—
150C2×#41C2×D26911yescharacteristic
151C22×D4—2691no—
152#127—5382no—
153#127—5381no—
154#124—5382no—
155#124—5381no—
156#121—5384no—
157#121—5382no—
158#120—2694no—
159#120—2698no—
160#117—2691no—
161#117C2×D26912yescomplemented
162#99—2698no—
163#99C2×D26914yescomplemented
164C24—2692no—
165#41—22no—
166#41—22no—
167#41—5384no—
168#41—5384no—
169#41—5388no—
170#41—5384no—
171#41—5384no—
172#41—5382no—
173#41—5382no—
174#41—5382no—
175#41—5382no—
176#41—5384no—
177C2×Q8—2692no—
178C2×Q8—5382no—
179C2×Q8—5384no—
180C2×Q8D107611yescharacteristic, complemented
181C2×D4—2694no—
182C2×D4—2692no—
183C2×D4—5382no—
184C2×D4—5388no—
185C2×D4—5388no—
186C2×D4—5384no—
187C2×D4—5384no—
188C2×D4—5384no—
189C2×D4—5382no—
190C2×D4—5382no—
191C2×D4—5381no—
192C2×D4—5381no—
193C2×D4D107611yescharacteristic, complemented
194C2×D4C22×D26912yes—
195C22×C4—2694no—
196C22×C4—5384no—
197C22×C4D107612yescomplemented
198C22×C4C22×D26911yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—5381no—
201C22⋊C4—2692no—
202C22⋊C4—5388no—
203C22⋊C4—5384no—
204C22⋊C4C22×D26914yes—
205C42—22no—
206C42—5381no—
207C23—2694no—
208C23—5384no—
209C23—5384no—
210C23—5382no—
211C23—21522no—
212C23C2×D107612yes—
213C23C23×D26911yescharacteristic
214Q8—22no—
215Q8—5384no—
216Q8—5382no—
217Q8—5382no—
218D4—24no—
219D4—22no—
220D4—5384no—
221D4—5388no—
222D4—5384no—
223D4—5384no—
224D4—5382no—
225D4—5382no—
226D4—10761no—
227D4—10761no—
228D4—10762no—
229D4—21528no—
230D4—21524no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—2694no—
235C2×C4—5388no—
236C2×C4—5384no—
237C2×C4—5384no—
238C2×C4—5384no—
239C2×C4—5382no—
240C2×C4D4×D26912yes—
241C2×C4D4×D26912yes—
242C2×C4C2×D107612yes—
243C2×C4C2×D107611yescharacteristic
244C2×C4C2×D107611yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—5384no—
249C22—10764no—
250C22—10768no—
251C22—10764no—
252C22—10762no—
253C22—21524no—
254C22C2×D4×D26912yes—
255C22C22×D107611yescharacteristic
256C4—22no—
257C4—44no—
258C4—5384no—
259C4—10762no—
260C4#2257542311yescharacteristic
261C4#2257542311yescharacteristic
262C4#2257542312yes—
263C2—22no—
264C2—21no—
265C2—42no—
266C2—10764no—
267C2—21522no—
268C2C2×#2257542311yescharacteristic
subgroupG/H
centerC2C2×#22575423
commutator, cosocleC23×C269C24
FrattiniC23C23×D269
Fitting#410×C269C2
socleC2×C269C2×#120
hypercenter#410D269
hypocenterC269#3027
LCSG ▷ C23×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #31771441C2semidirect
2. #31771174C2semidirect
3. #31771169C2semidirect
4. #31770793C2semidirect
5. #31770578C2semidirect
6. #410×C269C2semidirect
15. C22×D1076C22semidirect
22. #22575423C22semidirect
23. #22575421C22semidirect
24. #22575412C22semidirect
25. #22575367C22semidirect
27. #117×C269C22semidirect
28. #99×C269C22semidirect
41. C2×Dic538D4semidirect
49. C2×D1076D4semidirect
56. C2×C4×D269D4semidirect
64. C2×Q8×C269D4semidirect
65. C2×D4×C269D4semidirect
67. C22×C4×C269D4semidirect
134. C269#3027semidirect
141. #410D269semidirect
161. #117C2×D269semidirect
163. #99C2×D269semidirect
180. C2×Q8D1076semidirect
193. C2×D4D1076semidirect
197. C22×C4D1076semidirect

Maximal subgroups

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

subgroupindex
#317714412
#317711742
#317711692
#317707932
#317705782
#410×C2692
#3027269