Finite Group Databasebeta

← #46821237

Subgroups — #46821237

Order 34432.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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#31771236C211yesmaximal, characteristic, complemented
2#31771172C211yesmaximal, characteristic
3#31770968C212yesmaximal, complemented
4#31770963C212yesmaximal, complemented
5#31770958C212yesmaximal, complemented
6#31770950C212yesmaximal, complemented
7#31770610C211yesmaximal, characteristic, complemented
8#31770604C211yesmaximal, characteristic, complemented
9#31770602C211yesmaximal, characteristic, complemented
10#31770560C211yesmaximal, characteristic, complemented
11C2×#102×C269C211yesmaximal, characteristic, complemented
12C2×#16864402C2211yescharacteristic
13C2×#16864386C2211yescharacteristic
14#22575452—21no—
15#22575452C2212yescomplemented
16#22575449C2212yes—
17#22575444—21no—
18#22575444C2212yescomplemented
19#22575440C2212yes—
20#22575432—22no—
21#22575424—21no—
22#22575424—21no—
23#22575422—22no—
24#22575412C2211yescharacteristic
25#22575412C2211yescharacteristic
26#22575378C2211yescharacteristic, complemented
27#22575378C2211yescharacteristic, complemented
28#22575376C2212yescomplemented
29#22575375C2211yescharacteristic, complemented
30#22575375C2211yescharacteristic, complemented
31#22575374C2212yes—
32#22575374C2212yes—
33#22575373C2212yescomplemented
34#22575363C2212yescomplemented
35C8×Dic269C2212yescomplemented
36C22×D4×C269C2211yescharacteristic
37C22×C8×C269C2211yescharacteristic
38C2×C4⋊C4×C269C2211yescharacteristic
39#102×C269C2212yes—
40#102×C269C2212yes—
41#16864402—22no—
42#16864402—22no—
43C2×Dic538—22no—
44C2×Dic538C2311yescharacteristic
45C2×D1076C2311yescharacteristic
46C2×C22⋊D269—22no—
47C2×C22⋊D269—22no—
48C2×C22⋊D269—21no—
49C2×C4×D269—21no—
50#16864386—22no—
51C2×#14524130C2312yes—
52C4⋊Dic269—22no—
53C4⋊Dic269D411yescharacteristic
54C4⋊Dic269D411yescharacteristic
55C4⋊Dic269C2311yescharacteristic
56C4⋊Dic269C2311yescharacteristic
57#16864381—22no—
58#16864381—22no—
59C22⋊Dic269—22no—
60C22⋊Dic269—22no—
61C22⋊Dic269—21no—
62C22⋊Dic269D412yes—
63C4×Dic269—21no—
64C4×Dic269C2312yes—
65C24×C269—21no—
66C2×D4×C269—21no—
67C2×D4×C269—22no—
68C2×D4×C269—22no—
69C2×D4×C269C2312yes—
70C22×C4×C269—21no—
71C22×C4×C269C2311yescharacteristic
72C2×C8×C269—22no—
73C2×C8×C269C2311yescharacteristic
74C2×C8×C269C2311yescharacteristic
75C4⋊C4×C269—21no—
76C4⋊C4×C269C2312yes—
77C22×D269—21no—
78Dic538—21no—
79Dic538—21no—
80Dic538—82no—
81D1076—21no—
82D1076—21no—
83C22⋊D269—42no—
84C22⋊D269—82no—
85C22⋊D269—82no—
86C2×Dic269—21no—
87C2×Dic269—21no—
88C2×Dic269—21no—
89C2×Dic269C2×D412yes—
90C2×Dic269C2×D412yes—
91C4×D269—42no—
92#14524130—22no—
93C23×C269—22no—
94C23×C269—42no—
95C23×C269—81no—
96C23×C269—81no—
97C23×C269C2×D411yescharacteristic
98D4×C269—22no—
99D4×C269—22no—
100D4×C269—42no—
101C2×C4×C269—22no—
102C2×C4×C269—81no—
103C2×C4×C269C2×D411yescharacteristic
104C2×C4×C269#4112yes—
105C2×C4×C269#4112yes—
106C2×C4×C269C2411yescharacteristic
107C8×C269—21no—
108C8×C269—21no—
109C22×C269—21no—
110C22×C269—21no—
111C22×C269—42no—
112C22×C269—42no—
113C22×C269—42no—
114C22×C269—84no—
115C22×C269#12412yes—
116C22×C269C22×D411yescharacteristic
117C2×D269—41no—
118C2×D269—41no—
119C2×D269—41no—
120C4×C269—42no—
121C4×C269#12412yes—
122C4×C269C2×#4111yescharacteristic
123C4×C269C2×#4111yescharacteristic
124Dic269—22no—
125Dic269—22no—
126Dic269—41no—
127Dic269—41no—
128Dic269—41no—
129C2×C269—22no—
130C2×C269—82no—
131C2×C269C2×#12411yescharacteristic
132C2×C269#57611yescharacteristic
133C2×C269#57711yescharacteristic
134D269—81no—
135C269#313611yescharacteristic, complemented, Sylow
136#3136—2691nomaximal, Sylow
137#544—2691no—
138#535—2691no—
139#490—2692no—
140#489—2692no—
141#488—2692no—
142#486—2692no—
143#418—2691no—
144#417—2691no—
145#416—2691no—
146C2×#102D26911yescharacteristic, complemented
147#405—2691no—
148C2×#41—2691no—
149C22×D4C2×D26911yescharacteristic
150C2×M16—2691no—
151C22×C8C2×D26911yescharacteristic
152C4⋊Q8—2692no—
153C4⋊Q8—5381no—
154#127—2692no—
155#125—2692no—
156#125—5381no—
157#124—2692no—
158C22⋊Q8—5382no—
159#121—5381no—
160#121—5381no—
161#120—5382no—
162#117—2691no—
163#117—2691no—
164C2×C4⋊C4C2×D26911yescharacteristic
165#103—2691no—
166#103—2691no—
167#103—2692no—
168#102—2692no—
169#102—2692no—
170#102—2691no—
171#102—2691no—
172#102—2692no—
173#102C2×D26912yes—
174#102C2×D26912yes—
175#97—2692no—
176C4×C8—2692no—
177C24—21no—
178#41—5382no—
179#41—5382no—
180C2×Q8—2691no—
181C2×Q8—5382no—
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—2691no—
186C2×D4—5382no—
187C2×D4—5382no—
188C2×D4—5381no—
189C2×D4C22×D26912yes—
190C22×C4—21no—
191C22×C4—5381no—
192C22×C4C22×D26911yescharacteristic
193M16—5382no—
194C2×C8—22no—
195C2×C8—2692no—
196C2×C8C22×D26911yescharacteristic
197C2×C8C22×D26911yescharacteristic
198C4⋊C4—21no—
199C4⋊C4—2691no—
200C4⋊C4—2691no—
201C4⋊C4—2691no—
202C4⋊C4—2691no—
203C4⋊C4—5382no—
204C4⋊C4—5382no—
205C4⋊C4—5382no—
206C4⋊C4C22×D26912yes—
207C22⋊C4—2692no—
208C22⋊C4—5382no—
209C22⋊C4—5382no—
210C22⋊C4—5381no—
211C42—2692no—
212C42—5381no—
213C23—22no—
214C23—42no—
215C23—81no—
216C23—81no—
217C23—5381no—
218C23D4×D26911yescharacteristic
219Q8—5381no—
220Q8—5381no—
221Q8—21522no—
222D4—22no—
223D4—22no—
224D4—42no—
225D4—5381no—
226D4—5381no—
227D4—10762no—
228D4—21522no—
229D4—21522no—
230C2×C4—22no—
231C2×C4—81no—
232C2×C4—2692no—
233C2×C4—2692no—
234C2×C4—5381no—
235C2×C4—5381no—
236C2×C4—5381no—
237C2×C4—10762no—
238C2×C4D4×D26911yescharacteristic
239C2×C4#1686440112yes—
240C2×C4#1686440212yes—
241C2×C4C23×D26911yescharacteristic
242C8—21no—
243C8—21no—
244C8—5382no—
245C22—21no—
246C22—21no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—84no—
251C22—10761no—
252C22—10761no—
253C22—10761no—
254C22#2257543912yes—
255C22C2×D4×D26911yescharacteristic
256C4—42no—
257C4—5382no—
258C4—5382no—
259C4—10761no—
260C4—10761no—
261C4—10761no—
262C4#2257543912yes—
263C4C2×#1686440111yescharacteristic
264C4C2×#1686440211yescharacteristic
265C2—22no—
266C2—82no—
267C2—21521no—
268C2C2×#2257543911yescharacteristic
269C2#3177140611yescharacteristic
270C2#3177141111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#102×C269C2
socleC22×C269C22×D4
prime radical#31771172C2
hypercenterC2×#102D269
hypocenterC269#3136
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #31771236C2semidirect
3. #31770968C2semidirect
4. #31770963C2semidirect
5. #31770958C2semidirect
6. #31770950C2semidirect
7. #31770610C2semidirect
8. #31770604C2semidirect
9. #31770602C2semidirect
10. #31770560C2semidirect
11. C2×#102×C269C2semidirect
15. #22575452C22semidirect
18. #22575444C22semidirect
26. #22575378C22semidirect
27. #22575378C22semidirect
28. #22575376C22semidirect
29. #22575375C22semidirect
30. #22575375C22semidirect
33. #22575373C22semidirect
34. #22575363C22semidirect
35. C8×Dic269C22semidirect
135. C269#3136semidirect
146. C2×#102D269semidirect

Maximal subgroups

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

subgroupindex
#317712362
#317711722
#317709682
#317709632
#317709582
#317709502
#317706102
#317706042
#317706022
#317705602
C2×#102×C2692
#3136269