Finite Group Databasebeta

← #46820681

Subgroups — #46820681

Order 34432.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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#31771158C211yesmaximal, characteristic, complemented
2C2×#22575415C211yesmaximal, characteristic
3#31770908C212yesmaximal, complemented
4#31770904C212yesmaximal, complemented
5#31770894C212yesmaximal, complemented
6#31770884C212yesmaximal, complemented
7#31770645C211yesmaximal, characteristic, complemented
8#31770628C211yesmaximal, characteristic, complemented
9#31770611C211yesmaximal, characteristic
10#31770609C211yesmaximal, characteristic
11#428×C269C211yesmaximal, characteristic, complemented
12C22×D1076C2211yescharacteristic
13C22×C4×D269—21no—
14C2×#16864386C2211yescharacteristic
15#22575452C2211yescharacteristic, complemented
16#22575452C2211yescharacteristic, complemented
17#22575444C2211yescharacteristic, complemented
18#22575444C2211yescharacteristic, complemented
19#22575430—21no—
20#22575430—21no—
21C4×Dic538—21no—
22C4×Dic538—21no—
23#22575415—21no—
24#22575415—21no—
25#22575415C2212yes—
26#22575415C2212yes—
27#22575412C2211yescharacteristic
28C2×#16864379—21no—
29C2×C4×Dic269C2211yescharacteristic
30#22575386C2212yescomplemented
31#22575384C2212yescomplemented
32#22575382C2212yescomplemented
33#22575382C2212yescomplemented
34#22575375C2212yes—
35#22575375C2212yes—
36#22575373C2212yes—
37#22575373C2212yes—
38C2×M16×C269C2211yescharacteristic
39#117×C269C2211yescharacteristic
40C2×C4⋊C4×C269C2211yescharacteristic
41#107×C269C2212yescomplemented
42#106×C269C2212yescomplemented
43C23×D269—21no—
44C2×Dic538—21no—
45C2×Dic538—21no—
46C2×D1076—21no—
47C2×D1076—22no—
48C2×D1076Q814yescomplemented
49C2×D1076C2312yes—
50C22×Dic269D411yescharacteristic
51C22×Dic269D411yescharacteristic
52C2×C4×D269—22no—
53C2×C4×D269—22no—
54C2×C4×D269—44no—
55#16864386—22no—
56C2×#14524130C2312yes—
57C4⋊Dic269—21no—
58C4⋊Dic269—21no—
59C4⋊Dic269C2311yescharacteristic
60C4⋊Dic269C2311yescharacteristic
61#16864381—21no—
62#16864381—21no—
63#16864381—21no—
64#16864381—21no—
65C22⋊Dic269—21no—
66#16864379—22no—
67#16864379—22no—
68C4×Dic269—21no—
69C4×Dic269D411yescharacteristic
70C4×Dic269D411yescharacteristic
71C4×Dic269C2311yescharacteristic
72C4×Dic269C2311yescharacteristic
73C22×C4×C269—21no—
74C22×C4×C269C2311yescharacteristic
75M16×C269—22no—
76C2×C8×C269C2312yes—
77C4⋊C4×C269—21no—
78C4⋊C4×C269C2311yescharacteristic
79C4⋊C4×C269C2311yescharacteristic
80C4⋊C4×C269C2312yes—
81C22⋊C4×C269—21no—
82C42×C269—21no—
83C22×D269—24no—
84C22×D269—22no—
85C22×D269—42no—
86C22×D269—42no—
87Dic538—42no—
88Dic538—42no—
89D1076—24no—
90D1076—22no—
91D1076C2×Q814yes—
92C2×Dic269—22no—
93C2×Dic269—21no—
94C2×Dic269—21no—
95C2×Dic269—41no—
96C2×Dic269C2×D411yescharacteristic
97C2×Dic269C2×D411yescharacteristic
98C2×Dic269C2×D411yescharacteristic
99C2×Dic269C2×D411yescharacteristic
100C4×D269—44no—
101C4×D269—44no—
102#14524130—22no—
103C23×C269C2×D411yescharacteristic
104C2×C4×C269—21no—
105C2×C4×C269—21no—
106C2×C4×C269—22no—
107C2×C4×C269—42no—
108C2×C4×C269C2×D411yescharacteristic
109C2×C4×C269C2×Q812yes—
110C2×C4×C269#4112yes—
111C2×C4×C269C2411yescharacteristic
112C8×C269—22no—
113C22×C269—21no—
114C22×C269—21no—
115C22×C269C22⋊Q812yes—
116C22×C269C22×D411yescharacteristic
117C2×D269—24no—
118C2×D269—48no—
119C2×D269—42no—
120C2×D269—42no—
121C4×C269—42no—
122C4×C269—41no—
123C4×C269—41no—
124C4×C269C22⋊Q812yes—
125C4×C269C22×Q811yescharacteristic
126C4×C269C2×#4111yescharacteristic
127Dic269—21no—
128Dic269—21no—
129Dic269—21no—
130Dic269—41no—
131Dic269—41no—
132Dic269AGL(1,ℤ/8)12yes—
133C2×C269—22no—
134C2×C269C2×C22⋊Q811yescharacteristic
135C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C269#57511yescharacteristic
137D269—44no—
138C269#307711yescharacteristic, complemented, Sylow
139#3077—2691nomaximal, Sylow
140#533—2691no—
141C2×C4×D4—2691no—
142#478—2692no—
143#478—2692no—
144#476—2692no—
145#474—2692no—
146#428—2691no—
147#428D26911yescharacteristic, complemented
148#423—2691no—
149#418—2691no—
150#418—2691no—
151C22×D4—2691no—
152C23×C4—5381no—
153C2×M16—2691no—
154C2×M16C2×D26911yescharacteristic
155C4⋊Q8—2691no—
156C4⋊Q8—2691no—
157#125—2691no—
158#125—2691no—
159C22⋊Q8—5381no—
160C22⋊Q8—5381no—
161C4×Q8—5381no—
162C4×Q8—5381no—
163C4×D4—2692no—
164C4×D4—2692no—
165C4×D4—5381no—
166C4×D4—5381no—
167#117—2691no—
168#117C2×D26911yescharacteristic
169C2×C4⋊C4C2×D26911yescharacteristic
170C2×C22⋊C4—5381no—
171C2×C42—2691no—
172#107—2692no—
173#107C2×D26912yescomplemented
174#106—2692no—
175#106C2×D26912yescomplemented
176C4⋊C8—2692no—
177C4⋊C8—2692no—
178#102—2692no—
179#102—2692no—
180#102—2692no—
181#102—2692no—
182C24—5381no—
183C2×Q8—5381no—
184C2×Q8—5381no—
185C2×D4—2694no—
186C2×D4—2692no—
187C2×D4—5381no—
188C2×D4—5382no—
189C22×C4—21no—
190C22×C4—2691no—
191C22×C4—2691no—
192C22×C4—5382no—
193C22×C4—5382no—
194C22×C4—10764no—
195C22×C4C22×D26911yescharacteristic
196M16—22no—
197M16—5382no—
198C2×C8—2692no—
199C2×C8C22×D26912yes—
200C4⋊C4—21no—
201C4⋊C4—2691no—
202C4⋊C4—2691no—
203C4⋊C4—5381no—
204C4⋊C4—5381no—
205C4⋊C4—5381no—
206C4⋊C4—5381no—
207C4⋊C4—5381no—
208C4⋊C4—5381no—
209C4⋊C4C22×D26911yescharacteristic
210C4⋊C4C22×D26911yescharacteristic
211C4⋊C4C22×D26912yes—
212C22⋊C4—21no—
213C22⋊C4—5382no—
214C22⋊C4—5382no—
215C22⋊C4—5381no—
216C42—21no—
217C42—2691no—
218C42—2691no—
219C42—2691no—
220C42—2691no—
221C42—5381no—
222C23—5384no—
223C23—5382no—
224C23—10762no—
225C23—10762no—
226C23D4×D26911yescharacteristic
227Q8—10762no—
228Q8—10762no—
229D4—2694no—
230D4—5384no—
231D4—5382no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—2691no—
237C2×C4—2691no—
238C2×C4—2691no—
239C2×C4—2691no—
240C2×C4—5382no—
241C2×C4—5381no—
242C2×C4—5381no—
243C2×C4—10764no—
244C2×C4—10764no—
245C2×C4—10761no—
246C2×C4D4×D26911yescharacteristic
247C2×C4Q8×D26912yes—
248C2×C4#1686440212yes—
249C2×C4C23×D26911yescharacteristic
250C8—22no—
251C8—5382no—
252C22—21no—
253C22—21no—
254C22—5384no—
255C22—10768no—
256C22—10762no—
257C22—10762no—
258C22#2257543112yes—
259C22C2×D4×D26911yescharacteristic
260C4—41no—
261C4—42no—
262C4—41no—
263C4—2692no—
264C4—5381no—
265C4—5381no—
266C4—5381no—
267C4—10761no—
268C4—10761no—
269C4#2257543112yes—
270C4C2×Q8×D26911yescharacteristic
271C4C2×#1686440211yescharacteristic
272C2—22no—
273C2—10764no—
274C2C2×#2257543111yescharacteristic
275C2AGL(1,ℤ/8)×D26911yescharacteristic
276C2#3177140411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#428×C269C2
socleC22×C269C22×D4
prime radicalC22×D1076C22
hypercenter#428D269
hypocenterC269#3077
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #31771158C2semidirect
3. #31770908C2semidirect
4. #31770904C2semidirect
5. #31770894C2semidirect
6. #31770884C2semidirect
7. #31770645C2semidirect
8. #31770628C2semidirect
11. #428×C269C2semidirect
15. #22575452C22semidirect
16. #22575452C22semidirect
17. #22575444C22semidirect
18. #22575444C22semidirect
30. #22575386C22semidirect
31. #22575384C22semidirect
32. #22575382C22semidirect
33. #22575382C22semidirect
41. #107×C269C22semidirect
42. #106×C269C22semidirect
48. C2×D1076Q8semidirect
138. C269#3077semidirect
147. #428D269semidirect
173. #107C2×D269semidirect
175. #106C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317711582
C2×#225754152
#317709082
#317709042
#317708942
#317708842
#317706452
#317706282
#317706112
#317706092
#428×C2692
#3077269