Finite Group Databasebeta

← #489×D269 #46821211

Subgroups — #489×D269 #46821211

Order 34432.

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

normal subgroups only

This group has 436 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
1C4⋊Q8×D269C211yesmaximal, characteristic, complemented
2#124×D269C211yesmaximal, characteristic
3#31770967C211yesmaximal, characteristic, complemented
4#31770966C212yesmaximal, complemented
5#31770965C211yesmaximal, characteristic, complemented
6#31770964C211yesmaximal, characteristic, complemented
7#31770963C212yesmaximal, complemented
8#103×D269C212yesmaximal, complemented
9#102×D269C212yesmaximal
10#97×D269C211yesmaximal, characteristic, complemented
11#489×C269C211yesmaximal, characteristic, complemented
12C2×Q8×D269—21no—
13C2×Q8×D269C2211yescharacteristic
14C2×D4×D269C2211yescharacteristic
15C2×C8×D269C2212yes—
16#22575453C2211yescharacteristic, complemented
17#22575452—21no—
18#22575452C2212yescomplemented
19#22575451—21no—
20#22575442C2211yescharacteristic
21#22575441C2211yescharacteristic
22#22575440C2211yescharacteristic
23#22575439—21no—
24#22575439—21no—
25C4⋊C4×D269—21no—
26C4⋊C4×D269C2212yes—
27C22⋊C4×D269—21no—
28C22⋊C4×D269—21no—
29C42×D269C2211yescharacteristic
30#22575378C2212yescomplemented
31#22575377C2212yescomplemented
32#22575376C2212yescomplemented
33#22575375C2212yes—
34#22575374C2212yes—
35#22575373C2212yes—
36#22575364C2211yescharacteristic, complemented
37#22575363C2212yescomplemented
38C4⋊Q8×C269C2211yescharacteristic, complemented
39#124×C269C2211yescharacteristic
40#103×C269C2212yescomplemented
41#102×C269C2212yes—
42#97×C269C2211yescharacteristic, complemented
43C23×D269—21no—
44C2×Dic538—21no—
45C2×Dic538—22no—
46C2×Dic538—21no—
47C2×Dic538C2311yescharacteristic
48Q8×D269—24no—
49Q8×D269—44no—
50C2×D1076C2311yescharacteristic
51C2×C22⋊D269—21no—
52D4×D269—24no—
53C2×C4×D269—21no—
54C2×C4×D269—22no—
55C2×C4×D269D411yescharacteristic
56C2×C4×D269D411yescharacteristic
57C2×C4×D269C2311yescharacteristic
58C2×#14524130C2312yes—
59C8×D269—24no—
60C4⋊Dic269—21no—
61C4⋊Dic269C2312yes—
62#16864381—22no—
63#16864381—22no—
64C22⋊Dic269—21no—
65C22⋊Dic269—21no—
66#16864379—21no—
67#16864379—21no—
68#16864379—21no—
69#16864379—21no—
70C4×Dic269D411yescharacteristic
71C4×Dic269D411yescharacteristic
72C4×Dic269C2311yescharacteristic
73C2×Q8×C269—21no—
74C2×Q8×C269C2311yescharacteristic
75C2×D4×C269C2311yescharacteristic
76C2×C8×C269C2312yes—
77C4⋊C4×C269—21no—
78C4⋊C4×C269C2312yes—
79C22⋊C4×C269—21no—
80C22⋊C4×C269—21no—
81C42×C269C2311yescharacteristic
82C22×D269—21no—
83C22×D269—42no—
84C22×D269—42no—
85C22×D269—42no—
86C22×D269C2×D411yescharacteristic
87Dic538—22no—
88Dic538—44no—
89Dic538—42no—
90Dic538—42no—
91D1076—22no—
92C22⋊D269—42no—
93C2×Dic269—21no—
94C2×Dic269—22no—
95C2×Dic269C2×D411yescharacteristic
96C2×Dic269C2×D411yescharacteristic
97C2×Dic269C2×D411yescharacteristic
98C4×D269—22no—
99C4×D269—22no—
100C4×D269—42no—
101C4×D269—44no—
102C4×D269#4114yes—
103#14524130—22no—
104C23×C269—21no—
105Q8×C269—22no—
106Q8×C269—42no—
107D4×C269—22no—
108C2×C4×C269—21no—
109C2×C4×C269—22no—
110C2×C4×C269C2×D411yescharacteristic
111C2×C4×C269C2×D411yescharacteristic
112C2×C4×C269C2411yescharacteristic
113C8×C269—22no—
114C22×C269—41no—
115C22×C269—41no—
116C22×C269—41no—
117C22×C269C22×D411yescharacteristic
118C2×D269—41no—
119C2×D269—41no—
120C2×D269—41no—
121C2×D269—84no—
122C2×D269#12412yes—
123C2×D269AGL(1,ℤ/8)12yes—
124C2×D269#13712yes—
125C4×C269—21no—
126C4×C269—21no—
127C4×C269—41no—
128C4×C269—42no—
129C4×C269C2×#4112yes—
130Dic269—21no—
131Dic269—21no—
132Dic269—41no—
133Dic269—42no—
134Dic269#12412yes—
135C2×C269—81no—
136C2×C269C2×#12411yescharacteristic
137C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C269C2×#13711yescharacteristic
139D269—81no—
140D269#48914yesdirect product factor
141C269C2×#48911yescharacteristic, complemented, Sylow
142C2×#489—2691nomaximal, Sylow
143C2×C4⋊Q8—2691no—
144C2×#124—2691no—
145#489—2691no—
146#489—2691no—
147#489—2692no—
148#489—2692no—
149#489—2691no—
150#489D26911yescharacteristic, direct product factor
151C2×#103—2692no—
152C2×#102—2692no—
153C2×#97—2691no—
154C22×Q8—2691no—
155C22×Q8—5381no—
156C22×D4—2691no—
157C22×C8—2692no—
158C4⋊Q8—2692no—
159C4⋊Q8—2691no—
160C4⋊Q8—5381no—
161C4⋊Q8—5381no—
162C4⋊Q8C2×D26911yescharacteristic, complemented
163#124—2691no—
164#124—2691no—
165#124—2691no—
166#124—5381no—
167#124—5381no—
168#124C2×D26911yescharacteristic
169C2×C4⋊C4—2692no—
170C2×C4⋊C4—5381no—
171C2×C22⋊C4—5381no—
172C2×C22⋊C4—5381no—
173C2×C42—2691no—
174#103—2692no—
175#103—2692no—
176#103—2692no—
177#103C2×D26912yescomplemented
178#102—2692no—
179#102—2692no—
180#102—2692no—
181#102C2×D26912yes—
182#97—2692no—
183#97—2691no—
184#97C2×D26911yescharacteristic, complemented
185C24—5381no—
186C2×Q8—21no—
187C2×Q8—2691no—
188C2×Q8—5381no—
189C2×Q8—5382no—
190C2×Q8—5384no—
191C2×Q8—5381no—
192C2×Q8—10764no—
193C2×Q8C22×D26911yescharacteristic
194C2×D4—2691no—
195C2×D4—5384no—
196C2×D4—5381no—
197C2×D4C22×D26911yescharacteristic
198C22×C4—2691no—
199C22×C4—2691no—
200C22×C4—2691no—
201C22×C4—5381no—
202C22×C4—5382no—
203C2×C8—2692no—
204C2×C8—5384no—
205C2×C8C22×D26912yes—
206C4⋊C4—21no—
207C4⋊C4—2692no—
208C4⋊C4—5382no—
209C4⋊C4—5382no—
210C4⋊C4—5381no—
211C4⋊C4C22×D26912yes—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—5381no—
215C22⋊C4—5381no—
216C22⋊C4—5381no—
217C22⋊C4—5381no—
218C22⋊C4—5381no—
219C22⋊C4—5381no—
220C42—2691no—
221C42—2691no—
222C42—2691no—
223C42C22×D26911yescharacteristic
224C23—21no—
225C23—2691no—
226C23—5381no—
227C23—10762no—
228C23—10762no—
229C23—10762no—
230Q8—22no—
231Q8—42no—
232Q8—5382no—
233Q8—10762no—
234Q8—10762no—
235Q8—10764no—
236D4—22no—
237D4—5382no—
238D4—10762no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—2691no—
242C2×C4—2694no—
243C2×C4—2691no—
244C2×C4—2691no—
245C2×C4—5382no—
246C2×C4—5382no—
247C2×C4—5381no—
248C2×C4—5382no—
249C2×C4—10762no—
250C2×C4—10764no—
251C2×C4D4×D26911yescharacteristic
252C2×C4D4×D26911yescharacteristic
253C2×C4C23×D26911yescharacteristic
254C8—22no—
255C8—5382no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—2692no—
260C22—2692no—
261C22—2692no—
262C22—10761no—
263C22—10761no—
264C22—10761no—
265C22—21524no—
266C22C2×D4×D26911yescharacteristic
267C4—21no—
268C4—21no—
269C4—41no—
270C4—42no—
271C4—2692no—
272C4—5381no—
273C4—5381no—
274C4—10761no—
275C4—10762no—
276C4#41×D26912yes—
277C2—81no—
278C2—2694no—
279C2—21521no—
280C2#124×D26911yescharacteristic
281C2AGL(1,ℤ/8)×D26911yescharacteristic
282C2#137×D26911yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#489×C269C2
socleC22×C269C22×D4
prime radicalC2×D4×D269C22
hypercenter#489D269
hypocenterC269C2×#489
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D269C2semidirect
3. #31770967C2semidirect
4. #31770966C2semidirect
5. #31770965C2semidirect
6. #31770964C2semidirect
7. #31770963C2semidirect
8. #103×D269C2semidirect
10. #97×D269C2semidirect
11. #489×C269C2semidirect
16. #22575453C22semidirect
18. #22575452C22semidirect
30. #22575378C22semidirect
31. #22575377C22semidirect
32. #22575376C22semidirect
36. #22575364C22semidirect
37. #22575363C22semidirect
38. C4⋊Q8×C269C22semidirect
40. #103×C269C22semidirect
42. #97×C269C22semidirect
140. D269#489direct (N × H)
141. C269C2×#489semidirect
150. #489D269direct (N × H)
162. C4⋊Q8C2×D269semidirect
177. #103C2×D269semidirect
184. #97C2×D269semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D2692
#124×D2692
#317709672
#317709662
#317709652
#317709642
#317709632
#103×D2692
#102×D2692
#97×D2692
#489×C2692
C2×#489269