Finite Group Databasebeta

← #46822564

Subgroups — #46822564

Order 34432.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×D8×D269C211yesmaximal, characteristic, complemented
2#31771243C212yesmaximal, complemented
3#31770990C211yesmaximal, characteristic, complemented
4#31770843C211yesmaximal, characteristic, complemented
5#31770841C211yesmaximal, characteristic, complemented
6#31770822C212yesmaximal, complemented
7#31770754C212yesmaximal, complemented
8#31770750C212yesmaximal, complemented
9D8×Dic269C211yesmaximal, characteristic, complemented
10C8×C22⋊D269C211yesmaximal, characteristic, complemented
11C2×#135×C269C211yesmaximal, characteristic, complemented
12C2×D4×D269C2212yes—
13C2×#16864389C2212yescomplemented
14C2×D2152C2211yescharacteristic
15C2×#16864387C2212yes—
16D8×D269—24no—
17C2×C8×D269C2211yescharacteristic
18#22575442C2212yescomplemented
19#22575436—22no—
20#22575429C2212yes—
21#22575427C2212yescomplemented
22#22575425—22no—
23#22575421—22no—
24D4×Dic269C2212yescomplemented
25C4×C22⋊D269C2211yescharacteristic
26C2×#16864379—22no—
27#22575387C2211yescharacteristic, complemented
28#22575382C2211yescharacteristic, complemented
29#22575377C2212yescomplemented
30#22575375C2212yescomplemented
31#22575374C2212yescomplemented
32C22⋊#14524130C2211yescharacteristic, complemented
33#22575365C2211yescharacteristic
34C8×Dic269C2211yescharacteristic, complemented
35C2×#41×C269C2212yescomplemented
36#135×C269—24no—
37C2×Q16×C269C2211yescharacteristic, complemented
38C2×SD16×C269C2212yescomplemented
39C2×D8×C269C2211yescharacteristic
40C22×C8×C269C2211yescharacteristic, complemented
41C23×D269—22no—
42C2×D1076C2312yes—
43C2×C22⋊D269—22no—
44C2×C22⋊D269—22no—
45C2×C22⋊D269D411yescharacteristic
46D4×D269—24no—
47D4×D269—42no—
48C22×Dic269—22no—
49C2×C4×D269C2311yescharacteristic
50#16864389—42no—
51D2152—22no—
52#16864387—24no—
53C2×#14524130C2311yescharacteristic
54C2×#14524130C2311yescharacteristic
55C8×D269—22no—
56C4⋊Dic269C2312yes—
57#16864381—22no—
58#16864381D411yescharacteristic
59C22⋊Dic269—22no—
60C22⋊Dic269D411yescharacteristic
61#16864379—22no—
62#16864379—22no—
63#16864379—22no—
64#16864379—22no—
65#16864379D411yescharacteristic
66C4×Dic269C2311yescharacteristic
67#41×C269—24no—
68#41×C269—42no—
69C2×Q8×C269C2312yes—
70C2×D4×C269—22no—
71C2×D4×C269C2312yes—
72C22×C4×C269—22no—
73C22×C4×C269C2311yescharacteristic
74Q16×C269—22no—
75SD16×C269—24no—
76D8×C269D414yescomplemented
77C2×C8×C269—22no—
78C2×C8×C269C2311yescharacteristic
79C2×C8×C269C2311yescharacteristic
80C22×D269—22no—
81C22×D269—44no—
82C22×D269—44no—
83C22×D269—82no—
84C22×D269C2×D411yescharacteristic
85D1076—22no—
86D1076—22no—
87C22⋊D269—22no—
88C22⋊D269—44no—
89C22⋊D269—82no—
90C2×Dic269—22no—
91C2×Dic269—82no—
92C2×Dic269C2×D411yescharacteristic
93C2×Dic269C2×D411yescharacteristic
94C2×Dic269C2×D411yescharacteristic
95C4×D269—21no—
96C4×D269—21no—
97#14524130—21no—
98#14524130—21no—
99C23×C269—22no—
100C23×C269C2×D411yescharacteristic
101Q8×C269—22no—
102Q8×C269—22no—
103D4×C269—22no—
104D4×C269—44no—
105D4×C269C2×D414yes—
106C2×C4×C269—21no—
107C2×C4×C269—21no—
108C2×C4×C269—22no—
109C2×C4×C269—44no—
110C2×C4×C269C2×D411yescharacteristic
111C2×C4×C269C2411yescharacteristic
112C8×C269—21no—
113C8×C269C2×D412yes—
114C22×C269—21no—
115C22×C269—24no—
116C22×C269—21no—
117C22×C269—21no—
118C22×C269—42no—
119C22×C269—42no—
120C22×C269C22×D411yescharacteristic
121C2×D269—21no—
122C2×D269—21no—
123C2×D269—42no—
124C2×D269—42no—
125C2×D269—42no—
126C2×D269—88no—
127C2×D269#13512yescomplemented
128C4×C269—21no—
129C4×C269—42no—
130C4×C269C22×D411yescharacteristic
131C4×C269H(5,𝔽2)11yescharacteristic
132Dic269—21no—
133Dic269—21no—
134Dic269—21no—
135Dic269—42no—
136C2×C269—44no—
137C2×C269—41no—
138C2×C269D4211yescharacteristic
139C2×C269C2×#13511yescharacteristic
140C2×C269#57611yescharacteristic
141D269—22no—
142D269—82no—
143C269#328011yescharacteristic, complemented, Sylow
144#3280—2691nomaximal, Sylow
145C2×#135D26911yescharacteristic, complemented
146C22×D8—2691no—
147#546—2692no—
148#495—2691no—
149#467—2691no—
150#466—2691no—
151#463—2692no—
152#450—2692no—
153#449—2692no—
154C4×D8—2691no—
155C8×D4—2691no—
156C2×#41C2×D26912yescomplemented
157C22×D4—2692no—
158#135—24no—
159C2×Q16C2×D26911yescharacteristic, complemented
160C2×SD16—2692no—
161C2×SD16C2×D26912yescomplemented
162C2×D8—2692no—
163C2×D8—2691no—
164C2×D8—5384no—
165C2×D8C2×D26911yescharacteristic
166C22×C8—2691no—
167C22×C8C2×D26911yescharacteristic, complemented
168#124—2692no—
169#123—5382no—
170C22⋊Q8—2692no—
171#121—2692no—
172#121—5382no—
173#120—5382no—
174C4×D4—2691no—
175C4×D4—2692no—
176C2×C22⋊C4—5382no—
177#107—2691no—
178C4⋊C8—2691no—
179#103—2692no—
180#102—2692no—
181#102—2692no—
182C22⋊C8—2691no—
183C22⋊C8—2691no—
184C4×C8—2691no—
185C24—5382no—
186#41—24no—
187#41—42no—
188C2×Q8C22×D26912yes—
189C2×D4—22no—
190C2×D4—2691no—
191C2×D4—2692no—
192C2×D4—5382no—
193C2×D4—5384no—
194C2×D4—5382no—
195C2×D4—10762no—
196C2×D4C22×D26912yes—
197C22×C4—22no—
198C22×C4—2691no—
199C22×C4—5382no—
200C22×C4C22×D26911yescharacteristic
201Q16—22no—
202SD16—24no—
203SD16—10762no—
204D8—5384no—
205D8—5382no—
206D8C22⋊D26914yescomplemented
207C2×C8—22no—
208C2×C8—2691no—
209C2×C8—2691no—
210C2×C8—5382no—
211C2×C8C22×D26911yescharacteristic
212C2×C8C22×D26911yescharacteristic
213C4⋊C4—2691no—
214C4⋊C4—2692no—
215C4⋊C4—5382no—
216C22⋊C4—2691no—
217C22⋊C4—2691no—
218C22⋊C4—5382no—
219C22⋊C4—5382no—
220C22⋊C4—5382no—
221C22⋊C4—5382no—
222C22⋊C4—5382no—
223C42—2691no—
224C23—22no—
225C23—2691no—
226C23—5382no—
227C23—10764no—
228C23—10764no—
229C23—21522no—
230C23D4×D26911yescharacteristic
231Q8—22no—
232Q8—22no—
233D4—22no—
234D4—44no—
235D4—5382no—
236D4—5382no—
237D4—5382no—
238D4—10764no—
239D4—21522no—
240D4C2×C22⋊D26914yes—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—44no—
245C2×C4—2691no—
246C2×C4—2691no—
247C2×C4—2691no—
248C2×C4—5381no—
249C2×C4—5381no—
250C2×C4—5382no—
251C2×C4—21522no—
252C2×C4D4×D26911yescharacteristic
253C2×C4C23×D26911yescharacteristic
254C8—21no—
255C8—5381no—
256C8—5381no—
257C8C2×C22⋊D26912yes—
258C22—21no—
259C22—21no—
260C22—24no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—2692no—
265C22—5381no—
266C22—5381no—
267C22—10762no—
268C22—10762no—
269C22—10762no—
270C22—21528no—
271C22C2×D4×D26911yescharacteristic
272C4—21no—
273C4—42no—
274C4—5381no—
275C4—5381no—
276C4—5381no—
277C4—10762no—
278C4C22×C22⋊D26911yescharacteristic
279C4#2257549811yescharacteristic
280C2—41no—
281C2—44no—
282C2—5382no—
283C2—21522no—
284C2D4×C22⋊D26911yescharacteristic
285C2#135×D26911yescharacteristic
286C2#3177140711yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#135×C269C2
socleC22×C269C22×D4
hypercenterC2×#135D269
hypocenterC269#3280
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×D8×D269C2semidirect
2. #31771243C2semidirect
3. #31770990C2semidirect
4. #31770843C2semidirect
5. #31770841C2semidirect
6. #31770822C2semidirect
7. #31770754C2semidirect
8. #31770750C2semidirect
9. D8×Dic269C2semidirect
10. C8×C22⋊D269C2semidirect
11. C2×#135×C269C2semidirect
13. C2×#16864389C22semidirect
18. #22575442C22semidirect
21. #22575427C22semidirect
24. D4×Dic269C22semidirect
27. #22575387C22semidirect
28. #22575382C22semidirect
29. #22575377C22semidirect
30. #22575375C22semidirect
31. #22575374C22semidirect
32. C22⋊#14524130C22semidirect
34. C8×Dic269C22semidirect
35. C2×#41×C269C22semidirect
37. C2×Q16×C269C22semidirect
38. C2×SD16×C269C22semidirect
40. C22×C8×C269C22semidirect
76. D8×C269D4semidirect
127. C2×D269#135semidirect
143. C269#3280semidirect
145. C2×#135D269semidirect
156. C2×#41C2×D269semidirect
159. C2×Q16C2×D269semidirect
161. C2×SD16C2×D269semidirect
167. C22×C8C2×D269semidirect
206. D8C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D2692
#317712432
#317709902
#317708432
#317708412
#317708222
#317707542
#317707502
D8×Dic2692
C8×C22⋊D2692
C2×#135×C2692
#3280269