Finite Group Databasebeta

← #46822571

Subgroups — #46822571

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×#22575472C211yesmaximal, characteristic, complemented
2C2×D8×D269C211yesmaximal, characteristic, complemented
3#31771250C212yesmaximal, complemented
4#31770989C211yesmaximal, characteristic, complemented
5#31770847C211yesmaximal, characteristic, complemented
6#31770842C211yesmaximal, characteristic, complemented
7#31770820C212yesmaximal, complemented
8#31770757C212yesmaximal, complemented
9#31770751C212yesmaximal, complemented
10C8×D1076C211yesmaximal, characteristic, complemented
11C4×D8×C269C211yesmaximal, characteristic, complemented
12C2×#16864401C2212yescomplemented
13C2×D4×D269C2212yes—
14#22575472—24no—
15C2×Dic1076C2211yescharacteristic, complemented
16C2×#16864391C2212yescomplemented
17C2×#16864390C2212yescomplemented
18C2×D2152C2211yescharacteristic
19C2×#16864387C2212yes—
20D8×D269—24no—
21C2×C8×D269C2211yescharacteristic
22C2×C8×D269C2211yescharacteristic, complemented
23#22575441C2212yescomplemented
24#22575437—22no—
25#22575433C2212yes—
26#22575428C2212yescomplemented
27#22575427—22no—
28#22575423—22no—
29C4×D1076C2211yescharacteristic
30C2×#16864379—22no—
31C4⋊#14524130C2211yescharacteristic, complemented
32#22575376C2212yescomplemented
33#22575373C2212yescomplemented
34#22575365C2211yescharacteristic
35#22575365C2211yescharacteristic, complemented
36C2×D8×C269C2211yescharacteristic
37C4×D4×C269C2212yescomplemented
38#107×C269C2211yescharacteristic, complemented
39#102×C269C2212yescomplemented
40C4×C8×C269C2211yescharacteristic, complemented
41C23×D269—22no—
42#16864401—24no—
43#16864401—42no—
44C2×Dic538C2312yes—
45C2×D1076—22no—
46C2×D1076D411yescharacteristic
47C2×D1076C2312yes—
48C2×C22⋊D269—22no—
49C2×C22⋊D269—22no—
50D4×D269—24no—
51D4×D269—42no—
52C22×Dic269—22no—
53C2×C4×D269C2311yescharacteristic
54C2×C4×D269C2311yescharacteristic
55Dic1076—22no—
56#16864391—42no—
57#16864390—24no—
58D2152—22no—
59#16864387—24no—
60C2×#14524130C2311yescharacteristic
61C2×#14524130C2311yescharacteristic
62C8×D269—22no—
63C8×D269—22no—
64C4⋊Dic269—22no—
65C4⋊Dic269D411yescharacteristic
66#16864379—22no—
67#16864379—22no—
68#16864379—22no—
69#16864379—22no—
70#16864379D411yescharacteristic
71#16864379D411yescharacteristic
72C2×D4×C269C2312yes—
73C22×C4×C269—22no—
74D8×C269D414yescomplemented
75C2×C8×C269C2311yescharacteristic
76C2×C8×C269C2311yescharacteristic
77C4⋊C4×C269C2312yes—
78C22⋊C4×C269—22no—
79C42×C269C2311yescharacteristic
80C22×D269—22no—
81C22×D269—44no—
82C22×D269—44no—
83C22×D269—82no—
84C22×D269C2×D411yescharacteristic
85C22×D269C2×D411yescharacteristic
86Dic538—22no—
87Dic538—22no—
88D1076—22no—
89D1076—22no—
90D1076—22no—
91D1076—82no—
92C22⋊D269—44no—
93C22⋊D269—44no—
94C2×Dic269—22no—
95C2×Dic269—44no—
96C2×Dic269C2×D411yescharacteristic
97C2×Dic269C2×D411yescharacteristic
98C4×D269—21no—
99C4×D269—21no—
100C4×D269—21no—
101C4×D269—21no—
102#14524130—21no—
103#14524130—21no—
104C23×C269—22no—
105D4×C269—22no—
106D4×C269C2×D414yes—
107C2×C4×C269—22no—
108C2×C4×C269—82no—
109C2×C4×C269C2×D411yescharacteristic
110C2×C4×C269C2×D411yescharacteristic
111C2×C4×C269C2411yescharacteristic
112C8×C269—21no—
113C8×C269C2×D412yes—
114C22×C269—24no—
115C22×C269—42no—
116C22×C269—42no—
117C22×C269C22×D411yescharacteristic
118C2×D269—21no—
119C2×D269—21no—
120C2×D269—21no—
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—21no—
130C4×C269—42no—
131C4×C269C22×D411yescharacteristic
132C4×C269H(5,𝔽2)11yescharacteristic
133Dic269—21no—
134Dic269—21no—
135Dic269—42no—
136C2×C269—44no—
137C2×C269D4211yescharacteristic
138C2×C269C2×#13511yescharacteristic
139C2×C269#57611yescharacteristic
140D269—22no—
141D269—41no—
142D269—82no—
143C269#328011yescharacteristic, complemented, Sylow
144#3280—2691nomaximal, Sylow
145C2×#135—2691no—
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×D8D26911yescharacteristic, complemented
155C8×D4—2691no—
156C2×#41—2692no—
157C22×D4—2692no—
158#135—5384no—
159C2×Q16—2691no—
160C2×SD16—2692no—
161C2×SD16—2692no—
162C2×D8—2692no—
163C2×D8—2691no—
164C2×D8—5384no—
165C2×D8C2×D26911yescharacteristic
166C22×C8—2691no—
167C22×C8—2691no—
168#124—2692no—
169#123—5382no—
170C22⋊Q8—2692no—
171#121—2692no—
172#121—5382no—
173#120—5382no—
174C4×D4—2691no—
175C4×D4C2×D26912yescomplemented
176C2×C22⋊C4—5382no—
177#107C2×D26911yescharacteristic, complemented
178C4⋊C8—2691no—
179#103—2692no—
180#102—2692no—
181#102C2×D26912yescomplemented
182C22⋊C8—2691no—
183C22⋊C8—2691no—
184C4×C8C2×D26911yescharacteristic, complemented
185C24—5382no—
186#41—5384no—
187#41—10762no—
188C2×Q8—2692no—
189C2×D4—2691no—
190C2×D4—2692no—
191C2×D4—5382no—
192C2×D4—5382no—
193C2×D4—5384no—
194C2×D4—5382no—
195C2×D4—10762no—
196C2×D4C22×D26912yes—
197C22×C4—22no—
198C22×C4—2691no—
199C22×C4—2691no—
200C22×C4—5382no—
201Q16—5382no—
202SD16—5384no—
203SD16—10762no—
204D8—5384no—
205D8—5382no—
206D8D107614yescomplemented
207C2×C8—2691no—
208C2×C8—2691no—
209C2×C8—5382no—
210C2×C8—5382no—
211C2×C8C22×D26911yescharacteristic
212C2×C8C22×D26911yescharacteristic
213C4⋊C4—2691no—
214C4⋊C4—5382no—
215C4⋊C4C22×D26912yes—
216C22⋊C4—22no—
217C22⋊C4—2691no—
218C22⋊C4—2691no—
219C22⋊C4—5382no—
220C22⋊C4—5382no—
221C22⋊C4—5382no—
222C22⋊C4—5382no—
223C42C22×D26911yescharacteristic
224C23—22no—
225C23—2691no—
226C23—2691no—
227C23—5382no—
228C23—10764no—
229C23—10764no—
230C23—21522no—
231Q8—5382no—
232Q8—5382no—
233D4—22no—
234D4—5382no—
235D4—5382no—
236D4—5382no—
237D4—10764no—
238D4—10764no—
239D4—21522no—
240D4C2×D107614yes—
241C2×C4—22no—
242C2×C4—82no—
243C2×C4—2691no—
244C2×C4—2691no—
245C2×C4—5381no—
246C2×C4—5381no—
247C2×C4—5381no—
248C2×C4—5381no—
249C2×C4—5382no—
250C2×C4—10764no—
251C2×C4D4×D26911yescharacteristic
252C2×C4D4×D26911yescharacteristic
253C2×C4C23×D26911yescharacteristic
254C8—21no—
255C8—5381no—
256C8—5381no—
257C8C2×D107612yes—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—2692no—
262C22—5381no—
263C22—5381no—
264C22—5381no—
265C22—5381no—
266C22—5381no—
267C22—10762no—
268C22—10762no—
269C22—10762no—
270C22—21528no—
271C22C2×D4×D26911yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—5381no—
276C4—5381no—
277C4—10762no—
278C4C22×D107611yescharacteristic
279C4#2257549911yescharacteristic
280C2—44no—
281C2—5382no—
282C2—10761no—
283C2—21522no—
284C2D4×D107611yescharacteristic
285C2#135×D26911yescharacteristic
286C2#3177140611yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC4×D8×C269C2
socleC22×C269C22×D4
hypercenterC4×D8D269
hypocenterC269#3280
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#22575472C2semidirect
2. C2×D8×D269C2semidirect
3. #31771250C2semidirect
4. #31770989C2semidirect
5. #31770847C2semidirect
6. #31770842C2semidirect
7. #31770820C2semidirect
8. #31770757C2semidirect
9. #31770751C2semidirect
10. C8×D1076C2semidirect
11. C4×D8×C269C2semidirect
12. C2×#16864401C22semidirect
15. C2×Dic1076C22semidirect
16. C2×#16864391C22semidirect
17. C2×#16864390C22semidirect
22. C2×C8×D269C22semidirect
23. #22575441C22semidirect
26. #22575428C22semidirect
31. C4⋊#14524130C22semidirect
32. #22575376C22semidirect
33. #22575373C22semidirect
35. #22575365C22semidirect
37. C4×D4×C269C22semidirect
38. #107×C269C22semidirect
39. #102×C269C22semidirect
40. C4×C8×C269C22semidirect
74. D8×C269D4semidirect
127. C2×D269#135semidirect
143. C269#3280semidirect
154. C4×D8D269semidirect
175. C4×D4C2×D269semidirect
177. #107C2×D269semidirect
181. #102C2×D269semidirect
184. C4×C8C2×D269semidirect
206. D8D1076semidirect

Maximal subgroups

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

subgroupindex
C2×#225754722
C2×D8×D2692
#317712502
#317709892
#317708472
#317708422
#317708202
#317707572
#317707512
C8×D10762
C4×D8×C2692
#3280269