Finite Group Databasebeta

← #46820165

Subgroups — #46820165

Order 34432.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22575499C211yesmaximal, characteristic
2C2×#22575483C211yesmaximal, characteristic, complemented
3C22×#16864391C211yesmaximal, characteristic, complemented
4#31771181C211yesmaximal, characteristic, complemented
5#31770770C214yesmaximal, complemented
6#31770757C214yesmaximal, complemented
7#31770609C211yesmaximal, characteristic, complemented
8#31770571C211yesmaximal, characteristic, complemented
9#416×C269C211yesmaximal, characteristic, complemented
10#22575499—24no—
11#22575499—24no—
12C2×#16864403C2212yes—
13C2×#16864402—21no—
14C2×#16864402C2211yescharacteristic
15C22×Dic538C2211yescharacteristic, complemented
16C22×D1076—21no—
17C22×D1076C2211yescharacteristic
18C2×D4×D269—22no—
19C2×D4×D269C2212yes—
20#22575483—24no—
21C2×#16864392C2212yescomplemented
22C2×#16864391—24no—
23C2×#16864391C2214yescomplemented
24C2×#16864390C2212yescomplemented
25C2×#16864386C2211yescharacteristic, complemented
26#22575453—21no—
27#22575441—21no—
28#22575437—22no—
29#22575433C2214yescomplemented
30#22575373C2214yescomplemented
31#22575365C2214yescomplemented
32C2×#41×C269C2211yescharacteristic
33C22×C8×C269C2211yescharacteristic, complemented
34#117×C269C2211yescharacteristic, complemented
35#103×C269C2212yescomplemented
36#102×C269C2212yescomplemented
37C23×D269—22no—
38C23×D269—21no—
39#16864403—24no—
40#16864403—24no—
41#16864402—28no—
42#16864402—22no—
43#16864402—22no—
44C2×Dic538—21no—
45C2×Dic538—22no—
46C2×Dic538—22no—
47C2×Dic538D411yescharacteristic
48C2×Dic538C2312yes—
49C2×D1076—28no—
50C2×D1076—22no—
51C2×D1076—21no—
52C2×D1076—22no—
53C2×D1076D411yescharacteristic
54C2×D1076D414yes—
55C2×D1076C2312yes—
56C2×C22⋊D269—22no—
57C2×C22⋊D269D412yes—
58D4×D269—216no—
59D4×D269—24no—
60D4×D269—24no—
61C22×Dic269—21no—
62C2×C4×D269—22no—
63C2×C4×D269C2312yes—
64#16864392—24no—
65#16864391—28no—
66#16864390—24no—
67#16864386—22no—
68C2×#14524130C2312yes—
69C4⋊Dic269—24no—
70#16864379—24no—
71#41×C269—22no—
72#41×C269D414yes—
73C2×Q8×C269C2311yescharacteristic
74C2×D4×C269—21no—
75C2×D4×C269C2311yescharacteristic
76C22×C4×C269—21no—
77C22×C4×C269C2311yescharacteristic
78C2×C8×C269—22no—
79C2×C8×C269C2312yes—
80C4⋊C4×C269C2312yes—
81C22⋊C4×C269—21no—
82C42×C269—21no—
83C22×D269—22no—
84C22×D269—28no—
85C22×D269—24no—
86C22×D269—84no—
87C22×D269—82no—
88C22×D269C2×D412yes—
89Dic538—22no—
90Dic538—22no—
91Dic538—22no—
92Dic538—42no—
93D1076—28no—
94D1076—22no—
95D1076—24no—
96D1076—22no—
97D1076C2×D414yes—
98C22⋊D269—28no—
99C22⋊D269—24no—
100C2×Dic269—22no—
101C2×Dic269—42no—
102C2×Dic269C2×D412yes—
103C4×D269—28no—
104C4×D269—22no—
105C4×D269—22no—
106#14524130—22no—
107C23×C269—21no—
108C23×C269C2×D411yescharacteristic
109Q8×C269—21no—
110Q8×C269C2×D412yes—
111D4×C269—24no—
112D4×C269—21no—
113D4×C269C2×D412yes—
114C2×C4×C269—21no—
115C2×C4×C269—24no—
116C2×C4×C269—22no—
117C2×C4×C269C2×D411yescharacteristic
118C2×C4×C269C2×D412yes—
119C2×C4×C269C2×D412yes—
120C2×C4×C269C2411yescharacteristic
121C8×C269—22no—
122C22×C269—22no—
123C22×C269—22no—
124C22×C269—42no—
125C22×C269#12012yes—
126C22×C269C22×D411yescharacteristic
127C2×D269—22no—
128C2×D269—24no—
129C2×D269—24no—
130C2×D269—44no—
131C2×D269—84no—
132C2×D269—88no—
133C4×C269—22no—
134C4×C269—42no—
135C4×C269#12012yes—
136C4×C269C22×D411yescharacteristic
137C4×C269C22×D411yescharacteristic
138Dic269—22no—
139Dic269—42no—
140C2×C269—22no—
141C2×C269—42no—
142C2×C269C2×#12011yescharacteristic
143C2×C269#57712yes—
144D269—42no—
145D269—44no—
146C269#300911yescharacteristic, complemented, Sylow
147#3009—2691nomaximal, Sylow
148C2×H(5,𝔽2)—2691no—
149C2×#137—2691no—
150C22×SD16—2691no—
151#536—2691no—
152#452—2694no—
153#450—2694no—
154#418—2691no—
155#416D26911yescharacteristic, complemented
156#408—2691no—
157H(5,𝔽2)—5384no—
158H(5,𝔽2)—5384no—
159C2×#41—2691no—
160C2×#41—2692no—
161C2×#41—5381no—
162C2×#41C2×D26911yescharacteristic
163C22×Q8—2691no—
164C22×D4—2692no—
165C22×D4—2691no—
166C22×D4—5381no—
167C22×D4—5382no—
168#137—5384no—
169C2×Q16—2692no—
170C2×SD16—2692no—
171C2×SD16—2694no—
172C2×SD16—5384no—
173C2×M16—2691no—
174C22×C8C2×D26911yescharacteristic, complemented
175C4⋊Q8—5381no—
176#124—5381no—
177#123—5382no—
178C22⋊Q8—2694no—
179#117C2×D26911yescharacteristic, complemented
180#103C2×D26912yescomplemented
181#102—2694no—
182#102C2×D26912yescomplemented
183C22⋊C8—2694no—
184C24—5382no—
185C24—5381no—
186#41—22no—
187#41—5388no—
188#41—5384no—
189#41—5384no—
190#41—5382no—
191#41—5382no—
192#41D107614yes—
193C2×Q8—2691no—
194C2×Q8—2692no—
195C2×Q8—5381no—
196C2×Q8—5382no—
197C2×Q8—5382no—
198C2×Q8C22×D26911yescharacteristic
199C2×D4—21no—
200C2×D4—2692no—
201C2×D4—2694no—
202C2×D4—2691no—
203C2×D4—2692no—
204C2×D4—5382no—
205C2×D4—5388no—
206C2×D4—5382no—
207C2×D4—5384no—
208C2×D4—53816no—
209C2×D4—5384no—
210C2×D4—5381no—
211C2×D4—5382no—
212C2×D4C22×D26911yescharacteristic
213C22×C4—21no—
214C22×C4—2692no—
215C22×C4—5382no—
216C22×C4—5381no—
217C22×C4C22×D26911yescharacteristic
218Q16—5384no—
219SD16—5388no—
220SD16—5384no—
221M16—5382no—
222C2×C8—22no—
223C2×C8—2692no—
224C2×C8C22×D26912yes—
225C4⋊C4—5384no—
226C4⋊C4C22×D26912yes—
227C22⋊C4—21no—
228C22⋊C4—5384no—
229C42—21no—
230C23—21no—
231C23—2692no—
232C23—5388no—
233C23—5382no—
234C23—5384no—
235C23—21522no—
236C23—21524no—
237C23D4×D26911yescharacteristic
238Q8—21no—
239Q8—5382no—
240Q8—5382no—
241Q8—5382no—
242Q8—10762no—
243Q8C2×D107612yes—
244D4—24no—
245D4—21no—
246D4—2694no—
247D4—5388no—
248D4—5388no—
249D4—5384no—
250D4—5384no—
251D4—5382no—
252D4—5382no—
253D4C2×D107612yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—2692no—
258C2×C4—5388no—
259C2×C4—5382no—
260C2×C4—5382no—
261C2×C4—5382no—
262C2×C4—10762no—
263C2×C4D4×D26911yescharacteristic
264C2×C4D4×D26912yes—
265C2×C4C2×D107612yes—
266C2×C4C23×D26911yescharacteristic
267C8—22no—
268C8—5382no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—5384no—
273C22—5384no—
274C22—5382no—
275C22—10764no—
276C22—21524no—
277C22—21528no—
278C22#2257542312yes—
279C22C2×D4×D26911yescharacteristic
280C4—22no—
281C4—42no—
282C4—5382no—
283C4—10762no—
284C4#2257542312yes—
285C4C2×D4×D26911yescharacteristic
286C4C22×D107611yescharacteristic
287C2—22no—
288C2—42no—
289C2—10762no—
290C2—10764no—
291C2C2×#2257542311yescharacteristic
292C2#3177141212yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#416×C269C2
socleC22×C269C22×D4
prime radicalC2×#22575499C2
hypercenter#416D269
hypocenterC269#3009
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#22575483C2semidirect
3. C22×#16864391C2semidirect
4. #31771181C2semidirect
5. #31770770C2semidirect
6. #31770757C2semidirect
7. #31770609C2semidirect
8. #31770571C2semidirect
9. #416×C269C2semidirect
15. C22×Dic538C22semidirect
21. C2×#16864392C22semidirect
23. C2×#16864391C22semidirect
24. C2×#16864390C22semidirect
25. C2×#16864386C22semidirect
29. #22575433C22semidirect
30. #22575373C22semidirect
31. #22575365C22semidirect
33. C22×C8×C269C22semidirect
34. #117×C269C22semidirect
35. #103×C269C22semidirect
36. #102×C269C22semidirect
146. C269#3009semidirect
155. #416D269semidirect
174. C22×C8C2×D269semidirect
179. #117C2×D269semidirect
180. #103C2×D269semidirect
182. #102C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754992
C2×#225754832
C22×#168643912
#317711812
#317707702
#317707572
#317706092
#317705712
#416×C2692
#3009269