Finite Group Databasebeta

← #46822724

Subgroups — #46822724

Order 34432.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771297C211yesmaximal, characteristic, complemented
2#31771249C211yesmaximal, characteristic
3#31770958C211yesmaximal, characteristic, complemented
4#31770936C211yesmaximal, characteristic, complemented
5#31770928C211yesmaximal, characteristic, complemented
6#31770907C211yesmaximal, characteristic, complemented
7#31770844C211yesmaximal, characteristic, complemented
8#31770831C211yesmaximal, characteristic, complemented
9#31770825C211yesmaximal, characteristic, complemented
10#31770764C211yesmaximal, characteristic, complemented
11SD16×Dic269C211yesmaximal, characteristic, complemented
12#31770668C211yesmaximal, characteristic, complemented
13#31770598C211yesmaximal, characteristic, complemented
14C2×#22575374C211yesmaximal, characteristic, complemented
15#450×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic
17C2×#16864392C2211yescharacteristic, complemented
18C22×#14524130C2211yescharacteristic
19C2×C8×D269C2211yescharacteristic, complemented
20#22575448—21no—
21#22575444—21no—
22#22575444C2211yescharacteristic, complemented
23#22575440C2211yescharacteristic
24#22575437—21no—
25#22575434—21no—
26#22575432C2211yescharacteristic
27#22575429C2211yescharacteristic, complemented
28#22575425—21no—
29#22575424C2211yescharacteristic
30#22575422—21no—
31Q8×Dic269C2211yescharacteristic, complemented
32D4×Dic269C2211yescharacteristic
33C4×C22⋊D269C2211yescharacteristic
34#22575411C2211yescharacteristic, complemented
35#22575409—21no—
36C2×C4⋊Dic269C2211yescharacteristic
37C2×C22⋊Dic269—21no—
38#22575386C2211yescharacteristic, complemented
39#22575385C2211yescharacteristic, complemented
40#22575385C2211yescharacteristic, complemented
41#22575382C2211yescharacteristic, complemented
42#22575378C2211yescharacteristic, complemented
43#22575378C2211yescharacteristic, complemented
44#22575377C2211yescharacteristic, complemented
45#22575376C2211yescharacteristic, complemented
46#22575374C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575374C2211yescharacteristic
49#22575374C2211yescharacteristic
50#22575365C2211yescharacteristic, complemented
51C8×Dic269C2211yescharacteristic, complemented
52C22×D4×C269C2211yescharacteristic
53C2×SD16×C269C2211yescharacteristic
54C2×SD16×C269C2211yescharacteristic
55C22⋊Q8×C269C2211yescharacteristic
56#102×C269C2211yescharacteristic
57#102×C269C2211yescharacteristic
58C22⋊C8×C269C2211yescharacteristic
59#16864401—22no—
60#16864401—41no—
61C2×Dic538C2311yescharacteristic
62C2×C22⋊D269—21no—
63C2×C22⋊D269—21no—
64C2×C22⋊D269D411yescharacteristic
65C22×Dic269—21no—
66C22×Dic269—21no—
67C22×Dic269—21no—
68C2×C4×D269C2311yescharacteristic
69#16864392—41no—
70C2×#14524130—41no—
71C2×#14524130C2311yescharacteristic
72C2×#14524130C2311yescharacteristic
73C8×D269—41no—
74C4⋊Dic269—21no—
75C4⋊Dic269—21no—
76C4⋊Dic269—21no—
77C4⋊Dic269C2311yescharacteristic
78C4⋊Dic269C2311yescharacteristic
79C4⋊Dic269C2311yescharacteristic
80#16864381—21no—
81#16864381—21no—
82#16864381—21no—
83#16864381D411yescharacteristic
84C22⋊Dic269—21no—
85C22⋊Dic269—21no—
86C22⋊Dic269—21no—
87C22⋊Dic269—21no—
88C22⋊Dic269—21no—
89C22⋊Dic269D411yescharacteristic
90#16864379—21no—
91#16864379—21no—
92#16864379D411yescharacteristic
93C4×Dic269—21no—
94C4×Dic269—21no—
95C4×Dic269C2311yescharacteristic
96C24×C269—21no—
97C2×Q8×C269C2311yescharacteristic
98C2×D4×C269—22no—
99C2×D4×C269—21no—
100C2×D4×C269—41no—
101C2×D4×C269C2311yescharacteristic
102C2×D4×C269C2311yescharacteristic
103C22×C4×C269C2311yescharacteristic
104SD16×C269—22no—
105SD16×C269—41no—
106C2×C8×C269C2311yescharacteristic
107C2×C8×C269C2311yescharacteristic
108C4⋊C4×C269—21no—
109C4⋊C4×C269C2311yescharacteristic
110C22⋊C4×C269—21no—
111C22×D269C2×D411yescharacteristic
112Dic538—21no—
113Dic538—21no—
114C22⋊D269—22no—
115C22⋊D269—42no—
116C22⋊D269—81no—
117C2×Dic269—21no—
118C2×Dic269—21no—
119C2×Dic269—21no—
120C2×Dic269—21no—
121C2×Dic269—42no—
122C2×Dic269—42no—
123C2×Dic269—81no—
124C2×Dic269C2×D411yescharacteristic
125C2×Dic269C2×D411yescharacteristic
126C2×Dic269C2×D411yescharacteristic
127C4×D269—21no—
128C4×D269—21no—
129#14524130—21no—
130#14524130—21no—
131C23×C269—21no—
132C23×C269—21no—
133C23×C269—42no—
134C23×C269—42no—
135C23×C269—81no—
136C23×C269C2×D411yescharacteristic
137Q8×C269—21no—
138Q8×C269—21no—
139D4×C269—21no—
140D4×C269—21no—
141D4×C269—21no—
142D4×C269—42no—
143D4×C269#4112yes—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269—21no—
148C2×C4×C269C2×D411yescharacteristic
149C2×C4×C269C2411yescharacteristic
150C8×C269—21no—
151C8×C269—21no—
152C22×C269—21no—
153C22×C269—21no—
154C22×C269—22no—
155C22×C269—41no—
156C22×C269—41no—
157C22×C269—41no—
158C22×C269—41no—
159C22×C269—41no—
160C22×C269—84no—
161C22×C269#13512yes—
162C22×C269C22×D411yescharacteristic
163C2×D269—21no—
164C2×D269—21no—
165C2×D269—21no—
166C4×C269—21no—
167C4×C269—41no—
168C4×C269—41no—
169C4×C269C2×#4111yescharacteristic
170C4×C269H(5,𝔽2)11yescharacteristic
171Dic269—21no—
172Dic269—21no—
173Dic269—21no—
174Dic269—41no—
175Dic269—41no—
176Dic269—41no—
177Dic269—41no—
178C2×C269—22no—
179C2×C269—42no—
180C2×C269—81no—
181C2×C269#54611yescharacteristic
182C2×C269C2×#13511yescharacteristic
183C2×C269#57711yescharacteristic
184D269—41no—
185C269#329611yescharacteristic, complemented, Sylow
186#3296—2691nomaximal, Sylow
187#552—2691no—
188#546—2691no—
189#488—2691no—
190#483—2691no—
191#482—2691no—
192#478—2691no—
193#467—2691no—
194#465—2691no—
195#464—2691no—
196#452—2691no—
197#450D26911yescharacteristic, complemented
198C4×SD16—2691no—
199C8×D4—2691no—
200#416—2691no—
201C2×#102—2691no—
202C2×#41—2691no—
203C22×D4C2×D26911yescharacteristic
204C2×Q16—2691no—
205C2×SD16C2×D26911yescharacteristic
206C2×SD16C2×D26911yescharacteristic
207C22×C8—2691no—
208C22×C8—2691no—
209#126—5381no—
210#125—2691no—
211#125—5381no—
212#124—2691no—
213#123—5381no—
214#123—5381no—
215C22⋊Q8—2691no—
216C22⋊Q8—2691no—
217C22⋊Q8C2×D26911yescharacteristic
218#121—2691no—
219#121—5381no—
220#120—5381no—
221C4×Q8—2691no—
222C4×D4—2691no—
223C4×D4—2691no—
224#117—2691no—
225#117—5381no—
226C2×C4⋊C4—2691no—
227C2×C22⋊C4—5381no—
228#107—2691no—
229#106—2691no—
230#106—2691no—
231C4⋊C8—2691no—
232#103—2691no—
233#103—2691no—
234#103—2691no—
235#103—2691no—
236#102—2691no—
237#102—2691no—
238#102—2691no—
239#102—2691no—
240#102C2×D26911yescharacteristic
241#102C2×D26911yescharacteristic
242C22⋊C8—2691no—
243C22⋊C8C2×D26911yescharacteristic
244C4×C8—2691no—
245C24—21no—
246#41—5382no—
247#41—10761no—
248C2×Q8—2691no—
249C2×Q8C22×D26911yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—2691no—
254C2×D4—5381no—
255C2×D4—5381no—
256C2×D4C22×D26911yescharacteristic
257C2×D4C22×D26911yescharacteristic
258C22×C4—2691no—
259C22×C4—5381no—
260C22×C4—5381no—
261C22×C4—5381no—
262C22×C4C22×D26911yescharacteristic
263Q16—10761no—
264SD16—22no—
265SD16—41no—
266C2×C8—2691no—
267C2×C8—2691no—
268C2×C8—10761no—
269C2×C8—10761no—
270C2×C8C22×D26911yescharacteristic
271C2×C8C22×D26911yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2691no—
274C4⋊C4—2691no—
275C4⋊C4—2691no—
276C4⋊C4—2691no—
277C4⋊C4—5381no—
278C4⋊C4—5381no—
279C4⋊C4—5381no—
280C4⋊C4—5381no—
281C4⋊C4—5381no—
282C4⋊C4—5381no—
283C4⋊C4C22×D26911yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—2691no—
286C22⋊C4—2691no—
287C22⋊C4—5381no—
288C22⋊C4—5381no—
289C22⋊C4—5381no—
290C22⋊C4—5381no—
291C22⋊C4—5381no—
292C22⋊C4—5381no—
293C22⋊C4—5381no—
294C42—2691no—
295C42—5381no—
296C42—5381no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—2691no—
303C23D4×D26911yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5381no—
307Q8—5381no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—5382no—
313D4—10762no—
314D4—21521no—
315D4#1686440112yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—2691no—
321C2×C4—2691no—
322C2×C4—2691no—
323C2×C4—5381no—
324C2×C4—5381no—
325C2×C4—5381no—
326C2×C4—5381no—
327C2×C4—5381no—
328C2×C4—5381no—
329C2×C4—10762no—
330C2×C4—10762no—
331C2×C4—21521no—
332C2×C4D4×D26911yescharacteristic
333C2×C4C23×D26911yescharacteristic
334C8—21no—
335C8—21no—
336C8—5381no—
337C8—5381no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—5381no—
348C22—5381no—
349C22—5381no—
350C22#2257547412yes—
351C22C2×D4×D26911yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—5381no—
356C4—5381no—
357C4—5381no—
358C4—10761no—
359C4—10761no—
360C4—10761no—
361C4—10761no—
362C4C2×#1686440111yescharacteristic
363C4#2257549811yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—10761no—
368C2#3177124211yescharacteristic
369C2C2×#2257547411yescharacteristic
370C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#450×C269C2
socleC22×C269C22×D4
prime radical#31771249C2
hypercenter#450D269
hypocenterC269#3296
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #31771297C2semidirect
3. #31770958C2semidirect
4. #31770936C2semidirect
5. #31770928C2semidirect
6. #31770907C2semidirect
7. #31770844C2semidirect
8. #31770831C2semidirect
9. #31770825C2semidirect
10. #31770764C2semidirect
11. SD16×Dic269C2semidirect
12. #31770668C2semidirect
13. #31770598C2semidirect
14. C2×#22575374C2semidirect
15. #450×C269C2semidirect
17. C2×#16864392C22semidirect
19. C2×C8×D269C22semidirect
22. #22575444C22semidirect
27. #22575429C22semidirect
31. Q8×Dic269C22semidirect
34. #22575411C22semidirect
38. #22575386C22semidirect
39. #22575385C22semidirect
40. #22575385C22semidirect
41. #22575382C22semidirect
42. #22575378C22semidirect
43. #22575378C22semidirect
44. #22575377C22semidirect
45. #22575376C22semidirect
50. #22575365C22semidirect
51. C8×Dic269C22semidirect
185. C269#3296semidirect
197. #450D269semidirect

Maximal subgroups

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

subgroupindex
#317712972
#317712492
#317709582
#317709362
#317709282
#317709072
#317708442
#317708312
#317708252
#317707642
SD16×Dic2692
#317706682
#317705982
C2×#225753742
#450×C2692
#3296269