Finite Group Databasebeta

← #46822721

Subgroups — #46822721

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#31771294C211yesmaximal, characteristic, complemented
2#31771243C211yesmaximal, characteristic
3#31770961C211yesmaximal, characteristic, complemented
4#31770934C211yesmaximal, characteristic, complemented
5#31770929C211yesmaximal, characteristic, complemented
6#31770907C211yesmaximal, characteristic, complemented
7#31770845C211yesmaximal, characteristic, complemented
8#31770834C211yesmaximal, characteristic, complemented
9#31770830C211yesmaximal, characteristic, complemented
10#31770767C211yesmaximal, characteristic, complemented
11#31770757C211yesmaximal, characteristic, complemented
12#31770696C211yesmaximal, characteristic, complemented
13C8×C22⋊D269C211yesmaximal, characteristic, complemented
14#102×D269C211yesmaximal, characteristic, complemented
15#416×C269C211yesmaximal, characteristic, complemented
16C2×D4×D269C2211yescharacteristic
17C2×#16864392C2211yescharacteristic, complemented
18C2×#16864391C2211yescharacteristic
19C2×#16864390C2211yescharacteristic
20C2×C8×D269C2211yescharacteristic
21#22575446—21no—
22#22575445—21no—
23#22575444C2211yescharacteristic, complemented
24#22575442C2211yescharacteristic
25#22575436—21no—
26#22575435—21no—
27#22575433C2211yescharacteristic
28#22575432C2211yescharacteristic, complemented
29#22575429C2211yescharacteristic
30#22575427C2211yescharacteristic
31#22575425—21no—
32#22575421—21no—
33#22575418C2211yescharacteristic, complemented
34D4×Dic269C2211yescharacteristic
35C4×C22⋊D269C2211yescharacteristic
36#22575410—21no—
37C4⋊C4×D269C2211yescharacteristic
38C2×#16864379—21no—
39#22575386C2211yescharacteristic, complemented
40#22575385C2211yescharacteristic, complemented
41#22575384C2211yescharacteristic, complemented
42#22575382C2211yescharacteristic, complemented
43#22575378C2211yescharacteristic, complemented
44#22575377C2211yescharacteristic, complemented
45#22575375C2211yescharacteristic
46#22575374C2211yescharacteristic
47#22575373C2211yescharacteristic
48#22575373C2211yescharacteristic
49C22⋊#14524130C2211yescharacteristic, complemented
50#22575365C2211yescharacteristic
51C8×Dic269C2211yescharacteristic, complemented
52C2×#41×C269C2211yescharacteristic
53C22×C8×C269C2211yescharacteristic, complemented
54#117×C269C2211yescharacteristic, complemented
55#103×C269C2211yescharacteristic, complemented
56#103×C269C2211yescharacteristic, complemented
57#102×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic
59C23×D269—21no—
60C2×Dic538C2311yescharacteristic
61C2×D1076C2311yescharacteristic
62C2×C22⋊D269—21no—
63C2×C22⋊D269—21no—
64C2×C22⋊D269D411yescharacteristic
65D4×D269—22no—
66D4×D269—41no—
67C22×Dic269—21no—
68C2×C4×D269—21no—
69C2×C4×D269C2311yescharacteristic
70#16864392—41no—
71#16864391—41no—
72#16864390—22no—
73C2×#14524130C2311yescharacteristic
74C2×#14524130C2311yescharacteristic
75C8×D269—41no—
76C4⋊Dic269—21no—
77C4⋊Dic269—21no—
78C4⋊Dic269C2311yescharacteristic
79C4⋊Dic269C2311yescharacteristic
80#16864381—21no—
81#16864381—21no—
82#16864381—21no—
83#16864381—21no—
84#16864381—21no—
85#16864381D411yescharacteristic
86C22⋊Dic269—21no—
87C22⋊Dic269—21no—
88C22⋊Dic269D411yescharacteristic
89#16864379—21no—
90#16864379—21no—
91#16864379—21no—
92#16864379—21no—
93#16864379—21no—
94#16864379D411yescharacteristic
95C4×Dic269—21no—
96C4×Dic269C2311yescharacteristic
97#41×C269—22no—
98#41×C269—41no—
99C2×Q8×C269C2311yescharacteristic
100C2×D4×C269—21no—
101C2×D4×C269C2311yescharacteristic
102C22×C4×C269—21no—
103C22×C4×C269C2311yescharacteristic
104C2×C8×C269—41no—
105C2×C8×C269C2311yescharacteristic
106C2×C8×C269C2311yescharacteristic
107C4⋊C4×C269C2311yescharacteristic
108C4⋊C4×C269C2311yescharacteristic
109C22⋊C4×C269—21no—
110C42×C269—21no—
111C22×D269—21no—
112C22×D269—42no—
113C22×D269—42no—
114C22×D269—81no—
115C22×D269C2×D411yescharacteristic
116Dic538—21no—
117Dic538—21no—
118D1076—21no—
119D1076—21no—
120C22⋊D269—22no—
121C22⋊D269—42no—
122C22⋊D269—81no—
123C2×Dic269—21no—
124C2×Dic269—21no—
125C2×Dic269—21no—
126C2×Dic269—81no—
127C2×Dic269C2×D411yescharacteristic
128C2×Dic269C2×D411yescharacteristic
129C2×Dic269C2×D411yescharacteristic
130C4×D269—21no—
131C4×D269—21no—
132C4×D269—42no—
133#14524130—21no—
134#14524130—21no—
135C23×C269—21no—
136C23×C269C2×D411yescharacteristic
137Q8×C269—21no—
138Q8×C269—21no—
139D4×C269—21no—
140D4×C269—42no—
141D4×C269#4112yes—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269—42no—
148C2×C4×C269C2×D411yescharacteristic
149C2×C4×C269C2411yescharacteristic
150C8×C269—21no—
151C8×C269—21no—
152C22×C269—21no—
153C22×C269—22no—
154C22×C269—21no—
155C22×C269—21no—
156C22×C269—41no—
157C22×C269—41no—
158C22×C269C22×D411yescharacteristic
159C2×D269—21no—
160C2×D269—21no—
161C2×D269—41no—
162C2×D269—41no—
163C2×D269—41no—
164C2×D269—84no—
165C2×D269#13512yes—
166C4×C269—21no—
167C4×C269—41no—
168C4×C269—41no—
169C4×C269—41no—
170C4×C269C2×#4111yescharacteristic
171C4×C269H(5,𝔽2)11yescharacteristic
172Dic269—21no—
173Dic269—21no—
174Dic269—21no—
175Dic269—41no—
176Dic269—41no—
177Dic269—41no—
178C2×C269—42no—
179C2×C269—41no—
180C2×C269#54611yescharacteristic
181C2×C269C2×#13511yescharacteristic
182C2×C269#57711yescharacteristic
183D269—22no—
184D269—81no—
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#450—2691no—
198C4×SD16—2691no—
199C8×D4—2691no—
200#416D26911yescharacteristic, complemented
201C2×#102—2691no—
202C2×#41C2×D26911yescharacteristic
203C22×D4—2691no—
204C2×Q16—2691no—
205C2×SD16—2691no—
206C2×SD16—2691no—
207C22×C8—2691no—
208C22×C8C2×D26911yescharacteristic, complemented
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⋊Q8—2691no—
218#121—2691no—
219#121—5381no—
220#120—5381no—
221C4×Q8—2691no—
222C4×D4—2691no—
223C4×D4—2691no—
224#117—5381no—
225#117C2×D26911yescharacteristic, complemented
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#103C2×D26911yescharacteristic, complemented
235#103C2×D26911yescharacteristic, complemented
236#102—2691no—
237#102—2691no—
238#102—2691no—
239#102—2691no—
240#102C2×D26911yescharacteristic
241#102C2×D26911yescharacteristic
242C22⋊C8—2691no—
243C22⋊C8—2691no—
244C4×C8—2691no—
245C24—5381no—
246#41—22no—
247#41—41no—
248C2×Q8—2691no—
249C2×Q8C22×D26911yescharacteristic
250C2×D4—21no—
251C2×D4—2691no—
252C2×D4—2691no—
253C2×D4—5381no—
254C2×D4—5382no—
255C2×D4—5381no—
256C2×D4—10761no—
257C2×D4C22×D26911yescharacteristic
258C22×C4—21no—
259C22×C4—2691no—
260C22×C4—5381no—
261C22×C4—5381no—
262C22×C4C22×D26911yescharacteristic
263Q16—10761no—
264SD16—5382no—
265SD16—10761no—
266C2×C8—41no—
267C2×C8—2691no—
268C2×C8—2691no—
269C2×C8—10761no—
270C2×C8C22×D26911yescharacteristic
271C2×C8C22×D26911yescharacteristic
272C4⋊C4—2691no—
273C4⋊C4—2691no—
274C4⋊C4—2691no—
275C4⋊C4—5381no—
276C4⋊C4—5381no—
277C4⋊C4—5381no—
278C4⋊C4—5381no—
279C4⋊C4—5381no—
280C4⋊C4—5381no—
281C4⋊C4—5381no—
282C4⋊C4C22×D26911yescharacteristic
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—21no—
295C42—2691no—
296C42—5381no—
297C23—21no—
298C23—2691no—
299C23—5381no—
300C23—10762no—
301C23—10762no—
302C23—21521no—
303C23D4×D26911yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5381no—
307Q8—5381no—
308D4—21no—
309D4—42no—
310D4—5382no—
311D4—5381no—
312D4—5381no—
313D4—10762no—
314D4—21521no—
315D4#1686440212yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—2691no—
323C2×C4—2691no—
324C2×C4—2691no—
325C2×C4—5381no—
326C2×C4—5381no—
327C2×C4—5381no—
328C2×C4—5381no—
329C2×C4—5381no—
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—21no—
342C22—41no—
343C22—41no—
344C22—2692no—
345C22—5381no—
346C22—5381no—
347C22—10761no—
348C22—10761no—
349C22—10761no—
350C22—21524no—
351C22C2×D4×D26911yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—5381no—
357C4—5381no—
358C4—5381no—
359C4—10761no—
360C4—10761no—
361C4—10761no—
362C4C2×#1686440211yescharacteristic
363C4#2257549811yescharacteristic
364C2—41no—
365C2—42no—
366C2—5382no—
367C2—21521no—
368C2#3177124511yescharacteristic
369C2#135×D26911yescharacteristic
370C2#3177141211yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#416×C269C2
socleC22×C269C22×D4
prime radical#31771243C2
hypercenter#416D269
hypocenterC269#3296
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
1. #31771294C2semidirect
3. #31770961C2semidirect
4. #31770934C2semidirect
5. #31770929C2semidirect
6. #31770907C2semidirect
7. #31770845C2semidirect
8. #31770834C2semidirect
9. #31770830C2semidirect
10. #31770767C2semidirect
11. #31770757C2semidirect
12. #31770696C2semidirect
13. C8×C22⋊D269C2semidirect
14. #102×D269C2semidirect
15. #416×C269C2semidirect
17. C2×#16864392C22semidirect
23. #22575444C22semidirect
28. #22575432C22semidirect
33. #22575418C22semidirect
39. #22575386C22semidirect
40. #22575385C22semidirect
41. #22575384C22semidirect
42. #22575382C22semidirect
43. #22575378C22semidirect
44. #22575377C22semidirect
49. C22⋊#14524130C22semidirect
51. C8×Dic269C22semidirect
53. C22×C8×C269C22semidirect
54. #117×C269C22semidirect
55. #103×C269C22semidirect
56. #103×C269C22semidirect
185. C269#3296semidirect
200. #416D269semidirect
208. C22×C8C2×D269semidirect
225. #117C2×D269semidirect
234. #103C2×D269semidirect
235. #103C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712942
#317712432
#317709612
#317709342
#317709292
#317709072
#317708452
#317708342
#317708302
#317707672
#317707572
#317706962
C8×C22⋊D2692
#102×D2692
#416×C2692
#3296269