Finite Group Databasebeta

← #46822714

Subgroups — #46822714

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#31771251C211yesmaximal, characteristic
2#31770962C211yesmaximal, characteristic, complemented
3#31770935C211yesmaximal, characteristic, complemented
4#31770929C211yesmaximal, characteristic, complemented
5#31770902C211yesmaximal, characteristic, complemented
6#31770844C211yesmaximal, characteristic, complemented
7#31770836C211yesmaximal, characteristic, complemented
8#31770827C211yesmaximal, characteristic, complemented
9#31770765C211yesmaximal, characteristic, complemented
10#31770755C211yesmaximal, characteristic, complemented
11C4×#16864389C211yesmaximal, characteristic, complemented
12D4×#14524130C211yesmaximal, characteristic, complemented
13#31770599C211yesmaximal, characteristic, complemented
14C2×#22575373C211yesmaximal, characteristic, complemented
15#552×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic
17C22×D1076C2211yescharacteristic
18C2×#16864392C2211yescharacteristic, complemented
19C2×#16864389C2211yescharacteristic
20C2×#16864389C2211yescharacteristic
21C22×#14524130C2211yescharacteristic
22C22×#14524130C2211yescharacteristic, complemented
23#22575441C2211yescharacteristic
24#22575438—21no—
25#22575432C2211yescharacteristic
26#22575427C2211yescharacteristic
27#22575426—21no—
28#22575421—21no—
29C4×D1076C2211yescharacteristic
30C22⋊C4×D269—21no—
31#22575386C2211yescharacteristic, complemented
32#22575384C2211yescharacteristic, complemented
33#22575384C2211yescharacteristic, complemented
34C4⋊#14524130C2211yescharacteristic, complemented
35#22575377C2211yescharacteristic, complemented
36#22575376C2211yescharacteristic, complemented
37#22575376C2211yescharacteristic, complemented
38#22575376C2211yescharacteristic, complemented
39#22575373C2211yescharacteristic
40#22575373C2211yescharacteristic
41#22575373C2211yescharacteristic
42#22575373C2211yescharacteristic
43#22575373C2211yescharacteristic
44#22575373C2211yescharacteristic
45C22⋊#14524130C2211yescharacteristic
46C22⋊#14524130C2211yescharacteristic, complemented
47C4×#14524130C2211yescharacteristic, complemented
48#126×C269—21no—
49#125×C269—21no—
50#125×C269C2211yescharacteristic, complemented
51#123×C269—21no—
52C22⋊Q8×C269C2211yescharacteristic
53C22⋊Q8×C269C2211yescharacteristic, complemented
54C4×Q8×C269C2211yescharacteristic, complemented
55C4×D4×C269C2211yescharacteristic
56#117×C269—21no—
57#117×C269C2211yescharacteristic, complemented
58C2×C4⋊C4×C269C2211yescharacteristic
59C23×D269—21no—
60#16864402—22no—
61#16864402—41no—
62C2×Dic538C2311yescharacteristic
63C2×D1076—22no—
64C2×D1076—21no—
65C2×D1076—41no—
66C2×D1076C2311yescharacteristic
67C2×D1076C2311yescharacteristic
68C2×C22⋊D269—21no—
69C2×C22⋊D269—21no—
70C2×C4×D269—21no—
71C2×C4×D269—21no—
72#16864392—41no—
73#16864389—22no—
74#16864389—41no—
75C2×#14524130—41no—
76C2×#14524130—41no—
77C2×#14524130C2311yescharacteristic
78C2×#14524130C2311yescharacteristic
79C2×#14524130C2311yescharacteristic
80C2×#14524130C2311yescharacteristic
81C4⋊Dic269C2311yescharacteristic
82#16864381—21no—
83C22⋊Dic269—21no—
84#16864379—21no—
85#16864379—21no—
86#16864379—21no—
87#16864379—21no—
88C2×Q8×C269C2311yescharacteristic
89C2×D4×C269D411yescharacteristic
90C22×C4×C269—21no—
91C22×C4×C269C2311yescharacteristic
92C22×C4×C269C2311yescharacteristic
93C4⋊C4×C269—21no—
94C4⋊C4×C269—21no—
95C4⋊C4×C269—21no—
96C4⋊C4×C269—21no—
97C4⋊C4×C269—21no—
98C4⋊C4×C269—21no—
99C4⋊C4×C269D411yescharacteristic
100C4⋊C4×C269C2311yescharacteristic
101C4⋊C4×C269C2311yescharacteristic
102C4⋊C4×C269C2311yescharacteristic
103C22⋊C4×C269—21no—
104C22⋊C4×C269—21no—
105C22⋊C4×C269—21no—
106C22⋊C4×C269D411yescharacteristic
107C22⋊C4×C269D411yescharacteristic
108C42×C269—21no—
109C42×C269—21no—
110C42×C269C2311yescharacteristic
111C22×D269—21no—
112C22×D269—21no—
113C22×D269—42no—
114C22×D269—42no—
115C22×D269—81no—
116Dic538—21no—
117Dic538—21no—
118D1076—21no—
119D1076—21no—
120D1076—21no—
121D1076—42no—
122D1076#4112yes—
123C22⋊D269—42no—
124C22⋊D269—81no—
125C2×Dic269—21no—
126C2×Dic269—21no—
127C4×D269—42no—
128C4×D269—81no—
129#14524130—21no—
130#14524130—21no—
131#14524130—21no—
132#14524130—21no—
133C23×C269C2×D411yescharacteristic
134C23×C269C2×D411yescharacteristic
135Q8×C269—21no—
136Q8×C269—21no—
137D4×C269—22no—
138C2×C4×C269—21no—
139C2×C4×C269—21no—
140C2×C4×C269—21no—
141C2×C4×C269—21no—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—42no—
147C2×C4×C269C2×D411yescharacteristic
148C2×C4×C269C2×D411yescharacteristic
149C2×C4×C269C2×D411yescharacteristic
150C2×C4×C269C2×D411yescharacteristic
151C2×C4×C269C2411yescharacteristic
152C22×C269—21no—
153C22×C269—21no—
154C22×C269—21no—
155C22×C269—21no—
156C22×C269—21no—
157C22×C269#13512yes—
158C22×C269C22×D411yescharacteristic
159C2×D269—22no—
160C2×D269—41no—
161C2×D269—41no—
162C2×D269—41no—
163C2×D269—41no—
164C2×D269—41no—
165C2×D269—84no—
166C4×C269—21no—
167C4×C269—21no—
168C4×C269—21no—
169C4×C269—21no—
170C4×C269—41no—
171C4×C269—41no—
172C4×C269—41no—
173C4×C269—41no—
174C4×C269C2×#4111yescharacteristic
175C4×C269H(5,𝔽2)11yescharacteristic
176Dic269—41no—
177Dic269—41no—
178C2×C269—22no—
179C2×C269—41no—
180C2×C269#54611yescharacteristic
181C2×C269C2×#13511yescharacteristic
182C2×C269#57711yescharacteristic
183D269—42no—
184D269—81no—
185C269#329611yescharacteristic, complemented, Sylow
186#3296—2691nomaximal, Sylow
187#552D26911yescharacteristic, complemented
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#416—2691no—
201C2×#102—2691no—
202C2×#41—2691no—
203C22×D4—2691no—
204C2×Q16—2691no—
205C2×SD16—2691no—
206C2×SD16—2691no—
207C22×C8—2691no—
208C22×C8—2691no—
209#126—21no—
210#125—21no—
211#125C2×D26911yescharacteristic, complemented
212#124—2691no—
213#123—21no—
214#123—5381no—
215C22⋊Q8—2691no—
216C22⋊Q8C2×D26911yescharacteristic
217C22⋊Q8C2×D26911yescharacteristic, complemented
218#121—2691no—
219#121—5381no—
220#120—5381no—
221C4×Q8C2×D26911yescharacteristic, complemented
222C4×D4—2691no—
223C4×D4C2×D26911yescharacteristic
224#117—21no—
225#117C2×D26911yescharacteristic, complemented
226C2×C4⋊C4C2×D26911yescharacteristic
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#102—2691no—
241#102—2691no—
242C22⋊C8—2691no—
243C22⋊C8—2691no—
244C4×C8—2691no—
245C24—5381no—
246#41—5382no—
247#41—10761no—
248C2×Q8—2691no—
249C2×Q8C22×D26911yescharacteristic
250C2×D4—2691no—
251C2×D4—2691no—
252C2×D4—5382no—
253C2×D4—5381no—
254C2×D4—5381no—
255C2×D4—5381no—
256C2×D4—10761no—
257C2×D4C22⋊D26911yescharacteristic
258C22×C4—21no—
259C22×C4—5381no—
260C22×C4—5381no—
261C22×C4C22×D26911yescharacteristic
262C22×C4C22×D26911yescharacteristic
263Q16—10761no—
264SD16—5382no—
265SD16—10761no—
266C2×C8—2691no—
267C2×C8—2691no—
268C2×C8—2691no—
269C2×C8—2691no—
270C2×C8—10761no—
271C2×C8—10761no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—2691no—
279C4⋊C4—5381no—
280C4⋊C4C22⋊D26911yescharacteristic
281C4⋊C4C22×D26911yescharacteristic
282C4⋊C4C22×D26911yescharacteristic
283C4⋊C4C22×D26911yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—5381no—
288C22⋊C4—5381no—
289C22⋊C4—5381no—
290C22⋊C4—5381no—
291C22⋊C4—5381no—
292C22⋊C4C22⋊D26911yescharacteristic
293C22⋊C4C22⋊D26911yescharacteristic
294C42—21no—
295C42—21no—
296C42C22×D26911yescharacteristic
297C23—5381no—
298C23—5381no—
299C23—10762no—
300C23—10762no—
301C23—21521no—
302C23C2×C22⋊D26911yescharacteristic
303C23C2×C22⋊D26911yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5381no—
307Q8—5381no—
308D4—22no—
309D4—2692no—
310D4—5381no—
311D4—5381no—
312D4—5381no—
313D4—10762no—
314D4—10762no—
315D4—21521no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—5381no—
326C2×C4—5381no—
327C2×C4—10762no—
328C2×C4—21521no—
329C2×C4C2×C22⋊D26911yescharacteristic
330C2×C4C2×C22⋊D26911yescharacteristic
331C2×C4C2×C22⋊D26911yescharacteristic
332C2×C4C2×C22⋊D26911yescharacteristic
333C2×C4C23×D26911yescharacteristic
334C8—5381no—
335C8—5381no—
336C8—5381no—
337C8—5381no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—5382no—
344C22—10761no—
345C22—10761no—
346C22—10761no—
347C22—10761no—
348C22—10761no—
349C22—21524no—
350C22#2257547312yes—
351C22C22×C22⋊D26911yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—10761no—
361C4—10761no—
362C4#41×D26911yescharacteristic
363C4#2257549911yescharacteristic
364C2—22no—
365C2—41no—
366C2—10762no—
367C2—21521no—
368C2#3177124311yescharacteristic
369C2C2×#2257547311yescharacteristic
370C2#3177141311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#552×C269C2
socleC22×C269C22×D4
prime radical#31771251C2
hypercenter#552D269
hypocenterC269#3296
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #31770962C2semidirect
3. #31770935C2semidirect
4. #31770929C2semidirect
5. #31770902C2semidirect
6. #31770844C2semidirect
7. #31770836C2semidirect
8. #31770827C2semidirect
9. #31770765C2semidirect
10. #31770755C2semidirect
11. C4×#16864389C2semidirect
12. D4×#14524130C2semidirect
13. #31770599C2semidirect
14. C2×#22575373C2semidirect
15. #552×C269C2semidirect
18. C2×#16864392C22semidirect
22. C22×#14524130C22semidirect
31. #22575386C22semidirect
32. #22575384C22semidirect
33. #22575384C22semidirect
34. C4⋊#14524130C22semidirect
35. #22575377C22semidirect
36. #22575376C22semidirect
37. #22575376C22semidirect
38. #22575376C22semidirect
46. C22⋊#14524130C22semidirect
47. C4×#14524130C22semidirect
50. #125×C269C22semidirect
53. C22⋊Q8×C269C22semidirect
54. C4×Q8×C269C22semidirect
57. #117×C269C22semidirect
185. C269#3296semidirect
187. #552D269semidirect
211. #125C2×D269semidirect
217. C22⋊Q8C2×D269semidirect
221. C4×Q8C2×D269semidirect
225. #117C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712512
#317709622
#317709352
#317709292
#317709022
#317708442
#317708362
#317708272
#317707652
#317707552
C4×#168643892
D4×#145241302
#317705992
C2×#225753732
#552×C2692
#3296269