Finite Group Databasebeta

← #46822720

Subgroups — #46822720

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#31771295C211yesmaximal, characteristic, complemented
2#31770959C211yesmaximal, characteristic, complemented
3#31770933C211yesmaximal, characteristic, complemented
4#31770926C211yesmaximal, characteristic, complemented
5#31770905C211yesmaximal, characteristic, complemented
6#31770844C211yesmaximal, characteristic, complemented
7#31770832C211yesmaximal, characteristic, complemented
8#31770824C211yesmaximal, characteristic, complemented
9#31770767C211yesmaximal, characteristic, complemented
10#31770756C211yesmaximal, characteristic, complemented
11C4×#16864390C211yesmaximal, characteristic, complemented
12D4×#14524130C211yesmaximal, characteristic, complemented
13#31770601C211yesmaximal, characteristic, complemented
14C2×#22575374C211yesmaximal, characteristic, complemented
15#546×C269C211yesmaximal, characteristic
16C2×#16864392C2211yescharacteristic, complemented
17C2×#16864390C2211yescharacteristic
18C2×#16864390C2211yescharacteristic
19C22×#14524130C2211yescharacteristic
20C22×#14524130C2211yescharacteristic, complemented
21#22575446—21no—
22#22575445C2211yescharacteristic, complemented
23#22575443—21no—
24#22575434—21no—
25#22575432C2211yescharacteristic
26#22575432C2211yescharacteristic, complemented
27C4×Dic538C2211yescharacteristic, complemented
28#22575412C2211yescharacteristic, complemented
29#22575409—21no—
30C2×C4⋊Dic269C2211yescharacteristic
31#22575386C2211yescharacteristic, complemented
32#22575384C2211yescharacteristic, complemented
33#22575384C2211yescharacteristic, complemented
34C4⋊#14524130C2211yescharacteristic, complemented
35#22575377C2211yescharacteristic, complemented
36#22575377C2211yescharacteristic, complemented
37#22575377C2211yescharacteristic, complemented
38#22575376C2211yescharacteristic, complemented
39#22575374C2211yescharacteristic
40#22575374C2211yescharacteristic
41#22575374C2211yescharacteristic
42#22575374C2211yescharacteristic
43#22575374C2211yescharacteristic
44#22575374C2211yescharacteristic
45C22⋊#14524130C2211yescharacteristic
46C22⋊#14524130C2211yescharacteristic, complemented
47C4×#14524130C2211yescharacteristic, complemented
48C2×#41×C269C2211yescharacteristic
49C22×D4×C269C2211yescharacteristic
50#124×C269C2211yescharacteristic
51#123×C269—21no—
52C22⋊Q8×C269C2211yescharacteristic
53#121×C269—21no—
54#121×C269C2211yescharacteristic
55#120×C269—21no—
56C4×D4×C269C2211yescharacteristic
57C4×D4×C269C2211yescharacteristic
58C2×C22⋊C4×C269—21no—
59C2×Dic538C2311yescharacteristic
60C22×Dic269—21no—
61#16864392—41no—
62#16864390—22no—
63#16864390—41no—
64C2×#14524130—41no—
65C2×#14524130—41no—
66C2×#14524130C2311yescharacteristic
67C2×#14524130C2311yescharacteristic
68C2×#14524130C2311yescharacteristic
69C2×#14524130C2311yescharacteristic
70C4⋊Dic269—21no—
71C4⋊Dic269C2311yescharacteristic
72C4⋊Dic269C2311yescharacteristic
73C4⋊Dic269C2311yescharacteristic
74#16864381—21no—
75#16864381—21no—
76#16864381—21no—
77#16864381—21no—
78#16864381—21no—
79C22⋊Dic269—21no—
80C22⋊Dic269—21no—
81C22⋊Dic269—21no—
82C4×Dic269—21no—
83C4×Dic269—21no—
84C24×C269—21no—
85#41×C269—22no—
86#41×C269—41no—
87C2×Q8×C269C2311yescharacteristic
88C2×D4×C269—22no—
89C2×D4×C269—21no—
90C2×D4×C269—21no—
91C2×D4×C269—21no—
92C2×D4×C269—41no—
93C2×D4×C269D411yescharacteristic
94C2×D4×C269C2311yescharacteristic
95C2×D4×C269C2311yescharacteristic
96C22×C4×C269—21no—
97C22×C4×C269—21no—
98C22×C4×C269C2311yescharacteristic
99C22×C4×C269C2311yescharacteristic
100C4⋊C4×C269—21no—
101C4⋊C4×C269D411yescharacteristic
102C4⋊C4×C269C2311yescharacteristic
103C22⋊C4×C269—21no—
104C22⋊C4×C269—21no—
105C22⋊C4×C269—21no—
106C22⋊C4×C269—21no—
107C22⋊C4×C269—21no—
108C22⋊C4×C269D411yescharacteristic
109C22⋊C4×C269D411yescharacteristic
110C42×C269C2311yescharacteristic
111Dic538—21no—
112Dic538—21no—
113C2×Dic269—21no—
114C2×Dic269—21no—
115C2×Dic269—21no—
116C2×Dic269—21no—
117C2×Dic269—42no—
118#14524130—21no—
119#14524130—21no—
120#14524130—21no—
121#14524130—21no—
122C23×C269—21no—
123C23×C269—21no—
124C23×C269—42no—
125C23×C269—42no—
126C23×C269—81no—
127C23×C269C2×D411yescharacteristic
128C23×C269C2×D411yescharacteristic
129Q8×C269—21no—
130Q8×C269—21no—
131D4×C269—22no—
132D4×C269—21no—
133D4×C269—21no—
134D4×C269—21no—
135D4×C269—42no—
136D4×C269—42no—
137D4×C269—81no—
138D4×C269#4112yes—
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—42no—
146C2×C4×C269—81no—
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—22no—
156C22×C269—21no—
157C22×C269—21no—
158C22×C269—41no—
159C22×C269—41no—
160C22×C269—41no—
161C22×C269—41no—
162C22×C269—41no—
163C22×C269—84no—
164C22×C269#13512yes—
165C22×C269C22×D411yescharacteristic
166C4×C269—21no—
167C4×C269—21no—
168C4×C269—21no—
169C4×C269—21no—
170C4×C269—41no—
171C4×C269—41no—
172C4×C269C2×#4111yescharacteristic
173C4×C269H(5,𝔽2)11yescharacteristic
174Dic269—41no—
175Dic269—41no—
176Dic269—41no—
177Dic269—41no—
178C2×C269—22no—
179C2×C269—41no—
180C2×C269—42no—
181C2×C269—81no—
182C2×C269#54611yescharacteristic
183C2×C269C2×#13511yescharacteristic
184C2×C269#57711yescharacteristic
185C269#329611yescharacteristic, complemented, Sylow
186#3296—2691nomaximal, Sylow
187#552—2691no—
188#546D26911yescharacteristic
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×#41C2×D26911yescharacteristic
203C22×D4C2×D26911yescharacteristic
204C2×Q16—2691no—
205C2×SD16—2691no—
206C2×SD16—2691no—
207C22×C8—2691no—
208C22×C8—2691no—
209#126—5381no—
210#125—2691no—
211#125—5381no—
212#124C2×D26911yescharacteristic
213#123—21no—
214#123—5381no—
215C22⋊Q8—2691no—
216C22⋊Q8—2691no—
217C22⋊Q8C2×D26911yescharacteristic
218#121—21no—
219#121C2×D26911yescharacteristic
220#120—21no—
221C4×Q8—2691no—
222C4×D4C2×D26911yescharacteristic
223C4×D4C2×D26911yescharacteristic
224#117—2691no—
225#117—5381no—
226C2×C4⋊C4—2691no—
227C2×C22⋊C4—21no—
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—21no—
246#41—22no—
247#41—41no—
248C2×Q8—2691no—
249C2×Q8C22×D26911yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4C22⋊D26911yescharacteristic
256C2×D4C22×D26911yescharacteristic
257C2×D4C22×D26911yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
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—2691no—
274C4⋊C4—2691no—
275C4⋊C4—2691no—
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—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—5381no—
290C22⋊C4—5381no—
291C22⋊C4—5381no—
292C22⋊C4C22⋊D26911yescharacteristic
293C22⋊C4C22⋊D26911yescharacteristic
294C42—5381no—
295C42—5381no—
296C42C22×D26911yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D26911yescharacteristic
303C23C2×C22⋊D26911yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5381no—
307Q8—5381no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#1686440112yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—5381no—
325C2×C4—5381no—
326C2×C4—5381no—
327C2×C4—5381no—
328C2×C4—10762no—
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—22no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22#2257547312yes—
351C22C22×C22⋊D26911yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—10761no—
359C4—10761no—
360C4—10761no—
361C4—10761no—
362C4C2×#1686440111yescharacteristic
363C4#2257549811yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#3177124911yescharacteristic
369C2C2×#2257547311yescharacteristic
370C2#3177141611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting, prime radical#546×C269C2
socleC22×C269C22×D4
hypercenter#546D269
hypocenterC269#3296
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #31771295C2semidirect
2. #31770959C2semidirect
3. #31770933C2semidirect
4. #31770926C2semidirect
5. #31770905C2semidirect
6. #31770844C2semidirect
7. #31770832C2semidirect
8. #31770824C2semidirect
9. #31770767C2semidirect
10. #31770756C2semidirect
11. C4×#16864390C2semidirect
12. D4×#14524130C2semidirect
13. #31770601C2semidirect
14. C2×#22575374C2semidirect
16. C2×#16864392C22semidirect
20. C22×#14524130C22semidirect
22. #22575445C22semidirect
26. #22575432C22semidirect
27. C4×Dic538C22semidirect
28. #22575412C22semidirect
31. #22575386C22semidirect
32. #22575384C22semidirect
33. #22575384C22semidirect
34. C4⋊#14524130C22semidirect
35. #22575377C22semidirect
36. #22575377C22semidirect
37. #22575377C22semidirect
38. #22575376C22semidirect
46. C22⋊#14524130C22semidirect
47. C4×#14524130C22semidirect
185. C269#3296semidirect

Maximal subgroups

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

subgroupindex
#317712952
#317709592
#317709332
#317709262
#317709052
#317708442
#317708322
#317708242
#317707672
#317707562
C4×#168643902
D4×#145241302
#317706012
C2×#225753742
#546×C2692
#3296269