Finite Group Databasebeta

← #46822960

Subgroups — #46822960

Order 34432.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771251C211yesmaximal, characteristic
2#31770967C211yesmaximal, characteristic, complemented
3#31770940C211yesmaximal, characteristic, complemented
4#31770918C211yesmaximal, characteristic, complemented
5#31770882C211yesmaximal, characteristic, complemented
6#31770853C211yesmaximal, characteristic, complemented
7#31770844C211yesmaximal, characteristic, complemented
8#31770827C211yesmaximal, characteristic, complemented
9#31770765C211yesmaximal, characteristic, complemented
10#31770755C211yesmaximal, characteristic, complemented
11#31770703C211yesmaximal, characteristic, complemented
12#31770677C211yesmaximal, characteristic, complemented
13#31770604C211yesmaximal, characteristic, complemented
14C2×#22575373C211yesmaximal, characteristic, complemented
15#554×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic
17C22×D1076C2211yescharacteristic
18C2×#16864392C2211yescharacteristic, complemented
19C2×#16864389C2211yescharacteristic
20C2×#16864389C2211yescharacteristic
21C2×#16864386C2211yescharacteristic, complemented
22C22×#14524130C2211yescharacteristic
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#22575386C2211yescharacteristic, complemented
33#22575386C2211yescharacteristic, 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
47#22575364C2211yescharacteristic, complemented
48C4⋊Q8×C269C2211yescharacteristic, complemented
49#125×C269—21no—
50C22⋊Q8×C269—21no—
51C22⋊Q8×C269—21no—
52C22⋊Q8×C269C2211yescharacteristic
53C22⋊Q8×C269C2211yescharacteristic, complemented
54C4×Q8×C269C2211yescharacteristic, complemented
55C4×D4×C269—21no—
56C4×D4×C269C2211yescharacteristic
57C2×C4⋊C4×C269C2211yescharacteristic
58C2×C4⋊C4×C269C2211yescharacteristic, complemented
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—
75#16864386—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×C269—21no—
89C2×Q8×C269C2311yescharacteristic
90C2×D4×C269D411yescharacteristic
91C22×C4×C269—21no—
92C22×C4×C269—21no—
93C22×C4×C269C2311yescharacteristic
94C22×C4×C269C2311yescharacteristic
95C4⋊C4×C269—21no—
96C4⋊C4×C269—21no—
97C4⋊C4×C269—21no—
98C4⋊C4×C269—21no—
99C4⋊C4×C269—21no—
100C4⋊C4×C269—21no—
101C4⋊C4×C269D411yescharacteristic
102C4⋊C4×C269C2311yescharacteristic
103C4⋊C4×C269C2311yescharacteristic
104C4⋊C4×C269C2311yescharacteristic
105C22⋊C4×C269—21no—
106C22⋊C4×C269—21no—
107C22⋊C4×C269D411yescharacteristic
108C22⋊C4×C269D411yescharacteristic
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—
137Q8×C269—81no—
138D4×C269—22no—
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—21no—
147C2×C4×C269—81no—
148C2×C4×C269—81no—
149C2×C4×C269C2×D411yescharacteristic
150C2×C4×C269C2×D411yescharacteristic
151C2×C4×C269C2×D411yescharacteristic
152C2×C4×C269C2×D411yescharacteristic
153C2×C4×C269C2411yescharacteristic
154C22×C269—21no—
155C22×C269—21no—
156C22×C269—21no—
157C22×C269—21no—
158C22×C269—21no—
159C22×C269#13712yes—
160C22×C269C22×D411yescharacteristic
161C2×D269—22no—
162C2×D269—41no—
163C2×D269—41no—
164C2×D269—41no—
165C2×D269—41no—
166C2×D269—41no—
167C2×D269—84no—
168C4×C269—21no—
169C4×C269—21no—
170C4×C269—21no—
171C4×C269—21no—
172C4×C269—41no—
173C4×C269—41no—
174C4×C269—41no—
175C4×C269—41no—
176C4×C269C2×#4111yescharacteristic
177C4×C269H(5,𝔽2)11yescharacteristic
178Dic269—41no—
179Dic269—41no—
180C2×C269—22no—
181C2×C269—41no—
182C2×C269#54611yescharacteristic
183C2×C269C2×#13711yescharacteristic
184C2×C269#57611yescharacteristic
185D269—42no—
186D269—81no—
187C269#331411yescharacteristic, complemented, Sylow
188#3314—2691nomaximal, Sylow
189#554D26911yescharacteristic, complemented
190#546—2691no—
191#489—2691no—
192#484—2691no—
193#480—2691no—
194#474—2691no—
195#468—2691no—
196#467—2691no—
197#464—2691no—
198#452—2691no—
199#450—2691no—
200#440—2691no—
201#435—2691no—
202#417—2691no—
203C2×#102—2691no—
204C2×#41—2691no—
205C22×D4—2691no—
206C2×Q16—2691no—
207C2×SD16—2691no—
208C2×SD16—2691no—
209C2×M16—2691no—
210C22×C8—2691no—
211C4⋊Q8C2×D26911yescharacteristic, complemented
212#125—21no—
213#124—2691no—
214#123—5381no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—2691no—
218C22⋊Q8C2×D26911yescharacteristic
219C22⋊Q8C2×D26911yescharacteristic, complemented
220#121—2691no—
221#121—5381no—
222#120—5381no—
223C4×Q8C2×D26911yescharacteristic, complemented
224C4×D4—21no—
225C4×D4—2691no—
226C4×D4C2×D26911yescharacteristic
227C2×C4⋊C4C2×D26911yescharacteristic
228C2×C4⋊C4C2×D26911yescharacteristic, complemented
229C2×C22⋊C4—5381no—
230#107—2691no—
231#107—2691no—
232#107—2691no—
233C4⋊C8—2691no—
234#103—2691no—
235#103—2691no—
236#103—2691no—
237#103—2691no—
238#102—2691no—
239#102—2691no—
240#102—2691no—
241#102—2691no—
242#102—2691no—
243#102—2691no—
244C22⋊C8—2691no—
245C22⋊C8—2691no—
246#97—2691no—
247C24—5381no—
248#41—5382no—
249#41—10761no—
250C2×Q8—21no—
251C2×Q8—2691no—
252C2×Q8C22×D26911yescharacteristic
253C2×D4—2691no—
254C2×D4—2691no—
255C2×D4—5382no—
256C2×D4—5381no—
257C2×D4—5381no—
258C2×D4—5381no—
259C2×D4—10761no—
260C2×D4C22⋊D26911yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—5381no—
264C22×C4—5381no—
265C22×C4C22×D26911yescharacteristic
266C22×C4C22×D26911yescharacteristic
267Q16—10761no—
268SD16—5382no—
269SD16—10761no—
270M16—10761no—
271C2×C8—2691no—
272C2×C8—2691no—
273C2×C8—2691no—
274C2×C8—2691no—
275C2×C8—10761no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—2691no—
283C4⋊C4—5381no—
284C4⋊C4C22⋊D26911yescharacteristic
285C4⋊C4C22×D26911yescharacteristic
286C4⋊C4C22×D26911yescharacteristic
287C4⋊C4C22×D26911yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—5381no—
291C22⋊C4—5381no—
292C22⋊C4—5381no—
293C22⋊C4—5381no—
294C22⋊C4—5381no—
295C22⋊C4C22⋊D26911yescharacteristic
296C22⋊C4C22⋊D26911yescharacteristic
297C42—21no—
298C42C22×D26911yescharacteristic
299C23—5381no—
300C23—5381no—
301C23—10762no—
302C23—10762no—
303C23—21521no—
304C23C2×C22⋊D26911yescharacteristic
305C23C2×C22⋊D26911yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—81no—
309Q8—5381no—
310Q8—5381no—
311D4—22no—
312D4—2692no—
313D4—5381no—
314D4—5381no—
315D4—5381no—
316D4—10762no—
317D4—10762no—
318D4—21521no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—5381no—
330C2×C4—5381no—
331C2×C4—10762no—
332C2×C4—21521no—
333C2×C4C2×C22⋊D26911yescharacteristic
334C2×C4C2×C22⋊D26911yescharacteristic
335C2×C4C2×C22⋊D26911yescharacteristic
336C2×C4C2×C22⋊D26911yescharacteristic
337C2×C4C23×D26911yescharacteristic
338C8—5381no—
339C8—5381no—
340C8—5381no—
341C8—5381no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—5382no—
348C22—10761no—
349C22—10761no—
350C22—10761no—
351C22—10761no—
352C22—10761no—
353C22—21524no—
354C22#2257548412yes—
355C22C22×C22⋊D26911yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—10761no—
365C4—10761no—
366C4#41×D26911yescharacteristic
367C4#2257549911yescharacteristic
368C2—22no—
369C2—41no—
370C2—10762no—
371C2—21521no—
372C2#3177124311yescharacteristic
373C2C2×#2257548411yescharacteristic
374C2#3177140811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#554×C269C2
socleC22×C269C22×D4
prime radical#31771251C2
hypercenter#554D269
hypocenterC269#3314
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #31770967C2semidirect
3. #31770940C2semidirect
4. #31770918C2semidirect
5. #31770882C2semidirect
6. #31770853C2semidirect
7. #31770844C2semidirect
8. #31770827C2semidirect
9. #31770765C2semidirect
10. #31770755C2semidirect
11. #31770703C2semidirect
12. #31770677C2semidirect
13. #31770604C2semidirect
14. C2×#22575373C2semidirect
15. #554×C269C2semidirect
18. C2×#16864392C22semidirect
21. C2×#16864386C22semidirect
31. #22575386C22semidirect
32. #22575386C22semidirect
33. #22575386C22semidirect
34. C4⋊#14524130C22semidirect
35. #22575377C22semidirect
36. #22575376C22semidirect
37. #22575376C22semidirect
38. #22575376C22semidirect
46. C22⋊#14524130C22semidirect
47. #22575364C22semidirect
48. C4⋊Q8×C269C22semidirect
53. C22⋊Q8×C269C22semidirect
54. C4×Q8×C269C22semidirect
58. C2×C4⋊C4×C269C22semidirect
187. C269#3314semidirect
189. #554D269semidirect
211. C4⋊Q8C2×D269semidirect
219. C22⋊Q8C2×D269semidirect
223. C4×Q8C2×D269semidirect
228. C2×C4⋊C4C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712512
#317709672
#317709402
#317709182
#317708822
#317708532
#317708442
#317708272
#317707652
#317707552
#317707032
#317706772
#317706042
C2×#225753732
#554×C2692
#3314269