Finite Group Databasebeta

← #49170624

Subgroups — #49170624

Order 37504.

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#34036821C211yesmaximal, characteristic, complemented
2#34036773C211yesmaximal, characteristic
3#34036482C211yesmaximal, characteristic, complemented
4#34036460C211yesmaximal, characteristic, complemented
5#34036452C211yesmaximal, characteristic, complemented
6#34036431C211yesmaximal, characteristic, complemented
7#34036368C211yesmaximal, characteristic, complemented
8#34036355C211yesmaximal, characteristic, complemented
9#34036349C211yesmaximal, characteristic, complemented
10#34036288C211yesmaximal, characteristic, complemented
11SD16×Dic293C211yesmaximal, characteristic, complemented
12#34036192C211yesmaximal, characteristic, complemented
13#34036122C211yesmaximal, characteristic, complemented
14C2×#22627545C211yesmaximal, characteristic, complemented
15#450×C293C211yesmaximal, characteristic, complemented
16C2×#17989801C2211yescharacteristic
17C2×#17989792C2211yescharacteristic, complemented
18C22×#14556873C2211yescharacteristic
19C2×C8×D293C2211yescharacteristic, complemented
20#22627619—21no—
21#22627615—21no—
22#22627615C2211yescharacteristic, complemented
23#22627611C2211yescharacteristic
24#22627608—21no—
25#22627605—21no—
26#22627603C2211yescharacteristic
27#22627600C2211yescharacteristic, complemented
28#22627596—21no—
29#22627595C2211yescharacteristic
30#22627593—21no—
31Q8×Dic293C2211yescharacteristic, complemented
32D4×Dic293C2211yescharacteristic
33C4×C22⋊D293C2211yescharacteristic
34#22627582C2211yescharacteristic, complemented
35#22627580—21no—
36C2×C4⋊Dic293C2211yescharacteristic
37C2×C22⋊Dic293—21no—
38#22627557C2211yescharacteristic, complemented
39#22627556C2211yescharacteristic, complemented
40#22627556C2211yescharacteristic, complemented
41#22627553C2211yescharacteristic, complemented
42#22627549C2211yescharacteristic, complemented
43#22627549C2211yescharacteristic, complemented
44#22627548C2211yescharacteristic, complemented
45#22627547C2211yescharacteristic, complemented
46#22627545C2211yescharacteristic
47#22627545C2211yescharacteristic
48#22627545C2211yescharacteristic
49#22627545C2211yescharacteristic
50#22627536C2211yescharacteristic, complemented
51C8×Dic293C2211yescharacteristic, complemented
52C22×D4×C293C2211yescharacteristic
53C2×SD16×C293C2211yescharacteristic
54C2×SD16×C293C2211yescharacteristic
55C22⋊Q8×C293C2211yescharacteristic
56#102×C293C2211yescharacteristic
57#102×C293C2211yescharacteristic
58C22⋊C8×C293C2211yescharacteristic
59#17989801—22no—
60#17989801—41no—
61C2×Dic586C2311yescharacteristic
62C2×C22⋊D293—21no—
63C2×C22⋊D293—21no—
64C2×C22⋊D293D411yescharacteristic
65C22×Dic293—21no—
66C22×Dic293—21no—
67C22×Dic293—21no—
68C2×C4×D293C2311yescharacteristic
69#17989792—41no—
70C2×#14556873—41no—
71C2×#14556873C2311yescharacteristic
72C2×#14556873C2311yescharacteristic
73C8×D293—41no—
74C4⋊Dic293—21no—
75C4⋊Dic293—21no—
76C4⋊Dic293—21no—
77C4⋊Dic293C2311yescharacteristic
78C4⋊Dic293C2311yescharacteristic
79C4⋊Dic293C2311yescharacteristic
80#17989781—21no—
81#17989781—21no—
82#17989781—21no—
83#17989781D411yescharacteristic
84C22⋊Dic293—21no—
85C22⋊Dic293—21no—
86C22⋊Dic293—21no—
87C22⋊Dic293—21no—
88C22⋊Dic293—21no—
89C22⋊Dic293D411yescharacteristic
90#17989779—21no—
91#17989779—21no—
92#17989779D411yescharacteristic
93C4×Dic293—21no—
94C4×Dic293—21no—
95C4×Dic293C2311yescharacteristic
96C24×C293—21no—
97C2×Q8×C293C2311yescharacteristic
98C2×D4×C293—22no—
99C2×D4×C293—21no—
100C2×D4×C293—41no—
101C2×D4×C293C2311yescharacteristic
102C2×D4×C293C2311yescharacteristic
103C22×C4×C293C2311yescharacteristic
104SD16×C293—22no—
105SD16×C293—41no—
106C2×C8×C293C2311yescharacteristic
107C2×C8×C293C2311yescharacteristic
108C4⋊C4×C293—21no—
109C4⋊C4×C293C2311yescharacteristic
110C22⋊C4×C293—21no—
111C22×D293C2×D411yescharacteristic
112Dic586—21no—
113Dic586—21no—
114C22⋊D293—22no—
115C22⋊D293—42no—
116C22⋊D293—81no—
117C2×Dic293—21no—
118C2×Dic293—21no—
119C2×Dic293—21no—
120C2×Dic293—21no—
121C2×Dic293—42no—
122C2×Dic293—42no—
123C2×Dic293—81no—
124C2×Dic293C2×D411yescharacteristic
125C2×Dic293C2×D411yescharacteristic
126C2×Dic293C2×D411yescharacteristic
127C4×D293—21no—
128C4×D293—21no—
129#14556873—21no—
130#14556873—21no—
131C23×C293—21no—
132C23×C293—21no—
133C23×C293—42no—
134C23×C293—42no—
135C23×C293—81no—
136C23×C293C2×D411yescharacteristic
137Q8×C293—21no—
138Q8×C293—21no—
139D4×C293—21no—
140D4×C293—21no—
141D4×C293—21no—
142D4×C293—42no—
143D4×C293#4112yes—
144C2×C4×C293—21no—
145C2×C4×C293—21no—
146C2×C4×C293—21no—
147C2×C4×C293—21no—
148C2×C4×C293C2×D411yescharacteristic
149C2×C4×C293C2411yescharacteristic
150C8×C293—21no—
151C8×C293—21no—
152C22×C293—21no—
153C22×C293—21no—
154C22×C293—22no—
155C22×C293—41no—
156C22×C293—41no—
157C22×C293—41no—
158C22×C293—41no—
159C22×C293—41no—
160C22×C293—84no—
161C22×C293#13512yes—
162C22×C293C22×D411yescharacteristic
163C2×D293—21no—
164C2×D293—21no—
165C2×D293—21no—
166C4×C293—21no—
167C4×C293—41no—
168C4×C293—41no—
169C4×C293C2×#4111yescharacteristic
170C4×C293H(5,𝔽2)11yescharacteristic
171Dic293—21no—
172Dic293—21no—
173Dic293—21no—
174Dic293—41no—
175Dic293—41no—
176Dic293—41no—
177Dic293—41no—
178C2×C293—22no—
179C2×C293—42no—
180C2×C293—81no—
181C2×C293#54611yescharacteristic
182C2×C293C2×#13511yescharacteristic
183C2×C293#57711yescharacteristic
184D293—41no—
185C293#329611yescharacteristic, complemented, Sylow
186#3296—2931nomaximal, Sylow
187#552—2931no—
188#546—2931no—
189#488—2931no—
190#483—2931no—
191#482—2931no—
192#478—2931no—
193#467—2931no—
194#465—2931no—
195#464—2931no—
196#452—2931no—
197#450D29311yescharacteristic, complemented
198C4×SD16—2931no—
199C8×D4—2931no—
200#416—2931no—
201C2×#102—2931no—
202C2×#41—2931no—
203C22×D4C2×D29311yescharacteristic
204C2×Q16—2931no—
205C2×SD16C2×D29311yescharacteristic
206C2×SD16C2×D29311yescharacteristic
207C22×C8—2931no—
208C22×C8—2931no—
209#126—5861no—
210#125—2931no—
211#125—5861no—
212#124—2931no—
213#123—5861no—
214#123—5861no—
215C22⋊Q8—2931no—
216C22⋊Q8—2931no—
217C22⋊Q8C2×D29311yescharacteristic
218#121—2931no—
219#121—5861no—
220#120—5861no—
221C4×Q8—2931no—
222C4×D4—2931no—
223C4×D4—2931no—
224#117—2931no—
225#117—5861no—
226C2×C4⋊C4—2931no—
227C2×C22⋊C4—5861no—
228#107—2931no—
229#106—2931no—
230#106—2931no—
231C4⋊C8—2931no—
232#103—2931no—
233#103—2931no—
234#103—2931no—
235#103—2931no—
236#102—2931no—
237#102—2931no—
238#102—2931no—
239#102—2931no—
240#102C2×D29311yescharacteristic
241#102C2×D29311yescharacteristic
242C22⋊C8—2931no—
243C22⋊C8C2×D29311yescharacteristic
244C4×C8—2931no—
245C24—21no—
246#41—5862no—
247#41—11721no—
248C2×Q8—2931no—
249C2×Q8C22×D29311yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—2931no—
254C2×D4—5861no—
255C2×D4—5861no—
256C2×D4C22×D29311yescharacteristic
257C2×D4C22×D29311yescharacteristic
258C22×C4—2931no—
259C22×C4—5861no—
260C22×C4—5861no—
261C22×C4—5861no—
262C22×C4C22×D29311yescharacteristic
263Q16—11721no—
264SD16—22no—
265SD16—41no—
266C2×C8—2931no—
267C2×C8—2931no—
268C2×C8—11721no—
269C2×C8—11721no—
270C2×C8C22×D29311yescharacteristic
271C2×C8C22×D29311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2931no—
274C4⋊C4—2931no—
275C4⋊C4—2931no—
276C4⋊C4—2931no—
277C4⋊C4—5861no—
278C4⋊C4—5861no—
279C4⋊C4—5861no—
280C4⋊C4—5861no—
281C4⋊C4—5861no—
282C4⋊C4—5861no—
283C4⋊C4C22×D29311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—2931no—
286C22⋊C4—2931no—
287C22⋊C4—5861no—
288C22⋊C4—5861no—
289C22⋊C4—5861no—
290C22⋊C4—5861no—
291C22⋊C4—5861no—
292C22⋊C4—5861no—
293C22⋊C4—5861no—
294C42—2931no—
295C42—5861no—
296C42—5861no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—2931no—
303C23D4×D29311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5861no—
307Q8—5861no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—23441no—
313D4—5862no—
314D4—11722no—
315D4#1798980112yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—2931no—
321C2×C4—2931no—
322C2×C4—2931no—
323C2×C4—23441no—
324C2×C4—5861no—
325C2×C4—5861no—
326C2×C4—5861no—
327C2×C4—5861no—
328C2×C4—5861no—
329C2×C4—5861no—
330C2×C4—11722no—
331C2×C4—11722no—
332C2×C4D4×D29311yescharacteristic
333C2×C4C23×D29311yescharacteristic
334C8—21no—
335C8—21no—
336C8—5861no—
337C8—5861no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—5861no—
348C22—5861no—
349C22—5861no—
350C22#2262764512yes—
351C22C2×D4×D29311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—5861no—
356C4—5861no—
357C4—5861no—
358C4—11721no—
359C4—11721no—
360C4—11721no—
361C4—11721no—
362C4C2×#1798980111yescharacteristic
363C4#2262766911yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—11721no—
368C2#3403676611yescharacteristic
369C2C2×#2262764511yescharacteristic
370C2#3403693911yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#450×C293C2
socleC22×C293C22×D4
prime radical#34036773C2
hypercenter#450D293
hypocenterC293#3296
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #34036821C2semidirect
3. #34036482C2semidirect
4. #34036460C2semidirect
5. #34036452C2semidirect
6. #34036431C2semidirect
7. #34036368C2semidirect
8. #34036355C2semidirect
9. #34036349C2semidirect
10. #34036288C2semidirect
11. SD16×Dic293C2semidirect
12. #34036192C2semidirect
13. #34036122C2semidirect
14. C2×#22627545C2semidirect
15. #450×C293C2semidirect
17. C2×#17989792C22semidirect
19. C2×C8×D293C22semidirect
22. #22627615C22semidirect
27. #22627600C22semidirect
31. Q8×Dic293C22semidirect
34. #22627582C22semidirect
38. #22627557C22semidirect
39. #22627556C22semidirect
40. #22627556C22semidirect
41. #22627553C22semidirect
42. #22627549C22semidirect
43. #22627549C22semidirect
44. #22627548C22semidirect
45. #22627547C22semidirect
50. #22627536C22semidirect
51. C8×Dic293C22semidirect
185. C293#3296semidirect
197. #450D293semidirect

Maximal subgroups

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

subgroupindex
#340368212
#340367732
#340364822
#340364602
#340364522
#340364312
#340363682
#340363552
#340363492
#340362882
SD16×Dic2932
#340361922
#340361222
C2×#226275452
#450×C2932
#3296293