Finite Group Databasebeta

← #38754529

Subgroups — #38754529

Order 25216.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161787C211yesmaximal, characteristic, complemented
2D4×C22⋊D197C211yesmaximal, characteristic
3#27161424C211yesmaximal, characteristic, complemented
4#27161398C211yesmaximal, characteristic, complemented
5#27161396C211yesmaximal, characteristic, complemented
6#27161367C211yesmaximal, characteristic, complemented
7#27161310C211yesmaximal, characteristic, complemented
8#27161305C211yesmaximal, characteristic, complemented
9#27161277C211yesmaximal, characteristic, complemented
10#27161228C211yesmaximal, characteristic, complemented
11SD16×Dic197C211yesmaximal, characteristic, complemented
12#27161142C211yesmaximal, characteristic, complemented
13C2×#20263247C211yesmaximal, characteristic, complemented
14#102×D197C211yesmaximal, characteristic, complemented
15#450×C197C211yesmaximal, characteristic, complemented
16C22×C22⋊D197—21no—
17C2×D4×D197C2211yescharacteristic
18C2×#15691345C2211yescharacteristic, complemented
19C22×#13096802C2211yescharacteristic, complemented
20C2×C8×D197C2211yescharacteristic, complemented
21#20263325C2211yescharacteristic, complemented
22#20263322C2211yescharacteristic
23#20263316—21no—
24#20263306—21no—
25#20263303—21no—
26#20263302C2211yescharacteristic, complemented
27#20263300C2211yescharacteristic
28#20263298—21no—
29#20263297C2211yescharacteristic
30#20263295—21no—
31#20263294—21no—
32Q8×Dic197C2211yescharacteristic, complemented
33D4×Dic197C2211yescharacteristic
34C4×C22⋊D197C2211yescharacteristic
35#20263286—21no—
36C2×C4⋊Dic197C2211yescharacteristic, complemented
37C4⋊C4×D197C2211yescharacteristic, complemented
38#20263258C2211yescharacteristic, complemented
39#20263258C2211yescharacteristic, complemented
40#20263257C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263250C2211yescharacteristic, complemented
43#20263248C2211yescharacteristic, complemented
44#20263248C2211yescharacteristic, complemented
45#20263247C2211yescharacteristic
46#20263247C2211yescharacteristic
47#20263247C2211yescharacteristic, complemented
48#20263247C2211yescharacteristic, complemented
49#20263246C2211yescharacteristic, complemented
50#20263238C2211yescharacteristic, complemented
51C8×Dic197C2211yescharacteristic, complemented
52C22×D4×C197C2211yescharacteristic
53C2×SD16×C197C2211yescharacteristic
54C2×SD16×C197C2211yescharacteristic, complemented
55C22⋊Q8×C197C2211yescharacteristic, complemented
56#102×C197C2211yescharacteristic
57#102×C197C2211yescharacteristic, complemented
58C22⋊C8×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C2×Dic394—21no—
61C2×D788C2311yescharacteristic
62C2×C22⋊D197—21no—
63C2×C22⋊D197—21no—
64C2×C22⋊D197—21no—
65C2×C22⋊D197—21no—
66C2×C22⋊D197—21no—
67C2×C22⋊D197—44no—
68C2×C22⋊D197D411yescharacteristic
69D4×D197—22no—
70D4×D197—41no—
71C22×Dic197—21no—
72C22×Dic197—21no—
73C2×C4×D197—21no—
74C2×C4×D197C2311yescharacteristic
75#15691345—41no—
76C2×#13096802—41no—
77C2×#13096802C2311yescharacteristic
78C2×#13096802C2311yescharacteristic
79C8×D197—41no—
80C4⋊Dic197—21no—
81C4⋊Dic197—21no—
82C4⋊Dic197C2311yescharacteristic
83C4⋊Dic197C2311yescharacteristic
84C4⋊Dic197C2311yescharacteristic
85#15691337—21no—
86#15691337—21no—
87#15691337—21no—
88#15691337D411yescharacteristic
89C22⋊Dic197—21no—
90C22⋊Dic197—21no—
91C22⋊Dic197D411yescharacteristic
92#15691335—21no—
93#15691335—21no—
94#15691335D411yescharacteristic
95C4×Dic197—21no—
96C4×Dic197C2311yescharacteristic
97C24×C197—21no—
98C2×Q8×C197C2311yescharacteristic
99C2×D4×C197—22no—
100C2×D4×C197—21no—
101C2×D4×C197—41no—
102C2×D4×C197C2311yescharacteristic
103C2×D4×C197C2311yescharacteristic
104C22×C4×C197C2311yescharacteristic
105SD16×C197—22no—
106SD16×C197—41no—
107C2×C8×C197C2311yescharacteristic
108C2×C8×C197C2311yescharacteristic
109C4⋊C4×C197—21no—
110C4⋊C4×C197C2311yescharacteristic
111C22⋊C4×C197—21no—
112C22×D197—21no—
113C22×D197—42no—
114C22×D197—42no—
115C22×D197—81no—
116C22×D197C2×D411yescharacteristic
117Dic394—81no—
118D788—21no—
119D788—21no—
120C22⋊D197—42no—
121C22⋊D197—42no—
122C22⋊D197—42no—
123C22⋊D197—42no—
124C22⋊D197—81no—
125C22⋊D197SD1614yescomplemented
126C2×Dic197—21no—
127C2×Dic197—21no—
128C2×Dic197—21no—
129C2×Dic197—42no—
130C2×Dic197—81no—
131C2×Dic197C2×D411yescharacteristic
132C2×Dic197C2×D411yescharacteristic
133C2×Dic197C2×D411yescharacteristic
134C4×D197—21no—
135C4×D197—21no—
136C4×D197—81no—
137#13096802—21no—
138#13096802—21no—
139C23×C197—21no—
140C23×C197—21no—
141C23×C197—42no—
142C23×C197—42no—
143C23×C197—81no—
144C23×C197C2×D411yescharacteristic
145Q8×C197—21no—
146Q8×C197—21no—
147D4×C197—21no—
148D4×C197—21no—
149D4×C197—21no—
150D4×C197—42no—
151D4×C197#4112yes—
152C2×C4×C197—21no—
153C2×C4×C197—21no—
154C2×C4×C197—21no—
155C2×C4×C197—21no—
156C2×C4×C197C2×D411yescharacteristic
157C2×C4×C197C2411yescharacteristic
158C8×C197—21no—
159C8×C197—21no—
160C22×C197—21no—
161C22×C197—21no—
162C22×C197—22no—
163C22×C197—41no—
164C22×C197—41no—
165C22×C197—41no—
166C22×C197—41no—
167C22×C197—41no—
168C22×C197—84no—
169C22×C197C2×SD1612yes—
170C22×C197C22×D411yescharacteristic
171C2×D197—21no—
172C2×D197—21no—
173C2×D197—41no—
174C2×D197—41no—
175C2×D197—41no—
176C2×D197—84no—
177C2×D197C2×SD1612yes—
178C4×C197—21no—
179C4×C197—41no—
180C4×C197—41no—
181C4×C197C2×#4111yescharacteristic
182C4×C197H(5,𝔽2)11yescharacteristic
183Dic197—21no—
184Dic197—21no—
185Dic197—41no—
186Dic197—41no—
187Dic197—41no—
188Dic197C2×SD1612yes—
189C2×C197—22no—
190C2×C197—42no—
191C2×C197—81no—
192C2×C197#54611yescharacteristic
193C2×C197C22×SD1611yescharacteristic
194C2×C197#57611yescharacteristic
195D197—22no—
196D197—81no—
197C197#329511yescharacteristic, complemented, Sylow
198#3295—1971nomaximal, Sylow
199#554—1971no—
200D42—1971no—
201#486—1971no—
202#481—1971no—
203#481—1971no—
204#476—1971no—
205#465—1971no—
206#465—1971no—
207#460—1971no—
208#450—1971no—
209#450D19711yescharacteristic, complemented
210C4×SD16—1971no—
211C8×D4—1971no—
212C2×#102—1971no—
213C2×#102—1971no—
214C22×D4—3941no—
215C22×D4—1971no—
216C22×D4C2×D19711yescharacteristic
217C2×SD16—1971no—
218C2×SD16C2×D19711yescharacteristic
219C2×SD16C2×D19711yescharacteristic, complemented
220C22×C8—1971no—
221C22×C8—1971no—
222C4⋊Q8—1971no—
223#127—1971no—
224#125—3941no—
225C22⋊Q8—3941no—
226C22⋊Q8—3941no—
227C22⋊Q8—1971no—
228C22⋊Q8C2×D19711yescharacteristic, complemented
229#121—3941no—
230#121—1971no—
231#121—1971no—
232#120—3941no—
233#120—3941no—
234C4×Q8—1971no—
235C4×D4—3941no—
236C4×D4—1971no—
237C4×D4—1971no—
238C2×C4⋊C4—1971no—
239C2×C4⋊C4—1971no—
240#106—1971no—
241#106—1971no—
242#106—1971no—
243C4⋊C8—1971no—
244#103—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248#102—1971no—
249#102—1971no—
250#102—1971no—
251#102—1971no—
252#102C2×D19711yescharacteristic
253#102C2×D19711yescharacteristic, complemented
254C22⋊C8—1971no—
255C22⋊C8C2×D19711yescharacteristic, complemented
256C4×C8—1971no—
257C24—21no—
258C24—3941no—
259C2×Q8—3941no—
260C2×Q8C22×D19711yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—7884no—
265C2×D4—7881no—
266C2×D4—3941no—
267C2×D4—3941no—
268C2×D4—3942no—
269C2×D4—3941no—
270C2×D4—3941no—
271C2×D4—3941no—
272C2×D4—1971no—
273C2×D4—1971no—
274C2×D4C22×D19711yescharacteristic
275C2×D4C22×D19711yescharacteristic
276C22×C4—3941no—
277C22×C4—3941no—
278C22×C4—3941no—
279C22×C4—1971no—
280C22×C4C22×D19711yescharacteristic
281SD16—22no—
282SD16—41no—
283SD16—7881no—
284C2×C8—7881no—
285C2×C8—7881no—
286C2×C8—1971no—
287C2×C8—1971no—
288C2×C8C22×D19711yescharacteristic
289C2×C8C22×D19711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—3941no—
292C4⋊C4—3941no—
293C4⋊C4—3941no—
294C4⋊C4—3941no—
295C4⋊C4—3941no—
296C4⋊C4—1971no—
297C4⋊C4—1971no—
298C4⋊C4—1971no—
299C4⋊C4—1971no—
300C4⋊C4C22×D19711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—3941no—
303C22⋊C4—3941no—
304C22⋊C4—3941no—
305C22⋊C4—3941no—
306C22⋊C4—1971no—
307C22⋊C4—1971no—
308C42—3941no—
309C42—1971no—
310C23—21no—
311C23—21no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—7882no—
316C23—7882no—
317C23—15761no—
318C23—3941no—
319C23—1971no—
320C23D4×D19711yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—15761no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—7882no—
329D4—7882no—
330D4—7882no—
331D4—7882no—
332D4—15761no—
333D4—3941no—
334D4—3941no—
335D4—1974no—
336D4#1569135712yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—7882no—
342C2×C4—15761no—
343C2×C4—15761no—
344C2×C4—3941no—
345C2×C4—3941no—
346C2×C4—3941no—
347C2×C4—3941no—
348C2×C4—3941no—
349C2×C4—1971no—
350C2×C4—1971no—
351C2×C4—1971no—
352C2×C4D4×D19711yescharacteristic
353C2×C4C23×D19711yescharacteristic
354C8—21no—
355C8—21no—
356C8—3941no—
357C8—3941no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—7881no—
368C22—7881no—
369C22—7881no—
370C22—15764no—
371C22—3941no—
372C22—3941no—
373C22—1972no—
374C22SD16×D19712yes—
375C22C2×D4×D19711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—7881no—
380C4—7881no—
381C4—7881no—
382C4—3941no—
383C4—3941no—
384C4—1972no—
385C4C2×#1569135711yescharacteristic
386C4#2026337111yescharacteristic
387C2—22no—
388C2—42no—
389C2—81no—
390C2—15761no—
391C2—3942no—
392C2#2716171611yescharacteristic
393C2C2×SD16×D19711yescharacteristic
394C2#2716188311yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#450×C197C2
socleC22×C197C22×D4
prime radicalD4×C22⋊D197C2
hypercenter#450D197
hypocenterC197#3295
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #27161787C2semidirect
3. #27161424C2semidirect
4. #27161398C2semidirect
5. #27161396C2semidirect
6. #27161367C2semidirect
7. #27161310C2semidirect
8. #27161305C2semidirect
9. #27161277C2semidirect
10. #27161228C2semidirect
11. SD16×Dic197C2semidirect
12. #27161142C2semidirect
13. C2×#20263247C2semidirect
14. #102×D197C2semidirect
15. #450×C197C2semidirect
18. C2×#15691345C22semidirect
19. C22×#13096802C22semidirect
20. C2×C8×D197C22semidirect
21. #20263325C22semidirect
26. #20263302C22semidirect
32. Q8×Dic197C22semidirect
36. C2×C4⋊Dic197C22semidirect
37. C4⋊C4×D197C22semidirect
38. #20263258C22semidirect
39. #20263258C22semidirect
40. #20263257C22semidirect
41. #20263255C22semidirect
42. #20263250C22semidirect
43. #20263248C22semidirect
44. #20263248C22semidirect
47. #20263247C22semidirect
48. #20263247C22semidirect
49. #20263246C22semidirect
50. #20263238C22semidirect
51. C8×Dic197C22semidirect
54. C2×SD16×C197C22semidirect
55. C22⋊Q8×C197C22semidirect
57. #102×C197C22semidirect
58. C22⋊C8×C197C22semidirect
125. C22⋊D197SD16semidirect
197. C197#3295semidirect
209. #450D197semidirect
219. C2×SD16C2×D197semidirect
228. C22⋊Q8C2×D197semidirect
253. #102C2×D197semidirect
255. C22⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617872
D4×C22⋊D1972
#271614242
#271613982
#271613962
#271613672
#271613102
#271613052
#271612772
#271612282
SD16×Dic1972
#271611422
C2×#202632472
#102×D1972
#450×C1972
#3295197