Finite Group Databasebeta

← #38754564

Subgroups — #38754564

Order 25216.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161775C211yesmaximal, characteristic, complemented
2#27161725C211yesmaximal, characteristic, complemented
3#27161434C211yesmaximal, characteristic, complemented
4#27161415C211yesmaximal, characteristic, complemented
5#27161404C211yesmaximal, characteristic, complemented
6#27161380C211yesmaximal, characteristic, complemented
7#27161315C211yesmaximal, characteristic, complemented
8#27161308C211yesmaximal, characteristic, complemented
9#27161295C211yesmaximal, characteristic, complemented
10#27161226C211yesmaximal, characteristic, complemented
11#27161215C211yesmaximal, characteristic, complemented
12C4×D1576C211yesmaximal, characteristic, complemented
13#27161076C211yesmaximal, characteristic, complemented
14C2×#20263248C211yesmaximal, characteristic, complemented
15C8×D4×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C2×#15691347C2211yescharacteristic, complemented
19C2×D1576C2211yescharacteristic
20C2×D1576C2211yescharacteristic
21#20263321—21no—
22#20263318C2211yescharacteristic, complemented
23#20263314C2211yescharacteristic
24#20263311—21no—
25#20263308—21no—
26#20263305C2211yescharacteristic
27#20263300C2211yescharacteristic
28#20263300C2211yescharacteristic
29#20263300C2211yescharacteristic, complemented
30#20263299—21no—
31#20263298—21no—
32#20263294—21no—
33C4×D788C2211yescharacteristic
34C4×D788C2211yescharacteristic, complemented
35#20263286—21no—
36#20263285C2211yescharacteristic, complemented
37C2×C4⋊Dic197C2211yescharacteristic
38C22⋊C4×D197—21no—
39#20263260C2211yescharacteristic, complemented
40#20263260C2211yescharacteristic, complemented
41#20263258C2211yescharacteristic, complemented
42#20263251C2211yescharacteristic, complemented
43#20263251C2211yescharacteristic, complemented
44#20263251C2211yescharacteristic, complemented
45#20263248C2211yescharacteristic
46#20263248C2211yescharacteristic
47#20263248C2211yescharacteristic
48#20263248C2211yescharacteristic
49#20263248C2211yescharacteristic
50#20263248C2211yescharacteristic
51#20263248C2211yescharacteristic, complemented
52C22×C8×C197C2211yescharacteristic
53C22×C8×C197C2211yescharacteristic, complemented
54C4×D4×C197C2211yescharacteristic
55C4⋊C8×C197C2211yescharacteristic, complemented
56C22⋊C8×C197C2211yescharacteristic
57C22⋊C8×C197C2211yescharacteristic, complemented
58C4×C8×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60#15691358—22no—
61#15691358—41no—
62C2×Dic394C2311yescharacteristic
63C2×D788—22no—
64C2×D788—21no—
65C2×D788—41no—
66C2×D788C2311yescharacteristic
67C2×D788C2311yescharacteristic
68C2×D788C2311yescharacteristic
69C2×C22⋊D197—21no—
70C2×C22⋊D197—21no—
71C2×C22⋊D197—21no—
72C2×C22⋊D197—21no—
73C22×Dic197—21no—
74C2×C4×D197—21no—
75C2×C4×D197—21no—
76C2×C4×D197—21no—
77#15691347—41no—
78D1576—22no—
79D1576—41no—
80C4⋊Dic197—21no—
81C4⋊Dic197C2311yescharacteristic
82C4⋊Dic197C2311yescharacteristic
83C4⋊Dic197C2311yescharacteristic
84C4⋊Dic197C2311yescharacteristic
85#15691337—21no—
86#15691337—21no—
87#15691337—21no—
88C22⋊Dic197—21no—
89C22⋊Dic197—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335—21no—
96#15691335—21no—
97C4×Dic197—21no—
98C2×D4×C197D411yescharacteristic
99C22×C4×C197C2311yescharacteristic
100C22×C4×C197C2311yescharacteristic
101C2×C8×C197—41no—
102C2×C8×C197—41no—
103C2×C8×C197C2311yescharacteristic
104C2×C8×C197C2311yescharacteristic
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C4⋊C4×C197D411yescharacteristic
108C22⋊C4×C197D411yescharacteristic
109C22⋊C4×C197D411yescharacteristic
110C42×C197C2311yescharacteristic
111C22×D197—21no—
112C22×D197—21no—
113C22×D197—21no—
114C22×D197—42no—
115C22×D197—42no—
116C22×D197—81no—
117Dic394—21no—
118Dic394—21no—
119D788—21no—
120D788—21no—
121D788—21no—
122D788—21no—
123D788—21no—
124D788—42no—
125D788#4112yes—
126C22⋊D197—42no—
127C22⋊D197—42no—
128C22⋊D197—81no—
129C22⋊D197—81no—
130C2×Dic197—21no—
131C2×Dic197—21no—
132C2×Dic197—21no—
133C2×Dic197—21no—
134C2×Dic197—21no—
135C2×Dic197—81no—
136C4×D197—42no—
137C4×D197—81no—
138C4×D197—81no—
139C23×C197C2×D411yescharacteristic
140C23×C197C2×D411yescharacteristic
141D4×C197—22no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2×D411yescharacteristic
149C2×C4×C197C2×D411yescharacteristic
150C2×C4×C197C2411yescharacteristic
151C8×C197—21no—
152C8×C197—21no—
153C8×C197—21no—
154C8×C197—21no—
155C22×C197—21no—
156C22×C197—21no—
157C22×C197—21no—
158C22×C197—21no—
159C22×C197—21no—
160C22×C197#13512yes—
161C22×C197C22×D411yescharacteristic
162C2×D197—22no—
163C2×D197—41no—
164C2×D197—41no—
165C2×D197—41no—
166C2×D197—41no—
167C2×D197—41no—
168C2×D197—41no—
169C2×D197—41no—
170C2×D197—41no—
171C2×D197—84no—
172C4×C197—21no—
173C4×C197—21no—
174C4×C197—21no—
175C4×C197—21no—
176C4×C197C2×#4111yescharacteristic
177C4×C197H(5,𝔽2)11yescharacteristic
178Dic197—41no—
179Dic197—41no—
180Dic197—41no—
181Dic197—41no—
182Dic197—41no—
183C2×C197—22no—
184C2×C197—41no—
185C2×C197#54611yescharacteristic
186C2×C197C2×#13511yescharacteristic
187C2×C197#57611yescharacteristic
188D197—42no—
189D197—81no—
190D197—81no—
191C197#329711yescharacteristic, complemented, Sylow
192#3297—1971nomaximal, Sylow
193#553—1971no—
194#546—1971no—
195#488—1971no—
196#484—1971no—
197#482—1971no—
198#478—1971no—
199#466—1971no—
200#465—1971no—
201#463—1971no—
202#449—1971no—
203#447—1971no—
204C4×D8—1971no—
205C8×D4D19711yescharacteristic, complemented
206#416—1971no—
207C2×#102—1971no—
208C2×#41—1971no—
209C22×D4—1971no—
210C2×SD16—1971no—
211C2×D8—1971no—
212C2×D8—1971no—
213C22×C8C2×D19711yescharacteristic
214C22×C8C2×D19711yescharacteristic, complemented
215#126—3941no—
216#125—1971no—
217#124—1971no—
218#123—3941no—
219#123—3941no—
220C22⋊Q8—1971no—
221#121—3941no—
222#121—3941no—
223#121—1971no—
224#121—1971no—
225#121—1971no—
226#120—3941no—
227C4×D4—3941no—
228C4×D4—1971no—
229C4×D4—1971no—
230C4×D4C2×D19711yescharacteristic
231#117—1971no—
232C2×C4⋊C4—1971no—
233C2×C22⋊C4—3941no—
234#107—1971no—
235#107—1971no—
236#106—1971no—
237C4⋊C8C2×D19711yescharacteristic, complemented
238#103—1971no—
239#103—1971no—
240#103—1971no—
241#102—1971no—
242#102—1971no—
243#102—1971no—
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248C22⋊C8C2×D19711yescharacteristic
249C22⋊C8C2×D19711yescharacteristic, complemented
250C4×C8C2×D19711yescharacteristic, complemented
251C24—3941no—
252#41—7881no—
253#41—3942no—
254C2×Q8—1971no—
255C2×D4—7881no—
256C2×D4—3942no—
257C2×D4—3941no—
258C2×D4—3941no—
259C2×D4—3941no—
260C2×D4—3941no—
261C2×D4—3941no—
262C2×D4—1971no—
263C2×D4—1971no—
264C2×D4—1971no—
265C2×D4D78811yescharacteristic
266C22×C4—3941no—
267C22×C4—3941no—
268C22×C4—3941no—
269C22×C4—3941no—
270C22×C4C22×D19711yescharacteristic
271C22×C4C22×D19711yescharacteristic
272SD16—7881no—
273D8—7881no—
274D8—3942no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22×D19711yescharacteristic
278C2×C8C22×D19711yescharacteristic
279C2×C8C22×D19711yescharacteristic
280C2×C8C22×D19711yescharacteristic
281C4⋊C4—3941no—
282C4⋊C4—3941no—
283C4⋊C4—3941no—
284C4⋊C4—3941no—
285C4⋊C4—1971no—
286C4⋊C4—1971no—
287C4⋊C4—1971no—
288C4⋊C4—1971no—
289C4⋊C4D78811yescharacteristic
290C22⋊C4—3941no—
291C22⋊C4—3941no—
292C22⋊C4—3941no—
293C22⋊C4—3941no—
294C22⋊C4—3941no—
295C22⋊C4—3941no—
296C22⋊C4—3941no—
297C22⋊C4—3941no—
298C22⋊C4—3941no—
299C22⋊C4D78811yescharacteristic
300C22⋊C4D78811yescharacteristic
301C42—3941no—
302C42C22×D19711yescharacteristic
303C23—7882no—
304C23—7882no—
305C23—15761no—
306C23—3941no—
307C23—3941no—
308C23—3941no—
309C23C2×D78811yescharacteristic
310C23C2×D78811yescharacteristic
311Q8—3941no—
312Q8—3941no—
313D4—22no—
314D4—7882no—
315D4—7882no—
316D4—7882no—
317D4—15761no—
318D4—15761no—
319D4—3941no—
320D4—3941no—
321D4—3941no—
322D4—3941no—
323D4—3941no—
324D4—1972no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—7882no—
330C2×C4—15761no—
331C2×C4—15761no—
332C2×C4—15761no—
333C2×C4—3941no—
334C2×C4—3941no—
335C2×C4—3941no—
336C2×C4—3941no—
337C2×C4—3941no—
338C2×C4C2×D78811yescharacteristic
339C2×C4C2×D78811yescharacteristic
340C2×C4C2×D78811yescharacteristic
341C2×C4C2×D78811yescharacteristic
342C2×C4C23×D19711yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—7881no—
353C22—7881no—
354C22—7881no—
355C22—7881no—
356C22—7881no—
357C22—7881no—
358C22—7881no—
359C22—7881no—
360C22—15764no—
361C22—3942no—
362C22#2026334812yes—
363C22C22×D78811yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—7881no—
369C4—7881no—
370C4—7881no—
371C4—7881no—
372C4—7881no—
373C4#41×D19711yescharacteristic
374C4#2026337111yescharacteristic
375C2—22no—
376C2—41no—
377C2—7882no—
378C2—15761no—
379C2—15761no—
380C2#2716172411yescharacteristic
381C2C2×#2026334811yescharacteristic
382C2#2716188411yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC8×D4×C197C2
socleC22×C197C22×D4
hypercenterC8×D4D197
hypocenterC197#3297
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #27161775C2semidirect
2. #27161725C2semidirect
3. #27161434C2semidirect
4. #27161415C2semidirect
5. #27161404C2semidirect
6. #27161380C2semidirect
7. #27161315C2semidirect
8. #27161308C2semidirect
9. #27161295C2semidirect
10. #27161226C2semidirect
11. #27161215C2semidirect
12. C4×D1576C2semidirect
13. #27161076C2semidirect
14. C2×#20263248C2semidirect
15. C8×D4×C197C2semidirect
18. C2×#15691347C22semidirect
22. #20263318C22semidirect
29. #20263300C22semidirect
34. C4×D788C22semidirect
36. #20263285C22semidirect
39. #20263260C22semidirect
40. #20263260C22semidirect
41. #20263258C22semidirect
42. #20263251C22semidirect
43. #20263251C22semidirect
44. #20263251C22semidirect
51. #20263248C22semidirect
53. C22×C8×C197C22semidirect
55. C4⋊C8×C197C22semidirect
57. C22⋊C8×C197C22semidirect
58. C4×C8×C197C22semidirect
191. C197#3297semidirect
205. C8×D4D197semidirect
214. C22×C8C2×D197semidirect
237. C4⋊C8C2×D197semidirect
249. C22⋊C8C2×D197semidirect
250. C4×C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617752
#271617252
#271614342
#271614152
#271614042
#271613802
#271613152
#271613082
#271612952
#271612262
#271612152
C4×D15762
#271610762
C2×#202632482
C8×D4×C1972
#3297197