Finite Group Databasebeta

← #38754735

Subgroups — #38754735

Order 25216.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161769C211yesmaximal, characteristic, complemented
2D4×D788C211yesmaximal, characteristic
3#27161444C211yesmaximal, characteristic, complemented
4#27161404C211yesmaximal, characteristic, complemented
5#27161393C211yesmaximal, characteristic, complemented
6#27161380C211yesmaximal, characteristic, complemented
7#27161325C211yesmaximal, characteristic, complemented
8#27161308C211yesmaximal, characteristic, complemented
9#27161280C211yesmaximal, characteristic, complemented
10#27161232C211yesmaximal, characteristic, complemented
11#27161232C211yesmaximal, characteristic, complemented
12#27161181C211yesmaximal, characteristic, complemented
13#27161085C211yesmaximal, characteristic, complemented
14C2×#20263248C211yesmaximal, characteristic, complemented
15#435×C197C211yesmaximal, characteristic, complemented
16C22×D788C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C2×D4×D197—21no—
19C2×#15691347C2211yescharacteristic
20C2×#15691347C2211yescharacteristic, complemented
21C2×#15691347C2211yescharacteristic, complemented
22#20263323C2211yescharacteristic
23#20263319—21no—
24#20263318C2211yescharacteristic, complemented
25#20263316—21no—
26#20263307—21no—
27#20263305C2211yescharacteristic, complemented
28#20263305C2211yescharacteristic, complemented
29#20263301—21no—
30#20263300C2211yescharacteristic
31#20263300C2211yescharacteristic
32#20263296—21no—
33#20263296—21no—
34C4×Dic394C2211yescharacteristic, complemented
35C4×D788C2211yescharacteristic
36#20263285C2211yescharacteristic, complemented
37#20263282—21no—
38C2×C4⋊Dic197C2211yescharacteristic, complemented
39#20263260C2211yescharacteristic, complemented
40#20263260C2211yescharacteristic, complemented
41#20263258C2211yescharacteristic, complemented
42#20263251C2211yescharacteristic, complemented
43#20263248C2211yescharacteristic
44#20263248C2211yescharacteristic
45#20263248C2211yescharacteristic
46#20263248C2211yescharacteristic, complemented
47#20263248C2211yescharacteristic, complemented
48#20263248C2211yescharacteristic, complemented
49#20263248C2211yescharacteristic, complemented
50#20263248C2211yescharacteristic, complemented
51#20263248C2211yescharacteristic, complemented
52C2×M16×C197C2211yescharacteristic, complemented
53C22×C8×C197C2211yescharacteristic, complemented
54C4×D4×C197C2211yescharacteristic
55C4⋊C8×C197C2211yescharacteristic, complemented
56C22⋊C8×C197C2211yescharacteristic, complemented
57C22⋊C8×C197C2211yescharacteristic, complemented
58#97×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C23×D197—21no—
61C2×Dic394C2311yescharacteristic
62C2×D788—22no—
63C2×D788—22no—
64C2×D788—21no—
65C2×D788—21no—
66C2×D788—21no—
67C2×D788—21no—
68C2×D788—41no—
69C2×D788—41no—
70C2×D788C2311yescharacteristic
71C2×D788C2311yescharacteristic
72C2×D788C2311yescharacteristic
73C2×C22⋊D197—21no—
74C2×C22⋊D197—21no—
75D4×D197—44no—
76C22×Dic197—21no—
77C2×C4×D197—21no—
78#15691347—22no—
79#15691347—41no—
80#15691347—41no—
81C4⋊Dic197—21no—
82C4⋊Dic197C2311yescharacteristic
83C4⋊Dic197C2311yescharacteristic
84C4⋊Dic197C2311yescharacteristic
85C4⋊Dic197C2311yescharacteristic
86#15691337—21no—
87#15691337—21no—
88#15691337—21no—
89#15691337—21no—
90#15691337—21no—
91C22⋊Dic197—21no—
92C22⋊Dic197—21no—
93C22⋊Dic197—21no—
94#15691335—21no—
95#15691335—21no—
96#15691335—21no—
97C4×Dic197—21no—
98C4×Dic197—21no—
99C2×D4×C197D411yescharacteristic
100C22×C4×C197C2311yescharacteristic
101C22×C4×C197C2311yescharacteristic
102M16×C197—41no—
103C2×C8×C197—41no—
104C2×C8×C197C2311yescharacteristic
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C2×C8×C197C2311yescharacteristic
108C4⋊C4×C197D411yescharacteristic
109C22⋊C4×C197D411yescharacteristic
110C22⋊C4×C197D411yescharacteristic
111C42×C197C2311yescharacteristic
112C22×D197—21no—
113C22×D197—21no—
114C22×D197—21no—
115C22×D197—42no—
116C22×D197—42no—
117C22×D197—42no—
118C22×D197—42no—
119C22×D197—81no—
120C22×D197—81no—
121Dic394—21no—
122Dic394—21no—
123D788—21no—
124D788—21no—
125D788—21no—
126D788—21no—
127D788—21no—
128D788—42no—
129D788—42no—
130D788—42no—
131D788—81no—
132D788#4112yes—
133C22⋊D197—42no—
134C22⋊D197—42no—
135C2×Dic197—21no—
136C2×Dic197—21no—
137C2×Dic197—21no—
138C2×Dic197—21no—
139C2×Dic197—21no—
140C2×Dic197—42no—
141C4×D197—42no—
142C23×C197C2×D411yescharacteristic
143C23×C197C2×D411yescharacteristic
144D4×C197—22no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197C2×D411yescharacteristic
150C2×C4×C197C2×D411yescharacteristic
151C2×C4×C197C2×D411yescharacteristic
152C2×C4×C197C2×D411yescharacteristic
153C2×C4×C197C2411yescharacteristic
154C8×C197—21no—
155C8×C197—21no—
156C8×C197—21no—
157C8×C197—21no—
158C22×C197—21no—
159C22×C197—21no—
160C22×C197—21no—
161C22×C197—21no—
162C22×C197—21no—
163C22×C197AGL(1,ℤ/8)12yes—
164C22×C197C22×D411yescharacteristic
165C2×D197—22no—
166C2×D197—41no—
167C2×D197—41no—
168C2×D197—41no—
169C2×D197—41no—
170C2×D197—41no—
171C2×D197—41no—
172C2×D197—41no—
173C2×D197—41no—
174C2×D197—84no—
175C2×D197—84no—
176C4×C197—21no—
177C4×C197—21no—
178C4×C197—21no—
179C4×C197—21no—
180C4×C197C2×#4111yescharacteristic
181C4×C197H(5,𝔽2)11yescharacteristic
182Dic197—41no—
183Dic197—41no—
184Dic197—41no—
185Dic197—41no—
186Dic197—41no—
187C2×C197—22no—
188C2×C197—41no—
189C2×C197#54611yescharacteristic
190C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C197#57711yescharacteristic
192D197—42no—
193D197—81no—
194D197—81no—
195C197#331011yescharacteristic, complemented, Sylow
196#3310—1971nomaximal, Sylow
197#552—1971no—
198D42—1971no—
199#490—1971no—
200#482—1971no—
201#480—1971no—
202#478—1971no—
203#468—1971no—
204#465—1971no—
205#460—1971no—
206#450—1971no—
207#450—1971no—
208#440—1971no—
209#435D19711yescharacteristic, complemented
210#418—1971no—
211C2×#102—1971no—
212C22×D4—3941no—
213C22×D4—1971no—
214C22×D4—1971no—
215C2×SD16—1971no—
216C2×SD16—1971no—
217C2×SD16—1971no—
218C2×M16C2×D19711yescharacteristic, complemented
219C22×C8C2×D19711yescharacteristic, complemented
220#127—1971no—
221#126—3941no—
222#125—3941no—
223#125—1971no—
224#123—3941no—
225C22⋊Q8—1971no—
226C22⋊Q8—1971no—
227#121—3941no—
228#121—1971no—
229#121—1971no—
230#120—3941no—
231#120—3941no—
232C4×Q8—1971no—
233C4×D4—1971no—
234C4×D4C2×D19711yescharacteristic
235#117—3941no—
236#117—1971no—
237C2×C4⋊C4—1971no—
238#107—1971no—
239#107—1971no—
240#106—1971no—
241C4⋊C8C2×D19711yescharacteristic, complemented
242#103—1971no—
243#102—1971no—
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248#102—1971no—
249#102—1971no—
250#102—1971no—
251#102—1971no—
252C22⋊C8C2×D19711yescharacteristic, complemented
253C22⋊C8C2×D19711yescharacteristic, complemented
254#97C2×D19711yescharacteristic, complemented
255C24—3941no—
256C24—3941no—
257C2×Q8—1971no—
258C2×D4—7881no—
259C2×D4—7881no—
260C2×D4—7884no—
261C2×D4—3942no—
262C2×D4—3942no—
263C2×D4—3941no—
264C2×D4—3941no—
265C2×D4—3941no—
266C2×D4—3941no—
267C2×D4—3941no—
268C2×D4—3941no—
269C2×D4—1971no—
270C2×D4—1971no—
271C2×D4—1971no—
272C2×D4D78811yescharacteristic
273C22×C4—3941no—
274C22×C4—3941no—
275C22×C4C22×D19711yescharacteristic
276C22×C4C22×D19711yescharacteristic
277SD16—7881no—
278SD16—7881no—
279SD16—3942no—
280M16—41no—
281C2×C8—41no—
282C2×C8C22×D19711yescharacteristic
283C2×C8C22×D19711yescharacteristic
284C2×C8C22×D19711yescharacteristic
285C2×C8C22×D19711yescharacteristic
286C4⋊C4—3941no—
287C4⋊C4—3941no—
288C4⋊C4—3941no—
289C4⋊C4—3941no—
290C4⋊C4—3941no—
291C4⋊C4—3941no—
292C4⋊C4—1971no—
293C4⋊C4—1971no—
294C4⋊C4—1971no—
295C4⋊C4—1971no—
296C4⋊C4D78811yescharacteristic
297C22⋊C4—3941no—
298C22⋊C4—3941no—
299C22⋊C4—3941no—
300C22⋊C4—3941no—
301C22⋊C4—3941no—
302C22⋊C4—3941no—
303C22⋊C4D78811yescharacteristic
304C22⋊C4D78811yescharacteristic
305C42—3941no—
306C42—3941no—
307C42C22×D19711yescharacteristic
308C23—7882no—
309C23—7882no—
310C23—7882no—
311C23—7882no—
312C23—15761no—
313C23—15761no—
314C23—3941no—
315C23—3941no—
316C23—3941no—
317C23C2×D78811yescharacteristic
318C23C2×D78811yescharacteristic
319Q8—3941no—
320Q8—3941no—
321D4—22no—
322D4—7882no—
323D4—7882no—
324D4—7882no—
325D4—7882no—
326D4—7882no—
327D4—15761no—
328D4—3941no—
329D4—3941no—
330D4—3941no—
331D4—3941no—
332D4—3941no—
333D4—1972no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—7882no—
339C2×C4—7882no—
340C2×C4—3941no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4—3941no—
345C2×C4C2×D78811yescharacteristic
346C2×C4C2×D78811yescharacteristic
347C2×C4C2×D78811yescharacteristic
348C2×C4C2×D78811yescharacteristic
349C2×C4C23×D19711yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—7881no—
360C22—7881no—
361C22—7881no—
362C22—7881no—
363C22—7881no—
364C22—7881no—
365C22—7881no—
366C22—7881no—
367C22—15764no—
368C22—15764no—
369C22—3942no—
370C22#2026335412yes—
371C22C22×D78811yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—7881no—
377C4—7881no—
378C4—7881no—
379C4—7881no—
380C4—7881no—
381C4#41×D19711yescharacteristic
382C4#2026337111yescharacteristic
383C2—22no—
384C2—41no—
385C2—7882no—
386C2—15761no—
387C2—15761no—
388C2#2716172411yescharacteristic
389C2C2×#2026335411yescharacteristic
390C2#2716189111yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#435×C197C2
socleC22×C197C22×D4
prime radicalD4×D788C2
hypercenter#435D197
hypocenterC197#3310
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #27161769C2semidirect
3. #27161444C2semidirect
4. #27161404C2semidirect
5. #27161393C2semidirect
6. #27161380C2semidirect
7. #27161325C2semidirect
8. #27161308C2semidirect
9. #27161280C2semidirect
10. #27161232C2semidirect
11. #27161232C2semidirect
12. #27161181C2semidirect
13. #27161085C2semidirect
14. C2×#20263248C2semidirect
15. #435×C197C2semidirect
20. C2×#15691347C22semidirect
21. C2×#15691347C22semidirect
24. #20263318C22semidirect
27. #20263305C22semidirect
28. #20263305C22semidirect
34. C4×Dic394C22semidirect
36. #20263285C22semidirect
38. C2×C4⋊Dic197C22semidirect
39. #20263260C22semidirect
40. #20263260C22semidirect
41. #20263258C22semidirect
42. #20263251C22semidirect
46. #20263248C22semidirect
47. #20263248C22semidirect
48. #20263248C22semidirect
49. #20263248C22semidirect
50. #20263248C22semidirect
51. #20263248C22semidirect
52. C2×M16×C197C22semidirect
53. C22×C8×C197C22semidirect
55. C4⋊C8×C197C22semidirect
56. C22⋊C8×C197C22semidirect
57. C22⋊C8×C197C22semidirect
58. #97×C197C22semidirect
195. C197#3310semidirect
209. #435D197semidirect
218. C2×M16C2×D197semidirect
219. C22×C8C2×D197semidirect
241. C4⋊C8C2×D197semidirect
252. C22⋊C8C2×D197semidirect
253. C22⋊C8C2×D197semidirect
254. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617692
D4×D7882
#271614442
#271614042
#271613932
#271613802
#271613252
#271613082
#271612802
#271612322
#271611812
#271610852
C2×#202632482
#435×C1972
#3310197