Finite Group Databasebeta

← #38754734

Subgroups — #38754734

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#27161768C211yesmaximal, characteristic, complemented
2D4×C22⋊D197C211yesmaximal, characteristic
3#27161442C211yesmaximal, characteristic, complemented
4#27161403C211yesmaximal, characteristic, complemented
5#27161391C211yesmaximal, characteristic, complemented
6#27161377C211yesmaximal, characteristic, complemented
7#27161325C211yesmaximal, characteristic, complemented
8#27161307C211yesmaximal, characteristic, complemented
9#27161278C211yesmaximal, characteristic, complemented
10#27161231C211yesmaximal, characteristic, complemented
11#27161230C211yesmaximal, characteristic, complemented
12#27161176C211yesmaximal, characteristic, complemented
13#27161148C211yesmaximal, characteristic, complemented
14#102×D197C211yesmaximal, characteristic, complemented
15#418×C197C211yesmaximal, characteristic, complemented
16C22×C22⋊D197—21no—
17C2×D4×D197C2211yescharacteristic
18C2×#15691347C2211yescharacteristic, complemented
19C2×#15691346C2211yescharacteristic
20C2×#15691346C2211yescharacteristic, complemented
21C2×C8×D197C2211yescharacteristic, complemented
22#20263322C2211yescharacteristic
23#20263319—21no—
24#20263318—21no—
25#20263317C2211yescharacteristic, complemented
26#20263308—21no—
27#20263306C2211yescharacteristic, complemented
28#20263305C2211yescharacteristic, complemented
29#20263300C2211yescharacteristic
30#20263298—21no—
31#20263297C2211yescharacteristic
32#20263295—21no—
33#20263294—21no—
34#20263291C2211yescharacteristic, complemented
35D4×Dic197C2211yescharacteristic
36C4×C22⋊D197C2211yescharacteristic
37#20263283—21no—
38C4⋊C4×D197C2211yescharacteristic, complemented
39#20263260C2211yescharacteristic, complemented
40#20263259C2211yescharacteristic, complemented
41#20263257C2211yescharacteristic, complemented
42#20263255C2211yescharacteristic, complemented
43#20263251C2211yescharacteristic, complemented
44#20263248C2211yescharacteristic, complemented
45#20263247C2211yescharacteristic
46#20263247C2211yescharacteristic, complemented
47#20263246C2211yescharacteristic, complemented
48#20263246C2211yescharacteristic, complemented
49C22⋊#13096802C2211yescharacteristic, complemented
50#20263238C2211yescharacteristic, complemented
51#20263236C2211yescharacteristic, complemented
52C22×D4×C197C2211yescharacteristic
53C2×M16×C197C2211yescharacteristic, complemented
54#117×C197C2211yescharacteristic, complemented
55#102×C197C2211yescharacteristic
56#102×C197C2211yescharacteristic
57#102×C197C2211yescharacteristic, complemented
58#102×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C2×Dic394C2311yescharacteristic
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—
72C2×C4×D197—21no—
73C2×C4×D197C2311yescharacteristic
74#15691347—41no—
75#15691346—22no—
76#15691346—41no—
77C2×#13096802C2311yescharacteristic
78C2×#13096802C2311yescharacteristic
79C8×D197—41no—
80C4⋊Dic197—21no—
81C4⋊Dic197—21no—
82C4⋊Dic197C2311yescharacteristic
83C4⋊Dic197C2311yescharacteristic
84#15691337—21no—
85#15691337—21no—
86#15691337—21no—
87#15691337—21no—
88#15691337D411yescharacteristic
89C22⋊Dic197—21no—
90C22⋊Dic197—21no—
91C22⋊Dic197—21no—
92C22⋊Dic197D411yescharacteristic
93#15691335—21no—
94#15691335—21no—
95#15691335D411yescharacteristic
96C4×Dic197—21no—
97C4×Dic197C2311yescharacteristic
98C24×C197—21no—
99C2×D4×C197—22no—
100C2×D4×C197—21no—
101C2×D4×C197—41no—
102C2×D4×C197C2311yescharacteristic
103C2×D4×C197C2311yescharacteristic
104C22×C4×C197C2311yescharacteristic
105M16×C197—41no—
106C2×C8×C197C2311yescharacteristic
107C2×C8×C197C2311yescharacteristic
108C4⋊C4×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C22⋊C4×C197—21no—
111C42×C197—21no—
112C22×D197—21no—
113C22×D197—42no—
114C22×D197—42no—
115C22×D197—81no—
116C22×D197C2×D411yescharacteristic
117Dic394—21no—
118Dic394—21no—
119D788—21no—
120D788—21no—
121C22⋊D197—22no—
122C22⋊D197—42no—
123C22⋊D197—42no—
124C22⋊D197—42no—
125C22⋊D197—42no—
126C22⋊D197—81no—
127C2×Dic197—21no—
128C2×Dic197—21no—
129C2×Dic197—21no—
130C2×Dic197—42no—
131C2×Dic197C2×D411yescharacteristic
132C2×Dic197C2×D411yescharacteristic
133C2×Dic197C2×D411yescharacteristic
134C4×D197—21no—
135C4×D197—21no—
136C4×D197—42no—
137#13096802—21no—
138#13096802—21no—
139C23×C197—21no—
140C23×C197—21no—
141C23×C197—42no—
142C23×C197—42no—
143C23×C197—81no—
144C23×C197C2×D411yescharacteristic
145D4×C197—21no—
146D4×C197—21no—
147D4×C197—21no—
148D4×C197—42no—
149D4×C197#4112yes—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197—21no—
154C2×C4×C197C2×D411yescharacteristic
155C2×C4×C197C2411yescharacteristic
156C8×C197—21no—
157C8×C197—21no—
158C22×C197—21no—
159C22×C197—22no—
160C22×C197—21no—
161C22×C197—21no—
162C22×C197—41no—
163C22×C197—41no—
164C22×C197—41no—
165C22×C197—41no—
166C22×C197—41no—
167C22×C197—84no—
168C22×C197C22×D411yescharacteristic
169C2×D197—21no—
170C2×D197—21no—
171C2×D197—41no—
172C2×D197—41no—
173C2×D197—41no—
174C2×D197—84no—
175C2×D197AGL(1,ℤ/8)12yes—
176C4×C197—21no—
177C4×C197—41no—
178C4×C197—41no—
179C4×C197C2×#4111yescharacteristic
180C4×C197H(5,𝔽2)11yescharacteristic
181Dic197—21no—
182Dic197—21no—
183Dic197—21no—
184Dic197—41no—
185Dic197—41no—
186Dic197—41no—
187C2×C197—42no—
188C2×C197—41no—
189C2×C197—81no—
190C2×C197#54611yescharacteristic
191C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C197#57711yescharacteristic
193D197—22no—
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#435—1971no—
210#418D19711yescharacteristic, complemented
211C2×#102—1971no—
212C22×D4—3941no—
213C22×D4—1971no—
214C22×D4C2×D19711yescharacteristic
215C2×SD16—1971no—
216C2×SD16—1971no—
217C2×SD16—1971no—
218C2×M16C2×D19711yescharacteristic, complemented
219C22×C8—1971no—
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×D4—1971no—
235#117—3941no—
236#117C2×D19711yescharacteristic, complemented
237C2×C4⋊C4—1971no—
238#107—1971no—
239#107—1971no—
240#106—1971no—
241C4⋊C8—1971no—
242#103—1971no—
243#102—1971no—
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248#102C2×D19711yescharacteristic
249#102C2×D19711yescharacteristic
250#102C2×D19711yescharacteristic, complemented
251#102C2×D19711yescharacteristic, complemented
252C22⋊C8—1971no—
253C22⋊C8—1971no—
254#97—1971no—
255C24—21no—
256C24—3941no—
257C2×Q8—1971no—
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—7884no—
262C2×D4—7881no—
263C2×D4—3941no—
264C2×D4—3941no—
265C2×D4—3942no—
266C2×D4—3941no—
267C2×D4—3941no—
268C2×D4—3941no—
269C2×D4—1971no—
270C2×D4—1971no—
271C2×D4C22×D19711yescharacteristic
272C2×D4C22×D19711yescharacteristic
273C22×C4—3941no—
274C22×C4—3941no—
275C22×C4—1971no—
276C22×C4C22×D19711yescharacteristic
277SD16—7881no—
278SD16—7881no—
279SD16—3942no—
280M16—41no—
281C2×C8—7881no—
282C2×C8—1971no—
283C2×C8—1971no—
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⋊C4C22×D19711yescharacteristic
296C4⋊C4C22×D19711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—3941no—
299C22⋊C4—3941no—
300C22⋊C4—3941no—
301C22⋊C4—3941no—
302C22⋊C4—3941no—
303C22⋊C4—1971no—
304C22⋊C4—1971no—
305C42—21no—
306C42—3941no—
307C42—1971no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—7882no—
314C23—7882no—
315C23—15761no—
316C23—3941no—
317C23—1971no—
318C23D4×D19711yescharacteristic
319Q8—3941no—
320Q8—3941no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—7882no—
326D4—7882no—
327D4—7882no—
328D4—7882no—
329D4—15761no—
330D4—3942no—
331D4—3941no—
332D4—3941no—
333D4#1569135812yes—
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×C4—1971no—
346C2×C4—1971no—
347C2×C4—1971no—
348C2×C4D4×D19711yescharacteristic
349C2×C4C23×D19711yescharacteristic
350C8—21no—
351C8—21no—
352C8—3941no—
353C8—3941no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—84no—
364C22—7881no—
365C22—7881no—
366C22—7881no—
367C22—15764no—
368C22—3941no—
369C22—3941no—
370C22—1972no—
371C22C2×D4×D19711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—7881no—
376C4—7881no—
377C4—7881no—
378C4—3941no—
379C4—3941no—
380C4—3941no—
381C4C2×#1569135811yescharacteristic
382C4#2026337111yescharacteristic
383C2—41no—
384C2—42no—
385C2—81no—
386C2—15761no—
387C2—3942no—
388C2#2716171911yescharacteristic
389C2AGL(1,ℤ/8)×D19711yescharacteristic
390C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#418×C197C2
socleC22×C197C22×D4
prime radicalD4×C22⋊D197C2
hypercenter#418D197
hypocenterC197#3310
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #27161768C2semidirect
3. #27161442C2semidirect
4. #27161403C2semidirect
5. #27161391C2semidirect
6. #27161377C2semidirect
7. #27161325C2semidirect
8. #27161307C2semidirect
9. #27161278C2semidirect
10. #27161231C2semidirect
11. #27161230C2semidirect
12. #27161176C2semidirect
13. #27161148C2semidirect
14. #102×D197C2semidirect
15. #418×C197C2semidirect
18. C2×#15691347C22semidirect
20. C2×#15691346C22semidirect
21. C2×C8×D197C22semidirect
25. #20263317C22semidirect
27. #20263306C22semidirect
28. #20263305C22semidirect
34. #20263291C22semidirect
38. C4⋊C4×D197C22semidirect
39. #20263260C22semidirect
40. #20263259C22semidirect
41. #20263257C22semidirect
42. #20263255C22semidirect
43. #20263251C22semidirect
44. #20263248C22semidirect
46. #20263247C22semidirect
47. #20263246C22semidirect
48. #20263246C22semidirect
49. C22⋊#13096802C22semidirect
50. #20263238C22semidirect
51. #20263236C22semidirect
53. C2×M16×C197C22semidirect
54. #117×C197C22semidirect
57. #102×C197C22semidirect
58. #102×C197C22semidirect
195. C197#3310semidirect
210. #418D197semidirect
218. C2×M16C2×D197semidirect
236. #117C2×D197semidirect
250. #102C2×D197semidirect
251. #102C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617682
D4×C22⋊D1972
#271614422
#271614032
#271613912
#271613772
#271613252
#271613072
#271612782
#271612312
#271612302
#271611762
#271611482
#102×D1972
#418×C1972
#3310197