Finite Group Databasebeta

← #38754614

Subgroups — #38754614

Order 25216.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161775C211yesmaximal, characteristic, complemented
2#27161433C211yesmaximal, characteristic, complemented
3#27161417C211yesmaximal, characteristic, complemented
4#27161395C211yesmaximal, characteristic, complemented
5#27161386C211yesmaximal, characteristic, complemented
6#27161313C211yesmaximal, characteristic, complemented
7#27161310C211yesmaximal, characteristic, complemented
8#27161291C211yesmaximal, characteristic, complemented
9#27161224C211yesmaximal, characteristic, complemented
10#27161218C211yesmaximal, characteristic, complemented
11C4×#15691345C211yesmaximal, characteristic, complemented
12D4×#13096802C211yesmaximal, characteristic, complemented
13#27161075C211yesmaximal, characteristic, complemented
14C2×#20263250C211yesmaximal, characteristic, complemented
15#548×C197C211yesmaximal, characteristic, complemented
16C2×#15691345C2211yescharacteristic
17C2×#15691345C2211yescharacteristic
18C2×#15691343C2211yescharacteristic
19C22×#13096802C2211yescharacteristic
20C22×#13096802C2211yescharacteristic
21#20263321—21no—
22#20263318C2211yescharacteristic
23#20263308—21no—
24#20263300C2211yescharacteristic
25#20263300C2211yescharacteristic
26#20263298—21no—
27C4×D788C2211yescharacteristic
28#20263286—21no—
29#20263285C2211yescharacteristic
30C2×C4⋊Dic197C2211yescharacteristic
31#20263259C2211yescharacteristic
32#20263257C2211yescharacteristic
33#20263257C2211yescharacteristic
34C4⋊#13096802C2211yescharacteristic
35#20263250C2211yescharacteristic
36#20263250C2211yescharacteristic
37#20263250C2211yescharacteristic
38#20263250C2211yescharacteristic
39#20263250C2211yescharacteristic
40#20263250C2211yescharacteristic
41#20263247C2211yescharacteristic
42#20263247C2211yescharacteristic
43#20263247C2211yescharacteristic
44#20263246C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic
47C4×#13096802C2211yescharacteristic
48C2×#41×C197C2211yescharacteristic
49C22×Q8×C197C2211yescharacteristic
50#124×C197—21no—
51#124×C197C2211yescharacteristic
52C22⋊Q8×C197—21no—
53C22⋊Q8×C197C2211yescharacteristic
54#121×C197—21no—
55#121×C197C2211yescharacteristic
56C4×Q8×C197C2211yescharacteristic
57C4×D4×C197—21no—
58C4×D4×C197C2211yescharacteristic
59C2×D788C2311yescharacteristic
60C2×C22⋊D197—21no—
61C2×C22⋊D197—21no—
62C22×Dic197—21no—
63C2×C4×D197—21no—
64#15691345—22no—
65#15691345—41no—
66#15691343—41no—
67C2×#13096802—41no—
68C2×#13096802—41no—
69C2×#13096802C2311yescharacteristic
70C2×#13096802C2311yescharacteristic
71C2×#13096802C2311yescharacteristic
72C2×#13096802C2311yescharacteristic
73C4⋊Dic197—21no—
74C4⋊Dic197C2311yescharacteristic
75C4⋊Dic197C2311yescharacteristic
76C4⋊Dic197C2311yescharacteristic
77#15691337—21no—
78#15691337—21no—
79C22⋊Dic197—21no—
80#15691335—21no—
81#15691335—21no—
82#15691335—21no—
83C4×Dic197—21no—
84#41×C197—22no—
85#41×C197—41no—
86C2×Q8×C197—21no—
87C2×Q8×C197—22no—
88C2×Q8×C197—41no—
89C2×Q8×C197C2311yescharacteristic
90C2×Q8×C197C2311yescharacteristic
91C2×D4×C197—21no—
92C2×D4×C197—21no—
93C2×D4×C197D411yescharacteristic
94C2×D4×C197C2311yescharacteristic
95C22×C4×C197—21no—
96C22×C4×C197—21no—
97C22×C4×C197C2311yescharacteristic
98C22×C4×C197C2311yescharacteristic
99C4⋊C4×C197—21no—
100C4⋊C4×C197—21no—
101C4⋊C4×C197D411yescharacteristic
102C4⋊C4×C197C2311yescharacteristic
103C22⋊C4×C197—21no—
104C22⋊C4×C197—21no—
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C22⋊C4×C197D411yescharacteristic
108C22⋊C4×C197D411yescharacteristic
109C42×C197—21no—
110C42×C197C2311yescharacteristic
111C22×D197—21no—
112D788—21no—
113D788—21no—
114C22⋊D197—42no—
115C22⋊D197—81no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C2×Dic197—21no—
119C2×Dic197—81no—
120C4×D197—81no—
121#13096802—21no—
122#13096802—21no—
123#13096802—21no—
124#13096802—21no—
125C23×C197—21no—
126C23×C197C2×D411yescharacteristic
127C23×C197C2×D411yescharacteristic
128Q8×C197—21no—
129Q8×C197—21no—
130Q8×C197—21no—
131Q8×C197—42no—
132Q8×C197#4112yes—
133D4×C197—22no—
134D4×C197—21no—
135D4×C197—21no—
136D4×C197—42no—
137D4×C197—42no—
138C2×C4×C197—21no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—42no—
146C2×C4×C197—42no—
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2×D411yescharacteristic
149C2×C4×C197C2×D411yescharacteristic
150C2×C4×C197C2×D411yescharacteristic
151C2×C4×C197C2411yescharacteristic
152C22×C197—21no—
153C22×C197—21no—
154C22×C197—21no—
155C22×C197—21no—
156C22×C197—21no—
157C22×C197—41no—
158C22×C197—41no—
159C22×C197—41no—
160C22×C197#13512yes—
161C22×C197C22×D411yescharacteristic
162C2×D197—41no—
163C2×D197—41no—
164C2×D197—41no—
165C4×C197—21no—
166C4×C197—21no—
167C4×C197—22no—
168C4×C197—21no—
169C4×C197—21no—
170C4×C197—41no—
171C4×C197—41no—
172C4×C197C2×#4111yescharacteristic
173C4×C197H(5,𝔽2)11yescharacteristic
174Dic197—41no—
175Dic197—41no—
176Dic197—41no—
177C2×C197—22no—
178C2×C197—41no—
179C2×C197—81no—
180C2×C197#54611yescharacteristic
181C2×C197C2×#13511yescharacteristic
182C2×C197#57711yescharacteristic
183D197—81no—
184C197#330111yescharacteristic, complemented, Sylow
185#3301—1971nomaximal, Sylow
186#553—1971no—
187#548D19711yescharacteristic, complemented
188#488—1971no—
189#485—1971no—
190#481—1971no—
191#479—1971no—
192#466—1971no—
193#465—1971no—
194#463—1971no—
195#449—1971no—
196#448—1971no—
197C4×SD16—1971no—
198C8×D4—1971no—
199#416—1971no—
200C2×#103—1971no—
201C2×#41C2×D19711yescharacteristic
202C22×Q8C2×D19711yescharacteristic
203C2×SD16—1971no—
204C2×SD16—1971no—
205C2×D8—1971no—
206C22×C8—1971no—
207C22×C8—1971no—
208#126—3941no—
209#125—1971no—
210#124—21no—
211#124C2×D19711yescharacteristic
212#123—3941no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D19711yescharacteristic
215#121—21no—
216#121—3941no—
217#121—1971no—
218#121—1971no—
219#121C2×D19711yescharacteristic
220C4×Q8C2×D19711yescharacteristic
221C4×D4—21no—
222C4×D4—3941no—
223C4×D4—1971no—
224C4×D4C2×D19711yescharacteristic
225#117—1971no—
226C2×C4⋊C4—1971no—
227#107—1971no—
228#106—1971no—
229#106—1971no—
230C4⋊C8—1971no—
231#103—1971no—
232#103—1971no—
233#103—1971no—
234#103—1971no—
235#103—1971no—
236#103—1971no—
237#102—1971no—
238#102—1971no—
239#102—1971no—
240#102—1971no—
241C22⋊C8—1971no—
242C22⋊C8—1971no—
243C4×C8—1971no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D19711yescharacteristic
250C2×Q8C22×D19711yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—3941no—
254C2×D4—3941no—
255C2×D4—1971no—
256C2×D4C22⋊D19711yescharacteristic
257C2×D4C22×D19711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—3941no—
261C22×C4—3941no—
262C22×C4C22×D19711yescharacteristic
263C22×C4C22×D19711yescharacteristic
264SD16—7881no—
265SD16—3942no—
266D8—7881no—
267C2×C8—7881no—
268C2×C8—7881no—
269C2×C8—1971no—
270C2×C8—1971no—
271C2×C8—1971no—
272C2×C8—1971no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—3941no—
276C4⋊C4—3941no—
277C4⋊C4—3941no—
278C4⋊C4—1971no—
279C4⋊C4—1971no—
280C4⋊C4—1971no—
281C4⋊C4C22⋊D19711yescharacteristic
282C4⋊C4C22×D19711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—3941no—
288C22⋊C4—3941no—
289C22⋊C4—3941no—
290C22⋊C4—3941no—
291C22⋊C4C22⋊D19711yescharacteristic
292C22⋊C4C22⋊D19711yescharacteristic
293C42—21no—
294C42—3941no—
295C42C22×D19711yescharacteristic
296C23—21no—
297C23—3941no—
298C23C2×C22⋊D19711yescharacteristic
299C23C2×C22⋊D19711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1569135712yes—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—7882no—
311D4—15761no—
312D4—3941no—
313D4—3941no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—15761no—
324C2×C4—15761no—
325C2×C4—3941no—
326C2×C4—3941no—
327C2×C4—3941no—
328C2×C4C2×C22⋊D19711yescharacteristic
329C2×C4C2×C22⋊D19711yescharacteristic
330C2×C4C2×C22⋊D19711yescharacteristic
331C2×C4C2×C22⋊D19711yescharacteristic
332C2×C4C23×D19711yescharacteristic
333C8—3941no—
334C8—3941no—
335C8—3941no—
336C8—3941no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—7881no—
346C22—7881no—
347C22—7881no—
348C22#2026334612yes—
349C22C22×C22⋊D19711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—7881no—
358C4—7881no—
359C4—7881no—
360C4C2×#1569135711yescharacteristic
361C4#2026337111yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—15761no—
366C2#2716172311yescharacteristic
367C2C2×#2026334611yescharacteristic
368C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#548×C197C2
socleC22×C197C22×D4
hypercenter#548D197
hypocenterC197#3301
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #27161775C2semidirect
2. #27161433C2semidirect
3. #27161417C2semidirect
4. #27161395C2semidirect
5. #27161386C2semidirect
6. #27161313C2semidirect
7. #27161310C2semidirect
8. #27161291C2semidirect
9. #27161224C2semidirect
10. #27161218C2semidirect
11. C4×#15691345C2semidirect
12. D4×#13096802C2semidirect
13. #27161075C2semidirect
14. C2×#20263250C2semidirect
15. #548×C197C2semidirect
184. C197#3301semidirect
187. #548D197semidirect

Maximal subgroups

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

subgroupindex
#271617752
#271614332
#271614172
#271613952
#271613862
#271613132
#271613102
#271612912
#271612242
#271612182
C4×#156913452
D4×#130968022
#271610752
C2×#202632502
#548×C1972
#3301197