Finite Group Databasebeta

← #38754608

Subgroups — #38754608

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#27161738C211yesmaximal, characteristic, complemented
2#27161436C211yesmaximal, characteristic, complemented
3#27161420C211yesmaximal, characteristic, complemented
4#27161397C211yesmaximal, characteristic, complemented
5#27161383C211yesmaximal, characteristic, complemented
6#27161313C211yesmaximal, characteristic, complemented
7#27161306C211yesmaximal, characteristic, complemented
8#27161294C211yesmaximal, characteristic, complemented
9#27161222C211yesmaximal, characteristic, complemented
10#27161217C211yesmaximal, characteristic, complemented
11C4×#15691346C211yesmaximal, characteristic, complemented
12D4×#13096802C211yesmaximal, characteristic, complemented
13#27161073C211yesmaximal, characteristic, complemented
14C2×#20263249C211yesmaximal, characteristic, complemented
15#553×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×Dic394C2211yescharacteristic
18C2×#15691346C2211yescharacteristic
19C2×#15691346C2211yescharacteristic
20C2×#15691343C2211yescharacteristic
21C22×#13096802C2211yescharacteristic
22C22×#13096802C2211yescharacteristic
23#20263314C2211yescharacteristic
24#20263312—21no—
25#20263305C2211yescharacteristic
26#20263304—21no—
27#20263300C2211yescharacteristic
28#20263298—21no—
29C4×Dic394C2211yescharacteristic
30#20263286—21no—
31#20263259C2211yescharacteristic
32#20263257C2211yescharacteristic
33#20263257C2211yescharacteristic
34C4⋊#13096802C2211yescharacteristic
35#20263249C2211yescharacteristic
36#20263249C2211yescharacteristic
37#20263249C2211yescharacteristic
38#20263249C2211yescharacteristic
39#20263249C2211yescharacteristic
40#20263249C2211yescharacteristic
41#20263247C2211yescharacteristic
42#20263246C2211yescharacteristic
43#20263246C2211yescharacteristic
44#20263246C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic
47C4×#13096802C2211yescharacteristic
48#126×C197—21no—
49#125×C197C2211yescharacteristic
50#123×C197—21no—
51#121×C197—21no—
52#121×C197C2211yescharacteristic
53#121×C197C2211yescharacteristic
54C4×D4×C197—21no—
55C4×D4×C197C2211yescharacteristic
56C4×D4×C197C2211yescharacteristic
57#117×C197C2211yescharacteristic
58C2×C4⋊C4×C197C2211yescharacteristic
59#15691358—22no—
60#15691358—41no—
61C2×Dic394—22no—
62C2×Dic394—21no—
63C2×Dic394—41no—
64C2×Dic394C2311yescharacteristic
65C2×Dic394C2311yescharacteristic
66C2×D788C2311yescharacteristic
67C2×C22⋊D197—21no—
68C2×C22⋊D197—21no—
69C22×Dic197—21no—
70C2×C4×D197—21no—
71#15691346—22no—
72#15691346—41no—
73#15691343—41no—
74C2×#13096802—41no—
75C2×#13096802—41no—
76C2×#13096802C2311yescharacteristic
77C2×#13096802C2311yescharacteristic
78C2×#13096802C2311yescharacteristic
79C2×#13096802C2311yescharacteristic
80C4⋊Dic197C2311yescharacteristic
81#15691337—21no—
82#15691337—21no—
83C22⋊Dic197—21no—
84#15691335—21no—
85#15691335—21no—
86#15691335—21no—
87C4×Dic197—21no—
88C2×D4×C197—21no—
89C2×D4×C197—21no—
90C2×D4×C197D411yescharacteristic
91C2×D4×C197C2311yescharacteristic
92C22×C4×C197—21no—
93C22×C4×C197—21no—
94C22×C4×C197C2311yescharacteristic
95C22×C4×C197C2311yescharacteristic
96C4⋊C4×C197—21no—
97C4⋊C4×C197—21no—
98C4⋊C4×C197—21no—
99C4⋊C4×C197D411yescharacteristic
100C4⋊C4×C197C2311yescharacteristic
101C4⋊C4×C197C2311yescharacteristic
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—
112Dic394—21no—
113Dic394—21no—
114Dic394—21no—
115Dic394—42no—
116Dic394#4112yes—
117D788—21no—
118D788—21no—
119C22⋊D197—42no—
120C22⋊D197—42no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197—21no—
124C2×Dic197—42no—
125C4×D197—42no—
126#13096802—21no—
127#13096802—21no—
128#13096802—21no—
129#13096802—21no—
130C23×C197—21no—
131C23×C197C2×D411yescharacteristic
132C23×C197C2×D411yescharacteristic
133D4×C197—22no—
134D4×C197—21no—
135D4×C197—21no—
136D4×C197—42no—
137D4×C197—81no—
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—81no—
146C2×C4×C197—81no—
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—21no—
168C4×C197—21no—
169C4×C197—41no—
170C4×C197—41no—
171C4×C197—41no—
172C4×C197C2×#4111yescharacteristic
173C4×C197H(5,𝔽2)11yescharacteristic
174Dic197—22no—
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#553D19711yescharacteristic, complemented
187#548—1971no—
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×#41—1971no—
202C22×Q8—1971no—
203C2×SD16—1971no—
204C2×SD16—1971no—
205C2×D8—1971no—
206C22×C8—1971no—
207C22×C8—1971no—
208#126—21no—
209#125C2×D19711yescharacteristic
210#124—3941no—
211#124—1971no—
212#123—21no—
213C22⋊Q8—3941no—
214C22⋊Q8—1971no—
215#121—21no—
216#121—3941no—
217#121—1971no—
218#121C2×D19711yescharacteristic
219#121C2×D19711yescharacteristic
220C4×Q8—1971no—
221C4×D4—21no—
222C4×D4—3941no—
223C4×D4C2×D19711yescharacteristic
224C4×D4C2×D19711yescharacteristic
225#117C2×D19711yescharacteristic
226C2×C4⋊C4C2×D19711yescharacteristic
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—7881no—
245#41—3942no—
246C2×Q8—7881no—
247C2×Q8—3942no—
248C2×Q8—3941no—
249C2×Q8—1971no—
250C2×Q8—1971no—
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—21no—
276C4⋊C4—3941no—
277C4⋊C4—3941no—
278C4⋊C4—1971no—
279C4⋊C4C22⋊D19711yescharacteristic
280C4⋊C4C22×D19711yescharacteristic
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—7882no—
301Q8—3941no—
302Q8—3941no—
303Q8—3941no—
304Q8—1972no—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—7882no—
311D4—7882no—
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—81no—
322C2×C4—81no—
323C2×C4—7882no—
324C2×C4—7882no—
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—41no—
355C4—41no—
356C4—41no—
357C4—7881no—
358C4—7881no—
359C4—3942no—
360C4#41×D19711yescharacteristic
361C4#2026337211yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—15761no—
366C2#2716171711yescharacteristic
367C2C2×#2026334611yescharacteristic
368C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#553×C197C2
socleC22×C197C22×D4
hypercenter#553D197
hypocenterC197#3301
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #27161738C2semidirect
2. #27161436C2semidirect
3. #27161420C2semidirect
4. #27161397C2semidirect
5. #27161383C2semidirect
6. #27161313C2semidirect
7. #27161306C2semidirect
8. #27161294C2semidirect
9. #27161222C2semidirect
10. #27161217C2semidirect
11. C4×#15691346C2semidirect
12. D4×#13096802C2semidirect
13. #27161073C2semidirect
14. C2×#20263249C2semidirect
15. #553×C197C2semidirect
184. C197#3301semidirect
186. #553D197semidirect

Maximal subgroups

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

subgroupindex
#271617382
#271614362
#271614202
#271613972
#271613832
#271613132
#271613062
#271612942
#271612222
#271612172
C4×#156913462
D4×#130968022
#271610732
C2×#202632492
#553×C1972
#3301197