Finite Group Databasebeta

← #38755287

Subgroups — #38755287

Order 25216.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161764C211yesmaximal, characteristic
2#27161763C211yesmaximal, characteristic
3#27161480C211yesmaximal, characteristic, complemented
4#27161421C211yesmaximal, characteristic, complemented
5#27161419C211yesmaximal, characteristic, complemented
6#27161416C211yesmaximal, characteristic, complemented
7#27161385C211yesmaximal, characteristic, complemented
8#27161384C211yesmaximal, characteristic, complemented
9#27161336C211yesmaximal, characteristic, complemented
10#27161335C211yesmaximal, characteristic, complemented
11#27161171C211yesmaximal, characteristic, complemented
12#27161154C211yesmaximal, characteristic
13#27161120C211yesmaximal, characteristic, complemented
14#27161119C211yesmaximal, characteristic, complemented
15#484×C197C211yesmaximal, characteristic, complemented
16C2×#15691348C2211yescharacteristic
17C2×#15691342C2211yescharacteristic
18C2×#15691341C2211yescharacteristic
19#20263321—21no—
20#20263319—21no—
21#20263317C2211yescharacteristic
22#20263317C2211yescharacteristic
23#20263316—21no—
24#20263316—21no—
25#20263309—21no—
26#20263308—21no—
27#20263306C2211yescharacteristic
28#20263305C2211yescharacteristic
29#20263302C2211yescharacteristic
30Q8×Dic197C2211yescharacteristic
31#20263291C2211yescharacteristic
32C4×C22⋊D197C2211yescharacteristic
33#20263285—21no—
34#20263285C2211yescharacteristic
35#20263284—21no—
36#20263284C2211yescharacteristic
37C4⋊C4×D197C2211yescharacteristic
38#20263260C2211yescharacteristic
39#20263259C2211yescharacteristic
40#20263259C2211yescharacteristic
41#20263258C2211yescharacteristic
42#20263257C2211yescharacteristic
43#20263257C2211yescharacteristic
44#20263257C2211yescharacteristic
45#20263255C2211yescharacteristic
46#20263251C2211yescharacteristic
47#20263251C2211yescharacteristic
48#20263250C2211yescharacteristic
49#20263249C2211yescharacteristic
50#20263238C2211yescharacteristic
51C8×Dic197C2211yescharacteristic
52C22⋊Q8×C197C2211yescharacteristic
53C2×C4⋊C4×C197C2211yescharacteristic
54#107×C197C2211yescharacteristic
55#107×C197C2211yescharacteristic
56#103×C197C2211yescharacteristic
57#103×C197C2211yescharacteristic
58C22⋊C8×C197C2211yescharacteristic
59C2×Dic394C2311yescharacteristic
60C2×C22⋊D197D411yescharacteristic
61C2×C4×D197—21no—
62C2×C4×D197C2311yescharacteristic
63#15691348—22no—
64#15691342—22no—
65#15691341—22no—
66C2×#13096802C2311yescharacteristic
67C2×#13096802C2311yescharacteristic
68C4⋊Dic197—21no—
69C4⋊Dic197—21no—
70C4⋊Dic197—21no—
71C4⋊Dic197—21no—
72C4⋊Dic197C2311yescharacteristic
73C4⋊Dic197C2311yescharacteristic
74C4⋊Dic197C2311yescharacteristic
75#15691337—21no—
76#15691337—21no—
77#15691337—21no—
78#15691337—21no—
79#15691337—21no—
80#15691337—21no—
81#15691337D411yescharacteristic
82C22⋊Dic197—21no—
83C22⋊Dic197—21no—
84C22⋊Dic197D411yescharacteristic
85#15691335—21no—
86#15691335—21no—
87#15691335—21no—
88#15691335D411yescharacteristic
89C4×Dic197—21no—
90C4×Dic197—21no—
91C4×Dic197—21no—
92C4×Dic197—21no—
93C4×Dic197C2311yescharacteristic
94C2×Q8×C197C2311yescharacteristic
95C22×C4×C197—21no—
96C22×C4×C197C2311yescharacteristic
97C2×C8×C197C2311yescharacteristic
98C2×C8×C197C2311yescharacteristic
99C4⋊C4×C197—21no—
100C4⋊C4×C197—21no—
101C4⋊C4×C197C2311yescharacteristic
102C4⋊C4×C197C2311yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C22⋊C4×C197—21no—
105C22×D197C2×D411yescharacteristic
106Dic394—21no—
107Dic394—21no—
108C22⋊D197—41no—
109C2×Dic197—21no—
110C2×Dic197—21no—
111C2×Dic197—21no—
112C2×Dic197—21no—
113C2×Dic197C2×D411yescharacteristic
114C2×Dic197C2×D411yescharacteristic
115C2×Dic197C2×D411yescharacteristic
116C4×D197—21no—
117C4×D197—21no—
118C4×D197—42no—
119#13096802—21no—
120#13096802#4112yes—
121C23×C197C2×D411yescharacteristic
122Q8×C197—21no—
123Q8×C197—21no—
124C2×C4×C197—21no—
125C2×C4×C197—21no—
126C2×C4×C197—21no—
127C2×C4×C197—21no—
128C2×C4×C197—21no—
129C2×C4×C197—21no—
130C2×C4×C197—42no—
131C2×C4×C197C2×D411yescharacteristic
132C2×C4×C197C2411yescharacteristic
133C8×C197—21no—
134C8×C197—21no—
135C22×C197—21no—
136C22×C197—21no—
137C22×C197—21no—
138C22×C197C22×D411yescharacteristic
139C2×D197—21no—
140C2×D197—21no—
141C2×D197—21no—
142C4×C197—21no—
143C4×C197—41no—
144C4×C197—41no—
145C4×C197—41no—
146C4×C197—41no—
147C4×C197C2×#4111yescharacteristic
148C4×C1972−1+411yescharacteristic
149Dic197—21no—
150Dic197—21no—
151Dic197—21no—
152Dic197—41no—
153Dic197—41no—
154Dic197—41no—
155Dic197—41no—
156C2×C197—41no—
157C2×C197#54711yescharacteristic
158C2×C197#57511yescharacteristic
159C2×C197#57811yescharacteristic
160D197—41no—
161C197#335211yescharacteristic, complemented, Sylow
162#3352—1971nomaximal, Sylow
163#552—1971no—
164#552—1971no—
165#499—1971no—
166#485—1971no—
167#485—1971no—
168#484—1971no—
169#484D19711yescharacteristic, complemented
170#479—1971no—
171#479—1971no—
172#470—1971no—
173#470—1971no—
174C4×Q16—1971no—
175#436—1971no—
176#428—1971no—
177#428—1971no—
178C2×Q16—1971no—
179C2×M16—1971no—
180C2×M16—1971no—
181#126—3941no—
182#126—3941no—
183#125—3941no—
184#125—3941no—
185#125—1971no—
186#125—1971no—
187#123—3941no—
188#123—3941no—
189C22⋊Q8—1971no—
190C22⋊Q8—1971no—
191C22⋊Q8—1971no—
192C22⋊Q8C2×D19711yescharacteristic
193C4×Q8—1971no—
194C4×Q8—1971no—
195C4×D4—1971no—
196#117—3941no—
197#117—3941no—
198#117—1971no—
199#117—1971no—
200C2×C4⋊C4—1971no—
201C2×C4⋊C4C2×D19711yescharacteristic
202#107—1971no—
203#107—1971no—
204#107—1971no—
205#107C2×D19711yescharacteristic
206#107C2×D19711yescharacteristic
207#106—1971no—
208#106—1971no—
209#106—1971no—
210#106—1971no—
211C4⋊C8—1971no—
212#103—1971no—
213#103—1971no—
214#103—1971no—
215#103—1971no—
216#103C2×D19711yescharacteristic
217#103C2×D19711yescharacteristic
218C22⋊C8—1971no—
219C22⋊C8C2×D19711yescharacteristic
220C4×C8—1971no—
221C2×Q8—1971no—
222C2×Q8C22×D19711yescharacteristic
223C2×D4—1971no—
224C22×C4—21no—
225C22×C4—3941no—
226C22×C4—1971no—
227C22×C4C22×D19711yescharacteristic
228Q16—3942no—
229M16—3942no—
230M16—3942no—
231C2×C8—1971no—
232C2×C8—1971no—
233C2×C8C22×D19711yescharacteristic
234C2×C8C22×D19711yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—3941no—
238C4⋊C4—3941no—
239C4⋊C4—3941no—
240C4⋊C4—3941no—
241C4⋊C4—3941no—
242C4⋊C4—3941no—
243C4⋊C4—3941no—
244C4⋊C4—3941no—
245C4⋊C4—3941no—
246C4⋊C4—3941no—
247C4⋊C4—1971no—
248C4⋊C4—1971no—
249C4⋊C4—1971no—
250C4⋊C4—1971no—
251C4⋊C4C22×D19711yescharacteristic
252C4⋊C4C22×D19711yescharacteristic
253C4⋊C4C22×D19711yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—3941no—
256C22⋊C4—3941no—
257C22⋊C4—3941no—
258C22⋊C4—3941no—
259C22⋊C4—3941no—
260C22⋊C4—1971no—
261C22⋊C4—1971no—
262C42—3941no—
263C42—3941no—
264C42—3941no—
265C42—3941no—
266C42—1971no—
267C23—1971no—
268C23D4×D19711yescharacteristic
269Q8—21no—
270Q8—21no—
271Q8—3941no—
272Q8—3941no—
273D4—7881no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—7882no—
282C2×C4—3941no—
283C2×C4—3941no—
284C2×C4—3941no—
285C2×C4—3941no—
286C2×C4—3941no—
287C2×C4—3941no—
288C2×C4—1971no—
289C2×C4—1971no—
290C2×C4—1971no—
291C2×C4D4×D19711yescharacteristic
292C2×C4C23×D19711yescharacteristic
293C8—21no—
294C8—21no—
295C8—3941no—
296C8—1972no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—3941no—
301C22—3941no—
302C22—3941no—
303C22C2×D4×D19711yescharacteristic
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—7881no—
310C4—7881no—
311C4—7881no—
312C4—7881no—
313C4—3941no—
314C4—3941no—
315C4—3941no—
316C4#41×D19711yescharacteristic
317C4#2026337311yescharacteristic
318C2—41no—
319C2—7881no—
320C2#2716172711yescharacteristic
321C2#2716187511yescharacteristic
322C2#2716189211yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#484×C197C2
socleC22×C197C22×D4
prime radicalC2×C22⋊D197D4
hypercenter#484D197
hypocenterC197#3352
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #27161480C2semidirect
4. #27161421C2semidirect
5. #27161419C2semidirect
6. #27161416C2semidirect
7. #27161385C2semidirect
8. #27161384C2semidirect
9. #27161336C2semidirect
10. #27161335C2semidirect
11. #27161171C2semidirect
13. #27161120C2semidirect
14. #27161119C2semidirect
15. #484×C197C2semidirect
161. C197#3352semidirect
169. #484D197semidirect

Maximal subgroups

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

subgroupindex
#271617642
#271617632
#271614802
#271614212
#271614192
#271614162
#271613852
#271613842
#271613362
#271613352
#271611712
#271611542
#271611202
#271611192
#484×C1972
#3352197