Finite Group Databasebeta

← #38751160

Subgroups — #38751160

Order 25216.

774 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D197C211yesmaximal, characteristic
2C2×#20263334C211yesmaximal, characteristic, complemented
3C2×#20263332C211yesmaximal, characteristic, complemented
4#27161046C211yesmaximal, characteristic, complemented
5#27161044C212yesmaximal, complemented
6#27161044C212yesmaximal, complemented
7#27161042C211yesmaximal, characteristic, complemented
8#27161041C212yesmaximal, complemented
9C2×#20263238C211yesmaximal, characteristic, complemented
10C22⋊C8×D197C212yesmaximal, complemented
11#408×C197C211yesmaximal, characteristic, complemented
12C2×#15691359C411yescharacteristic
13C2×#15691358C2212yes—
14C2×#15691358C2211yescharacteristic
15C2×#15691357C411yescharacteristic
16C2×#15691357C412yes—
17#41×D197—24no—
18#41×D197—24no—
19C2×Q8×D197C411yescharacteristic
20C2×D4×D197C411yescharacteristic
21C2×D4×D197C412yes—
22C22×C4×D197C2212yes—
23C22×C4×D197C2211yescharacteristic
24#20263334—24no—
25#20263332—24no—
26C2×#15691342C2211yescharacteristic, complemented
27C2×#15691341C2212yescomplemented
28C2×#15691341C2212yescomplemented
29C22×#13096802C2211yescharacteristic, complemented
30C2×C8×D197C2212yescomplemented
31C2×C8×D197C2212yescomplemented
32C22⋊#13096802C2212yescomplemented
33C22⋊#13096802C2212yescomplemented
34#20263238C2211yescharacteristic
35#20263238C2212yes—
36#20263238C2211yescharacteristic
37#20263238C2211yescharacteristic, complemented
38#20263238C2211yescharacteristic, complemented
39#20263238C2212yescomplemented
40C2×#41×C197C2211yescharacteristic
41C2×M16×C197C2211yescharacteristic, complemented
42C22×C8×C197C2211yescharacteristic, complemented
43C22⋊C8×C197C2212yescomplemented
44C22⋊C8×C197C2212yescomplemented
45C23×D197C2×C412yes—
46C23×D197C2×C411yescharacteristic
47#15691359—24no—
48#15691358—24no—
49#15691358—24no—
50#15691358D414yes—
51#15691358D414yes—
52#15691357—24no—
53#15691357—28no—
54C2×Dic394C2×C412yes—
55C2×Dic394C2×C411yescharacteristic
56Q8×D197—24no—
57C2×D788C2×C412yes—
58C2×D788C2×C411yescharacteristic
59C2×C22⋊D197C2×C412yes—
60C2×C22⋊D197C2×C412yes—
61C2×C22⋊D197C2×C412yes—
62D4×D197—24no—
63D4×D197—28no—
64C22×Dic197C2×C412yes—
65C22×Dic197C2×C411yescharacteristic
66C2×C4×D197—24no—
67C2×C4×D197—24no—
68C2×C4×D197—22no—
69C2×C4×D197—22no—
70C2×C4×D197C2×C412yes—
71C2×C4×D197C2×C411yescharacteristic
72C2×C4×D197C2×C412yes—
73C2×C4×D197C2×C411yescharacteristic
74C2×C4×D197C2311yescharacteristic
75C2×C4×D197C2311yescharacteristic
76C2×C4×D197C2312yes—
77#15691342—22no—
78#15691341—24no—
79#15691341—24no—
80C2×#13096802—22no—
81C2×#13096802C2311yescharacteristic
82C2×#13096802C2312yes—
83C2×#13096802C2311yescharacteristic
84C8×D197—24no—
85C8×D197—24no—
86#41×C197—22no—
87#41×C197—22no—
88C2×Q8×C197C2×C411yescharacteristic
89C2×D4×C197C2×C412yes—
90C2×D4×C197C2×C411yescharacteristic
91C22×C4×C197C2312yes—
92C22×C4×C197C2311yescharacteristic
93M16×C197—22no—
94C2×C8×C197—22no—
95C2×C8×C197C2311yescharacteristic
96C2×C8×C197C2311yescharacteristic
97C2×C8×C197C2312yes—
98C22×D197—22no—
99C22×D197—24no—
100C22×D197—24no—
101C22×D197—22no—
102C22×D197—42no—
103C22×D197—41no—
104C22×D197C22×C411yescharacteristic
105C22×D197C22×C412yes—
106C22×D197C22×C411yescharacteristic
107Dic394—24no—
108Dic394C22⋊C414yes—
109D788—24no—
110D788C22⋊C414yes—
111C22⋊D197—24no—
112C22⋊D197—24no—
113C22⋊D197C22⋊C418yes—
114C2×Dic197—22no—
115C2×Dic197—24no—
116C2×Dic197C22×C411yescharacteristic
117C2×Dic197C22×C411yescharacteristic
118C2×Dic197C22×C412yes—
119C4×D197—24no—
120C4×D197—22no—
121C4×D197—22no—
122C4×D197—21no—
123C4×D197—21no—
124C4×D197—21no—
125C4×D197—21no—
126C4×D197—24no—
127C4×D197C2×D414yes—
128C4×D197C2×D414yes—
129#13096802—21no—
130#13096802—22no—
131#13096802—21no—
132C23×C197C22×C412yes—
133C23×C197C22×C411yescharacteristic
134Q8×C197—22no—
135D4×C197—24no—
136D4×C197—22no—
137C2×C4×C197—22no—
138C2×C4×C197—22no—
139C2×C4×C197C22×C412yes—
140C2×C4×C197C22×C411yescharacteristic
141C2×C4×C197C2×D412yes—
142C2×C4×C197C2×D412yes—
143C2×C4×C197C2411yescharacteristic
144C8×C197—21no—
145C8×C197—21no—
146C8×C197—22no—
147C22×C197—21no—
148C22×C197—21no—
149C22×C197—22no—
150C22×C197—22no—
151C22×C197—22no—
152C22×C197C2×C22⋊C412yes—
153C22×C197C23×C411yescharacteristic
154C2×D197—21no—
155C2×D197—22no—
156C2×D197—21no—
157C2×D197—22no—
158C2×D197—21no—
159C2×D197—21no—
160C2×D197—21no—
161C2×D197—48no—
162C2×D197—44no—
163C2×D197C2×C22⋊C414yes—
164C2×D197#13112yescomplemented
165C4×C197—22no—
166C4×C197C2×C22⋊C412yes—
167C4×C197C22×D411yescharacteristic
168C4×C197C22×D411yescharacteristic
169Dic197—21no—
170Dic197—21no—
171Dic197C2×C22⋊C414yes—
172C2×C197—22no—
173C2×C197—42no—
174C2×C197C22×C22⋊C411yescharacteristic
175C2×C197C2×#13111yescharacteristic
176C2×C197#56811yescharacteristic
177D197—22no—
178D197—24no—
179D197—41no—
180C197#288011yescharacteristic, complemented, Sylow
181#2880—1971nomaximal, Sylow
182C22×#41—1971no—
183C2×#131—1971no—
184C2×#131—1971no—
185#408—1971no—
186#408—1972no—
187#408—1972no—
188#408D19711yescharacteristic, complemented
189#407—1971no—
190#407—1972no—
191C2×C22⋊C8—1971no—
192C2×C22⋊C8—1972no—
193C2×#41—3944no—
194C2×#41—3944no—
195C2×#41—1971no—
196C2×#41—1971no—
197C2×#41—1972no—
198C2×#41—1972no—
199C2×#41—1971no—
200C2×#41C2×D19711yescharacteristic
201C22×Q8—1971no—
202C22×D4—1971no—
203C22×D4—1972no—
204C23×C4—1972no—
205C23×C4—1971no—
206#131—3944no—
207#131—3944no—
208C2×M16—1971no—
209C2×M16—1972no—
210C2×M16—1972no—
211C2×M16C2×D19711yescharacteristic, complemented
212C22×C8—1971no—
213C22×C8—1972no—
214C22×C8—1972no—
215C22×C8C2×D19711yescharacteristic, complemented
216C22⋊C8—1972no—
217C22⋊C8—1972no—
218C22⋊C8—1971no—
219C22⋊C8—1972no—
220C22⋊C8—1971no—
221C22⋊C8—1971no—
222C22⋊C8—1971no—
223C22⋊C8—1972no—
224C22⋊C8C2×D19712yescomplemented
225C22⋊C8C2×D19712yescomplemented
226C24—1972no—
227C24—1971no—
228#41—22no—
229#41—22no—
230#41—3944no—
231#41—3944no—
232#41—3944no—
233#41—3948no—
234#41—3944no—
235#41—1974no—
236#41—1974no—
237C2×Q8—3944no—
238C2×Q8—1972no—
239C2×Q8—1971no—
240C2×Q8C4×D19711yescharacteristic
241C2×D4—3948no—
242C2×D4—3944no—
243C2×D4—1972no—
244C2×D4—1972no—
245C2×D4—1972no—
246C2×D4—1972no—
247C2×D4—1971no—
248C2×D4C4×D19712yes—
249C2×D4C4×D19711yescharacteristic
250C22×C4—3944no—
251C22×C4—3944no—
252C22×C4—3942no—
253C22×C4—3942no—
254C22×C4—1971no—
255C22×C4—1972no—
256C22×C4—1972no—
257C22×C4—1971no—
258C22×C4—1972no—
259C22×C4—1971no—
260C22×C4—1971no—
261C22×C4—1971no—
262C22×C4—1972no—
263C22×C4C22×D19712yes—
264C22×C4C22×D19711yescharacteristic
265M16—22no—
266M16—3944no—
267M16—3944no—
268M16—3942no—
269C2×C8—22no—
270C2×C8—3944no—
271C2×C8—3944no—
272C2×C8—3942no—
273C2×C8—1971no—
274C2×C8—1972no—
275C2×C8—1971no—
276C2×C8C22×D19711yescharacteristic
277C2×C8C22×D19712yes—
278C2×C8C22×D19711yescharacteristic
279C23—7881no—
280C23—7882no—
281C23—3944no—
282C23—3942no—
283C23—3944no—
284C23—3942no—
285C23—1971no—
286C23—1971no—
287C23—1972no—
288C23C2×C4×D19712yes—
289C23C2×C4×D19711yescharacteristic
290Q8—22no—
291Q8—3944no—
292Q8—1974no—
293D4—24no—
294D4—22no—
295D4—3944no—
296D4—3944no—
297D4—3944no—
298D4—1978no—
299D4—1974no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—3944no—
303C2×C4—3944no—
304C2×C4—3942no—
305C2×C4—3944no—
306C2×C4—3942no—
307C2×C4—3942no—
308C2×C4—3941no—
309C2×C4—3941no—
310C2×C4—3941no—
311C2×C4—3941no—
312C2×C4—1971no—
313C2×C4—1971no—
314C2×C4—1972no—
315C2×C4—1974no—
316C2×C4—1974no—
317C2×C4C2×C4×D19712yes—
318C2×C4C2×C4×D19711yescharacteristic
319C2×C4D4×D19712yes—
320C2×C4D4×D19712yes—
321C2×C4C23×D19711yescharacteristic
322C8—21no—
323C8—21no—
324C8—22no—
325C8—3941no—
326C8—3942no—
327C8—3941no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—7884no—
334C22—7888no—
335C22—3941no—
336C22—3941no—
337C22—3941no—
338C22—3941no—
339C22—3942no—
340C22—3941no—
341C22—3942no—
342C22—1974no—
343C22—1972no—
344C22C22⋊C4×D19712yes—
345C22C22×C4×D19711yescharacteristic
346C4—22no—
347C4—3941no—
348C4—3941no—
349C4—1974no—
350C4C22⋊C4×D19712yes—
351C4C2×D4×D19711yescharacteristic
352C4C2×D4×D19711yescharacteristic
353C2—22no—
354C2—42no—
355C2—7881no—
356C2—3942no—
357C2—3944no—
358C2C2×C22⋊C4×D19711yescharacteristic
359C2#131×D19711yescharacteristic
360C2#2716184411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D197
commutator, socleC22×C197C23×C4
Fitting#408×C197C2
prime radicalC2×#41×D197C2
cosocleC2×C4×C197C24
hypercenter#408D197
hypocenterC197#2880
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#20263334C2semidirect
3. C2×#20263332C2semidirect
4. #27161046C2semidirect
5. #27161044C2semidirect
6. #27161044C2semidirect
7. #27161042C2semidirect
8. #27161041C2semidirect
9. C2×#20263238C2semidirect
10. C22⋊C8×D197C2semidirect
11. #408×C197C2semidirect
26. C2×#15691342C22semidirect
27. C2×#15691341C22semidirect
28. C2×#15691341C22semidirect
29. C22×#13096802C22semidirect
30. C2×C8×D197C22semidirect
31. C2×C8×D197C22semidirect
32. C22⋊#13096802C22semidirect
33. C22⋊#13096802C22semidirect
37. #20263238C22semidirect
38. #20263238C22semidirect
39. #20263238C22semidirect
41. C2×M16×C197C22semidirect
42. C22×C8×C197C22semidirect
43. C22⋊C8×C197C22semidirect
44. C22⋊C8×C197C22semidirect
164. C2×D197#131semidirect
180. C197#2880semidirect
188. #408D197semidirect
211. C2×M16C2×D197semidirect
215. C22×C8C2×D197semidirect
224. C22⋊C8C2×D197semidirect
225. C22⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1972
C2×#202633342
C2×#202633322
#271610462
#271610442
#271610422
#271610412
C2×#202632382
C22⋊C8×D1972
#408×C1972
#2880197