Finite Group Databasebeta

← #38752021

Subgroups — #38752021

Order 25216.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263371C211yesmaximal, characteristic, complemented
2C2×#20263352C211yesmaximal, characteristic, complemented
3C2×#20263348C211yesmaximal, characteristic, complemented
4#27161660C211yesmaximal, characteristic, complemented
5#27161242C212yesmaximal, complemented
6#27161231C212yesmaximal, complemented
7#27161225C212yesmaximal, complemented
8#27161214C212yesmaximal, complemented
9#27161078C211yesmaximal, characteristic, complemented
10#27161045C211yesmaximal, characteristic, complemented
11C2×#102×C197C211yesmaximal, characteristic, complemented
12#20263371—24no—
13#20263371—24no—
14C2×#15691358C2211yescharacteristic
15C2×#15691358C2211yescharacteristic
16C2×#15691358C2211yescharacteristic, complemented
17C2×#15691357—21no—
18C2×#15691357C2212yes—
19C22×C22⋊D197—22no—
20C2×D4×D197—21no—
21C2×D4×D197C2212yes—
22#20263352—24no—
23#20263348—24no—
24C2×Dic788C2211yescharacteristic, complemented
25C2×#15691347C2211yescharacteristic, complemented
26C2×#15691347C2211yescharacteristic, complemented
27C2×#15691346C2212yescomplemented
28C2×D1576C2211yescharacteristic, complemented
29C2×#15691343C2212yescomplemented
30C2×#15691342C2211yescharacteristic, complemented
31#20263306—21no—
32#20263306C2212yescomplemented
33#20263301—21no—
34#20263301C2212yescomplemented
35#20263300—22no—
36#20263249C2212yescomplemented
37#20263246C2212yescomplemented
38#20263238C2212yescomplemented
39#20263238C2212yescomplemented
40C22×D4×C197C2211yescharacteristic
41C22×C8×C197C2211yescharacteristic, complemented
42C2×C4⋊C4×C197C2211yescharacteristic, complemented
43#102×C197C2212yescomplemented
44#102×C197C2212yescomplemented
45C23×D197—22no—
46#15691358—22no—
47#15691358—22no—
48#15691358—22no—
49#15691358—22no—
50#15691358—22no—
51#15691358D414yescomplemented
52#15691357—28no—
53#15691357—24no—
54#15691357—24no—
55C2×Dic394D411yescharacteristic
56C2×Dic394C2311yescharacteristic
57C2×Dic394C2311yescharacteristic
58C2×D788—22no—
59C2×D788D411yescharacteristic
60C2×D788C2311yescharacteristic
61C2×D788C2311yescharacteristic
62C2×C22⋊D197—216no—
63C2×C22⋊D197—22no—
64C2×C22⋊D197—22no—
65C2×C22⋊D197—21no—
66C2×C22⋊D197—21no—
67C2×C22⋊D197D412yes—
68D4×D197—28no—
69D4×D197—24no—
70D4×D197—24no—
71C22×Dic197—22no—
72C2×C4×D197—21no—
73C2×C4×D197—21no—
74C2×C4×D197C2312yes—
75Dic788—22no—
76#15691347—22no—
77#15691347—22no—
78#15691346—24no—
79D1576—22no—
80#15691343—24no—
81#15691342—22no—
82C2×#13096802C2312yes—
83C4⋊Dic197—22no—
84#15691335—22no—
85#15691335—22no—
86C24×C197—21no—
87C2×D4×C197—21no—
88C2×D4×C197—22no—
89C2×D4×C197D414yescomplemented
90C2×D4×C197C2312yes—
91C22×C4×C197—21no—
92C22×C4×C197C2311yescharacteristic
93C2×C8×C197—22no—
94C2×C8×C197C2311yescharacteristic
95C2×C8×C197C2311yescharacteristic
96C4⋊C4×C197—21no—
97C4⋊C4×C197C2312yes—
98C22×D197—21no—
99C22×D197—28no—
100C22×D197—21no—
101C22×D197—82no—
102C22×D197—82no—
103C22×D197C2×D412yes—
104Dic394—22no—
105Dic394—21no—
106Dic394—21no—
107Dic394—21no—
108Dic394C2×D412yes—
109D788—22no—
110D788—21no—
111D788—21no—
112D788—21no—
113D788—82no—
114D788C2×D412yes—
115C22⋊D197—28no—
116C22⋊D197—28no—
117C22⋊D197—24no—
118C22⋊D197—24no—
119C22⋊D197—42no—
120C2×Dic197—28no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197C2×D412yes—
124C4×D197—24no—
125C4×D197—22no—
126C4×D197—22no—
127C4×D197—42no—
128#13096802—22no—
129C23×C197—22no—
130C23×C197—24no—
131C23×C197—81no—
132C23×C197—81no—
133C23×C197C2×D411yescharacteristic
134D4×C197—24no—
135D4×C197—22no—
136D4×C197C2×D414yes—
137C2×C4×C197—22no—
138C2×C4×C197—81no—
139C2×C4×C197C2×D411yescharacteristic
140C2×C4×C197C2×D412yes—
141C2×C4×C197C2×D412yes—
142C2×C4×C197C2411yescharacteristic
143C8×C197—21no—
144C8×C197—21no—
145C22×C197—21no—
146C22×C197—21no—
147C22×C197—24no—
148C22×C197—42no—
149C22×C197—42no—
150C22×C197—84no—
151C22×C197#12012yes—
152C22×C197C22×D411yescharacteristic
153C2×D197—22no—
154C2×D197—22no—
155C2×D197—22no—
156C2×D197—22no—
157C2×D197—41no—
158C2×D197—41no—
159C2×D197—41no—
160C2×D197—41no—
161C2×D197—41no—
162C2×D197—88no—
163C4×C197—42no—
164C4×C197#12012yes—
165C4×C197C22×D411yescharacteristic
166C4×C197C22×D411yescharacteristic
167Dic197—22no—
168Dic197—22no—
169Dic197—41no—
170C2×C197—22no—
171C2×C197—44no—
172C2×C197C2×#12011yescharacteristic
173C2×C197#57611yescharacteristic
174C2×C197#57711yescharacteristic
175D197—42no—
176D197—42no—
177D197—81no—
178C197#301211yescharacteristic, complemented, Sylow
179#3012—1971nomaximal, Sylow
180C2×H(5,𝔽2)—1971no—
181C2×AGL(1,ℤ/8)—1971no—
182C2×#135—1971no—
183#537—1971no—
184#452—1972no—
185#450—1972no—
186#449—1972no—
187#447—1972no—
188#417—1971no—
189C2×#102D19711yescharacteristic, complemented
190#408—1971no—
191H(5,𝔽2)—3944no—
192H(5,𝔽2)—3944no—
193C2×#41—3941no—
194C2×#41—1972no—
195C2×#41—1971no—
196C2×#41—1971no—
197C2×#41—1971no—
198C22×D4—3941no—
199C22×D4—3942no—
200C22×D4—1972no—
201C22×D4C2×D19711yescharacteristic
202AGL(1,ℤ/8)—3944no—
203#135—3944no—
204C2×Q16—1971no—
205C2×SD16—1972no—
206C2×SD16—1971no—
207C2×SD16—1971no—
208C2×D8—1972no—
209C2×D8—1971no—
210C2×M16—1971no—
211C22×C8C2×D19711yescharacteristic, complemented
212C22⋊Q8—3941no—
213C22⋊Q8—1972no—
214#121—3941no—
215#121—3942no—
216#121—1972no—
217C2×C4⋊C4C2×D19711yescharacteristic, complemented
218#103—1972no—
219#102—1972no—
220#102C2×D19712yescomplemented
221#102C2×D19712yescomplemented
222C22⋊C8—1972no—
223C22⋊C8—1972no—
224C24—21no—
225C24—3942no—
226#41—3942no—
227#41—3942no—
228#41—3948no—
229#41—3944no—
230#41—3944no—
231#41—3942no—
232#41—3942no—
233#41—3942no—
234#41—1974no—
235C2×Q8—1971no—
236C2×Q8—1971no—
237C2×Q8—1971no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—3942no—
241C2×D4—3948no—
242C2×D4—3944no—
243C2×D4—3941no—
244C2×D4—39416no—
245C2×D4—3942no—
246C2×D4—3941no—
247C2×D4—3944no—
248C2×D4—3942no—
249C2×D4—1972no—
250C2×D4—1971no—
251C2×D4—1971no—
252C2×D4—1971no—
253C2×D4D78814yescomplemented
254C2×D4C22×D19712yes—
255C22×C4—21no—
256C22×C4—3941no—
257C22×C4—3942no—
258C22×C4—3941no—
259C22×C4—1972no—
260C22×C4C22×D19711yescharacteristic
261Q16—3942no—
262SD16—3942no—
263SD16—3942no—
264SD16—3944no—
265D8—3942no—
266D8—3944no—
267M16—3942no—
268C2×C8—22no—
269C2×C8—1972no—
270C2×C8C22×D19711yescharacteristic
271C2×C8C22×D19711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—3942no—
274C4⋊C4C22×D19712yes—
275C22⋊C4—3942no—
276C22⋊C4—3942no—
277C23—22no—
278C23—24no—
279C23—81no—
280C23—81no—
281C23—15762no—
282C23—15762no—
283C23—3948no—
284C23—3941no—
285C23—3941no—
286C23—1972no—
287C23D4×D19711yescharacteristic
288Q8—3941no—
289Q8—3941no—
290Q8—3942no—
291Q8—3941no—
292Q8—1972no—
293D4—24no—
294D4—22no—
295D4—7882no—
296D4—15762no—
297D4—3948no—
298D4—3948no—
299D4—3944no—
300D4—3944no—
301D4—3942no—
302D4—3941no—
303D4—3941no—
304D4—3941no—
305D4—1972no—
306D4C2×D78814yes—
307C2×C4—22no—
308C2×C4—81no—
309C2×C4—7882no—
310C2×C4—3941no—
311C2×C4—3948no—
312C2×C4—3941no—
313C2×C4—3944no—
314C2×C4—3942no—
315C2×C4—3942no—
316C2×C4—1972no—
317C2×C4D4×D19711yescharacteristic
318C2×C4D4×D19712yes—
319C2×C4C2×D78812yes—
320C2×C4C23×D19711yescharacteristic
321C8—21no—
322C8—21no—
323C8—3942no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—84no—
330C22—7881no—
331C22—7881no—
332C22—7881no—
333C22—7881no—
334C22—7881no—
335C22—15768no—
336C22—3942no—
337C22—3942no—
338C22—3942no—
339C22—3942no—
340C22#2026329612yes—
341C22C2×D4×D19711yescharacteristic
342C4—42no—
343C4—7881no—
344C4—3942no—
345C4—3942no—
346C4#2026329612yes—
347C4C2×D4×D19711yescharacteristic
348C4C22×D78811yescharacteristic
349C2—22no—
350C2—44no—
351C2—7882no—
352C2—7882no—
353C2—15761no—
354C2C2×#2026329611yescharacteristic
355C2#2716188011yescharacteristic
356C2#2716188511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#102×C197C2
socleC22×C197C22×D4
hypercenterC2×#102D197
hypocenterC197#3012
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#20263371C2semidirect
2. C2×#20263352C2semidirect
3. C2×#20263348C2semidirect
4. #27161660C2semidirect
5. #27161242C2semidirect
6. #27161231C2semidirect
7. #27161225C2semidirect
8. #27161214C2semidirect
9. #27161078C2semidirect
10. #27161045C2semidirect
11. C2×#102×C197C2semidirect
16. C2×#15691358C22semidirect
24. C2×Dic788C22semidirect
25. C2×#15691347C22semidirect
26. C2×#15691347C22semidirect
27. C2×#15691346C22semidirect
28. C2×D1576C22semidirect
29. C2×#15691343C22semidirect
30. C2×#15691342C22semidirect
32. #20263306C22semidirect
34. #20263301C22semidirect
36. #20263249C22semidirect
37. #20263246C22semidirect
38. #20263238C22semidirect
39. #20263238C22semidirect
41. C22×C8×C197C22semidirect
42. C2×C4⋊C4×C197C22semidirect
43. #102×C197C22semidirect
44. #102×C197C22semidirect
51. #15691358D4semidirect
89. C2×D4×C197D4semidirect
178. C197#3012semidirect
189. C2×#102D197semidirect
211. C22×C8C2×D197semidirect
217. C2×C4⋊C4C2×D197semidirect
220. #102C2×D197semidirect
221. #102C2×D197semidirect
253. C2×D4D788semidirect

Maximal subgroups

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

subgroupindex
C2×#202633712
C2×#202633522
C2×#202633482
#271616602
#271612422
#271612312
#271612252
#271612142
#271610782
#271610452
C2×#102×C1972
#3012197