Finite Group Databasebeta

← #38751962

Subgroups — #38751962

Order 25216.

806 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#15691358C211yesmaximal, characteristic, complemented
2C2×#20263352C212yesmaximal, complemented
3#27161222C214yesmaximal, complemented
4#27161217C212yesmaximal, complemented
5#27161212C212yesmaximal, complemented
6#27161078C212yesmaximal, complemented
7#27161043C211yesmaximal, characteristic, complemented
8C2×#121×C197C211yesmaximal, characteristic, complemented
9C2×#15691358—28no—
10C2×#15691358—21no—
11C2×#15691358—21no—
12C2×#15691358C2212yes—
13C2×#15691358C2212yescomplemented
14C22×Dic394C2211yescharacteristic, complemented
15C22×D788C2211yescharacteristic, complemented
16C22×C22⋊D197—21no—
17C22×C4×D197—21no—
18#20263352—28no—
19C2×#15691346C2214yescomplemented
20C2×#15691343C2214yescomplemented
21C2×#15691342C2212yescomplemented
22#20263249C2214yescomplemented
23#20263246C2214yescomplemented
24C22⋊#13096802C2212yescomplemented
25C22⋊#13096802C2212yescomplemented
26C22×D4×C197—21no—
27C22×D4×C197C2211yescharacteristic, complemented
28C23×C4×C197C2211yescharacteristic
29#121×C197—21no—
30#121×C197—21no—
31#121×C197C2212yes—
32#121×C197C2212yescomplemented
33C2×C4⋊C4×C197C2211yescharacteristic, complemented
34C2×C22⋊C4×C197—21no—
35C23×D197—21no—
36#15691358—28no—
37#15691358—28no—
38#15691358—24no—
39#15691358—24no—
40#15691358—24no—
41#15691358D418yescomplemented
42C2×Dic394—24no—
43C2×Dic394—21no—
44C2×Dic394C2312yes—
45C2×D788—24no—
46C2×D788—21no—
47C2×D788C2312yes—
48C2×C22⋊D197—28no—
49C2×C22⋊D197—22no—
50C2×C22⋊D197—22no—
51C22×Dic197—21no—
52C2×C4×D197—28no—
53C2×C4×D197—22no—
54C2×C4×D197—22no—
55#15691346—28no—
56#15691343—28no—
57#15691342—24no—
58C2×#13096802C2314yes—
59C24×C197—21no—
60C24×C197D411yescharacteristic
61C2×D4×C197—22no—
62C2×D4×C197—22no—
63C2×D4×C197—24no—
64C2×D4×C197—21no—
65C2×D4×C197—44no—
66C2×D4×C197C2312yes—
67C22×C4×C197—24no—
68C22×C4×C197—21no—
69C22×C4×C197D411yescharacteristic
70C22×C4×C197D411yescharacteristic
71C22×C4×C197D411yescharacteristic
72C22×C4×C197C2311yescharacteristic
73C22×C4×C197C2311yescharacteristic
74C22×C4×C197C2311yescharacteristic
75C4⋊C4×C197—21no—
76C4⋊C4×C197C2312yes—
77C22⋊C4×C197—22no—
78C22⋊C4×C197—22no—
79C22×D197—24no—
80C22×D197—22no—
81C22×D197—42no—
82C22×D197—42no—
83Dic394—24no—
84Dic394—22no—
85Dic394C2×D414yescomplemented
86D788—24no—
87D788—22no—
88D788C2×D414yescomplemented
89C22⋊D197—28no—
90C22⋊D197—28no—
91C2×Dic197—24no—
92C2×Dic197—22no—
93C4×D197—28no—
94C4×D197—28no—
95#13096802—24no—
96C23×C197—22no—
97C23×C197—22no—
98C23×C197—22no—
99C23×C197—41no—
100C23×C197—42no—
101C23×C197—42no—
102C23×C197—42no—
103C23×C197C2×D411yescharacteristic
104C23×C197C2×D411yescharacteristic
105C23×C197C2×D411yescharacteristic
106D4×C197—24no—
107D4×C197—44no—
108D4×C197—44no—
109D4×C197—42no—
110C2×C4×C197—22no—
111C2×C4×C197—22no—
112C2×C4×C197—22no—
113C2×C4×C197—22no—
114C2×C4×C197—22no—
115C2×C4×C197—22no—
116C2×C4×C197—42no—
117C2×C4×C197C2×D414yes—
118C2×C4×C197C2×D411yescharacteristic
119C2×C4×C197C2×D411yescharacteristic
120C2×C4×C197C2×D411yescharacteristic
121C2×C4×C197C2411yescharacteristic
122C22×C197—21no—
123C22×C197—21no—
124C22×C197—21no—
125C22×C197—21no—
126C22×C197—21no—
127C22×C197—21no—
128C22×C197—21no—
129C22×C197—44no—
130C22×C197—42no—
131C22×C197—42no—
132C22×C197—42no—
133C22×C197—84no—
134C22×C197#12012yes—
135C22×C197AGL(1,ℤ/8)12yescomplemented
136C22×C197C22×D411yescharacteristic
137C2×D197—24no—
138C2×D197—48no—
139C2×D197—42no—
140C2×D197—42no—
141C4×C197—21no—
142C4×C197—21no—
143C4×C197—42no—
144C4×C197#12012yes—
145C4×C197C22×D412yes—
146Dic197—24no—
147C2×C197—22no—
148C2×C197—22no—
149C2×C197—41no—
150C2×C197—82no—
151C2×C197C2×#12011yescharacteristic
152C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C197#57511yescharacteristic
154D197—44no—
155C197#300611yescharacteristic, complemented, Sylow
156#3006—1971nomaximal, Sylow
157C22×#41—1971no—
158C2×AGL(1,ℤ/8)—1972no—
159C2×#121D19711yescharacteristic, complemented
160#449—1974no—
161#448—1972no—
162#447—1972no—
163#417—1972no—
164#407—1971no—
165C2×#41—3948no—
166C2×#41—3941no—
167C2×#41—3941no—
168C2×#41—1972no—
169C2×#41—1972no—
170C22×Q8—1971no—
171C22×D4—21no—
172C22×D4—3941no—
173C22×D4—1971no—
174C22×D4C2×D19711yescharacteristic, complemented
175C23×C4—3941no—
176C23×C4C2×D19711yescharacteristic
177AGL(1,ℤ/8)—3948no—
178C2×SD16—1974no—
179C2×D8—1974no—
180C2×M16—1972no—
181#121—21no—
182#121—21no—
183#121C2×D19712yes—
184#121C2×D19712yescomplemented
185C2×C4⋊C4C2×D19711yescharacteristic, complemented
186C2×C22⋊C4—21no—
187#103—1974no—
188#102—1974no—
189C22⋊C8—1972no—
190C22⋊C8—1972no—
191C24—21no—
192C24—3941no—
193C24C22⋊D19711yescharacteristic
194#41—3948no—
195#41—3948no—
196#41—3944no—
197#41—3944no—
198#41—3944no—
199#41—1978no—
200C2×Q8—3944no—
201C2×Q8—3941no—
202C2×Q8—1972no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—21no—
207C2×D4—44no—
208C2×D4—3944no—
209C2×D4—3948no—
210C2×D4—3942no—
211C2×D4—3942no—
212C2×D4—3941no—
213C2×D4—1972no—
214C2×D4C22×D19712yes—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—3948no—
218C22×C4—3941no—
219C22×C4—3942no—
220C22×C4—3942no—
221C22×C4C22⋊D19711yescharacteristic
222C22×C4C22⋊D19711yescharacteristic
223C22×C4C22⋊D19711yescharacteristic
224C22×C4C22×D19711yescharacteristic
225C22×C4C22×D19711yescharacteristic
226C22×C4C22×D19711yescharacteristic
227SD16—3948no—
228D8—3948no—
229M16—3944no—
230C2×C8—1974no—
231C4⋊C4—21no—
232C4⋊C4C22×D19712yes—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—7882no—
243C23—7882no—
244C23—3944no—
245C23—3942no—
246C23C2×C22⋊D19711yescharacteristic
247C23C2×C22⋊D19711yescharacteristic
248C23C2×C22⋊D19711yescharacteristic
249Q8—3944no—
250Q8—3942no—
251Q8—1974no—
252D4—24no—
253D4—44no—
254D4—44no—
255D4—42no—
256D4—3944no—
257D4—3948no—
258D4—3948no—
259D4—3942no—
260D4—1974no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—3948no—
269C2×C4—3944no—
270C2×C4—3942no—
271C2×C4—3948no—
272C2×C4D4×D19714yes—
273C2×C4C2×C22⋊D19711yescharacteristic
274C2×C4C2×C22⋊D19711yescharacteristic
275C2×C4C2×C22⋊D19711yescharacteristic
276C2×C4C23×D19711yescharacteristic
277C8—3944no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—84no—
290C22—7888no—
291C22—7882no—
292C22—7882no—
293C22—3944no—
294C22#2026329412yes—
295C22#2026335212yescomplemented
296C22C22×C22⋊D19711yescharacteristic
297C4—21no—
298C4—21no—
299C4—42no—
300C4—3944no—
301C4#2026329412yes—
302C4C2×D4×D19712yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—82no—
307C2—7884no—
308C2C2×#2026329411yescharacteristic
309C2C2×#2026335211yescharacteristic
310C2#2716187711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#121×C197C2
socleC22×C197C22×D4
hypercenterC2×#121D197
hypocenterC197#3006
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#15691358C2semidirect
2. C2×#20263352C2semidirect
3. #27161222C2semidirect
4. #27161217C2semidirect
5. #27161212C2semidirect
6. #27161078C2semidirect
7. #27161043C2semidirect
8. C2×#121×C197C2semidirect
13. C2×#15691358C22semidirect
14. C22×Dic394C22semidirect
15. C22×D788C22semidirect
19. C2×#15691346C22semidirect
20. C2×#15691343C22semidirect
21. C2×#15691342C22semidirect
22. #20263249C22semidirect
23. #20263246C22semidirect
24. C22⋊#13096802C22semidirect
25. C22⋊#13096802C22semidirect
27. C22×D4×C197C22semidirect
32. #121×C197C22semidirect
33. C2×C4⋊C4×C197C22semidirect
41. #15691358D4semidirect
85. Dic394C2×D4semidirect
88. D788C2×D4semidirect
135. C22×C197AGL(1,ℤ/8)semidirect
155. C197#3006semidirect
159. C2×#121D197semidirect
174. C22×D4C2×D197semidirect
184. #121C2×D197semidirect
185. C2×C4⋊C4C2×D197semidirect
295. C22#20263352semidirect

Maximal subgroups

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

subgroupindex
C22×#156913582
C2×#202633522
#271612222
#271612172
#271612122
#271610782
#271610432
C2×#121×C1972
#3006197