Finite Group Databasebeta

← #38754312

Subgroups — #38754312

Order 25216.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#20263351C212yesmaximal, complemented
2D4×D788C211yesmaximal, characteristic, complemented
3#27161453C211yesmaximal, characteristic, complemented
4#27161327C212yesmaximal, complemented
5#27161281C211yesmaximal, characteristic, complemented
6#27161274C211yesmaximal, characteristic, complemented
7#27161229C212yesmaximal, complemented
8#27161224C212yesmaximal, complemented
9C4×#15691343C211yesmaximal, characteristic, complemented
10#27161157C211yesmaximal, characteristic, complemented
11#547×C197C211yesmaximal, characteristic, complemented
12C22×D788C2212yes—
13C2×D4×D197—21no—
14#20263351—28no—
15C2×#15691345C2212yes—
16C2×#15691345C2212yescomplemented
17C2×#15691345C2212yescomplemented
18C2×#15691343C2211yescharacteristic
19C2×#15691343C2212yescomplemented
20C2×#15691342C2212yescomplemented
21#20263323C2211yescharacteristic
22#20263301—21no—
23#20263300C2212yes—
24#20263296—22no—
25C4×D788C2211yescharacteristic
26#20263259C2211yescharacteristic, complemented
27C4⋊#13096802C2211yescharacteristic, complemented
28#20263250C2212yescomplemented
29#20263247C2211yescharacteristic, complemented
30#20263246C2211yescharacteristic
31#20263246C2212yescomplemented
32C22⋊#13096802C2212yescomplemented
33C4×#13096802C2211yescharacteristic, complemented
34C2×#41×C197C2212yescomplemented
35C4⋊Q8×C197C2211yescharacteristic, complemented
36#123×C197—22no—
37C22⋊Q8×C197C2212yescomplemented
38#121×C197—21no—
39C4×D4×C197C2211yescharacteristic
40C4×D4×C197C2211yescharacteristic, complemented
41C2×C4⋊C4×C197—21no—
42C23×D197—22no—
43C2×D788—24no—
44C2×D788—22no—
45C2×D788—21no—
46C2×D788—21no—
47C2×D788—42no—
48C2×D788C2312yes—
49C2×D788C2311yescharacteristic
50C2×C22⋊D197—22no—
51D4×D197—44no—
52C2×C4×D197—21no—
53#15691345—24no—
54#15691345—24no—
55#15691345—42no—
56#15691343—24no—
57#15691343D414yescomplemented
58#15691342—24no—
59C2×#13096802C2311yescharacteristic
60C2×#13096802C2312yes—
61C2×#13096802C2311yescharacteristic
62C4⋊Dic197C2311yescharacteristic
63#15691335—21no—
64#15691335—22no—
65#41×C197—24no—
66#41×C197—42no—
67C2×Q8×C197C2312yes—
68C2×D4×C197—22no—
69C2×D4×C197D411yescharacteristic
70C2×D4×C197C2311yescharacteristic
71C22×C4×C197—21no—
72C22×C4×C197—22no—
73C22×C4×C197C2312yes—
74C4⋊C4×C197—21no—
75C4⋊C4×C197—22no—
76C4⋊C4×C197—21no—
77C4⋊C4×C197D411yescharacteristic
78C4⋊C4×C197C2311yescharacteristic
79C22⋊C4×C197—22no—
80C22⋊C4×C197—21no—
81C22⋊C4×C197D412yes—
82C42×C197C2311yescharacteristic
83C22×D197—21no—
84C22×D197—22no—
85C22×D197—44no—
86C22×D197—44no—
87C22×D197—82no—
88D788—22no—
89D788—21no—
90D788—22no—
91D788—44no—
92D788—42no—
93D788—81no—
94D788C2×D412yes—
95C22⋊D197—44no—
96C2×Dic197—21no—
97C4×D197—42no—
98#13096802—22no—
99#13096802—21no—
100#13096802C2×D412yescomplemented
101C23×C197—21no—
102C23×C197C2×D412yes—
103Q8×C197—22no—
104Q8×C197—22no—
105D4×C197—22no—
106D4×C197—21no—
107D4×C197—44no—
108D4×C197C2×D412yescomplemented
109C2×C4×C197—22no—
110C2×C4×C197—22no—
111C2×C4×C197—21no—
112C2×C4×C197—22no—
113C2×C4×C197—44no—
114C2×C4×C197—42no—
115C2×C4×C197C2×D411yescharacteristic
116C2×C4×C197C2×D412yes—
117C2×C4×C197C2×D411yescharacteristic
118C2×C4×C197C2411yescharacteristic
119C22×C197—22no—
120C22×C197—22no—
121C22×C197—22no—
122C22×C197—22no—
123C22×C197—41no—
124C22×C197—41no—
125C22×C197C22×D411yescharacteristic
126C2×D197—22no—
127C2×D197—41no—
128C2×D197—41no—
129C2×D197—42no—
130C2×D197—42no—
131C2×D197—42no—
132C2×D197—88no—
133C4×C197—22no—
134C4×C197—21no—
135C4×C197—42no—
136C4×C197—41no—
137C4×C197AGL(1,ℤ/8)12yescomplemented
138C4×C197C22×D411yescharacteristic
139C4×C197H(5,𝔽2)11yescharacteristic
140Dic197—41no—
141C2×C197—42no—
142C2×C197—42no—
143C2×C197D4211yescharacteristic
144C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C197#57711yescharacteristic
146D197—42no—
147D197—82no—
148C197#327211yescharacteristic, complemented, Sylow
149#3272—1971nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1972no—
151#547D19711yescharacteristic, complemented
152D42—1971no—
153#492—1971no—
154#468—1972no—
155#461—1971no—
156#459—1971no—
157#450—1972no—
158#449—1972no—
159C4×D8—1971no—
160#436—1971no—
161C2×#41C2×D19712yescomplemented
162C22×D4—3941no—
163C22×D4—1972no—
164AGL(1,ℤ/8)—3948no—
165C2×SD16—1972no—
166C2×SD16—1972no—
167C2×SD16—1972no—
168C2×D8—1972no—
169C2×D8—1971no—
170C2×M16—1972no—
171C4⋊Q8C2×D19711yescharacteristic, complemented
172#127—1971no—
173#123—22no—
174C22⋊Q8C2×D19712yescomplemented
175#121—21no—
176#121—3941no—
177#121—1972no—
178#120—3942no—
179C4×D4—1971no—
180C4×D4C2×D19711yescharacteristic
181C4×D4C2×D19711yescharacteristic, complemented
182C2×C4⋊C4—21no—
183#107—1971no—
184C4⋊C8—1971no—
185#103—1972no—
186#102—1971no—
187#102—1971no—
188#102—1972no—
189C22⋊C8—1972no—
190C4×C8—1971no—
191C24—3942no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D19712yes—
195C2×D4—22no—
196C2×D4—7882no—
197C2×D4—7884no—
198C2×D4—3944no—
199C2×D4—3942no—
200C2×D4—3942no—
201C2×D4—3941no—
202C2×D4—3941no—
203C2×D4—1972no—
204C2×D4—1971no—
205C2×D4C22⋊D19711yescharacteristic
206C2×D4C22×D19711yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—3941no—
210C22×C4C22×D19712yes—
211SD16—7882no—
212SD16—3944no—
213SD16—3944no—
214D8—3944no—
215D8—1974no—
216M16—3944no—
217C2×C8—1971no—
218C2×C8—1972no—
219C2×C8—1971no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—1971no—
224C4⋊C4C22⋊D19711yescharacteristic
225C4⋊C4C22×D19711yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—3941no—
229C22⋊C4—3942no—
230C22⋊C4C22⋊D19712yes—
231C42C22×D19711yescharacteristic
232C23—21no—
233C23—7884no—
234C23—7884no—
235C23—15762no—
236C23—3942no—
237C23—3941no—
238C23C2×C22⋊D19712yes—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—21no—
243D4—44no—
244D4—7884no—
245D4—7884no—
246D4—7882no—
247D4—15761no—
248D4—3942no—
249D4—3942no—
250D4—3941no—
251D4—1972no—
252D4D4×D19712yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—7882no—
260C2×C4—3941no—
261C2×C4C2×C22⋊D19711yescharacteristic
262C2×C4C2×C22⋊D19712yes—
263C2×C4C2×C22⋊D19711yescharacteristic
264C2×C4C23×D19711yescharacteristic
265C8—3942no—
266C8—3941no—
267C8—1972no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—7881no—
275C22—7881no—
276C22—7882no—
277C22—7882no—
278C22—7882no—
279C22—15768no—
280C22—3942no—
281C22C22×C22⋊D19711yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—42no—
286C4—7881no—
287C4#2026335112yescomplemented
288C4C2×D4×D19711yescharacteristic
289C4#2026337111yescharacteristic
290C2—42no—
291C2—42no—
292C2—7882no—
293C2—15762no—
294C2D4×C22⋊D19711yescharacteristic
295C2C2×#2026335111yescharacteristic
296C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#547×C197C2
socleC22×C197C22×D4
hypercenter#547D197
hypocenterC197#3272
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#20263351C2semidirect
2. D4×D788C2semidirect
3. #27161453C2semidirect
4. #27161327C2semidirect
5. #27161281C2semidirect
6. #27161274C2semidirect
7. #27161229C2semidirect
8. #27161224C2semidirect
9. C4×#15691343C2semidirect
10. #27161157C2semidirect
11. #547×C197C2semidirect
16. C2×#15691345C22semidirect
17. C2×#15691345C22semidirect
19. C2×#15691343C22semidirect
20. C2×#15691342C22semidirect
26. #20263259C22semidirect
27. C4⋊#13096802C22semidirect
28. #20263250C22semidirect
29. #20263247C22semidirect
31. #20263246C22semidirect
32. C22⋊#13096802C22semidirect
33. C4×#13096802C22semidirect
34. C2×#41×C197C22semidirect
35. C4⋊Q8×C197C22semidirect
37. C22⋊Q8×C197C22semidirect
40. C4×D4×C197C22semidirect
57. #15691343D4semidirect
100. #13096802C2×D4semidirect
108. D4×C197C2×D4semidirect
137. C4×C197AGL(1,ℤ/8)semidirect
148. C197#3272semidirect
151. #547D197semidirect
161. C2×#41C2×D197semidirect
171. C4⋊Q8C2×D197semidirect
174. C22⋊Q8C2×D197semidirect
181. C4×D4C2×D197semidirect
252. D4D4×D197semidirect
287. C4#20263351semidirect

Maximal subgroups

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

subgroupindex
C2×#202633512
D4×D7882
#271614532
#271613272
#271612812
#271612742
#271612292
#271612242
C4×#156913432
#271611572
#547×C1972
#3272197