Finite Group Databasebeta

← #38745926

Subgroups — #38745926

Order 25216.

944 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 163 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×#15691357C211yesmaximal, characteristic, complemented
2C2×#20263288C211yesmaximal, characteristic, complemented
3C2×#20263286C212yesmaximal, complemented
4#27160999C211yesmaximal, characteristic, complemented
5#27160994C212yesmaximal, complemented
6#27160992C211yesmaximal, characteristic, complemented
7#27160988C212yesmaximal, complemented
8#27160987C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic197C212yesmaximal, complemented
10#27160926C211yesmaximal, characteristic, complemented
11#394×C197C211yesmaximal, characteristic, complemented
12C2×#15691357—28no—
13C2×#15691357—24no—
14C22×Dic394C2211yescharacteristic
15C22×D788C2211yescharacteristic
16C22×C22⋊D197C2212yes—
17C22×C22⋊D197C2212yes—
18C23×Dic197C2212yes—
19C22×C4×D197C2211yescharacteristic
20C22×C4×D197C2211yescharacteristic
21#20263288—24no—
22#20263286—28no—
23C2×#15691337C2212yes—
24C2×C22⋊Dic197C2212yes—
25C2×#15691335C2212yes—
26C2×#15691335C2211yescharacteristic
27C2×#15691335C2212yes—
28C2×#15691335C2211yescharacteristic
29C2×C4×Dic197C2212yes—
30C2×C4×Dic197C2211yescharacteristic
31#20263233C2211yescharacteristic
32#20263233C2212yes—
33#20263233C2212yes—
34C22×D4×C197C2211yescharacteristic
35C2×C4⋊C4×C197C2211yescharacteristic
36C2×C22⋊C4×C197C2212yes—
37C2×C22⋊C4×C197C2212yes—
38#95×C197C2211yescharacteristic
39C23×D197C2311yescharacteristic
40C23×D197C2311yescharacteristic
41#15691357—232no—
42C2×Dic394—24no—
43C2×Dic394—22no—
44C2×D788—22no—
45C2×D788—22no—
46C2×D788—22no—
47C2×C22⋊D197—24no—
48C2×C22⋊D197—24no—
49C2×C22⋊D197—24no—
50C2×C22⋊D197—24no—
51C2×C22⋊D197—24no—
52C2×C22⋊D197—24no—
53C22×Dic197—24no—
54C22×Dic197—24no—
55C22×Dic197D418yescomplemented
56C22×Dic197C2311yescharacteristic
57C22×Dic197C2312yes—
58C22×Dic197C2312yes—
59C22×Dic197C2311yescharacteristic
60C2×C4×D197—24no—
61C2×C4×D197—24no—
62C2×C4×D197—22no—
63C2×C4×D197D414yescomplemented
64#15691337—24no—
65C22⋊Dic197—24no—
66#15691335—22no—
67#15691335—24no—
68#15691335—24no—
69#15691335—22no—
70C4×Dic197—24no—
71C4×Dic197—22no—
72C24×C197C2312yes—
73C2×D4×C197—24no—
74C2×D4×C197—22no—
75C22×C4×C197C2312yes—
76C22×C4×C197C2311yescharacteristic
77C22×C4×C197C2311yescharacteristic
78C22×C4×C197C2311yescharacteristic
79C4⋊C4×C197—22no—
80C22⋊C4×C197—24no—
81C22⋊C4×C197—24no—
82C22×D197—22no—
83C22×D197—21no—
84C22×D197—22no—
85C22×D197—22no—
86C22×D197—22no—
87C22×D197—21no—
88C22×D197—21no—
89C22×D197—21no—
90C22×D197C2×D412yes—
91C22×D197#4112yescomplemented
92Dic394—28no—
93D788—28no—
94C22⋊D197—216no—
95C22⋊D197—216no—
96C2×Dic197—22no—
97C2×Dic197—22no—
98C2×Dic197—22no—
99C2×Dic197—22no—
100C2×Dic197—216no—
101C2×Dic197—21no—
102C2×Dic197—22no—
103C2×Dic197C2×D414yes—
104C2×Dic197C2×D414yes—
105C2×Dic197C2×D412yes—
106C2×Dic197#4114yescomplemented
107C4×D197—28no—
108C4×D197—44no—
109C23×C197—22no—
110C23×C197—22no—
111C23×C197—22no—
112C23×C197—22no—
113C23×C197—22no—
114C23×C197—22no—
115C23×C197C2×D414yes—
116C23×C197C2411yescharacteristic
117D4×C197—28no—
118C2×C4×C197—21no—
119C2×C4×C197—22no—
120C2×C4×C197—21no—
121C2×C4×C197—22no—
122C2×C4×C197—22no—
123C2×C4×C197—21no—
124C2×C4×C197—22no—
125C2×C4×C197—22no—
126C2×C4×C197C2×D412yes—
127C2×C4×C197#4112yescomplemented
128C22×C197—24no—
129C22×C197—24no—
130C22×C197—24no—
131C22×C197—42no—
132C22×C197—42no—
133C22×C197—42no—
134C22×C197—42no—
135C22×C197C22×D412yes—
136C22×C197C22×D411yescharacteristic
137C22×C197C2×#4111yescharacteristic
138C22×C197C2×#4112yes—
139C22×C197C2×#4111yescharacteristic
140C2×D197—22no—
141C2×D197—24no—
142C2×D197—22no—
143C2×D197—22no—
144C2×D197—22no—
145C2×D197—41no—
146C2×D197—42no—
147C2×D197—42no—
148C2×D197—42no—
149C2×D197—41no—
150C4×C197—22no—
151C4×C197—22no—
152C4×C197—41no—
153C4×C197—42no—
154Dic197—24no—
155Dic197—24no—
156Dic197—41no—
157Dic197#12014yes—
158C2×C197—44no—
159C2×C197C2×#12011yescharacteristic
160C2×C197#52512yes—
161C2×C197#52511yescharacteristic
162C2×C197#53212yes—
163C2×C197#53211yescharacteristic
164D197—42no—
165D197—42no—
166C197#238811yescharacteristic, complemented, Sylow
167#2388—1971nomaximal, Sylow
168C22×#41—1971no—
169C2×C4×D4—1971no—
170C2×C4×D4—1972no—
171#395—1971no—
172#394—1972no—
173#394D19711yescharacteristic, complemented
174#393—1971no—
175#392—1971no—
176#392—1972no—
177C4×C22⋊C4—1972no—
178C4×C22⋊C4—1971no—
179C2×#41—3948no—
180C2×#41—3944no—
181C22×Q8—1971no—
182C22×D4—1972no—
183C22×D4—1972no—
184C22×D4—1971no—
185C22×D4C2×D19711yescharacteristic
186C23×C4—1971no—
187C23×C4—1972no—
188C23×C4—1971no—
189C4×D4—3948no—
190C4×D4—3944no—
191C2×C4⋊C4—1972no—
192C2×C4⋊C4C2×D19711yescharacteristic
193C2×C22⋊C4—1972no—
194C2×C22⋊C4—1972no—
195C2×C22⋊C4—1971no—
196C2×C22⋊C4—1971no—
197C2×C22⋊C4—1972no—
198C2×C22⋊C4C2×D19712yes—
199C2×C22⋊C4C2×D19712yes—
200C2×C42—1971no—
201C2×C42—1972no—
202#95—1971no—
203#95—1972no—
204#95—1972no—
205#95C2×D19711yescharacteristic
206C24—1971no—
207C24—1971no—
208C24C22×D19712yes—
209#41—39432no—
210C2×Q8—3944no—
211C2×Q8—3942no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—3942no—
215C2×D4—3942no—
216C2×D4—3944no—
217C2×D4—3944no—
218C2×D4—3944no—
219C2×D4—3944no—
220C2×D4—3944no—
221C2×D4—3944no—
222C2×D4—3942no—
223C22×C4—3944no—
224C22×C4—3942no—
225C22×C4—3944no—
226C22×C4—3944no—
227C22×C4—3944no—
228C22×C4—1971no—
229C22×C4—1978no—
230C22×C4—1972no—
231C22×C4—1972no—
232C22×C4—1974no—
233C22×C4—1971no—
234C22×C4C22×D19712yes—
235C22×C4C22×D19711yescharacteristic
236C22×C4C22×D19711yescharacteristic
237C22×C4C22×D19711yescharacteristic
238C4⋊C4—22no—
239C4⋊C4—3944no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—3942no—
243C22⋊C4—3944no—
244C22⋊C4—3944no—
245C22⋊C4—3944no—
246C22⋊C4—3942no—
247C42—3942no—
248C42—3944no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—3942no—
256C23—3941no—
257C23—3942no—
258C23—3942no—
259C23—3942no—
260C23—3941no—
261C23—3941no—
262C23—3941no—
263C23—1972no—
264C23—1972no—
265C23D4×D19714yes—
266C23C23×D19711yescharacteristic
267Q8—3948no—
268D4—28no—
269D4—3948no—
270D4—39416no—
271D4—39416no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—7884no—
281C2×C4—3948no—
282C2×C4—3941no—
283C2×C4—3942no—
284C2×C4—3942no—
285C2×C4—3942no—
286C2×C4—3942no—
287C2×C4—3942no—
288C2×C4—39416no—
289C2×C4—1974no—
290C2×C4—1974no—
291C2×C4—1974no—
292C2×C4—1972no—
293C2×C4D4×D19712yes—
294C2×C4#1569135712yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—7882no—
303C22—7882no—
304C22—7881no—
305C22—7881no—
306C22—7882no—
307C22—3942no—
308C22—3942no—
309C22—3944no—
310C22—3942no—
311C22—3942no—
312C22C2×D4×D19712yes—
313C22C2×D4×D19711yescharacteristic
314C22#41×D19712yes—
315C22#41×D19711yescharacteristic
316C22C2×#1569135711yescharacteristic
317C4—22no—
318C4—22no—
319C4—42no—
320C4—41no—
321C4—7881no—
322C4—3944no—
323C4—3944no—
324C4—1974no—
325C2—44no—
326C2—7882no—
327C2—7882no—
328C2#120×D19711yescharacteristic
329C2#2716158712yes—
330C2#2716158911yescharacteristic
331C2#2716162412yes—
332C2#2716162811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#394×C197C2
hypercenter#394D197
hypocenterC197#2388
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#15691357C2semidirect
2. C2×#20263288C2semidirect
3. C2×#20263286C2semidirect
4. #27160999C2semidirect
5. #27160994C2semidirect
6. #27160992C2semidirect
7. #27160988C2semidirect
8. #27160987C2semidirect
9. C22⋊C4×Dic197C2semidirect
10. #27160926C2semidirect
11. #394×C197C2semidirect
55. C22×Dic197D4semidirect
63. C2×C4×D197D4semidirect
91. C22×D197#41semidirect
106. C2×Dic197#41semidirect
127. C2×C4×C197#41semidirect
166. C197#2388semidirect
173. #394D197semidirect
294. C2×C4#15691357semidirect

Maximal subgroups

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

subgroupindex
C22×#156913572
C2×#202632882
C2×#202632862
#271609992
#271609942
#271609922
#271609882
#271609872
C22⋊C4×Dic1972
#271609262
#394×C1972
#2388197