Finite Group Databasebeta

← #38751971

Subgroups — #38751971

Order 25216.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15691359C211yesmaximal, characteristic
2C2×#20263358C212yesmaximal, complemented
3C22⋊Q8×D197C211yesmaximal, characteristic, complemented
4#27161244C212yesmaximal, complemented
5#27161240C212yesmaximal, complemented
6#27161235C211yesmaximal, characteristic, complemented
7#27161232C211yesmaximal, characteristic, complemented
8#27161229C211yesmaximal, characteristic, complemented
9#27161079C212yesmaximal, complemented
10#27161041C211yesmaximal, characteristic, complemented
11#451×C197C211yesmaximal, characteristic, complemented
12C2×#15691359—28no—
13C2×#15691359—21no—
14C2×#15691359—21no—
15C2×#15691359C2212yes—
16C2×#15691359C2212yes—
17C2×Q8×D197C2211yescharacteristic, complemented
18C22×D788—21no—
19C22×D788C2211yescharacteristic
20C22×C4×D197—21no—
21C22×C4×D197C2211yescharacteristic
22#20263358—28no—
23C2×#15691348C2212yescomplemented
24C2×#15691347C2212yescomplemented
25C2×#15691345C2212yescomplemented
26C2×#15691341C2212yescomplemented
27#20263306C2211yescharacteristic, complemented
28#20263305C2211yescharacteristic
29#20263304—21no—
30#20263303—21no—
31#20263302C2211yescharacteristic, complemented
32C4⋊C4×D197—21no—
33C4⋊C4×D197C2211yescharacteristic, complemented
34C22⋊C4×D197—21no—
35#20263250C2212yescomplemented
36#20263248C2212yescomplemented
37#20263246C2212yescomplemented
38C22⋊#13096802C2211yescharacteristic
39#20263238C2212yescomplemented
40C22×Q8×C197C2211yescharacteristic
41C2×Q16×C197C2212yescomplemented
42C22⋊Q8×C197C2211yescharacteristic
43#103×C197C2212yescomplemented
44C22⋊C8×C197C2211yescharacteristic
45C23×D197—21no—
46C23×D197D411yescharacteristic
47#15691359—28no—
48#15691359—28no—
49#15691359—24no—
50#15691359—24no—
51#15691359—24no—
52#15691359D418yes—
53C2×Dic394—21no—
54C2×Dic394C2311yescharacteristic
55Q8×D197—24no—
56C2×D788—28no—
57C2×D788—22no—
58C2×D788—22no—
59C2×D788—24no—
60C2×D788—21no—
61C2×D788C2312yes—
62C22×Dic197D411yescharacteristic
63C2×C4×D197—28no—
64C2×C4×D197—22no—
65C2×C4×D197—22no—
66C2×C4×D197—24no—
67C2×C4×D197—21no—
68C2×C4×D197—21no—
69C2×C4×D197D411yescharacteristic
70C2×C4×D197D411yescharacteristic
71C2×C4×D197C2311yescharacteristic
72C2×C4×D197C2311yescharacteristic
73#15691348—24no—
74#15691347—24no—
75#15691345—24no—
76#15691341—24no—
77C2×#13096802C2312yes—
78C4⋊Dic197—21no—
79C4⋊Dic197C2311yescharacteristic
80#15691337—21no—
81#15691337—21no—
82#15691337—21no—
83C22⋊Dic197—21no—
84#15691335—21no—
85#15691335—21no—
86C2×Q8×C197—24no—
87C2×Q8×C197—21no—
88C2×Q8×C197C2311yescharacteristic
89C2×Q8×C197C2312yes—
90C22×C4×C197—21no—
91C22×C4×C197C2311yescharacteristic
92Q16×C197—24no—
93C2×C8×C197C2312yes—
94C4⋊C4×C197—21no—
95C4⋊C4×C197C2311yescharacteristic
96C22⋊C4×C197—21no—
97C22×D197—24no—
98C22×D197—22no—
99C22×D197—22no—
100C22×D197—22no—
101C22×D197—41no—
102C22×D197—42no—
103C22×D197—42no—
104C22×D197C2×D411yescharacteristic
105C22×D197C2×D411yescharacteristic
106Dic394—22no—
107Dic394—42no—
108D788—28no—
109D788—24no—
110D788—28no—
111D788—22no—
112D788C2×D414yes—
113C2×Dic197—22no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197C2×D411yescharacteristic
117C2×Dic197C2×D411yescharacteristic
118C4×D197—28no—
119C4×D197—22no—
120C4×D197—22no—
121C4×D197—28no—
122C4×D197—22no—
123C4×D197—42no—
124C4×D197—42no—
125C4×D197C2×D414yes—
126#13096802—22no—
127C23×C197C2×D411yescharacteristic
128Q8×C197—24no—
129Q8×C197—22no—
130Q8×C197—22no—
131Q8×C197C2×D414yes—
132C2×C4×C197—24no—
133C2×C4×C197—22no—
134C2×C4×C197—22no—
135C2×C4×C197—21no—
136C2×C4×C197—21no—
137C2×C4×C197C2×D411yescharacteristic
138C2×C4×C197C2411yescharacteristic
139C8×C197—22no—
140C22×C197—21no—
141C22×C197—21no—
142C22×C197#13712yescomplemented
143C22×C197C22×D411yescharacteristic
144C2×D197—21no—
145C2×D197—21no—
146C2×D197—21no—
147C2×D197—24no—
148C2×D197—21no—
149C2×D197—21no—
150C2×D197—44no—
151C2×D197—48no—
152C2×D197—42no—
153C2×D197—42no—
154C2×D197#12012yes—
155C4×C197—24no—
156C4×C197—21no—
157C4×C197—41no—
158C4×C197—41no—
159C4×C197C22×D412yes—
160Dic197—21no—
161Dic197—41no—
162Dic197—41no—
163Dic197#12012yes—
164C2×C197—22no—
165C2×C197C2×#12011yescharacteristic
166C2×C197C2×#13711yescharacteristic
167C2×C197#57511yescharacteristic
168D197—22no—
169D197—41no—
170D197—44no—
171C197#300711yescharacteristic, complemented, Sylow
172#3007—1971nomaximal, Sylow
173C22×#41—1971no—
174C2×#137—1972no—
175C2×C22⋊Q8—1971no—
176#452—1972no—
177#452—1972no—
178#451—1971no—
179#451D19711yescharacteristic, complemented
180#450—1971no—
181#450—1971no—
182#417—1972no—
183#407—1971no—
184C2×#41—3948no—
185C2×#41—3941no—
186C2×#41—3941no—
187C2×#41—1972no—
188C2×#41—1972no—
189C22×Q8—1971no—
190C22×Q8C2×D19711yescharacteristic
191C22×D4—3941no—
192C22×D4—1971no—
193C23×C4—3941no—
194C23×C4—1971no—
195#137—3948no—
196C2×Q16—1972no—
197C2×Q16C2×D19712yescomplemented
198C2×SD16—1972no—
199C2×SD16—1972no—
200C2×M16—1972no—
201C22⋊Q8—3941no—
202C22⋊Q8—3941no—
203C22⋊Q8—1971no—
204C22⋊Q8—1971no—
205C22⋊Q8—1971no—
206C22⋊Q8C2×D19711yescharacteristic
207C2×C4⋊C4—3941no—
208C2×C4⋊C4—1971no—
209C2×C22⋊C4—3941no—
210#103—1972no—
211#103C2×D19712yescomplemented
212#102—1972no—
213#102—1972no—
214C22⋊C8—1971no—
215C22⋊C8—1972no—
216C22⋊C8C2×D19711yescharacteristic
217C24—3941no—
218C24—1971no—
219#41—3948no—
220#41—3948no—
221#41—3944no—
222#41—3944no—
223#41—3944no—
224#41—1978no—
225C2×Q8—24no—
226C2×Q8—21no—
227C2×Q8—3941no—
228C2×Q8—3944no—
229C2×Q8—1971no—
230C2×Q8C22×D19711yescharacteristic
231C2×Q8C22×D19712yes—
232C2×D4—3948no—
233C2×D4—3942no—
234C2×D4—3942no—
235C2×D4—3944no—
236C2×D4—3941no—
237C2×D4—1972no—
238C22×C4—21no—
239C22×C4—3941no—
240C22×C4—3941no—
241C22×C4—3948no—
242C22×C4—3942no—
243C22×C4—3942no—
244C22×C4—3944no—
245C22×C4—1971no—
246C22×C4—1971no—
247C22×C4—1971no—
248C22×C4—1971no—
249C22×C4—1971no—
250C22×C4C22×D19711yescharacteristic
251Q16—24no—
252Q16—3944no—
253SD16—3944no—
254SD16—3944no—
255M16—3944no—
256C2×C8—1972no—
257C2×C8C22×D19712yes—
258C4⋊C4—21no—
259C4⋊C4—3941no—
260C4⋊C4—3941no—
261C4⋊C4—3941no—
262C4⋊C4—3941no—
263C4⋊C4—1971no—
264C4⋊C4C22×D19711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—3941no—
267C22⋊C4—3941no—
268C22⋊C4—3941no—
269C23—7882no—
270C23—7881no—
271C23—7882no—
272C23—3944no—
273C23—3942no—
274C23—3942no—
275C23—3942no—
276C23—1971no—
277C23—1971no—
278C23D4×D19711yescharacteristic
279Q8—24no—
280Q8—22no—
281Q8—22no—
282Q8—7882no—
283Q8—3942no—
284Q8D4×D19714yes—
285D4—3948no—
286D4—3944no—
287D4—3948no—
288D4—3942no—
289D4—1974no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—7882no—
296C2×C4—7882no—
297C2×C4—3948no—
298C2×C4—3942no—
299C2×C4—3948no—
300C2×C4—3942no—
301C2×C4—3941no—
302C2×C4—3941no—
303C2×C4—3942no—
304C2×C4—3942no—
305C2×C4—1971no—
306C2×C4—1971no—
307C2×C4—1974no—
308C2×C4D4×D19711yescharacteristic
309C2×C4C23×D19711yescharacteristic
310C8—22no—
311C8—3942no—
312C22—21no—
313C22—21no—
314C22—7888no—
315C22—7884no—
316C22—7882no—
317C22—7882no—
318C22—3941no—
319C22—3941no—
320C22—3941no—
321C22—3944no—
322C22—3941no—
323C22—3941no—
324C22—1972no—
325C22#2026335812yescomplemented
326C22C2×D4×D19711yescharacteristic
327C4—24no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—7881no—
332C4—7881no—
333C4—3941no—
334C4—1972no—
335C4C2×D4×D19712yes—
336C2—22no—
337C2—7884no—
338C2—7881no—
339C2—3942no—
340C2#120×D19711yescharacteristic
341C2C2×#2026335811yescharacteristic
342C2#2716187811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#451×C197C2
socleC22×C197C22×D4
prime radicalC22×#15691359C2
hypercenter#451D197
hypocenterC197#3007
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#20263358C2semidirect
3. C22⋊Q8×D197C2semidirect
4. #27161244C2semidirect
5. #27161240C2semidirect
6. #27161235C2semidirect
7. #27161232C2semidirect
8. #27161229C2semidirect
9. #27161079C2semidirect
10. #27161041C2semidirect
11. #451×C197C2semidirect
17. C2×Q8×D197C22semidirect
23. C2×#15691348C22semidirect
24. C2×#15691347C22semidirect
25. C2×#15691345C22semidirect
26. C2×#15691341C22semidirect
27. #20263306C22semidirect
31. #20263302C22semidirect
33. C4⋊C4×D197C22semidirect
35. #20263250C22semidirect
36. #20263248C22semidirect
37. #20263246C22semidirect
39. #20263238C22semidirect
41. C2×Q16×C197C22semidirect
43. #103×C197C22semidirect
142. C22×C197#137semidirect
171. C197#3007semidirect
179. #451D197semidirect
197. C2×Q16C2×D197semidirect
211. #103C2×D197semidirect
325. C22#20263358semidirect

Maximal subgroups

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

subgroupindex
C22×#156913592
C2×#202633582
C22⋊Q8×D1972
#271612442
#271612402
#271612352
#271612322
#271612292
#271610792
#271610412
#451×C1972
#3007197