Finite Group Databasebeta

← #38754601

Subgroups — #38754601

Order 25216.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161769C211yesmaximal, characteristic, complemented
2#27161738C211yesmaximal, characteristic
3#27161434C211yesmaximal, characteristic, complemented
4#27161421C211yesmaximal, characteristic, complemented
5#27161393C211yesmaximal, characteristic, complemented
6#27161387C211yesmaximal, characteristic, complemented
7#27161320C211yesmaximal, characteristic, complemented
8#27161308C211yesmaximal, characteristic, complemented
9#27161302C211yesmaximal, characteristic, complemented
10#27161243C211yesmaximal, characteristic, complemented
11#27161237C211yesmaximal, characteristic, complemented
12C4×Dic788C211yesmaximal, characteristic, complemented
13#27161076C211yesmaximal, characteristic, complemented
14C2×#20263251C211yesmaximal, characteristic, complemented
15C8×D4×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×Dic394C2211yescharacteristic
18C2×Dic788C2211yescharacteristic
19C2×Dic788C2211yescharacteristic
20C2×#15691347C2211yescharacteristic
21#20263319—21no—
22#20263318C2211yescharacteristic
23#20263316—21no—
24#20263314C2211yescharacteristic
25#20263312—21no—
26#20263307—21no—
27#20263305C2211yescharacteristic
28#20263305C2211yescharacteristic
29#20263305C2211yescharacteristic
30#20263304—21no—
31#20263300C2211yescharacteristic
32#20263298—21no—
33C4×Dic394C2211yescharacteristic
34C4×Dic394C2211yescharacteristic
35#20263286—21no—
36#20263285C2211yescharacteristic
37#20263282—21no—
38C2×C4⋊Dic197C2211yescharacteristic
39#20263260C2211yescharacteristic
40#20263260C2211yescharacteristic
41#20263258C2211yescharacteristic
42#20263251C2211yescharacteristic
43#20263251C2211yescharacteristic
44#20263251C2211yescharacteristic
45#20263251C2211yescharacteristic
46#20263251C2211yescharacteristic
47#20263251C2211yescharacteristic
48#20263251C2211yescharacteristic
49#20263248C2211yescharacteristic
50#20263248C2211yescharacteristic
51#20263248C2211yescharacteristic
52C22×C8×C197C2211yescharacteristic
53C22×C8×C197C2211yescharacteristic
54C4×D4×C197C2211yescharacteristic
55C4⋊C8×C197C2211yescharacteristic
56C22⋊C8×C197C2211yescharacteristic
57C22⋊C8×C197C2211yescharacteristic
58C4×C8×C197C2211yescharacteristic
59#15691358—22no—
60#15691358—41no—
61C2×Dic394—22no—
62C2×Dic394—21no—
63C2×Dic394—41no—
64C2×Dic394C2311yescharacteristic
65C2×Dic394C2311yescharacteristic
66C2×Dic394C2311yescharacteristic
67C2×D788C2311yescharacteristic
68C2×C22⋊D197—21no—
69C2×C22⋊D197—21no—
70C22×Dic197—21no—
71C22×Dic197—21no—
72C2×C4×D197—21no—
73Dic788—22no—
74Dic788—41no—
75#15691347—41no—
76C4⋊Dic197—21no—
77C4⋊Dic197C2311yescharacteristic
78C4⋊Dic197C2311yescharacteristic
79C4⋊Dic197C2311yescharacteristic
80C4⋊Dic197C2311yescharacteristic
81#15691337—21no—
82#15691337—21no—
83#15691337—21no—
84#15691337—21no—
85#15691337—21no—
86#15691337—21no—
87#15691337—21no—
88C22⋊Dic197—21no—
89C22⋊Dic197—21no—
90C22⋊Dic197—21no—
91C22⋊Dic197—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95C4×Dic197—21no—
96C4×Dic197—21no—
97C4×Dic197—21no—
98C2×D4×C197D411yescharacteristic
99C22×C4×C197C2311yescharacteristic
100C22×C4×C197C2311yescharacteristic
101C2×C8×C197—41no—
102C2×C8×C197—41no—
103C2×C8×C197C2311yescharacteristic
104C2×C8×C197C2311yescharacteristic
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C4⋊C4×C197D411yescharacteristic
108C22⋊C4×C197D411yescharacteristic
109C22⋊C4×C197D411yescharacteristic
110C42×C197C2311yescharacteristic
111C22×D197—21no—
112Dic394—21no—
113Dic394—21no—
114Dic394—21no—
115Dic394—21no—
116Dic394—21no—
117Dic394—42no—
118Dic394#4112yes—
119D788—21no—
120D788—21no—
121C22⋊D197—42no—
122C22⋊D197—42no—
123C2×Dic197—21no—
124C2×Dic197—21no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C2×Dic197—21no—
128C2×Dic197—21no—
129C2×Dic197—21no—
130C2×Dic197—42no—
131C2×Dic197—42no—
132C4×D197—42no—
133C23×C197C2×D411yescharacteristic
134C23×C197C2×D411yescharacteristic
135D4×C197—22no—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
138C2×C4×C197—21no—
139C2×C4×C197—21no—
140C2×C4×C197C2×D411yescharacteristic
141C2×C4×C197C2×D411yescharacteristic
142C2×C4×C197C2×D411yescharacteristic
143C2×C4×C197C2×D411yescharacteristic
144C2×C4×C197C2411yescharacteristic
145C8×C197—21no—
146C8×C197—21no—
147C8×C197—21no—
148C8×C197—21no—
149C22×C197—21no—
150C22×C197—21no—
151C22×C197—21no—
152C22×C197—21no—
153C22×C197—21no—
154C22×C197#13512yes—
155C22×C197C22×D411yescharacteristic
156C2×D197—41no—
157C2×D197—41no—
158C2×D197—41no—
159C4×C197—21no—
160C4×C197—21no—
161C4×C197—21no—
162C4×C197—21no—
163C4×C197C2×#4111yescharacteristic
164C4×C197H(5,𝔽2)11yescharacteristic
165Dic197—22no—
166Dic197—41no—
167Dic197—41no—
168Dic197—41no—
169Dic197—41no—
170Dic197—41no—
171Dic197—41no—
172C2×C197—22no—
173C2×C197—41no—
174C2×C197#54611yescharacteristic
175C2×C197C2×#13511yescharacteristic
176C2×C197#57811yescharacteristic
177D197—81no—
178C197#330011yescharacteristic, complemented, Sylow
179#3300—1971nomaximal, Sylow
180#552—1971no—
181#548—1971no—
182#488—1971no—
183#485—1971no—
184#480—1971no—
185#479—1971no—
186#467—1971no—
187#465—1971no—
188#464—1971no—
189#452—1971no—
190#451—1971no—
191C4×Q16—1971no—
192C8×D4D19711yescharacteristic, complemented
193#416—1971no—
194C2×#103—1971no—
195C2×#41—1971no—
196C22×Q8—1971no—
197C2×Q16—1971no—
198C2×Q16—1971no—
199C2×SD16—1971no—
200C22×C8C2×D19711yescharacteristic
201C22×C8C2×D19711yescharacteristic
202#126—3941no—
203#125—3941no—
204#125—1971no—
205#124—3941no—
206#124—1971no—
207#123—3941no—
208C22⋊Q8—3941no—
209C22⋊Q8—1971no—
210C22⋊Q8—1971no—
211C22⋊Q8—1971no—
212#121—3941no—
213#121—1971no—
214C4×Q8—1971no—
215C4×Q8—1971no—
216C4×D4—3941no—
217C4×D4C2×D19711yescharacteristic
218#117—3941no—
219#117—1971no—
220C2×C4⋊C4—1971no—
221#107—1971no—
222#107—1971no—
223#106—1971no—
224C4⋊C8C2×D19711yescharacteristic
225#103—1971no—
226#103—1971no—
227#103—1971no—
228#103—1971no—
229#103—1971no—
230#103—1971no—
231#103—1971no—
232#102—1971no—
233#102—1971no—
234#102—1971no—
235C22⋊C8C2×D19711yescharacteristic
236C22⋊C8C2×D19711yescharacteristic
237C4×C8C2×D19711yescharacteristic
238#41—7881no—
239#41—3942no—
240C2×Q8—7881no—
241C2×Q8—3942no—
242C2×Q8—3941no—
243C2×Q8—1971no—
244C2×Q8—1971no—
245C2×Q8—1971no—
246C2×D4—3941no—
247C2×D4—3941no—
248C2×D4—1971no—
249C2×D4D78811yescharacteristic
250C22×C4—3941no—
251C22×C4—3941no—
252C22×C4—3941no—
253C22×C4C22×D19711yescharacteristic
254C22×C4C22×D19711yescharacteristic
255Q16—7881no—
256Q16—3942no—
257SD16—7881no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8C22×D19711yescharacteristic
261C2×C8C22×D19711yescharacteristic
262C2×C8C22×D19711yescharacteristic
263C2×C8C22×D19711yescharacteristic
264C4⋊C4—3941no—
265C4⋊C4—3941no—
266C4⋊C4—3941no—
267C4⋊C4—3941no—
268C4⋊C4—3941no—
269C4⋊C4—3941no—
270C4⋊C4—3941no—
271C4⋊C4—3941no—
272C4⋊C4—1971no—
273C4⋊C4—1971no—
274C4⋊C4—1971no—
275C4⋊C4—1971no—
276C4⋊C4D78811yescharacteristic
277C22⋊C4—3941no—
278C22⋊C4—3941no—
279C22⋊C4—3941no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4—3941no—
284C22⋊C4D78811yescharacteristic
285C22⋊C4D78811yescharacteristic
286C42—3941no—
287C42—3941no—
288C42—3941no—
289C42C22×D19711yescharacteristic
290C23—3941no—
291C23C2×D78811yescharacteristic
292C23C2×D78811yescharacteristic
293Q8—7882no—
294Q8—3941no—
295Q8—3941no—
296Q8—3941no—
297Q8—3941no—
298Q8—3941no—
299Q8—1972no—
300D4—22no—
301D4—7882no—
302D4—7882no—
303D4—3941no—
304D4—3941no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—7882no—
310C2×C4—7882no—
311C2×C4—7882no—
312C2×C4—3941no—
313C2×C4—3941no—
314C2×C4—3941no—
315C2×C4—3941no—
316C2×C4—3941no—
317C2×C4—3941no—
318C2×C4—3941no—
319C2×C4C2×D78811yescharacteristic
320C2×C4C2×D78811yescharacteristic
321C2×C4C2×D78811yescharacteristic
322C2×C4C2×D78811yescharacteristic
323C2×C4C23×D19711yescharacteristic
324C8—21no—
325C8—21no—
326C8—21no—
327C8—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—7881no—
334C22—7881no—
335C22—7881no—
336C22#2026334812yes—
337C22C22×D78811yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—7881no—
343C4—7881no—
344C4—7881no—
345C4—7881no—
346C4—7881no—
347C4—7881no—
348C4—3942no—
349C4#41×D19711yescharacteristic
350C4#2026337111yescharacteristic
351C2—22no—
352C2—41no—
353C2—15761no—
354C2#2716172411yescharacteristic
355C2C2×#2026334811yescharacteristic
356C2#2716189611yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC8×D4×C197C2
socleC22×C197C22×D4
prime radical#27161738C2
hypercenterC8×D4D197
hypocenterC197#3300
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #27161769C2semidirect
3. #27161434C2semidirect
4. #27161421C2semidirect
5. #27161393C2semidirect
6. #27161387C2semidirect
7. #27161320C2semidirect
8. #27161308C2semidirect
9. #27161302C2semidirect
10. #27161243C2semidirect
11. #27161237C2semidirect
12. C4×Dic788C2semidirect
13. #27161076C2semidirect
14. C2×#20263251C2semidirect
15. C8×D4×C197C2semidirect
178. C197#3300semidirect
192. C8×D4D197semidirect

Maximal subgroups

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

subgroupindex
#271617692
#271617382
#271614342
#271614212
#271613932
#271613872
#271613202
#271613082
#271613022
#271612432
#271612372
C4×Dic7882
#271610762
C2×#202632512
C8×D4×C1972
#3300197