Finite Group Databasebeta

← #38754705

Subgroups — #38754705

Order 25216.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161738C211yesmaximal, characteristic, complemented
2#27161733C211yesmaximal, characteristic, complemented
3#27161439C211yesmaximal, characteristic, complemented
4#27161415C211yesmaximal, characteristic, complemented
5#27161393C211yesmaximal, characteristic, complemented
6#27161375C211yesmaximal, characteristic, complemented
7#27161325C211yesmaximal, characteristic, complemented
8#27161315C211yesmaximal, characteristic, complemented
9#27161295C211yesmaximal, characteristic, complemented
10#27161226C211yesmaximal, characteristic, complemented
11#27161220C211yesmaximal, characteristic, complemented
12#27161181C211yesmaximal, characteristic, complemented
13#27161081C211yesmaximal, characteristic, complemented
14C2×#20263251C211yesmaximal, characteristic, complemented
15#435×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C2×#15691357—21no—
18C22×Dic394C2211yescharacteristic
19C2×#15691347C2211yescharacteristic
20C2×#15691347C2211yescharacteristic
21C2×D1576C2211yescharacteristic
22#20263326C2211yescharacteristic
23#20263314C2211yescharacteristic
24#20263312—21no—
25#20263310—21no—
26#20263310—21no—
27#20263306—21no—
28#20263305C2211yescharacteristic
29#20263304—21no—
30#20263300C2211yescharacteristic
31#20263300C2211yescharacteristic
32#20263300C2211yescharacteristic
33#20263298—21no—
34C4×Dic394C2211yescharacteristic
35C4×D788C2211yescharacteristic
36#20263286—21no—
37C2×C4⋊Dic197C2211yescharacteristic
38C2×C4⋊Dic197C2211yescharacteristic
39#20263260C2211yescharacteristic
40#20263260C2211yescharacteristic
41#20263260C2211yescharacteristic
42#20263251C2211yescharacteristic
43#20263251C2211yescharacteristic
44#20263251C2211yescharacteristic
45#20263251C2211yescharacteristic
46#20263251C2211yescharacteristic
47#20263251C2211yescharacteristic
48#20263248C2211yescharacteristic
49#20263248C2211yescharacteristic
50#20263248C2211yescharacteristic
51#20263248C2211yescharacteristic
52C2×M16×C197C2211yescharacteristic
53C22×C8×C197C2211yescharacteristic
54C4×D4×C197C2211yescharacteristic
55C4⋊C8×C197C2211yescharacteristic
56C22⋊C8×C197C2211yescharacteristic
57C22⋊C8×C197C2211yescharacteristic
58#97×C197C2211yescharacteristic
59#15691358—22no—
60#15691358—41no—
61#15691357—44no—
62C2×Dic394—22no—
63C2×Dic394—21no—
64C2×Dic394—21no—
65C2×Dic394—41no—
66C2×Dic394C2311yescharacteristic
67C2×Dic394C2311yescharacteristic
68C2×D788C2311yescharacteristic
69C2×D788C2311yescharacteristic
70C2×C22⋊D197—21no—
71C2×C22⋊D197—21no—
72C2×C22⋊D197—21no—
73C2×C22⋊D197—21no—
74C22×Dic197—21no—
75C22×Dic197—21no—
76C22×Dic197—21no—
77C2×C4×D197—21no—
78C2×C4×D197—21no—
79#15691347—22no—
80#15691347—41no—
81D1576—41no—
82C4⋊Dic197—21no—
83C4⋊Dic197—21no—
84C4⋊Dic197—21no—
85C4⋊Dic197C2311yescharacteristic
86C4⋊Dic197C2311yescharacteristic
87C4⋊Dic197C2311yescharacteristic
88C4⋊Dic197C2311yescharacteristic
89#15691337—21no—
90#15691337—21no—
91C22⋊Dic197—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335—21no—
96#15691335—21no—
97#15691335—21no—
98C4×Dic197—21no—
99C2×D4×C197D411yescharacteristic
100C22×C4×C197C2311yescharacteristic
101C22×C4×C197C2311yescharacteristic
102M16×C197—41no—
103C2×C8×C197—41no—
104C2×C8×C197C2311yescharacteristic
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C2×C8×C197C2311yescharacteristic
108C4⋊C4×C197D411yescharacteristic
109C22⋊C4×C197D411yescharacteristic
110C22⋊C4×C197D411yescharacteristic
111C42×C197C2311yescharacteristic
112C22×D197—21no—
113C22×D197—21no—
114Dic394—21no—
115Dic394—21no—
116Dic394—21no—
117Dic394—42no—
118Dic394—42no—
119Dic394#4112yescomplemented
120D788—21no—
121D788—21no—
122D788—21no—
123D788—21no—
124C22⋊D197—42no—
125C22⋊D197—42no—
126C22⋊D197—42no—
127C22⋊D197—42no—
128C2×Dic197—21no—
129C2×Dic197—21no—
130C2×Dic197—21no—
131C2×Dic197—21no—
132C2×Dic197—21no—
133C2×Dic197—21no—
134C2×Dic197—42no—
135C2×Dic197—42no—
136C2×Dic197—42no—
137C4×D197—42no—
138C4×D197—42no—
139C23×C197C2×D411yescharacteristic
140C23×C197C2×D411yescharacteristic
141D4×C197—22no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2×D411yescharacteristic
149C2×C4×C197C2×D411yescharacteristic
150C2×C4×C197C2411yescharacteristic
151C8×C197—21no—
152C8×C197—21no—
153C8×C197—21no—
154C8×C197—21no—
155C22×C197—21no—
156C22×C197—21no—
157C22×C197—21no—
158C22×C197—21no—
159C22×C197—21no—
160C22×C197AGL(1,ℤ/8)12yes—
161C22×C197C22×D411yescharacteristic
162C2×D197—41no—
163C2×D197—41no—
164C2×D197—41no—
165C2×D197—41no—
166C2×D197—41no—
167C2×D197—41no—
168C4×C197—21no—
169C4×C197—21no—
170C4×C197—21no—
171C4×C197—21no—
172C4×C197C2×#4111yescharacteristic
173C4×C197H(5,𝔽2)11yescharacteristic
174Dic197—22no—
175Dic197—41no—
176Dic197—41no—
177Dic197—41no—
178Dic197—41no—
179Dic197—41no—
180C2×C197—22no—
181C2×C197—41no—
182C2×C197#54611yescharacteristic
183C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C197#57811yescharacteristic
185D197—81no—
186D197—81no—
187C197#330811yescharacteristic, complemented, Sylow
188#3308—1971nomaximal, Sylow
189#548—1971no—
190#547—1971no—
191#489—1971no—
192#484—1971no—
193#480—1971no—
194#477—1971no—
195#468—1971no—
196#466—1971no—
197#463—1971no—
198#449—1971no—
199#448—1971no—
200#440—1971no—
201#435D19711yescharacteristic, complemented
202#417—1971no—
203C2×#103—1971no—
204C2×#41—3941no—
205C2×#41—1971no—
206C22×Q8—1971no—
207C2×SD16—1971no—
208C2×SD16—1971no—
209C2×D8—1971no—
210C2×M16C2×D19711yescharacteristic
211C22×C8C2×D19711yescharacteristic
212C4⋊Q8—1971no—
213#124—3941no—
214#124—1971no—
215#123—3941no—
216#123—3941no—
217C22⋊Q8—3941no—
218C22⋊Q8—3941no—
219C22⋊Q8—1971no—
220#121—3941no—
221#121—1971no—
222#121—1971no—
223#121—1971no—
224C4×Q8—1971no—
225C4×D4—3941no—
226C4×D4—1971no—
227C4×D4C2×D19711yescharacteristic
228C2×C4⋊C4—1971no—
229C2×C4⋊C4—1971no—
230#107—1971no—
231#107—1971no—
232#107—1971no—
233C4⋊C8C2×D19711yescharacteristic
234#103—1971no—
235#103—1971no—
236#103—1971no—
237#103—1971no—
238#103—1971no—
239#103—1971no—
240#102—1971no—
241#102—1971no—
242#102—1971no—
243#102—1971no—
244C22⋊C8C2×D19711yescharacteristic
245C22⋊C8C2×D19711yescharacteristic
246#97C2×D19711yescharacteristic
247#41—7881no—
248#41—7884no—
249#41—3942no—
250C2×Q8—7881no—
251C2×Q8—3942no—
252C2×Q8—3941no—
253C2×Q8—3941no—
254C2×Q8—1971no—
255C2×Q8—1971no—
256C2×D4—3941no—
257C2×D4—3941no—
258C2×D4—3941no—
259C2×D4—3941no—
260C2×D4—1971no—
261C2×D4—1971no—
262C2×D4D78811yescharacteristic
263C22×C4—3941no—
264C22×C4—3941no—
265C22×C4—3941no—
266C22×C4—3941no—
267C22×C4—3941no—
268C22×C4C22×D19711yescharacteristic
269C22×C4C22×D19711yescharacteristic
270SD16—7881no—
271SD16—3942no—
272D8—7881no—
273M16—41no—
274C2×C8—41no—
275C2×C8C22×D19711yescharacteristic
276C2×C8C22×D19711yescharacteristic
277C2×C8C22×D19711yescharacteristic
278C2×C8C22×D19711yescharacteristic
279C4⋊C4—3941no—
280C4⋊C4—3941no—
281C4⋊C4—3941no—
282C4⋊C4—3941no—
283C4⋊C4—3941no—
284C4⋊C4—1971no—
285C4⋊C4—1971no—
286C4⋊C4—1971no—
287C4⋊C4—1971no—
288C4⋊C4D78811yescharacteristic
289C22⋊C4—3941no—
290C22⋊C4—3941no—
291C22⋊C4—3941no—
292C22⋊C4—3941no—
293C22⋊C4—3941no—
294C22⋊C4—3941no—
295C22⋊C4—3941no—
296C22⋊C4D78811yescharacteristic
297C22⋊C4D78811yescharacteristic
298C42—3941no—
299C42C22×D19711yescharacteristic
300C23—3941no—
301C23—3941no—
302C23C2×D78811yescharacteristic
303C23C2×D78811yescharacteristic
304Q8—7882no—
305Q8—7882no—
306Q8—3941no—
307Q8—3941no—
308Q8—3941no—
309Q8—1972no—
310D4—22no—
311D4—7882no—
312D4—7882no—
313D4—7882no—
314D4—7882no—
315D4—3941no—
316D4—3941no—
317D4—3941no—
318D4—3941no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—7882no—
324C2×C4—7882no—
325C2×C4—7882no—
326C2×C4—7882no—
327C2×C4—7882no—
328C2×C4—3941no—
329C2×C4—3941no—
330C2×C4—3941no—
331C2×C4—3941no—
332C2×C4—3941no—
333C2×C4—3941no—
334C2×C4C2×D78811yescharacteristic
335C2×C4C2×D78811yescharacteristic
336C2×C4C2×D78811yescharacteristic
337C2×C4C2×D78811yescharacteristic
338C2×C4C23×D19711yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—7881no—
349C22—7881no—
350C22—7881no—
351C22—7881no—
352C22—7881no—
353C22—7881no—
354C22#2026335412yes—
355C22C22×D78811yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—7881no—
361C4—7881no—
362C4—7881no—
363C4—7881no—
364C4—7881no—
365C4—3942no—
366C4#41×D19711yescharacteristic
367C4#2026337111yescharacteristic
368C2—22no—
369C2—41no—
370C2—15761no—
371C2—15761no—
372C2#2716172411yescharacteristic
373C2C2×#2026335411yescharacteristic
374C2#2716189611yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#435×C197C2
socleC22×C197C22×D4
hypercenter#435D197
hypocenterC197#3308
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #27161738C2semidirect
2. #27161733C2semidirect
3. #27161439C2semidirect
4. #27161415C2semidirect
5. #27161393C2semidirect
6. #27161375C2semidirect
7. #27161325C2semidirect
8. #27161315C2semidirect
9. #27161295C2semidirect
10. #27161226C2semidirect
11. #27161220C2semidirect
12. #27161181C2semidirect
13. #27161081C2semidirect
14. C2×#20263251C2semidirect
15. #435×C197C2semidirect
119. Dic394#41semidirect
187. C197#3308semidirect
201. #435D197semidirect

Maximal subgroups

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

subgroupindex
#271617382
#271617332
#271614392
#271614152
#271613932
#271613752
#271613252
#271613152
#271612952
#271612262
#271612202
#271611812
#271610812
C2×#202632512
#435×C1972
#3308197