Finite Group Databasebeta

← #38754676

Subgroups — #38754676

Order 25216.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161730C211yesmaximal, characteristic, complemented
2#27161726C211yesmaximal, characteristic, complemented
3#27161437C211yesmaximal, characteristic, complemented
4#27161411C211yesmaximal, characteristic, complemented
5#27161368C211yesmaximal, characteristic, complemented
6#27161330C211yesmaximal, characteristic, complemented
7#27161309C211yesmaximal, characteristic, complemented
8#27161293C211yesmaximal, characteristic, complemented
9#27161226C211yesmaximal, characteristic, complemented
10#27161214C211yesmaximal, characteristic, complemented
11#27161190C211yesmaximal, characteristic, complemented
12#27161153C211yesmaximal, characteristic, complemented
13#27161078C211yesmaximal, characteristic, complemented
14#102×D197C211yesmaximal, characteristic, complemented
15#481×C197C211yesmaximal, characteristic, complemented
16C2×#15691358—21no—
17C2×#15691358C2211yescharacteristic
18C2×D4×D197C2211yescharacteristic
19C2×#15691347C2211yescharacteristic, complemented
20C2×D1576C2211yescharacteristic
21C2×#15691343C2211yescharacteristic
22C2×#15691342C2211yescharacteristic, complemented
23C2×C8×D197C2211yescharacteristic
24#20263325C2211yescharacteristic, complemented
25#20263313C2211yescharacteristic
26#20263311—21no—
27#20263309—21no—
28#20263308—21no—
29#20263306C2211yescharacteristic
30#20263304—21no—
31#20263301C2211yescharacteristic
32#20263300C2211yescharacteristic, complemented
33#20263297—21no—
34#20263294—21no—
35#20263288C2211yescharacteristic
36#20263288C2211yescharacteristic, complemented
37C4×C22⋊D197C2211yescharacteristic
38C4⋊C4×D197C2211yescharacteristic
39C22⋊C4×D197—21no—
40#20263257C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263251C2211yescharacteristic, complemented
43#20263249C2211yescharacteristic, complemented
44#20263249C2211yescharacteristic, complemented
45#20263248C2211yescharacteristic
46#20263247C2211yescharacteristic
47#20263246C2211yescharacteristic
48#20263246C2211yescharacteristic
49#20263246C2211yescharacteristic, complemented
50#20263238C2211yescharacteristic
51#20263236C2211yescharacteristic, complemented
52#121×C197C2211yescharacteristic
53C2×C4⋊C4×C197C2211yescharacteristic, complemented
54#106×C197C2211yescharacteristic, complemented
55#106×C197C2211yescharacteristic, complemented
56#102×C197C2211yescharacteristic
57#102×C197C2211yescharacteristic
58C22⋊C8×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60#15691358—22no—
61#15691358—41no—
62#15691358—44no—
63C2×Dic394—21no—
64C2×Dic394C2311yescharacteristic
65C2×D788—21no—
66C2×D788C2311yescharacteristic
67C2×D788C2311yescharacteristic
68C2×C22⋊D197—21no—
69C2×C22⋊D197—21no—
70C2×C22⋊D197—21no—
71C2×C22⋊D197D411yescharacteristic
72D4×D197—22no—
73D4×D197—41no—
74C2×C4×D197—21no—
75C2×C4×D197—21no—
76C2×C4×D197—21no—
77C2×C4×D197—21no—
78C2×C4×D197C2311yescharacteristic
79#15691347—41no—
80D1576—22no—
81#15691343—41no—
82#15691342—41no—
83C2×#13096802C2311yescharacteristic
84C2×#13096802C2311yescharacteristic
85C8×D197—41no—
86C4⋊Dic197—21no—
87C4⋊Dic197C2311yescharacteristic
88#15691337—21no—
89#15691337—21no—
90#15691337D411yescharacteristic
91C22⋊Dic197—21no—
92C22⋊Dic197—21no—
93C22⋊Dic197D411yescharacteristic
94#15691335—21no—
95#15691335—21no—
96#15691335—21no—
97#15691335—21no—
98#15691335—21no—
99#15691335D411yescharacteristic
100C4×Dic197C2311yescharacteristic
101C2×D4×C197—21no—
102C2×D4×C197C2311yescharacteristic
103C22×C4×C197—21no—
104C22×C4×C197C2311yescharacteristic
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C4⋊C4×C197—21no—
108C4⋊C4×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C4⋊C4×C197C2311yescharacteristic
111C22⋊C4×C197—21no—
112C22×D197—21no—
113C22×D197—21no—
114C22×D197—42no—
115C22×D197—42no—
116C22×D197—81no—
117C22×D197C2×D411yescharacteristic
118Dic394—21no—
119Dic394—21no—
120Dic394—42no—
121D788—21no—
122D788—21no—
123D788—21no—
124D788—42no—
125D788#4112yescomplemented
126C22⋊D197—22no—
127C22⋊D197—42no—
128C22⋊D197—42no—
129C22⋊D197—42no—
130C2×Dic197—21no—
131C2×Dic197—21no—
132C2×Dic197C2×D411yescharacteristic
133C2×Dic197C2×D411yescharacteristic
134C2×Dic197C2×D411yescharacteristic
135C4×D197—21no—
136C4×D197—21no—
137C4×D197—42no—
138C4×D197—42no—
139C4×D197—42no—
140C4×D197—81no—
141#13096802—21no—
142#13096802—21no—
143C23×C197—21no—
144C23×C197C2×D411yescharacteristic
145D4×C197—21no—
146D4×C197—21no—
147D4×C197—81no—
148C2×C4×C197—21no—
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197—42no—
154C2×C4×C197C2×D411yescharacteristic
155C2×C4×C197C2411yescharacteristic
156C8×C197—21no—
157C8×C197—21no—
158C22×C197—21no—
159C22×C197—21no—
160C22×C197—21no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197—41no—
164C22×C197C22×D411yescharacteristic
165C2×D197—21no—
166C2×D197—21no—
167C2×D197—22no—
168C2×D197—41no—
169C2×D197—41no—
170C2×D197—41no—
171C2×D197—41no—
172C2×D197—41no—
173C2×D197—84no—
174C2×D197AGL(1,ℤ/8)12yes—
175C4×C197—21no—
176C4×C197—41no—
177C4×C197—41no—
178C4×C197—41no—
179C4×C197C2×#4111yescharacteristic
180C4×C197H(5,𝔽2)11yescharacteristic
181Dic197—21no—
182Dic197—21no—
183Dic197—21no—
184Dic197—41no—
185Dic197—41no—
186C2×C197—41no—
187C2×C197—81no—
188C2×C197#54611yescharacteristic
189C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C197#57711yescharacteristic
191D197—22no—
192D197—42no—
193D197—81no—
194C197#330611yescharacteristic, complemented, Sylow
195#3306—1971nomaximal, Sylow
196#547—1971no—
197#546—1971no—
198#489—1971no—
199#483—1971no—
200#481D19711yescharacteristic, complemented
201#476—1971no—
202#469—1971no—
203#465—1971no—
204#463—1971no—
205#449—1971no—
206#447—1971no—
207#442—1971no—
208#435—1971no—
209#417—1971no—
210C2×#102—1971no—
211C2×#41—3941no—
212C2×#41—1971no—
213C22×D4—1971no—
214C2×SD16—1971no—
215C2×D8—1971no—
216C2×D8—1971no—
217C2×M16—1971no—
218C22×C8—1971no—
219C4⋊Q8—1971no—
220#124—1971no—
221#123—3941no—
222#123—3941no—
223#123—3941no—
224C22⋊Q8—3941no—
225C22⋊Q8—1971no—
226#121—3941no—
227#121—1971no—
228#121—1971no—
229#121C2×D19711yescharacteristic
230#120—3941no—
231C4×D4—1971no—
232C4×D4—1971no—
233C4×D4—1971no—
234C2×C4⋊C4—1971no—
235C2×C4⋊C4C2×D19711yescharacteristic, complemented
236C2×C22⋊C4—3941no—
237#106—1971no—
238#106C2×D19711yescharacteristic, complemented
239#106C2×D19711yescharacteristic, complemented
240C4⋊C8—1971no—
241#103—1971no—
242#103—1971no—
243#103—1971no—
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248#102—1971no—
249#102C2×D19711yescharacteristic
250#102C2×D19711yescharacteristic
251C22⋊C8—1971no—
252C22⋊C8C2×D19711yescharacteristic, complemented
253#97—1971no—
254C24—3941no—
255#41—7881no—
256#41—7884no—
257#41—3942no—
258C2×Q8—3941no—
259C2×Q8—1971no—
260C2×D4—21no—
261C2×D4—7881no—
262C2×D4—3941no—
263C2×D4—3942no—
264C2×D4—3941no—
265C2×D4—3941no—
266C2×D4—3941no—
267C2×D4—1971no—
268C2×D4—1971no—
269C2×D4—1971no—
270C2×D4C22×D19711yescharacteristic
271C22×C4—21no—
272C22×C4—3941no—
273C22×C4—3941no—
274C22×C4—3941no—
275C22×C4—3941no—
276C22×C4—1971no—
277C22×C4C22×D19711yescharacteristic
278SD16—7881no—
279D8—7881no—
280D8—3942no—
281M16—7881no—
282C2×C8—7881no—
283C2×C8—1971no—
284C2×C8—1971no—
285C2×C8C22×D19711yescharacteristic
286C2×C8C22×D19711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—3941no—
289C4⋊C4—3941no—
290C4⋊C4—3941no—
291C4⋊C4—1971no—
292C4⋊C4—1971no—
293C4⋊C4C22×D19711yescharacteristic
294C4⋊C4C22×D19711yescharacteristic
295C4⋊C4C22×D19711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—3941no—
298C22⋊C4—3941no—
299C22⋊C4—3941no—
300C22⋊C4—3941no—
301C22⋊C4—3941no—
302C22⋊C4—3941no—
303C22⋊C4—3941no—
304C22⋊C4—1971no—
305C22⋊C4—1971no—
306C42—1971no—
307C23—21no—
308C23—7882no—
309C23—7882no—
310C23—15761no—
311C23—3941no—
312C23—3941no—
313C23—1971no—
314C23D4×D19711yescharacteristic
315Q8—7882no—
316Q8—3941no—
317Q8—3941no—
318D4—21no—
319D4—21no—
320D4—81no—
321D4—7882no—
322D4—7882no—
323D4—7882no—
324D4—7882no—
325D4—3942no—
326D4—3941no—
327D4—3941no—
328D4—3941no—
329D4—1972no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—7882no—
337C2×C4—7882no—
338C2×C4—7882no—
339C2×C4—15761no—
340C2×C4—3941no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4—1971no—
345C2×C4—1971no—
346C2×C4—1971no—
347C2×C4D4×D19711yescharacteristic
348C2×C4C23×D19711yescharacteristic
349C8—21no—
350C8—21no—
351C8—3941no—
352C8—3941no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—7881no—
360C22—7881no—
361C22—7881no—
362C22—7881no—
363C22—7881no—
364C22—15764no—
365C22—3942no—
366C22—3941no—
367C22—3941no—
368C22—1972no—
369C22C2×D4×D19711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—7881no—
375C4—7881no—
376C4—3941no—
377C4—3941no—
378C4—3941no—
379C4#41×D19711yescharacteristic
380C4#2026337211yescharacteristic
381C2—41no—
382C2—81no—
383C2—7882no—
384C2—15761no—
385C2—3942no—
386C2#2716172111yescharacteristic
387C2AGL(1,ℤ/8)×D19711yescharacteristic
388C2#2716188511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#481×C197C2
socleC22×C197C22×D4
hypercenter#481D197
hypocenterC197#3306
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #27161730C2semidirect
2. #27161726C2semidirect
3. #27161437C2semidirect
4. #27161411C2semidirect
5. #27161368C2semidirect
6. #27161330C2semidirect
7. #27161309C2semidirect
8. #27161293C2semidirect
9. #27161226C2semidirect
10. #27161214C2semidirect
11. #27161190C2semidirect
12. #27161153C2semidirect
13. #27161078C2semidirect
14. #102×D197C2semidirect
15. #481×C197C2semidirect
19. C2×#15691347C22semidirect
22. C2×#15691342C22semidirect
24. #20263325C22semidirect
32. #20263300C22semidirect
36. #20263288C22semidirect
40. #20263257C22semidirect
41. #20263255C22semidirect
42. #20263251C22semidirect
43. #20263249C22semidirect
44. #20263249C22semidirect
49. #20263246C22semidirect
51. #20263236C22semidirect
53. C2×C4⋊C4×C197C22semidirect
54. #106×C197C22semidirect
55. #106×C197C22semidirect
58. C22⋊C8×C197C22semidirect
125. D788#41semidirect
194. C197#3306semidirect
200. #481D197semidirect
235. C2×C4⋊C4C2×D197semidirect
238. #106C2×D197semidirect
239. #106C2×D197semidirect
252. C22⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617302
#271617262
#271614372
#271614112
#271613682
#271613302
#271613092
#271612932
#271612262
#271612142
#271611902
#271611532
#271610782
#102×D1972
#481×C1972
#3306197