Finite Group Databasebeta

← #38752895

Subgroups — #38752895

Order 25216.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161698C211yesmaximal, characteristic, complemented
2#27161551C211yesmaximal, characteristic, complemented
3#27161418C211yesmaximal, characteristic, complemented
4#27161402C211yesmaximal, characteristic, complemented
5#27161370C211yesmaximal, characteristic, complemented
6#27161359C211yesmaximal, characteristic, complemented
7#27161291C211yesmaximal, characteristic, complemented
8#27161228C211yesmaximal, characteristic, complemented
9#27161211C211yesmaximal, characteristic
10#27161191C211yesmaximal, characteristic, complemented
11#27161182C211yesmaximal, characteristic, complemented
12SD16×Dic197C211yesmaximal, characteristic, complemented
13#27161159C211yesmaximal, characteristic, complemented
14#27161139C211yesmaximal, characteristic, complemented
15#463×C197C211yesmaximal, characteristic, complemented
16C2×D4×D197C2211yescharacteristic
17C2×#15691345C2211yescharacteristic
18C2×#15691343C2211yescharacteristic
19#20263326C2211yescharacteristic, complemented
20#20263319—21no—
21#20263312—21no—
22#20263310—21no—
23#20263303—21no—
24#20263302C2211yescharacteristic
25#20263297C2211yescharacteristic
26Q8×Dic197C2211yescharacteristic, complemented
27C4×D788C2211yescharacteristic
28D4×Dic197C2211yescharacteristic
29D4×Dic197C2211yescharacteristic, complemented
30#20263288C2211yescharacteristic
31C4×C22⋊D197—21no—
32#20263286—21no—
33#20263284C2211yescharacteristic
34#20263283C2211yescharacteristic
35C2×#15691335—21no—
36C22⋊C4×D197—21no—
37#20263259C2211yescharacteristic, complemented
38#20263259C2211yescharacteristic, complemented
39#20263258C2211yescharacteristic, complemented
40#20263258C2211yescharacteristic, complemented
41C4⋊#13096802C2211yescharacteristic, complemented
42#20263250C2211yescharacteristic, complemented
43#20263248C2211yescharacteristic
44#20263248C2211yescharacteristic
45#20263247C2211yescharacteristic
46#20263247C2211yescharacteristic
47#20263247C2211yescharacteristic, complemented
48#20263238C2211yescharacteristic
49#20263238C2211yescharacteristic
50#20263236C2211yescharacteristic, complemented
51C8×Dic197C2211yescharacteristic, complemented
52C2×SD16×C197C2211yescharacteristic
53C2×D8×C197C2211yescharacteristic
54#124×C197C2211yescharacteristic, complemented
55C4×D4×C197C2211yescharacteristic
56C4⋊C8×C197C2211yescharacteristic, complemented
57#103×C197C2211yescharacteristic, complemented
58#102×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C2×Dic394—21no—
61C2×D788C2311yescharacteristic
62C2×C22⋊D197—21no—
63C2×C22⋊D197—21no—
64D4×D197—22no—
65D4×D197—22no—
66C22×Dic197—21no—
67C22×Dic197—21no—
68C2×C4×D197—21no—
69C2×C4×D197—21no—
70C2×C4×D197C2311yescharacteristic
71#15691345—22no—
72#15691343—22no—
73C2×#13096802C2311yescharacteristic
74C2×#13096802C2311yescharacteristic
75C4⋊Dic197—21no—
76C4⋊Dic197—21no—
77C4⋊Dic197C2311yescharacteristic
78C4⋊Dic197C2311yescharacteristic
79C4⋊Dic197C2311yescharacteristic
80#15691337—21no—
81#15691337D411yescharacteristic
82#15691337D411yescharacteristic
83C22⋊Dic197—21no—
84C22⋊Dic197—21no—
85C22⋊Dic197—21no—
86#15691335—21no—
87#15691335—21no—
88#15691335—21no—
89#15691335—21no—
90#15691335D411yescharacteristic
91#15691335D411yescharacteristic
92C4×Dic197—21no—
93C4×Dic197—21no—
94C4×Dic197C2311yescharacteristic
95C2×Q8×C197C2311yescharacteristic
96C2×D4×C197C2311yescharacteristic
97C2×D4×C197C2311yescharacteristic
98C22×C4×C197—21no—
99SD16×C197—22no—
100D8×C197—22no—
101C2×C8×C197C2311yescharacteristic
102C2×C8×C197C2311yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C22⋊C4×C197—21no—
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C42×C197C2311yescharacteristic
108C22×D197—21no—
109C22×D197—24no—
110C22×D197—81no—
111C22×D197—81no—
112C22×D197C2×D411yescharacteristic
113Dic394—81no—
114D788—21no—
115D788#4112yes—
116C22⋊D197—24no—
117C22⋊D197—81no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197—81no—
122C2×Dic197—81no—
123C2×Dic197C2×D411yescharacteristic
124C2×Dic197C2×D411yescharacteristic
125C2×Dic197C2×D411yescharacteristic
126C4×D197—21no—
127C4×D197—21no—
128C4×D197—81no—
129C4×D197—81no—
130#13096802—21no—
131#13096802—21no—
132C23×C197—21no—
133C23×C197—21no—
134Q8×C197—21no—
135Q8×C197—21no—
136D4×C197—21no—
137D4×C197—21no—
138D4×C197—21no—
139D4×C197#4112yes—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—81no—
143C2×C4×C197C2×D411yescharacteristic
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2411yescharacteristic
146C8×C197—21no—
147C8×C197—21no—
148C22×C197—22no—
149C22×C197—41no—
150C22×C197—41no—
151C22×C197—41no—
152C22×C197—41no—
153C22×C197—41no—
154C22×C197C22×D411yescharacteristic
155C2×D197—21no—
156C2×D197—22no—
157C2×D197—21no—
158C2×D197—21no—
159C2×D197—41no—
160C2×D197—41no—
161C2×D197—84no—
162C4×C197—21no—
163C4×C197—21no—
164C4×C197—41no—
165C4×C197—41no—
166C4×C197C2×#4111yescharacteristic
167C4×C197C2×#4111yescharacteristic
168Dic197—21no—
169Dic197—21no—
170Dic197—21no—
171Dic197—41no—
172Dic197—41no—
173Dic197—41no—
174C2×C197—42no—
175C2×C197—81no—
176C2×C197#52511yescharacteristic
177C2×C197#57611yescharacteristic
178C2×C197#57711yescharacteristic
179D197—41no—
180D197—42no—
181C197#311811yescharacteristic, complemented, Sylow
182#3118—1971nomaximal, Sylow
183#542—1971no—
184#518—1971no—
185#485—1971no—
186#482—1971no—
187#476—1971no—
188#474—1971no—
189#463—1971no—
190#463D19711yescharacteristic, complemented
191#450—1971no—
192#447—1971no—
193#442—1971no—
194#440—1971no—
195C4×SD16—1971no—
196C4×D8—1971no—
197#433—1971no—
198C22×D4—1971no—
199C2×SD16—1971no—
200C2×SD16C2×D19711yescharacteristic
201C2×D8—1971no—
202C2×D8C2×D19711yescharacteristic
203C4⋊Q8—1971no—
204#126—3941no—
205#124—3941no—
206#124C2×D19711yescharacteristic, complemented
207#123—3941no—
208C22⋊Q8—3941no—
209C22⋊Q8—1971no—
210#121—1971no—
211C4×Q8—1971no—
212C4×D4—3941no—
213C4×D4—3941no—
214C4×D4—1971no—
215C4×D4—1971no—
216C4×D4—1971no—
217C4×D4—1971no—
218C4×D4C2×D19711yescharacteristic
219#117—1971no—
220#117—1971no—
221C2×C22⋊C4—3941no—
222C2×C22⋊C4—3941no—
223#107—1971no—
224#107—1971no—
225#106—1971no—
226#106—1971no—
227C4⋊C8—1971no—
228C4⋊C8C2×D19711yescharacteristic, complemented
229#103—1971no—
230#103C2×D19711yescharacteristic, complemented
231#102—1971no—
232#102—1971no—
233#102—1971no—
234#102—1971no—
235#102—1971no—
236#102C2×D19711yescharacteristic, complemented
237C22⋊C8—1971no—
238C22⋊C8—1971no—
239#97—1971no—
240C4×C8—1971no—
241C24—3941no—
242C2×Q8—3941no—
243C2×Q8C22×D19711yescharacteristic
244C2×D4—3942no—
245C2×D4—3941no—
246C2×D4—3942no—
247C2×D4—3941no—
248C2×D4—1971no—
249C2×D4C22×D19711yescharacteristic
250C2×D4C22×D19711yescharacteristic
251C22×C4—21no—
252C22×C4—3941no—
253C22×C4—3941no—
254C22×C4—3941no—
255C22×C4—3941no—
256C22×C4—1971no—
257SD16—22no—
258SD16—3942no—
259D8—22no—
260D8—3942no—
261C2×C8—1971no—
262C2×C8—1971no—
263C2×C8C22×D19711yescharacteristic
264C2×C8C22×D19711yescharacteristic
265C4⋊C4—3941no—
266C4⋊C4—3941no—
267C4⋊C4—3941no—
268C4⋊C4—1971no—
269C4⋊C4—1971no—
270C4⋊C4—1971no—
271C4⋊C4—1971no—
272C4⋊C4—1971no—
273C4⋊C4C22×D19711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—3941no—
278C22⋊C4—3941no—
279C22⋊C4—3941no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4—3941no—
284C22⋊C4—1971no—
285C22⋊C4—1971no—
286C42—3941no—
287C42—3941no—
288C42—1971no—
289C42C22×D19711yescharacteristic
290C23—21no—
291C23—21no—
292C23—15761no—
293C23—15761no—
294C23—3944no—
295C23—3941no—
296C23—1971no—
297Q8—21no—
298Q8—21no—
299Q8—15761no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—15761no—
304D4—3944no—
305D4—3941no—
306D4—1972no—
307D4#1569135712yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—15761no—
312C2×C4—15761no—
313C2×C4—15761no—
314C2×C4—15761no—
315C2×C4—3941no—
316C2×C4—3941no—
317C2×C4—3941no—
318C2×C4—3941no—
319C2×C4—3941no—
320C2×C4—1971no—
321C2×C4—1971no—
322C2×C4—1971no—
323C2×C4D4×D19711yescharacteristic
324C2×C4D4×D19711yescharacteristic
325C2×C4C23×D19711yescharacteristic
326C8—21no—
327C8—21no—
328C8—3941no—
329C8—3941no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—7881no—
337C22—7881no—
338C22—15764no—
339C22—3941no—
340C22—3941no—
341C22—3941no—
342C22—3942no—
343C22C2×D4×D19711yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—7881no—
349C4—7881no—
350C4—7881no—
351C4—3941no—
352C4—3941no—
353C4—3941no—
354C4#41×D19711yescharacteristic
355C4C2×#1569135711yescharacteristic
356C2—42no—
357C2—81no—
358C2—7882no—
359C2—7881no—
360C2#2716158911yescharacteristic
361C2#2716188311yescharacteristic
362C2#2716188611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#463×C197C2
socleC22×C197C22×D4
prime radical#27161211C2
hypercenter#463D197
hypocenterC197#3118
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #27161698C2semidirect
2. #27161551C2semidirect
3. #27161418C2semidirect
4. #27161402C2semidirect
5. #27161370C2semidirect
6. #27161359C2semidirect
7. #27161291C2semidirect
8. #27161228C2semidirect
10. #27161191C2semidirect
11. #27161182C2semidirect
12. SD16×Dic197C2semidirect
13. #27161159C2semidirect
14. #27161139C2semidirect
15. #463×C197C2semidirect
19. #20263326C22semidirect
26. Q8×Dic197C22semidirect
29. D4×Dic197C22semidirect
37. #20263259C22semidirect
38. #20263259C22semidirect
39. #20263258C22semidirect
40. #20263258C22semidirect
41. C4⋊#13096802C22semidirect
42. #20263250C22semidirect
47. #20263247C22semidirect
50. #20263236C22semidirect
51. C8×Dic197C22semidirect
54. #124×C197C22semidirect
56. C4⋊C8×C197C22semidirect
57. #103×C197C22semidirect
58. #102×C197C22semidirect
181. C197#3118semidirect
190. #463D197semidirect
206. #124C2×D197semidirect
228. C4⋊C8C2×D197semidirect
230. #103C2×D197semidirect
236. #102C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616982
#271615512
#271614182
#271614022
#271613702
#271613592
#271612912
#271612282
#271612112
#271611912
#271611822
SD16×Dic1972
#271611592
#271611392
#463×C1972
#3118197