Finite Group Databasebeta

← #38752507

Subgroups — #38752507

Order 25216.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161635C211yesmaximal, characteristic, complemented
2C4×D4×D197C211yesmaximal, characteristic
3#27161380C211yesmaximal, characteristic, complemented
4#27161379C211yesmaximal, characteristic, complemented
5#27161376C211yesmaximal, characteristic, complemented
6#27161368C211yesmaximal, characteristic, complemented
7#27161367C211yesmaximal, characteristic, complemented
8#27161358C211yesmaximal, characteristic, complemented
9#27161357C211yesmaximal, characteristic, complemented
10#27161120C211yesmaximal, characteristic, complemented
11#27161118C211yesmaximal, characteristic, complemented
12#27161101C211yesmaximal, characteristic, complemented
13#27161082C211yesmaximal, characteristic
14#27161082C211yesmaximal, characteristic
15#478×C197C211yesmaximal, characteristic, complemented
16C2×D4×D197C2211yescharacteristic
17C22×C4×D197—21no—
18C2×#15691341C2211yescharacteristic
19C2×#15691341C2211yescharacteristic
20#20263325C2211yescharacteristic, complemented
21#20263325C2211yescharacteristic, complemented
22#20263318C2211yescharacteristic, complemented
23#20263304—21no—
24#20263304—21no—
25#20263291—21no—
26#20263291—21no—
27C4×D788C2211yescharacteristic
28D4×Dic197C2211yescharacteristic
29#20263288C2211yescharacteristic
30C4×C22⋊D197—21no—
31#20263286—21no—
32#20263284C2211yescharacteristic
33#20263284C2211yescharacteristic
34C4⋊C4×D197C2211yescharacteristic
35C22⋊C4×D197—21no—
36C42×D197C2211yescharacteristic
37#20263260C2211yescharacteristic, complemented
38#20263259C2211yescharacteristic, complemented
39#20263259C2211yescharacteristic, complemented
40#20263258C2211yescharacteristic, complemented
41#20263257C2211yescharacteristic, complemented
42#20263257C2211yescharacteristic, complemented
43C4⋊#13096802C2211yescharacteristic, complemented
44#20263255C2211yescharacteristic, complemented
45#20263255C2211yescharacteristic, complemented
46#20263248C2211yescharacteristic
47#20263248C2211yescharacteristic
48#20263247C2211yescharacteristic
49#20263247C2211yescharacteristic
50#20263246C2211yescharacteristic
51#20263246C2211yescharacteristic
52#125×C197C2211yescharacteristic, complemented
53C4×D4×C197C2211yescharacteristic
54#107×C197C2211yescharacteristic, complemented
55#106×C197C2211yescharacteristic, complemented
56C4⋊C8×C197C2211yescharacteristic, complemented
57#102×C197C2211yescharacteristic
58#102×C197C2211yescharacteristic
59C23×D197—21no—
60C2×Dic394—21no—
61C2×Dic394—21no—
62C2×D788C2311yescharacteristic
63C2×C22⋊D197—21no—
64D4×D197—22no—
65D4×D197Q814yescomplemented
66C22×Dic197—21no—
67C2×C4×D197—21no—
68C2×C4×D197—21no—
69C2×C4×D197—21no—
70C2×C4×D197—44no—
71C2×C4×D197D411yescharacteristic
72C2×C4×D197D411yescharacteristic
73C2×C4×D197C2311yescharacteristic
74#15691341—22no—
75#15691341—22no—
76C2×#13096802C2311yescharacteristic
77C2×#13096802C2311yescharacteristic
78C4⋊Dic197C2311yescharacteristic
79C4⋊Dic197C2311yescharacteristic
80C4⋊Dic197C2311yescharacteristic
81#15691337—21no—
82#15691337—21no—
83#15691337—21no—
84#15691337—21no—
85#15691337—21no—
86C22⋊Dic197—21no—
87#15691335—21no—
88#15691335—21no—
89#15691335—21no—
90#15691335—21no—
91C4×Dic197—21no—
92C4×Dic197—21no—
93C4×Dic197D411yescharacteristic
94C4×Dic197D411yescharacteristic
95C4×Dic197C2311yescharacteristic
96C2×D4×C197C2311yescharacteristic
97C22×C4×C197—21no—
98C2×C8×C197C2311yescharacteristic
99C2×C8×C197C2311yescharacteristic
100C4⋊C4×C197—21no—
101C4⋊C4×C197—21no—
102C4⋊C4×C197C2311yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C4⋊C4×C197C2311yescharacteristic
105C22⋊C4×C197—21no—
106C42×C197C2311yescharacteristic
107C22×D197—21no—
108C22×D197—24no—
109C22×D197—42no—
110C22×D197—42no—
111C22×D197C2×D411yescharacteristic
112Dic394—42no—
113Dic394—42no—
114D788—21no—
115D788C2×Q812yes—
116C22⋊D197—24no—
117C2×Dic197—21no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—42no—
121C2×Dic197C2×D411yescharacteristic
122C2×Dic197C2×D411yescharacteristic
123C2×Dic197C2×D411yescharacteristic
124C4×D197—22no—
125C4×D197—41no—
126C4×D197—42no—
127C4×D197—42no—
128C4×D197—42no—
129C4×D197C2×Q812yes—
130C4×D197#4112yes—
131#13096802—21no—
132#13096802—21no—
133C23×C197—21no—
134D4×C197—21no—
135D4×C197C2×Q812yes—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
138C2×C4×C197—21no—
139C2×C4×C197—42no—
140C2×C4×C197C2×D411yescharacteristic
141C2×C4×C197C2×D411yescharacteristic
142C2×C4×C197C2411yescharacteristic
143C8×C197—21no—
144C8×C197—21no—
145C22×C197—22no—
146C22×C197—41no—
147C22×C197—41no—
148C22×C197C22×D411yescharacteristic
149C2×D197—21no—
150C2×D197—21no—
151C2×D197—22no—
152C2×D197—48no—
153C2×D197—41no—
154C2×D197—41no—
155C2×D197C22⋊Q812yes—
156C4×C197—21no—
157C4×C197—21no—
158C4×C197—41no—
159C4×C197—41no—
160C4×C197—41no—
161C4×C197C22×Q811yescharacteristic
162C4×C197C2×#4111yescharacteristic
163Dic197—21no—
164Dic197—41no—
165Dic197—41no—
166Dic197—41no—
167Dic197C22⋊Q812yes—
168Dic197AGL(1,ℤ/8)12yes—
169C2×C197—42no—
170C2×C197C2×C22⋊Q811yescharacteristic
171C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C197#57511yescharacteristic
173D197—22no—
174D197—42no—
175C197#307711yescharacteristic, complemented, Sylow
176#3077—1971nomaximal, Sylow
177#533—1971no—
178C2×C4×D4—1971no—
179#478—1971no—
180#478—1971no—
181#478—1971no—
182#478D19711yescharacteristic, complemented
183#476—1971no—
184#476—1971no—
185#474—1971no—
186#474—1971no—
187#428—1971no—
188#428—1971no—
189#423—1971no—
190#418—1971no—
191#418—1971no—
192C22×D4—1971no—
193C23×C4—3941no—
194C2×M16—1971no—
195C2×M16—1971no—
196C4⋊Q8—1971no—
197C4⋊Q8—1971no—
198#125—1971no—
199#125C2×D19711yescharacteristic, complemented
200C22⋊Q8—3941no—
201C22⋊Q8—3941no—
202C4×Q8—3941no—
203C4×Q8—3941no—
204C4×D4—3941no—
205C4×D4—3941no—
206C4×D4—1971no—
207C4×D4—1971no—
208C4×D4—1971no—
209C4×D4C2×D19711yescharacteristic
210#117—1971no—
211#117—1971no—
212C2×C4⋊C4—1971no—
213C2×C22⋊C4—3941no—
214C2×C42—1971no—
215#107—1971no—
216#107—1971no—
217#107—1971no—
218#107C2×D19711yescharacteristic, complemented
219#106—1971no—
220#106—1971no—
221#106—1971no—
222#106C2×D19711yescharacteristic, complemented
223C4⋊C8—1971no—
224C4⋊C8—1971no—
225C4⋊C8—1971no—
226C4⋊C8C2×D19711yescharacteristic, complemented
227#102—1971no—
228#102—1971no—
229#102—1971no—
230#102—1971no—
231#102—1971no—
232#102—1971no—
233#102C2×D19711yescharacteristic
234#102C2×D19711yescharacteristic
235C24—3941no—
236C2×Q8—3941no—
237C2×Q8—3941no—
238C2×D4—3942no—
239C2×D4—3941no—
240C2×D4—1974no—
241C2×D4—1971no—
242C2×D4C22×D19711yescharacteristic
243C22×C4—21no—
244C22×C4—7884no—
245C22×C4—3941no—
246C22×C4—3941no—
247C22×C4—3941no—
248C22×C4—3941no—
249C22×C4—1971no—
250C22×C4—1971no—
251C22×C4—1971no—
252M16—3942no—
253M16—3942no—
254C2×C8—1971no—
255C2×C8—1971no—
256C2×C8C22×D19711yescharacteristic
257C2×C8C22×D19711yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—3941no—
261C4⋊C4—3941no—
262C4⋊C4—3941no—
263C4⋊C4—3941no—
264C4⋊C4—3941no—
265C4⋊C4—1971no—
266C4⋊C4—1971no—
267C4⋊C4—1971no—
268C4⋊C4C22×D19711yescharacteristic
269C4⋊C4C22×D19711yescharacteristic
270C4⋊C4C22×D19711yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—3941no—
273C22⋊C4—3941no—
274C22⋊C4—3941no—
275C22⋊C4—3941no—
276C22⋊C4—3941no—
277C42—3941no—
278C42—3941no—
279C42—1971no—
280C42—1971no—
281C42—1971no—
282C42C22×D19711yescharacteristic
283C23—21no—
284C23—7882no—
285C23—7882no—
286C23—3944no—
287C23—3941no—
288C23—1971no—
289Q8—7882no—
290Q8—7882no—
291D4—21no—
292D4—3944no—
293D4—3941no—
294D4—1972no—
295D4Q8×D19712yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—7882no—
301C2×C4—7881no—
302C2×C4—7882no—
303C2×C4—7882no—
304C2×C4—7882no—
305C2×C4—3942no—
306C2×C4—3941no—
307C2×C4—3941no—
308C2×C4—3941no—
309C2×C4—1971no—
310C2×C4—1971no—
311C2×C4—1971no—
312C2×C4—1972no—
313C2×C4—1972no—
314C2×C4D4×D19711yescharacteristic
315C2×C4D4×D19711yescharacteristic
316C2×C4C23×D19711yescharacteristic
317C8—21no—
318C8—21no—
319C8—3941no—
320C8—3941no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—7888no—
325C22—7881no—
326C22—7881no—
327C22—3942no—
328C22—3941no—
329C22—3941no—
330C22—1972no—
331C22C2×D4×D19711yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—7881no—
338C4—7881no—
339C4—7881no—
340C4—3941no—
341C4—1972no—
342C4—1972no—
343C4C2×Q8×D19711yescharacteristic
344C4#41×D19711yescharacteristic
345C2—42no—
346C2—7882no—
347C2—3942no—
348C2C22⋊Q8×D19711yescharacteristic
349C2AGL(1,ℤ/8)×D19711yescharacteristic
350C2#2716187611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#478×C197C2
socleC22×C197C22×D4
prime radicalC2×D4×D197C22
hypercenter#478D197
hypocenterC197#3077
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #27161635C2semidirect
3. #27161380C2semidirect
4. #27161379C2semidirect
5. #27161376C2semidirect
6. #27161368C2semidirect
7. #27161367C2semidirect
8. #27161358C2semidirect
9. #27161357C2semidirect
10. #27161120C2semidirect
11. #27161118C2semidirect
12. #27161101C2semidirect
15. #478×C197C2semidirect
20. #20263325C22semidirect
21. #20263325C22semidirect
22. #20263318C22semidirect
37. #20263260C22semidirect
38. #20263259C22semidirect
39. #20263259C22semidirect
40. #20263258C22semidirect
41. #20263257C22semidirect
42. #20263257C22semidirect
43. C4⋊#13096802C22semidirect
44. #20263255C22semidirect
45. #20263255C22semidirect
52. #125×C197C22semidirect
54. #107×C197C22semidirect
55. #106×C197C22semidirect
56. C4⋊C8×C197C22semidirect
65. D4×D197Q8semidirect
175. C197#3077semidirect
182. #478D197semidirect
199. #125C2×D197semidirect
218. #107C2×D197semidirect
222. #106C2×D197semidirect
226. C4⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616352
C4×D4×D1972
#271613802
#271613792
#271613762
#271613682
#271613672
#271613582
#271613572
#271611202
#271611182
#271611012
#271610822
#478×C1972
#3077197