Finite Group Databasebeta

← #38754727

Subgroups — #38754727

Order 25216.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D788C211yesmaximal, characteristic
2#27161445C211yesmaximal, characteristic, complemented
3#27161403C211yesmaximal, characteristic, complemented
4#27161392C211yesmaximal, characteristic, complemented
5#27161376C211yesmaximal, characteristic, complemented
6#27161327C211yesmaximal, characteristic, complemented
7#27161310C211yesmaximal, characteristic, complemented
8#27161279C211yesmaximal, characteristic, complemented
9#27161229C211yesmaximal, characteristic, complemented
10#27161229C211yesmaximal, characteristic, complemented
11#27161177C211yesmaximal, characteristic, complemented
12#27161151C211yesmaximal, characteristic, complemented
13#27161083C211yesmaximal, characteristic, complemented
14C2×#20263246C211yesmaximal, characteristic, complemented
15#552×C197C211yesmaximal, characteristic, complemented
16C22×D788C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C2×D4×D197—21no—
19C2×#15691345C2211yescharacteristic
20C2×#15691345C2211yescharacteristic, complemented
21C2×#15691345C2211yescharacteristic, complemented
22C2×#15691342C2211yescharacteristic, complemented
23C22×#13096802C2211yescharacteristic, complemented
24#20263323C2211yescharacteristic
25#20263301—21no—
26#20263300C2211yescharacteristic
27#20263300C2211yescharacteristic
28#20263296—21no—
29#20263296—21no—
30C4×D788C2211yescharacteristic
31#20263259C2211yescharacteristic, complemented
32#20263259C2211yescharacteristic, complemented
33#20263257C2211yescharacteristic, complemented
34C4⋊#13096802C2211yescharacteristic, complemented
35#20263250C2211yescharacteristic, complemented
36#20263246C2211yescharacteristic
37#20263246C2211yescharacteristic
38#20263246C2211yescharacteristic
39#20263246C2211yescharacteristic, complemented
40#20263246C2211yescharacteristic, complemented
41#20263246C2211yescharacteristic, complemented
42#20263246C2211yescharacteristic, complemented
43#20263246C2211yescharacteristic, complemented
44#20263246C2211yescharacteristic, complemented
45C22⋊#13096802C2211yescharacteristic, complemented
46C22⋊#13096802C2211yescharacteristic, complemented
47#20263237C2211yescharacteristic, complemented
48#126×C197—21no—
49#125×C197—21no—
50#125×C197C2211yescharacteristic, complemented
51#123×C197—21no—
52C22⋊Q8×C197C2211yescharacteristic, complemented
53C22⋊Q8×C197C2211yescharacteristic, complemented
54C4×Q8×C197C2211yescharacteristic, complemented
55C4×D4×C197C2211yescharacteristic
56#117×C197—21no—
57#117×C197C2211yescharacteristic, complemented
58C2×C4⋊C4×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C23×D197—21no—
61C2×D788—22no—
62C2×D788—22no—
63C2×D788—21no—
64C2×D788—21no—
65C2×D788—21no—
66C2×D788—21no—
67C2×D788—41no—
68C2×D788—41no—
69C2×D788C2311yescharacteristic
70C2×D788C2311yescharacteristic
71C2×D788C2311yescharacteristic
72C2×C22⋊D197—21no—
73C2×C22⋊D197—21no—
74D4×D197—44no—
75C2×C4×D197—21no—
76#15691345—22no—
77#15691345—41no—
78#15691345—41no—
79#15691342—41no—
80C2×#13096802—41no—
81C2×#13096802C2311yescharacteristic
82C2×#13096802C2311yescharacteristic
83C2×#13096802C2311yescharacteristic
84C2×#13096802C2311yescharacteristic
85C4⋊Dic197C2311yescharacteristic
86#15691335—21no—
87#15691335—21no—
88#15691335—21no—
89C2×Q8×C197C2311yescharacteristic
90C2×D4×C197D411yescharacteristic
91C22×C4×C197—21no—
92C22×C4×C197C2311yescharacteristic
93C22×C4×C197C2311yescharacteristic
94C4⋊C4×C197—21no—
95C4⋊C4×C197—21no—
96C4⋊C4×C197—21no—
97C4⋊C4×C197—21no—
98C4⋊C4×C197—21no—
99C4⋊C4×C197—21no—
100C4⋊C4×C197D411yescharacteristic
101C4⋊C4×C197C2311yescharacteristic
102C4⋊C4×C197C2311yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C22⋊C4×C197—21no—
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C22⋊C4×C197D411yescharacteristic
108C22⋊C4×C197D411yescharacteristic
109C42×C197—21no—
110C42×C197—21no—
111C42×C197C2311yescharacteristic
112C22×D197—21no—
113C22×D197—21no—
114C22×D197—21no—
115C22×D197—42no—
116C22×D197—42no—
117C22×D197—42no—
118C22×D197—42no—
119C22×D197—81no—
120C22×D197—81no—
121D788—21no—
122D788—21no—
123D788—21no—
124D788—21no—
125D788—21no—
126D788—42no—
127D788—42no—
128D788—42no—
129D788—81no—
130D788#4112yes—
131C22⋊D197—42no—
132C22⋊D197—42no—
133C2×Dic197—21no—
134C4×D197—42no—
135#13096802—21no—
136#13096802—21no—
137#13096802—21no—
138#13096802—21no—
139C23×C197C2×D411yescharacteristic
140C23×C197C2×D411yescharacteristic
141Q8×C197—21no—
142Q8×C197—21no—
143D4×C197—22no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—42no—
153C2×C4×C197C2×D411yescharacteristic
154C2×C4×C197C2×D411yescharacteristic
155C2×C4×C197C2×D411yescharacteristic
156C2×C4×C197C2×D411yescharacteristic
157C2×C4×C197C2411yescharacteristic
158C22×C197—21no—
159C22×C197—21no—
160C22×C197—21no—
161C22×C197—21no—
162C22×C197—21no—
163C22×C197AGL(1,ℤ/8)12yes—
164C22×C197C22×D411yescharacteristic
165C2×D197—22no—
166C2×D197—41no—
167C2×D197—41no—
168C2×D197—41no—
169C2×D197—41no—
170C2×D197—41no—
171C2×D197—41no—
172C2×D197—41no—
173C2×D197—41no—
174C2×D197—84no—
175C2×D197—84no—
176C4×C197—21no—
177C4×C197—21no—
178C4×C197—21no—
179C4×C197—21no—
180C4×C197—41no—
181C4×C197—41no—
182C4×C197—41no—
183C4×C197—41no—
184C4×C197C2×#4111yescharacteristic
185C4×C197H(5,𝔽2)11yescharacteristic
186Dic197—41no—
187C2×C197—22no—
188C2×C197—41no—
189C2×C197#54611yescharacteristic
190C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C197#57711yescharacteristic
192D197—42no—
193D197—81no—
194D197—81no—
195C197#331011yescharacteristic, complemented, Sylow
196#3310—1971nomaximal, Sylow
197#552D19711yescharacteristic, complemented
198D42—1971no—
199#490—1971no—
200#482—1971no—
201#480—1971no—
202#478—1971no—
203#468—1971no—
204#465—1971no—
205#460—1971no—
206#450—1971no—
207#450—1971no—
208#440—1971no—
209#435—1971no—
210#418—1971no—
211C2×#102—1971no—
212C22×D4—3941no—
213C22×D4—1971no—
214C22×D4—1971no—
215C2×SD16—1971no—
216C2×SD16—1971no—
217C2×SD16—1971no—
218C2×M16—1971no—
219C22×C8—1971no—
220#127—1971no—
221#126—21no—
222#125—21no—
223#125C2×D19711yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D19711yescharacteristic, complemented
226C22⋊Q8C2×D19711yescharacteristic, complemented
227#121—3941no—
228#121—1971no—
229#121—1971no—
230#120—3941no—
231#120—3941no—
232C4×Q8C2×D19711yescharacteristic, complemented
233C4×D4—1971no—
234C4×D4C2×D19711yescharacteristic
235#117—21no—
236#117C2×D19711yescharacteristic, complemented
237C2×C4⋊C4C2×D19711yescharacteristic, complemented
238#107—1971no—
239#107—1971no—
240#106—1971no—
241C4⋊C8—1971no—
242#103—1971no—
243#102—1971no—
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248#102—1971no—
249#102—1971no—
250#102—1971no—
251#102—1971no—
252C22⋊C8—1971no—
253C22⋊C8—1971no—
254#97—1971no—
255C24—3941no—
256C24—3941no—
257C2×Q8C22×D19711yescharacteristic
258C2×D4—7881no—
259C2×D4—7881no—
260C2×D4—7884no—
261C2×D4—3942no—
262C2×D4—3942no—
263C2×D4—3941no—
264C2×D4—3941no—
265C2×D4—3941no—
266C2×D4—3941no—
267C2×D4—3941no—
268C2×D4—3941no—
269C2×D4—1971no—
270C2×D4—1971no—
271C2×D4—1971no—
272C2×D4C22⋊D19711yescharacteristic
273C22×C4—21no—
274C22×C4—3941no—
275C22×C4C22×D19711yescharacteristic
276C22×C4C22×D19711yescharacteristic
277SD16—7881no—
278SD16—7881no—
279SD16—3942no—
280M16—7881no—
281C2×C8—7881no—
282C2×C8—1971no—
283C2×C8—1971no—
284C2×C8—1971no—
285C2×C8—1971no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—1971no—
293C4⋊C4C22⋊D19711yescharacteristic
294C4⋊C4C22×D19711yescharacteristic
295C4⋊C4C22×D19711yescharacteristic
296C4⋊C4C22×D19711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3941no—
301C22⋊C4—3941no—
302C22⋊C4—3941no—
303C22⋊C4C22⋊D19711yescharacteristic
304C22⋊C4C22⋊D19711yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D19711yescharacteristic
308C23—7882no—
309C23—7882no—
310C23—7882no—
311C23—7882no—
312C23—15761no—
313C23—15761no—
314C23—3941no—
315C23—3941no—
316C23—3941no—
317C23C2×C22⋊D19711yescharacteristic
318C23C2×C22⋊D19711yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—7882no—
323D4—7882no—
324D4—7882no—
325D4—7882no—
326D4—7882no—
327D4—15761no—
328D4—3941no—
329D4—3941no—
330D4—3941no—
331D4—3941no—
332D4—3941no—
333D4—1972no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—7882no—
344C2×C4—3941no—
345C2×C4C2×C22⋊D19711yescharacteristic
346C2×C4C2×C22⋊D19711yescharacteristic
347C2×C4C2×C22⋊D19711yescharacteristic
348C2×C4C2×C22⋊D19711yescharacteristic
349C2×C4C23×D19711yescharacteristic
350C8—3941no—
351C8—3941no—
352C8—3941no—
353C8—3941no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—7881no—
360C22—7881no—
361C22—7881no—
362C22—7881no—
363C22—7881no—
364C22—7881no—
365C22—7881no—
366C22—7881no—
367C22—15764no—
368C22—15764no—
369C22—3942no—
370C22#2026335112yes—
371C22C22×C22⋊D19711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—7881no—
381C4#41×D19711yescharacteristic
382C4#2026337211yescharacteristic
383C2—22no—
384C2—41no—
385C2—7882no—
386C2—15761no—
387C2—15761no—
388C2#2716171711yescharacteristic
389C2C2×#2026335111yescharacteristic
390C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#552×C197C2
socleC22×C197C22×D4
prime radicalD4×D788C2
hypercenter#552D197
hypocenterC197#3310
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #27161445C2semidirect
3. #27161403C2semidirect
4. #27161392C2semidirect
5. #27161376C2semidirect
6. #27161327C2semidirect
7. #27161310C2semidirect
8. #27161279C2semidirect
9. #27161229C2semidirect
10. #27161229C2semidirect
11. #27161177C2semidirect
12. #27161151C2semidirect
13. #27161083C2semidirect
14. C2×#20263246C2semidirect
15. #552×C197C2semidirect
20. C2×#15691345C22semidirect
21. C2×#15691345C22semidirect
22. C2×#15691342C22semidirect
23. C22×#13096802C22semidirect
31. #20263259C22semidirect
32. #20263259C22semidirect
33. #20263257C22semidirect
34. C4⋊#13096802C22semidirect
35. #20263250C22semidirect
39. #20263246C22semidirect
40. #20263246C22semidirect
41. #20263246C22semidirect
42. #20263246C22semidirect
43. #20263246C22semidirect
44. #20263246C22semidirect
45. C22⋊#13096802C22semidirect
46. C22⋊#13096802C22semidirect
47. #20263237C22semidirect
50. #125×C197C22semidirect
52. C22⋊Q8×C197C22semidirect
53. C22⋊Q8×C197C22semidirect
54. C4×Q8×C197C22semidirect
57. #117×C197C22semidirect
58. C2×C4⋊C4×C197C22semidirect
195. C197#3310semidirect
197. #552D197semidirect
223. #125C2×D197semidirect
225. C22⋊Q8C2×D197semidirect
226. C22⋊Q8C2×D197semidirect
232. C4×Q8C2×D197semidirect
236. #117C2×D197semidirect
237. C2×C4⋊C4C2×D197semidirect

Maximal subgroups

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

subgroupindex
D4×D7882
#271614452
#271614032
#271613922
#271613762
#271613272
#271613102
#271612792
#271612292
#271611772
#271611512
#271610832
C2×#202632462
#552×C1972
#3310197