Finite Group Databasebeta

← #38754728

Subgroups — #38754728

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
1#27161763C211yesmaximal, characteristic, complemented
2#27161713C211yesmaximal, characteristic
3#27161442C211yesmaximal, characteristic, complemented
4#27161400C211yesmaximal, characteristic, complemented
5#27161394C211yesmaximal, characteristic, complemented
6#27161382C211yesmaximal, characteristic, complemented
7#27161311C211yesmaximal, characteristic, complemented
8#27161277C211yesmaximal, characteristic, complemented
9#27161229C211yesmaximal, characteristic, complemented
10#27161228C211yesmaximal, characteristic, complemented
11#27161182C211yesmaximal, characteristic, complemented
12#27161148C211yesmaximal, characteristic, complemented
13#27161085C211yesmaximal, characteristic, complemented
14#102×D197C211yesmaximal, characteristic, complemented
15#468×C197C211yesmaximal, characteristic, complemented
16C22×D788C2211yescharacteristic
17C2×D4×D197—21no—
18C2×D4×D197C2211yescharacteristic
19C2×#15691345C2211yescharacteristic
20C2×#15691345C2211yescharacteristic, complemented
21C2×C8×D197C2211yescharacteristic, complemented
22#20263322C2211yescharacteristic
23#20263319—21no—
24#20263317C2211yescharacteristic, complemented
25#20263316—21no—
26#20263308—21no—
27#20263302C2211yescharacteristic, complemented
28#20263301C2211yescharacteristic
29#20263299—21no—
30#20263296—21no—
31#20263294—21no—
32Q8×Dic197C2211yescharacteristic, complemented
33#20263288C2211yescharacteristic
34C4×C22⋊D197C2211yescharacteristic
35#20263285C2211yescharacteristic, complemented
36#20263284—21no—
37C4⋊C4×D197C2211yescharacteristic, complemented
38#20263259C2211yescharacteristic, complemented
39#20263257C2211yescharacteristic, complemented
40#20263255C2211yescharacteristic, complemented
41#20263248C2211yescharacteristic
42#20263248C2211yescharacteristic
43#20263248C2211yescharacteristic, complemented
44#20263248C2211yescharacteristic, complemented
45#20263247C2211yescharacteristic, complemented
46#20263247C2211yescharacteristic, complemented
47#20263246C2211yescharacteristic
48#20263246C2211yescharacteristic, complemented
49C22⋊#13096802C2211yescharacteristic, complemented
50#20263238C2211yescharacteristic, complemented
51#20263236C2211yescharacteristic, complemented
52C2×SD16×C197C2211yescharacteristic, complemented
53C2×M16×C197C2211yescharacteristic, complemented
54C22⋊Q8×C197C2211yescharacteristic, complemented
55#121×C197C2211yescharacteristic
56#107×C197C2211yescharacteristic, complemented
57#103×C197C2211yescharacteristic, complemented
58#102×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C23×D197—21no—
61C2×D788—22no—
62C2×D788—21no—
63C2×D788—21no—
64C2×D788—41no—
65C2×D788C2311yescharacteristic
66C2×D788C2311yescharacteristic
67C2×C22⋊D197—21no—
68C2×C22⋊D197—21no—
69C2×C22⋊D197—21no—
70C2×C22⋊D197D411yescharacteristic
71D4×D197—22no—
72D4×D197—41no—
73D4×D197—44no—
74C2×C4×D197—21no—
75C2×C4×D197—21no—
76C2×C4×D197C2311yescharacteristic
77#15691345—22no—
78#15691345—41no—
79C2×#13096802C2311yescharacteristic
80C2×#13096802C2311yescharacteristic
81C8×D197—41no—
82C4⋊Dic197—21no—
83C4⋊Dic197—21no—
84C4⋊Dic197C2311yescharacteristic
85C4⋊Dic197C2311yescharacteristic
86#15691337—21no—
87#15691337—21no—
88#15691337—21no—
89#15691337D411yescharacteristic
90C22⋊Dic197—21no—
91C22⋊Dic197D411yescharacteristic
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335D411yescharacteristic
96C4×Dic197—21no—
97C4×Dic197—21no—
98C4×Dic197C2311yescharacteristic
99C2×Q8×C197C2311yescharacteristic
100C2×D4×C197—21no—
101C2×D4×C197C2311yescharacteristic
102C22×C4×C197C2311yescharacteristic
103SD16×C197—41no—
104M16×C197—41no—
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C4⋊C4×C197—21no—
108C4⋊C4×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C22⋊C4×C197—21no—
111C22⋊C4×C197—21no—
112C22×D197—21no—
113C22×D197—21no—
114C22×D197—42no—
115C22×D197—42no—
116C22×D197—42no—
117C22×D197—42no—
118C22×D197—81no—
119C22×D197—81no—
120C22×D197C2×D411yescharacteristic
121D788—21no—
122D788—21no—
123D788—21no—
124D788—42no—
125D788—42no—
126D788#4112yes—
127C22⋊D197—22no—
128C22⋊D197—42no—
129C22⋊D197—42no—
130C22⋊D197—81no—
131C2×Dic197—21no—
132C2×Dic197—21no—
133C2×Dic197C2×D411yescharacteristic
134C2×Dic197C2×D411yescharacteristic
135C2×Dic197C2×D411yescharacteristic
136C4×D197—21no—
137C4×D197—21no—
138C4×D197—42no—
139C4×D197—42no—
140#13096802—21no—
141#13096802—21no—
142C23×C197—21no—
143C23×C197C2×D411yescharacteristic
144Q8×C197—21no—
145Q8×C197—21no—
146D4×C197—21no—
147D4×C197—21no—
148D4×C197—42no—
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197—21no—
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×D197—84no—
175C2×D197AGL(1,ℤ/8)12yes—
176C4×C197—21no—
177C4×C197—41no—
178C4×C197—41no—
179C4×C197—41no—
180C4×C197C2×#4111yescharacteristic
181C4×C197H(5,𝔽2)11yescharacteristic
182Dic197—21no—
183Dic197—21no—
184Dic197—21no—
185Dic197—41no—
186Dic197—41no—
187C2×C197—41no—
188C2×C197—81no—
189C2×C197#54611yescharacteristic
190C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C197#57711yescharacteristic
192D197—22no—
193D197—42no—
194D197—81no—
195C197#331011yescharacteristic, complemented, Sylow
196#3310—1971nomaximal, Sylow
197#552—1971no—
198D42—1971no—
199#490—1971no—
200#482—1971no—
201#480—1971no—
202#478—1971no—
203#468D19711yescharacteristic, complemented
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×SD16C2×D19711yescharacteristic, complemented
218C2×M16C2×D19711yescharacteristic, complemented
219C22×C8—1971no—
220#127—1971no—
221#126—3941no—
222#125—3941no—
223#125—1971no—
224#123—3941no—
225C22⋊Q8—1971no—
226C22⋊Q8C2×D19711yescharacteristic, complemented
227#121—3941no—
228#121—1971no—
229#121C2×D19711yescharacteristic
230#120—3941no—
231#120—3941no—
232C4×Q8—1971no—
233C4×D4—1971no—
234C4×D4—1971no—
235#117—3941no—
236#117—1971no—
237C2×C4⋊C4—1971no—
238#107—1971no—
239#107C2×D19711yescharacteristic, complemented
240#106—1971no—
241C4⋊C8—1971no—
242#103C2×D19711yescharacteristic, complemented
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#102C2×D19711yescharacteristic, complemented
252C22⋊C8—1971no—
253C22⋊C8—1971no—
254#97—1971no—
255C24—3941no—
256C24—3941no—
257C2×Q8C22×D19711yescharacteristic
258C2×D4—21no—
259C2×D4—7881no—
260C2×D4—7884no—
261C2×D4—7881no—
262C2×D4—3941no—
263C2×D4—3942no—
264C2×D4—3942no—
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—3941no—
274C22×C4—3941no—
275C22×C4—1971no—
276C22×C4C22×D19711yescharacteristic
277SD16—41no—
278SD16—7881no—
279SD16—3942no—
280M16—41no—
281C2×C8—7881no—
282C2×C8—1971no—
283C2×C8—1971no—
284C2×C8C22×D19711yescharacteristic
285C2×C8C22×D19711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—3941no—
288C4⋊C4—3941no—
289C4⋊C4—3941no—
290C4⋊C4—3941no—
291C4⋊C4—3941no—
292C4⋊C4—1971no—
293C4⋊C4—1971no—
294C4⋊C4—1971no—
295C4⋊C4C22×D19711yescharacteristic
296C4⋊C4C22×D19711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—3941no—
300C22⋊C4—3941no—
301C22⋊C4—3941no—
302C22⋊C4—3941no—
303C22⋊C4—1971no—
304C22⋊C4—1971no—
305C42—3941no—
306C42—3941no—
307C42—1971no—
308C23—21no—
309C23—7882no—
310C23—7882no—
311C23—7882no—
312C23—7882no—
313C23—15761no—
314C23—15761no—
315C23—3941no—
316C23—3941no—
317C23—1971no—
318C23D4×D19711yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—7882no—
325D4—7882no—
326D4—7882no—
327D4—7882no—
328D4—15761no—
329D4—3942no—
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—7882no—
340C2×C4—7882no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4—3941no—
345C2×C4—1971no—
346C2×C4—1971no—
347C2×C4—1971no—
348C2×C4D4×D19711yescharacteristic
349C2×C4C23×D19711yescharacteristic
350C8—21no—
351C8—21no—
352C8—3941no—
353C8—3941no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—7881no—
361C22—7881no—
362C22—7881no—
363C22—7881no—
364C22—7881no—
365C22—15764no—
366C22—15764no—
367C22—3942no—
368C22—3941no—
369C22—3941no—
370C22—1972no—
371C22C2×D4×D19711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—7881no—
377C4—7881no—
378C4—3941no—
379C4—3941no—
380C4—3941no—
381C4#41×D19711yescharacteristic
382C4#2026337111yescharacteristic
383C2—41no—
384C2—81no—
385C2—7882no—
386C2—15761no—
387C2—3942no—
388C2#2716171811yescharacteristic
389C2AGL(1,ℤ/8)×D19711yescharacteristic
390C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#468×C197C2
socleC22×C197C22×D4
prime radical#27161713C2
hypercenter#468D197
hypocenterC197#3310
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #27161763C2semidirect
3. #27161442C2semidirect
4. #27161400C2semidirect
5. #27161394C2semidirect
6. #27161382C2semidirect
7. #27161311C2semidirect
8. #27161277C2semidirect
9. #27161229C2semidirect
10. #27161228C2semidirect
11. #27161182C2semidirect
12. #27161148C2semidirect
13. #27161085C2semidirect
14. #102×D197C2semidirect
15. #468×C197C2semidirect
20. C2×#15691345C22semidirect
21. C2×C8×D197C22semidirect
24. #20263317C22semidirect
27. #20263302C22semidirect
32. Q8×Dic197C22semidirect
35. #20263285C22semidirect
37. C4⋊C4×D197C22semidirect
38. #20263259C22semidirect
39. #20263257C22semidirect
40. #20263255C22semidirect
43. #20263248C22semidirect
44. #20263248C22semidirect
45. #20263247C22semidirect
46. #20263247C22semidirect
48. #20263246C22semidirect
49. C22⋊#13096802C22semidirect
50. #20263238C22semidirect
51. #20263236C22semidirect
52. C2×SD16×C197C22semidirect
53. C2×M16×C197C22semidirect
54. C22⋊Q8×C197C22semidirect
56. #107×C197C22semidirect
57. #103×C197C22semidirect
58. #102×C197C22semidirect
195. C197#3310semidirect
203. #468D197semidirect
217. C2×SD16C2×D197semidirect
218. C2×M16C2×D197semidirect
226. C22⋊Q8C2×D197semidirect
239. #107C2×D197semidirect
242. #103C2×D197semidirect
251. #102C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617632
#271617132
#271614422
#271614002
#271613942
#271613822
#271613112
#271612772
#271612292
#271612282
#271611822
#271611482
#271610852
#102×D1972
#468×C1972
#3310197