Finite Group Databasebeta

← #38753196

Subgroups — #38753196

Order 25216.

596 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263353C211yesmaximal, characteristic, complemented
2C2×#20263350C211yesmaximal, characteristic, complemented
3C2×#20263347C211yesmaximal, characteristic, complemented
4C2×D8×D197C211yesmaximal, characteristic, complemented
5#27161647C211yesmaximal, characteristic, complemented
6#27161642C211yesmaximal, characteristic, complemented
7#27161468C211yesmaximal, characteristic, complemented
8#27161467C211yesmaximal, characteristic, complemented
9#27161465C211yesmaximal, characteristic, complemented
10#27161461C211yesmaximal, characteristic, complemented
11#27161460C211yesmaximal, characteristic, complemented
12#27161455C211yesmaximal, characteristic, complemented
13#27161454C211yesmaximal, characteristic, complemented
14#27161036C211yesmaximal, characteristic, complemented
15#496×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×#15691357C2211yescharacteristic
18C2×D4×D197C2211yescharacteristic
19C2×D4×D197C2211yescharacteristic
20#20263353—24no—
21#20263350—24no—
22#20263347—24no—
23C2×#15691348C2211yescharacteristic, complemented
24C2×#15691347C2211yescharacteristic, complemented
25C2×#15691347C2211yescharacteristic, complemented
26C2×#15691346C2211yescharacteristic, complemented
27C2×#15691345C2211yescharacteristic, complemented
28C2×D1576C2211yescharacteristic
29C2×D1576C2211yescharacteristic, complemented
30C2×#15691343C2211yescharacteristic
31C2×#15691343C2211yescharacteristic
32C2×#15691343C2211yescharacteristic, complemented
33C2×#15691343C2211yescharacteristic, complemented
34C2×#15691343C2211yescharacteristic, complemented
35D8×D197—24no—
36C2×#15691341C2211yescharacteristic
37C2×C8×D197C2211yescharacteristic
38#20263323C2211yescharacteristic, complemented
39#20263322C2211yescharacteristic, complemented
40#20263322C2211yescharacteristic, complemented
41#20263315C2211yescharacteristic, complemented
42#20263314C2211yescharacteristic, complemented
43#20263313C2211yescharacteristic, complemented
44#20263299—22no—
45#20263298—22no—
46#20263296—22no—
47#20263294—22no—
48#20263283C2211yescharacteristic
49#20263236C2211yescharacteristic, complemented
50C4×#13096802C2211yescharacteristic, complemented
51C8×Dic197C2211yescharacteristic, complemented
52C2×SD16×C197C2211yescharacteristic, complemented
53C2×SD16×C197C2211yescharacteristic, complemented
54C2×D8×C197C2211yescharacteristic
55C2×D8×C197C2211yescharacteristic, complemented
56#127×C197C2211yescharacteristic, complemented
57#124×C197C2211yescharacteristic, complemented
58#97×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C23×D197—21no—
61#15691359—22no—
62#15691359—22no—
63#15691357—22no—
64#15691357—22no—
65C2×Dic394C2311yescharacteristic
66C2×D788—21no—
67C2×D788—22no—
68C2×D788C2311yescharacteristic
69C2×D788C2311yescharacteristic
70C2×D788C2311yescharacteristic
71C2×C22⋊D197—21no—
72C2×C22⋊D197—21no—
73C2×C22⋊D197—21no—
74C2×C22⋊D197—22no—
75C2×C22⋊D197—22no—
76D4×D197—22no—
77D4×D197—22no—
78D4×D197—22no—
79D4×D197—22no—
80C22×Dic197—21no—
81C2×C4×D197—21no—
82C2×C4×D197C2311yescharacteristic
83#15691348—22no—
84#15691347—22no—
85#15691347—22no—
86#15691346—22no—
87#15691345—22no—
88D1576—22no—
89D1576—22no—
90#15691343—22no—
91#15691343—22no—
92#15691343—22no—
93#15691343—22no—
94#15691343—22no—
95#15691341D414yescomplemented
96C2×#13096802C2311yescharacteristic
97C2×#13096802C2311yescharacteristic
98C8×D197D414yescomplemented
99#15691337D412yes—
100C22⋊Dic197—22no—
101#15691335—22no—
102#15691335—22no—
103#15691335D412yes—
104C4×Dic197C2311yescharacteristic
105C2×Q8×C197C2311yescharacteristic
106C2×D4×C197—22no—
107C2×D4×C197C2311yescharacteristic
108C2×D4×C197C2311yescharacteristic
109C2×D4×C197C2311yescharacteristic
110SD16×C197—22no—
111SD16×C197—22no—
112D8×C197—22no—
113D8×C197—22no—
114C2×C8×C197C2311yescharacteristic
115C2×C8×C197C2311yescharacteristic
116C22⋊C4×C197—22no—
117C42×C197C2311yescharacteristic
118C22×D197—21no—
119C22×D197—21no—
120C22×D197—21no—
121C22×D197—42no—
122C22×D197—42no—
123C22×D197—82no—
124C22×D197—82no—
125C22×D197C2×D411yescharacteristic
126Dic394—21no—
127Dic394—21no—
128D788—21no—
129D788—21no—
130D788—21no—
131D788—21no—
132D788—21no—
133D788—21no—
134D788—42no—
135D788—82no—
136C22⋊D197—42no—
137C22⋊D197—42no—
138C22⋊D197—42no—
139C22⋊D197—82no—
140C22⋊D197—82no—
141C2×Dic197—21no—
142C2×Dic197—42no—
143C2×Dic197C2×D412yes—
144C2×Dic197C2×D411yescharacteristic
145C4×D197—42no—
146C4×D197C2×D412yes—
147C4×D197C2×D412yes—
148#13096802C2×D412yes—
149#13096802C2×D412yes—
150C23×C197—21no—
151C23×C197—21no—
152C23×C197—21no—
153Q8×C197—21no—
154Q8×C197—21no—
155D4×C197—21no—
156D4×C197—21no—
157D4×C197—21no—
158D4×C197—21no—
159D4×C197—21no—
160D4×C197—21no—
161D4×C197—82no—
162C2×C4×C197—21no—
163C2×C4×C197C2×D412yes—
164C2×C4×C197C2411yescharacteristic
165C8×C197C2×D412yes—
166C8×C197C2×D412yes—
167C22×C197—41no—
168C22×C197—41no—
169C22×C197—41no—
170C22×C197—42no—
171C22×C197—42no—
172C22×C197—42no—
173C22×C197C22×D411yescharacteristic
174C2×D197—22no—
175C2×D197—41no—
176C2×D197—41no—
177C2×D197—41no—
178C2×D197—42no—
179C2×D197—42no—
180C2×D197—42no—
181C2×D197—84no—
182C2×D197—84no—
183C2×D197#12712yes—
184C4×C197—22no—
185C4×C197—41no—
186C4×C197C22×D411yescharacteristic
187C4×C197C22×D411yescharacteristic
188Dic197—22no—
189Dic197—41no—
190Dic197#12712yes—
191C2×C197—81no—
192C2×C197—81no—
193C2×C197—81no—
194C2×C197C2×#12711yescharacteristic
195C2×C197#57612yes—
196D197—22no—
197D197—81no—
198D197—81no—
199D197—81no—
200C197#315511yescharacteristic, complemented, Sylow
201#3155—1971nomaximal, Sylow
202C2×AGL(1,ℤ/8)—1971no—
203C2×AGL(1,ℤ/8)—1971no—
204C2×#135—1971no—
205C22×D8—1971no—
206#535—1971no—
207#535—1971no—
208#496—1971no—
209#496—1971no—
210#496—1971no—
211#496D19711yescharacteristic, complemented
212#495—1971no—
213#495—1971no—
214#493—1971no—
215#493—1971no—
216#405—1971no—
217C2×#41—1971no—
218C2×#41—1971no—
219C22×D4—1971no—
220C22×D4—1971no—
221AGL(1,ℤ/8)—3944no—
222AGL(1,ℤ/8)—3944no—
223#135—3944no—
224C2×Q16—1971no—
225C2×SD16—1971no—
226C2×SD16—1971no—
227C2×SD16—1971no—
228C2×SD16—1971no—
229C2×SD16C2×D19711yescharacteristic, complemented
230C2×SD16C2×D19711yescharacteristic, complemented
231C2×D8—3944no—
232C2×D8—1971no—
233C2×D8—1971no—
234C2×D8—1971no—
235C2×D8—1971no—
236C2×D8—1971no—
237C2×D8—1971no—
238C2×D8—1971no—
239C2×D8C2×D19711yescharacteristic
240C2×D8C2×D19711yescharacteristic, complemented
241C2×M16—1971no—
242C22×C8—1971no—
243#127—1971no—
244#127—1971no—
245#127—1971no—
246#127C2×D19711yescharacteristic, complemented
247#124—1971no—
248#124—1971no—
249#124—1971no—
250#124C2×D19711yescharacteristic, complemented
251#121—3942no—
252#121—3942no—
253#120—3942no—
254#120—3942no—
255#117—1971no—
256#97—1971no—
257#97C2×D19711yescharacteristic, complemented
258C4×C8—1971no—
259C4×C8—1971no—
260C24—3941no—
261C24—3941no—
262#41—3942no—
263#41—3942no—
264#41—3942no—
265#41—3942no—
266C2×Q8—1971no—
267C2×Q8C22×D19711yescharacteristic
268C2×D4—22no—
269C2×D4—3941no—
270C2×D4—3941no—
271C2×D4—3942no—
272C2×D4—3942no—
273C2×D4—3942no—
274C2×D4—3942no—
275C2×D4—3941no—
276C2×D4—3941no—
277C2×D4—3942no—
278C2×D4—3942no—
279C2×D4—3942no—
280C2×D4—1971no—
281C2×D4—1971no—
282C2×D4—1971no—
283C2×D4C22×D19711yescharacteristic
284C2×D4C22×D19711yescharacteristic
285C2×D4C22×D19711yescharacteristic
286C22×C4—3941no—
287C22×C4—3941no—
288C22×C4—1971no—
289Q16—3942no—
290SD16—22no—
291SD16—22no—
292SD16—3942no—
293SD16—3942no—
294SD16—3942no—
295SD16—3942no—
296D8—22no—
297D8—22no—
298D8—3942no—
299D8—3942no—
300D8—3942no—
301D8—3942no—
302D8—3942no—
303D8—3942no—
304D8—3942no—
305M16—1974no—
306C2×C8—1971no—
307C2×C8—1971no—
308C2×C8—1974no—
309C2×C8C22×D19711yescharacteristic
310C2×C8C22×D19711yescharacteristic
311C4⋊C4—1972no—
312C22⋊C4—22no—
313C22⋊C4—3942no—
314C22⋊C4—3942no—
315C22⋊C4—3942no—
316C22⋊C4—1972no—
317C42—1971no—
318C42C22×D19711yescharacteristic
319C23—21no—
320C23—21no—
321C23—21no—
322C23—7882no—
323C23—7882no—
324C23—15762no—
325C23—15762no—
326C23—3941no—
327C23—3941no—
328C23—3941no—
329C23—1971no—
330Q8—21no—
331Q8—21no—
332Q8—3941no—
333Q8—3941no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—82no—
341D4—7882no—
342D4—7882no—
343D4—7882no—
344D4—7882no—
345D4—15762no—
346D4—15762no—
347D4—15762no—
348D4—3941no—
349D4—3941no—
350D4—3941no—
351D4—3941no—
352D4—3941no—
353D4—3941no—
354C2×C4—21no—
355C2×C4—7882no—
356C2×C4—7882no—
357C2×C4—3941no—
358C2×C4—1972no—
359C2×C4—1971no—
360C2×C4—1972no—
361C2×C4—1972no—
362C2×C4D4×D19712yes—
363C2×C4C23×D19711yescharacteristic
364C8—1972no—
365C8—1972no—
366C8D4×D19712yes—
367C8D4×D19712yes—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—7881no—
375C22—7881no—
376C22—7881no—
377C22—7882no—
378C22—7882no—
379C22—7882no—
380C22—15764no—
381C22—15764no—
382C22—3942no—
383C22—1972no—
384C22C2×D4×D19711yescharacteristic
385C4—22no—
386C4—41no—
387C4—7881no—
388C4—3942no—
389C4—1972no—
390C4C2×D4×D19711yescharacteristic
391C4C2×D4×D19711yescharacteristic
392C2—81no—
393C2—81no—
394C2—81no—
395C2—15761no—
396C2—15761no—
397C2—15761no—
398C2—3942no—
399C2#127×D19711yescharacteristic
400C2#2716188312yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#496×C197C2
socleC22×C197C22×D4
hypercenter#496D197
hypocenterC197#3155
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#20263353C2semidirect
2. C2×#20263350C2semidirect
3. C2×#20263347C2semidirect
4. C2×D8×D197C2semidirect
5. #27161647C2semidirect
6. #27161642C2semidirect
7. #27161468C2semidirect
8. #27161467C2semidirect
9. #27161465C2semidirect
10. #27161461C2semidirect
11. #27161460C2semidirect
12. #27161455C2semidirect
13. #27161454C2semidirect
14. #27161036C2semidirect
15. #496×C197C2semidirect
23. C2×#15691348C22semidirect
24. C2×#15691347C22semidirect
25. C2×#15691347C22semidirect
26. C2×#15691346C22semidirect
27. C2×#15691345C22semidirect
29. C2×D1576C22semidirect
32. C2×#15691343C22semidirect
33. C2×#15691343C22semidirect
34. C2×#15691343C22semidirect
38. #20263323C22semidirect
39. #20263322C22semidirect
40. #20263322C22semidirect
41. #20263315C22semidirect
42. #20263314C22semidirect
43. #20263313C22semidirect
49. #20263236C22semidirect
50. C4×#13096802C22semidirect
51. C8×Dic197C22semidirect
52. C2×SD16×C197C22semidirect
53. C2×SD16×C197C22semidirect
55. C2×D8×C197C22semidirect
56. #127×C197C22semidirect
57. #124×C197C22semidirect
58. #97×C197C22semidirect
95. #15691341D4semidirect
98. C8×D197D4semidirect
200. C197#3155semidirect
211. #496D197semidirect
229. C2×SD16C2×D197semidirect
230. C2×SD16C2×D197semidirect
240. C2×D8C2×D197semidirect
246. #127C2×D197semidirect
250. #124C2×D197semidirect
257. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633532
C2×#202633502
C2×#202633472
C2×D8×D1972
#271616472
#271616422
#271614682
#271614672
#271614652
#271614612
#271614602
#271614552
#271614542
#271610362
#496×C1972
#3155197