Finite Group Databasebeta

← #38754257

Subgroups — #38754257

Order 25216.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263354C211yesmaximal, characteristic, complemented
2C2×#20263348C211yesmaximal, characteristic, complemented
3#27161750C211yesmaximal, characteristic, complemented
4#27161725C211yesmaximal, characteristic, complemented
5#27161468C211yesmaximal, characteristic, complemented
6#27161325C211yesmaximal, characteristic, complemented
7#27161315C211yesmaximal, characteristic, complemented
8#27161302C211yesmaximal, characteristic, complemented
9#27161275C211yesmaximal, characteristic, complemented
10#27161243C211yesmaximal, characteristic, complemented
11#27161226C211yesmaximal, characteristic, complemented
12#27161226C211yesmaximal, characteristic, complemented
13#27161215C211yesmaximal, characteristic, complemented
14#27161181C211yesmaximal, characteristic, complemented
15#435×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C2×#15691358C2211yescharacteristic
18C2×#15691358C2211yescharacteristic, complemented
19C22×D788C2211yescharacteristic
20#20263354—24no—
21#20263348—24no—
22C2×Dic788C2211yescharacteristic, complemented
23C2×#15691347C2211yescharacteristic
24C2×#15691347C2211yescharacteristic
25C2×#15691347C2211yescharacteristic, complemented
26C2×#15691347C2211yescharacteristic, complemented
27C2×D1576C2211yescharacteristic
28C2×D1576C2211yescharacteristic
29C2×D1576C2211yescharacteristic
30C2×D1576C2211yescharacteristic
31#20263323C2211yescharacteristic, complemented
32#20263322—21no—
33#20263314C2211yescharacteristic
34#20263311—21no—
35#20263305C2211yescharacteristic
36#20263300C2211yescharacteristic
37#20263300C2211yescharacteristic
38#20263300C2211yescharacteristic, complemented
39#20263299—21no—
40#20263299—21no—
41#20263299—21no—
42#20263294—21no—
43C4×Dic394C2211yescharacteristic, complemented
44C4×D788C2211yescharacteristic
45#20263288—21no—
46C22⋊C4×D197—21no—
47#20263260C2211yescharacteristic, complemented
48#20263251C2211yescharacteristic, complemented
49#20263251C2211yescharacteristic, complemented
50#20263248C2211yescharacteristic
51#20263248C2211yescharacteristic
52#20263248C2211yescharacteristic, complemented
53#20263248C2211yescharacteristic, complemented
54C2×M16×C197C2211yescharacteristic
55C22×C8×C197C2211yescharacteristic, complemented
56C4×D4×C197C2211yescharacteristic
57C4⋊C8×C197C2211yescharacteristic, complemented
58C22⋊C8×C197C2211yescharacteristic
59C22⋊C8×C197C2211yescharacteristic, complemented
60#97×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62#15691358—22no—
63#15691358—22no—
64#15691358—22no—
65#15691358—41no—
66#15691358—41no—
67#15691358—41no—
68C2×Dic394C2311yescharacteristic
69C2×Dic394C2311yescharacteristic
70C2×D788—22no—
71C2×D788—21no—
72C2×D788—21no—
73C2×D788—21no—
74C2×D788—41no—
75C2×D788C2311yescharacteristic
76C2×D788C2311yescharacteristic
77C2×D788C2311yescharacteristic
78C2×D788C2311yescharacteristic
79C2×C22⋊D197—21no—
80C2×C22⋊D197—21no—
81C2×C22⋊D197—21no—
82C2×C22⋊D197—21no—
83C2×C22⋊D197—21no—
84C2×C22⋊D197—21no—
85C2×C4×D197—21no—
86C2×C4×D197—21no—
87C2×C4×D197—21no—
88C2×C4×D197—21no—
89Dic788—22no—
90#15691347—22no—
91#15691347—22no—
92#15691347—41no—
93#15691347D414yescomplemented
94D1576—22no—
95D1576—22no—
96D1576—22no—
97D1576—41no—
98C4⋊Dic197C2311yescharacteristic
99C4⋊Dic197C2311yescharacteristic
100#15691337—21no—
101#15691337—21no—
102C22⋊Dic197—21no—
103#15691335—21no—
104#15691335—21no—
105#15691335—21no—
106#15691335—21no—
107#15691335—21no—
108#15691335—21no—
109C4×Dic197—21no—
110C2×D4×C197D411yescharacteristic
111C22×C4×C197C2311yescharacteristic
112C22×C4×C197C2311yescharacteristic
113M16×C197—22no—
114C2×C8×C197—22no—
115C2×C8×C197C2311yescharacteristic
116C2×C8×C197C2311yescharacteristic
117C2×C8×C197C2311yescharacteristic
118C2×C8×C197C2311yescharacteristic
119C4⋊C4×C197D411yescharacteristic
120C22⋊C4×C197D411yescharacteristic
121C22⋊C4×C197D411yescharacteristic
122C42×C197C2311yescharacteristic
123C22×D197—21no—
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—21no—
127C22×D197—42no—
128C22×D197—42no—
129C22×D197—81no—
130Dic394—21no—
131Dic394—21no—
132Dic394—21no—
133Dic394C2×D412yes—
134D788—21no—
135D788—21no—
136D788—21no—
137D788—21no—
138D788—21no—
139D788—21no—
140D788—21no—
141D788—42no—
142D788—42no—
143D788—81no—
144D788C2×D412yes—
145C22⋊D197—42no—
146C22⋊D197—42no—
147C22⋊D197—42no—
148C22⋊D197—81no—
149C22⋊D197—81no—
150C22⋊D197—81no—
151C2×Dic197—21no—
152C2×Dic197—21no—
153C2×Dic197—21no—
154C2×Dic197—21no—
155C4×D197—42no—
156C4×D197—42no—
157C4×D197—42no—
158C4×D197—81no—
159C23×C197C2×D411yescharacteristic
160C23×C197C2×D411yescharacteristic
161D4×C197—41no—
162C2×C4×C197—21no—
163C2×C4×C197—21no—
164C2×C4×C197—21no—
165C2×C4×C197—21no—
166C2×C4×C197C2×D411yescharacteristic
167C2×C4×C197C2×D411yescharacteristic
168C2×C4×C197C2×D411yescharacteristic
169C2×C4×C197C2×D411yescharacteristic
170C2×C4×C197C2411yescharacteristic
171C8×C197—21no—
172C8×C197—21no—
173C8×C197—21no—
174C8×C197C2×D412yes—
175C22×C197—21no—
176C22×C197—21no—
177C22×C197—21no—
178C22×C197—21no—
179C22×C197—21no—
180C22×C197—21no—
181C22×C197C22×D411yescharacteristic
182C2×D197—22no—
183C2×D197—41no—
184C2×D197—41no—
185C2×D197—41no—
186C2×D197—41no—
187C2×D197—41no—
188C2×D197—41no—
189C2×D197—41no—
190C2×D197—41no—
191C2×D197—41no—
192C2×D197—41no—
193C2×D197—41no—
194C2×D197—84no—
195C4×C197—21no—
196C4×C197—21no—
197C4×C197—21no—
198C4×C197—21no—
199C4×C197C22×D411yescharacteristic
200C4×C197H(5,𝔽2)11yescharacteristic
201Dic197—22no—
202Dic197—41no—
203Dic197—41no—
204Dic197—41no—
205C2×C197—41no—
206C2×C197—41no—
207C2×C197D4211yescharacteristic
208C2×C197#57511yescharacteristic
209C2×C197#57611yescharacteristic
210D197—42no—
211D197—81no—
212D197—81no—
213D197—81no—
214C197#326811yescharacteristic, complemented, Sylow
215#3268—1971nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1971no—
217C2×#135—1971no—
218#550—1971no—
219#546—1971no—
220#496—1971no—
221#468—1971no—
222#466—1971no—
223#464—1971no—
224#459—1971no—
225#452—1971no—
226#449—1971no—
227#449—1971no—
228#447—1971no—
229#440—1971no—
230#435D19711yescharacteristic, complemented
231C2×#41—1971no—
232C2×#41—1971no—
233C2×#41—1971no—
234C22×D4—1971no—
235AGL(1,ℤ/8)—3944no—
236#135—3944no—
237C2×Q16—1971no—
238C2×SD16—1971no—
239C2×SD16—1971no—
240C2×SD16—1971no—
241C2×SD16—1971no—
242C2×D8—1971no—
243C2×D8—1971no—
244C2×D8—1971no—
245C2×D8—1971no—
246C2×M16C2×D19711yescharacteristic
247C22×C8C2×D19711yescharacteristic, complemented
248#127—3941no—
249#127—1971no—
250#124—1971no—
251#123—3941no—
252C22⋊Q8—1971no—
253#121—3941no—
254#121—3941no—
255#121—3941no—
256#121—1971no—
257#121—1971no—
258#121—1971no—
259#120—3941no—
260C4×Q8—1971no—
261C4×D4—3941no—
262C4×D4—1971no—
263C4×D4C2×D19711yescharacteristic
264C2×C22⋊C4—3941no—
265#107—1971no—
266C4⋊C8C2×D19711yescharacteristic, complemented
267#103—1971no—
268#103—1971no—
269#102—1971no—
270#102—1971no—
271#102—1971no—
272#102—1971no—
273C22⋊C8C2×D19711yescharacteristic
274C22⋊C8C2×D19711yescharacteristic, complemented
275#97C2×D19711yescharacteristic, complemented
276C24—3941no—
277#41—7881no—
278#41—7881no—
279#41—7881no—
280#41—3942no—
281#41—3942no—
282#41—3942no—
283C2×Q8—1971no—
284C2×Q8—1971no—
285C2×D4—7881no—
286C2×D4—3942no—
287C2×D4—3941no—
288C2×D4—3941no—
289C2×D4—3941no—
290C2×D4—3941no—
291C2×D4—3941no—
292C2×D4—3941no—
293C2×D4—3941no—
294C2×D4—3941no—
295C2×D4—3941no—
296C2×D4—1971no—
297C2×D4—1971no—
298C2×D4—1971no—
299C2×D4—1971no—
300C2×D4D78811yescharacteristic
301C22×C4—3941no—
302C22×C4—3941no—
303C22×C4—3941no—
304C22×C4—3941no—
305C22×C4C22×D19711yescharacteristic
306C22×C4C22×D19711yescharacteristic
307Q16—3942no—
308SD16—7881no—
309SD16—3942no—
310SD16—3942no—
311SD16—1974no—
312D8—7881no—
313D8—3942no—
314D8—3942no—
315D8—3942no—
316M16—22no—
317C2×C8—22no—
318C2×C8C22×D19711yescharacteristic
319C2×C8C22×D19711yescharacteristic
320C2×C8C22×D19711yescharacteristic
321C2×C8C22×D19711yescharacteristic
322C4⋊C4—3941no—
323C4⋊C4—3941no—
324C4⋊C4—1971no—
325C4⋊C4—1971no—
326C4⋊C4D78811yescharacteristic
327C22⋊C4—3941no—
328C22⋊C4—3941no—
329C22⋊C4—3941no—
330C22⋊C4—3941no—
331C22⋊C4—3941no—
332C22⋊C4—3941no—
333C22⋊C4—3941no—
334C22⋊C4D78811yescharacteristic
335C22⋊C4D78811yescharacteristic
336C42—3941no—
337C42C22×D19711yescharacteristic
338C23—7882no—
339C23—7882no—
340C23—15761no—
341C23—3941no—
342C23—3941no—
343C23—3941no—
344C23—3941no—
345C23C2×D78811yescharacteristic
346C23C2×D78811yescharacteristic
347Q8—3941no—
348Q8—3941no—
349Q8—3941no—
350Q8—1972no—
351D4—41no—
352D4—7882no—
353D4—7882no—
354D4—7882no—
355D4—7882no—
356D4—7882no—
357D4—15761no—
358D4—15761no—
359D4—15761no—
360D4—15761no—
361D4—3941no—
362D4—3941no—
363D4—3941no—
364D4—3941no—
365D4—3941no—
366D4—3941no—
367D4—3941no—
368D4—1972no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—7882no—
374C2×C4—7882no—
375C2×C4—7882no—
376C2×C4—15761no—
377C2×C4—3941no—
378C2×C4—3941no—
379C2×C4—3941no—
380C2×C4—3941no—
381C2×C4C2×D78811yescharacteristic
382C2×C4C2×D78811yescharacteristic
383C2×C4C2×D78811yescharacteristic
384C2×C4C2×D78811yescharacteristic
385C2×C4C23×D19711yescharacteristic
386C8—21no—
387C8—21no—
388C8—21no—
389C8D4×D19712yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—7881no—
397C22—7881no—
398C22—7881no—
399C22—7881no—
400C22—7881no—
401C22—7881no—
402C22—7881no—
403C22—7881no—
404C22—7881no—
405C22—7881no—
406C22—7881no—
407C22—15764no—
408C22—3942no—
409C22C22×D78811yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—7881no—
415C4—7881no—
416C4—7881no—
417C4—3942no—
418C4C2×D4×D19711yescharacteristic
419C4#2026337211yescharacteristic
420C2—41no—
421C2—41no—
422C2—7882no—
423C2—15761no—
424C2—15761no—
425C2—15761no—
426C2D4×D78811yescharacteristic
427C2#2716187911yescharacteristic
428C2#2716188411yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#435×C197C2
socleC22×C197C22×D4
hypercenter#435D197
hypocenterC197#3268
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#20263354C2semidirect
2. C2×#20263348C2semidirect
3. #27161750C2semidirect
4. #27161725C2semidirect
5. #27161468C2semidirect
6. #27161325C2semidirect
7. #27161315C2semidirect
8. #27161302C2semidirect
9. #27161275C2semidirect
10. #27161243C2semidirect
11. #27161226C2semidirect
12. #27161226C2semidirect
13. #27161215C2semidirect
14. #27161181C2semidirect
15. #435×C197C2semidirect
18. C2×#15691358C22semidirect
22. C2×Dic788C22semidirect
25. C2×#15691347C22semidirect
26. C2×#15691347C22semidirect
31. #20263323C22semidirect
38. #20263300C22semidirect
43. C4×Dic394C22semidirect
47. #20263260C22semidirect
48. #20263251C22semidirect
49. #20263251C22semidirect
52. #20263248C22semidirect
53. #20263248C22semidirect
55. C22×C8×C197C22semidirect
57. C4⋊C8×C197C22semidirect
59. C22⋊C8×C197C22semidirect
60. #97×C197C22semidirect
93. #15691347D4semidirect
214. C197#3268semidirect
230. #435D197semidirect
247. C22×C8C2×D197semidirect
266. C4⋊C8C2×D197semidirect
274. C22⋊C8C2×D197semidirect
275. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633542
C2×#202633482
#271617502
#271617252
#271614682
#271613252
#271613152
#271613022
#271612752
#271612432
#271612262
#271612152
#271611812
#435×C1972
#3268197