Finite Group Databasebeta

← #38754318

Subgroups — #38754318

Order 25216.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263353C211yesmaximal, characteristic, complemented
2C2×#20263350C211yesmaximal, characteristic, complemented
3#27161724C211yesmaximal, characteristic, complemented
4#27161722C211yesmaximal, characteristic, complemented
5#27161461C211yesmaximal, characteristic, complemented
6#27161328C211yesmaximal, characteristic, complemented
7#27161324C211yesmaximal, characteristic, complemented
8#27161295C211yesmaximal, characteristic, complemented
9#27161244C211yesmaximal, characteristic, complemented
10#27161238C211yesmaximal, characteristic, complemented
11#27161214C211yesmaximal, characteristic, complemented
12#27161211C211yesmaximal, characteristic, complemented
13C4×#15691343C211yesmaximal, characteristic, complemented
14#27161156C211yesmaximal, characteristic, complemented
15#463×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×#15691357C2211yescharacteristic
18C2×D4×D197C2211yescharacteristic, complemented
19C2×D4×D197C2211yescharacteristic, complemented
20#20263353—24no—
21#20263350—24no—
22C2×#15691348C2211yescharacteristic, complemented
23C2×#15691347C2211yescharacteristic, complemented
24C2×#15691346C2211yescharacteristic, complemented
25C2×#15691345C2211yescharacteristic, complemented
26C2×D1576C2211yescharacteristic
27C2×#15691343C2211yescharacteristic
28C2×#15691343C2211yescharacteristic
29C2×#15691341C2211yescharacteristic, complemented
30C2×#15691341C2211yescharacteristic, complemented
31#20263314C2211yescharacteristic, complemented
32#20263310—21no—
33#20263308—21no—
34#20263306C2211yescharacteristic, complemented
35#20263302C2211yescharacteristic, complemented
36#20263301C2211yescharacteristic
37#20263300—21no—
38#20263299—21no—
39#20263297C2211yescharacteristic
40#20263296—21no—
41#20263296—21no—
42C4×D788C2211yescharacteristic
43C4×D788C2211yescharacteristic, complemented
44C2×#15691335—21no—
45C22⋊C4×D197—21no—
46#20263259C2211yescharacteristic, complemented
47#20263251C2211yescharacteristic, complemented
48#20263248C2211yescharacteristic, complemented
49#20263247C2211yescharacteristic, complemented
50#20263246C2211yescharacteristic, complemented
51#20263238C2211yescharacteristic, complemented
52#20263238C2211yescharacteristic, complemented
53C4×#13096802C2211yescharacteristic, complemented
54C2×SD16×C197C2211yescharacteristic, complemented
55C2×D8×C197C2211yescharacteristic
56#124×C197C2211yescharacteristic, complemented
57C4×D4×C197C2211yescharacteristic, complemented
58C4⋊C8×C197C2211yescharacteristic, complemented
59#103×C197C2211yescharacteristic, complemented
60#102×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62C23×D197—21no—
63#15691359—22no—
64#15691359—41no—
65#15691357—22no—
66#15691357—41no—
67C2×Dic394C2311yescharacteristic
68C2×D788—21no—
69C2×D788—21no—
70C2×D788D411yescharacteristic
71C2×D788C2311yescharacteristic
72C2×D788C2311yescharacteristic
73C2×C22⋊D197—21no—
74C2×C22⋊D197—21no—
75C2×C22⋊D197—21no—
76C2×C22⋊D197—21no—
77D4×D197—22no—
78D4×D197—22no—
79D4×D197—41no—
80D4×D197—41no—
81C22×Dic197—21no—
82C2×C4×D197—21no—
83C2×C4×D197—21no—
84C2×C4×D197C2311yescharacteristic
85C2×C4×D197C2311yescharacteristic
86#15691348—41no—
87#15691347—22no—
88#15691346—22no—
89#15691345—22no—
90D1576—22no—
91#15691343—41no—
92#15691343D414yescomplemented
93#15691341—22no—
94#15691341—22no—
95C2×#13096802C2311yescharacteristic
96C2×#13096802C2311yescharacteristic
97C4⋊Dic197—21no—
98C4⋊Dic197D411yescharacteristic
99C4⋊Dic197C2311yescharacteristic
100#15691337—21no—
101C22⋊Dic197—21no—
102#15691335—21no—
103#15691335—21no—
104#15691335—21no—
105#15691335—21no—
106#15691335—21no—
107#15691335—21no—
108#15691335D411yescharacteristic
109#15691335D411yescharacteristic
110C2×Q8×C197C2311yescharacteristic
111C2×D4×C197C2311yescharacteristic
112C2×D4×C197C2311yescharacteristic
113C22×C4×C197—21no—
114SD16×C197—22no—
115D8×C197—22no—
116C2×C8×C197C2311yescharacteristic
117C2×C8×C197C2311yescharacteristic
118C4⋊C4×C197C2311yescharacteristic
119C22⋊C4×C197—21no—
120C22⋊C4×C197—21no—
121C22⋊C4×C197—21no—
122C42×C197C2311yescharacteristic
123C22×D197—21no—
124C22×D197—21no—
125C22×D197—42no—
126C22×D197—42no—
127C22×D197—42no—
128C22×D197—42no—
129C22×D197—81no—
130C22×D197—81no—
131C22×D197C2×D411yescharacteristic
132C22×D197C2×D411yescharacteristic
133Dic394—21no—
134Dic394—21no—
135D788—21no—
136D788—21no—
137D788—21no—
138D788—41no—
139D788—42no—
140D788—81no—
141D788C2×D412yes—
142C22⋊D197—42no—
143C22⋊D197—42no—
144C22⋊D197—42no—
145C22⋊D197—81no—
146C2×Dic197—21no—
147C2×Dic197—21no—
148C2×Dic197—42no—
149C2×Dic197C2×D411yescharacteristic
150C2×Dic197C2×D411yescharacteristic
151C4×D197—21no—
152C4×D197—21no—
153C4×D197—21no—
154C4×D197—21no—
155C4×D197—42no—
156C4×D197—81no—
157#13096802—21no—
158#13096802C2×D412yes—
159C23×C197—21no—
160C23×C197—21no—
161Q8×C197—21no—
162Q8×C197—21no—
163D4×C197—21no—
164D4×C197—21no—
165D4×C197—21no—
166D4×C197C2×D412yes—
167C2×C4×C197—21no—
168C2×C4×C197—21no—
169C2×C4×C197—81no—
170C2×C4×C197C2×D411yescharacteristic
171C2×C4×C197C2×D411yescharacteristic
172C2×C4×C197C2411yescharacteristic
173C8×C197—21no—
174C8×C197—21no—
175C22×C197—22no—
176C22×C197—41no—
177C22×C197—41no—
178C22×C197—41no—
179C22×C197—41no—
180C22×C197—41no—
181C22×C197C22×D411yescharacteristic
182C2×D197—21no—
183C2×D197—21no—
184C2×D197—22no—
185C2×D197—21no—
186C2×D197—21no—
187C2×D197—21no—
188C2×D197—21no—
189C2×D197—41no—
190C2×D197—41no—
191C2×D197—41no—
192C2×D197—41no—
193C2×D197—41no—
194C2×D197—84no—
195C2×D197—84no—
196C4×C197—21no—
197C4×C197—21no—
198C4×C197—41no—
199C4×C197—41no—
200C4×C197C22×D411yescharacteristic
201C4×C197H(5,𝔽2)11yescharacteristic
202Dic197—21no—
203Dic197—21no—
204Dic197—41no—
205Dic197—41no—
206C2×C197—42no—
207C2×C197—81no—
208C2×C197D4211yescharacteristic
209C2×C197#57511yescharacteristic
210C2×C197#57611yescharacteristic
211D197—41no—
212D197—41no—
213D197—42no—
214D197—81no—
215C197#327311yescharacteristic, complemented, Sylow
216#3273—1971nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1971no—
218C2×AGL(1,ℤ/8)—1971no—
219#546—1971no—
220#546—1971no—
221#495—1971no—
222#468—1971no—
223#468—1971no—
224#463—1971no—
225#463D19711yescharacteristic, complemented
226#452—1971no—
227#452—1971no—
228#447—1971no—
229#447—1971no—
230C4×D8—1971no—
231#436—1971no—
232C2×#41—1971no—
233C2×#41—1971no—
234C22×D4—1971no—
235C22×D4—1971no—
236AGL(1,ℤ/8)—3944no—
237AGL(1,ℤ/8)—3944no—
238C2×Q16—1971no—
239C2×SD16—1971no—
240C2×SD16—1971no—
241C2×SD16—1971no—
242C2×SD16C2×D19711yescharacteristic, complemented
243C2×D8—1971no—
244C2×D8—1971no—
245C2×D8—1971no—
246C2×D8C2×D19711yescharacteristic
247C2×M16—1971no—
248C2×M16—1971no—
249#124—1971no—
250#124C2×D19711yescharacteristic, complemented
251#123—3941no—
252#123—3941no—
253C22⋊Q8—1971no—
254C22⋊Q8—1971no—
255#121—3941no—
256#121—3941no—
257#121—1971no—
258#121—1971no—
259#120—3941no—
260#120—3941no—
261C4×D4—1971no—
262C4×D4—1971no—
263C4×D4C2×D19711yescharacteristic, complemented
264C2×C22⋊C4—3941no—
265C2×C22⋊C4—3941no—
266#107—1971no—
267C4⋊C8C2×D19711yescharacteristic, complemented
268#103—1971no—
269#103C2×D19711yescharacteristic, complemented
270#102—1971no—
271#102—1971no—
272#102—1971no—
273#102C2×D19711yescharacteristic, complemented
274C22⋊C8—1971no—
275C22⋊C8—1971no—
276C4×C8—1971no—
277C24—3941no—
278C24—3941no—
279#41—7881no—
280#41—7881no—
281#41—3942no—
282#41—3942no—
283C2×Q8—1971no—
284C2×Q8C22×D19711yescharacteristic
285C2×D4—7881no—
286C2×D4—7881no—
287C2×D4—3941no—
288C2×D4—3941no—
289C2×D4—3942no—
290C2×D4—3941no—
291C2×D4—3942no—
292C2×D4—3941no—
293C2×D4—3941no—
294C2×D4—3941no—
295C2×D4—1971no—
296C2×D4—1971no—
297C2×D4—1971no—
298C2×D4C22×D19711yescharacteristic
299C2×D4C22×D19711yescharacteristic
300C22×C4—21no—
301C22×C4—3941no—
302C22×C4—3941no—
303C22×C4—3941no—
304C22×C4—1971no—
305C22×C4—1971no—
306Q16—7881no—
307SD16—22no—
308SD16—3942no—
309SD16—3942no—
310SD16—3942no—
311D8—22no—
312D8—7881no—
313D8—3942no—
314D8—1974no—
315M16—3942no—
316M16—3942no—
317C2×C8—1971no—
318C2×C8—1971no—
319C2×C8C22×D19711yescharacteristic
320C2×C8C22×D19711yescharacteristic
321C4⋊C4—3941no—
322C4⋊C4—3941no—
323C4⋊C4—1971no—
324C4⋊C4—1971no—
325C4⋊C4C22×D19711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—3941no—
330C22⋊C4—3941no—
331C22⋊C4—3941no—
332C22⋊C4—3941no—
333C22⋊C4—3941no—
334C22⋊C4—3941no—
335C22⋊C4—3941no—
336C22⋊C4—1971no—
337C22⋊C4—1971no—
338C42C22×D19711yescharacteristic
339C23—21no—
340C23—21no—
341C23—7882no—
342C23—7882no—
343C23—7882no—
344C23—7882no—
345C23—15761no—
346C23—15761no—
347C23—3941no—
348C23—3941no—
349C23—1971no—
350C23—1971no—
351Q8—21no—
352Q8—21no—
353Q8—3941no—
354Q8—3941no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—7881no—
359D4—7882no—
360D4—7882no—
361D4—7882no—
362D4—7882no—
363D4—15761no—
364D4—15761no—
365D4—3941no—
366D4—3941no—
367D4—3941no—
368D4—1972no—
369D4D4×D19712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—81no—
373C2×C4—7882no—
374C2×C4—7882no—
375C2×C4—15761no—
376C2×C4—3941no—
377C2×C4—3941no—
378C2×C4—3941no—
379C2×C4—3941no—
380C2×C4—3941no—
381C2×C4—3941no—
382C2×C4—1971no—
383C2×C4—1971no—
384C2×C4D4×D19711yescharacteristic
385C2×C4D4×D19711yescharacteristic
386C2×C4C23×D19711yescharacteristic
387C8—21no—
388C8—21no—
389C8—3941no—
390C8—1972no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—7881no—
398C22—7881no—
399C22—7881no—
400C22—7881no—
401C22—7881no—
402C22—15764no—
403C22—15764no—
404C22—3941no—
405C22—3941no—
406C22—3941no—
407C22—3941no—
408C22—3942no—
409C22—3941no—
410C22—3941no—
411C22C2×D4×D19711yescharacteristic
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—7881no—
417C4—7881no—
418C4—3941no—
419C4—3941no—
420C4C2×D4×D19711yescharacteristic
421C4#2026337211yescharacteristic
422C2—42no—
423C2—81no—
424C2—7882no—
425C2—7881no—
426C2—7881no—
427C2—15761no—
428C2#2716171311yescharacteristic
429C2#2716187611yescharacteristic
430C2#2716188311yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#463×C197C2
socleC22×C197C22×D4
hypercenter#463D197
hypocenterC197#3273
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#20263353C2semidirect
2. C2×#20263350C2semidirect
3. #27161724C2semidirect
4. #27161722C2semidirect
5. #27161461C2semidirect
6. #27161328C2semidirect
7. #27161324C2semidirect
8. #27161295C2semidirect
9. #27161244C2semidirect
10. #27161238C2semidirect
11. #27161214C2semidirect
12. #27161211C2semidirect
13. C4×#15691343C2semidirect
14. #27161156C2semidirect
15. #463×C197C2semidirect
18. C2×D4×D197C22semidirect
19. C2×D4×D197C22semidirect
22. C2×#15691348C22semidirect
23. C2×#15691347C22semidirect
24. C2×#15691346C22semidirect
25. C2×#15691345C22semidirect
29. C2×#15691341C22semidirect
30. C2×#15691341C22semidirect
31. #20263314C22semidirect
34. #20263306C22semidirect
35. #20263302C22semidirect
43. C4×D788C22semidirect
46. #20263259C22semidirect
47. #20263251C22semidirect
48. #20263248C22semidirect
49. #20263247C22semidirect
50. #20263246C22semidirect
51. #20263238C22semidirect
52. #20263238C22semidirect
53. C4×#13096802C22semidirect
54. C2×SD16×C197C22semidirect
56. #124×C197C22semidirect
57. C4×D4×C197C22semidirect
58. C4⋊C8×C197C22semidirect
59. #103×C197C22semidirect
60. #102×C197C22semidirect
92. #15691343D4semidirect
215. C197#3273semidirect
225. #463D197semidirect
242. C2×SD16C2×D197semidirect
250. #124C2×D197semidirect
263. C4×D4C2×D197semidirect
267. C4⋊C8C2×D197semidirect
269. #103C2×D197semidirect
273. #102C2×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
#271617242
#271617222
#271614612
#271613282
#271613242
#271612952
#271612442
#271612382
#271612142
#271612112
C4×#156913432
#271611562
#463×C1972
#3273197