Finite Group Databasebeta

← #38751961

Subgroups — #38751961

Order 25216.

806 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D197C211yesmaximal, characteristic, complemented
2C2×#20263353C211yesmaximal, characteristic, complemented
3C2×#20263350C211yesmaximal, characteristic, complemented
4#121×D197C211yesmaximal, characteristic, complemented
5#27161226C211yesmaximal, characteristic, complemented
6#27161224C211yesmaximal, characteristic, complemented
7#27161222C211yesmaximal, characteristic, complemented
8#27161219C211yesmaximal, characteristic, complemented
9#27161216C211yesmaximal, characteristic, complemented
10#27161214C211yesmaximal, characteristic, complemented
11#27161211C211yesmaximal, characteristic, complemented
12#27161079C211yesmaximal, characteristic, complemented
13#27161077C211yesmaximal, characteristic, complemented
14#27161041C211yesmaximal, characteristic, complemented
15#449×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×#15691358—21no—
18C2×#15691358C2211yescharacteristic, complemented
19C2×#15691357—21no—
20C2×#15691357C2211yescharacteristic
21#41×D197—24no—
22#41×D197—24no—
23C2×Q8×D197C2211yescharacteristic, complemented
24C2×D4×D197—21no—
25C2×D4×D197—21no—
26C2×D4×D197C2211yescharacteristic, complemented
27C2×D4×D197C2211yescharacteristic, complemented
28C22×C4×D197—21no—
29C22×C4×D197C2211yescharacteristic
30#20263353—24no—
31#20263350—24no—
32C2×#15691347C2211yescharacteristic, complemented
33C2×#15691346C2211yescharacteristic, complemented
34C2×#15691345C2211yescharacteristic, complemented
35C2×D1576C2211yescharacteristic, complemented
36C2×#15691343C2211yescharacteristic, complemented
37C2×#15691343C2211yescharacteristic, complemented
38C2×#15691341C2211yescharacteristic, complemented
39C2×#15691341C2211yescharacteristic, complemented
40#20263301C2211yescharacteristic
41#20263300C2211yescharacteristic, complemented
42#20263299—21no—
43#20263298—21no—
44#20263297C2211yescharacteristic
45C4⋊C4×D197C2211yescharacteristic, complemented
46C22⋊C4×D197—21no—
47#20263251C2211yescharacteristic, complemented
48#20263250C2211yescharacteristic, complemented
49#20263249C2211yescharacteristic, complemented
50#20263248C2211yescharacteristic, complemented
51#20263247C2211yescharacteristic, complemented
52#20263246C2211yescharacteristic, complemented
53C22⋊#13096802C2211yescharacteristic, complemented
54#20263238C2211yescharacteristic, complemented
55#20263238C2211yescharacteristic, complemented
56C2×#41×C197C2211yescharacteristic, complemented
57C2×SD16×C197C2211yescharacteristic, complemented
58C2×D8×C197C2211yescharacteristic, complemented
59#121×C197C2211yescharacteristic, complemented
60#103×C197C2211yescharacteristic, complemented
61#102×C197C2211yescharacteristic, complemented
62C22⋊C8×C197C2211yescharacteristic, complemented
63C23×D197—21no—
64C23×D197—21no—
65C23×D197D411yescharacteristic
66#15691359—22no—
67#15691359D414yescomplemented
68#15691358—28no—
69#15691358—22no—
70#15691358—22no—
71#15691357—28no—
72#15691357—22no—
73#15691357D414yescomplemented
74C2×Dic394—21no—
75C2×Dic394C2311yescharacteristic
76Q8×D197—22no—
77Q8×D197—22no—
78C2×D788—21no—
79C2×D788—21no—
80C2×D788C2311yescharacteristic
81C2×D788C2311yescharacteristic
82C2×C22⋊D197—21no—
83C2×C22⋊D197—21no—
84C2×C22⋊D197—21no—
85C2×C22⋊D197—21no—
86C2×C22⋊D197—21no—
87C2×C22⋊D197—21no—
88D4×D197—28no—
89D4×D197—22no—
90D4×D197—22no—
91D4×D197—22no—
92D4×D197—22no—
93D4×D197—44no—
94C22×Dic197—21no—
95C22×Dic197D411yescharacteristic
96C2×C4×D197—28no—
97C2×C4×D197—21no—
98C2×C4×D197—21no—
99C2×C4×D197—22no—
100C2×C4×D197—22no—
101C2×C4×D197—21no—
102C2×C4×D197—21no—
103C2×C4×D197D411yescharacteristic
104C2×C4×D197D411yescharacteristic
105C2×C4×D197C2311yescharacteristic
106C2×C4×D197C2311yescharacteristic
107#15691347—22no—
108#15691346—22no—
109#15691345—22no—
110D1576—22no—
111#15691343—22no—
112#15691343—22no—
113#15691341—22no—
114#15691341—22no—
115C2×#13096802C2311yescharacteristic
116C2×#13096802C2311yescharacteristic
117C4⋊Dic197C2311yescharacteristic
118#15691337—21no—
119C22⋊Dic197—21no—
120#15691335—21no—
121#15691335—21no—
122#41×C197—22no—
123#41×C197—22no—
124C2×Q8×C197C2311yescharacteristic
125C2×D4×C197—21no—
126C2×D4×C197—21no—
127C2×D4×C197C2311yescharacteristic
128C2×D4×C197C2311yescharacteristic
129C22×C4×C197—21no—
130C22×C4×C197C2311yescharacteristic
131SD16×C197—22no—
132D8×C197—22no—
133C2×C8×C197C2311yescharacteristic
134C2×C8×C197C2311yescharacteristic
135C4⋊C4×C197C2311yescharacteristic
136C22⋊C4×C197—21no—
137C22×D197—24no—
138C22×D197—21no—
139C22×D197—22no—
140C22×D197—22no—
141C22×D197—21no—
142C22×D197—41no—
143C22×D197—42no—
144C22×D197—42no—
145C22×D197—42no—
146C22×D197—42no—
147C22×D197—42no—
148C22×D197C2×D411yescharacteristic
149C22×D197C2×D411yescharacteristic
150Dic394—24no—
151Dic394—21no—
152Dic394C2×D412yescomplemented
153D788—24no—
154D788—21no—
155D788—21no—
156D788—21no—
157D788—42no—
158D788C2×D412yescomplemented
159C22⋊D197—24no—
160C22⋊D197—24no—
161C22⋊D197—24no—
162C22⋊D197—42no—
163C22⋊D197—42no—
164C22⋊D197—42no—
165C2×Dic197—21no—
166C2×Dic197—24no—
167C2×Dic197—22no—
168C2×Dic197—21no—
169C2×Dic197C2×D411yescharacteristic
170C2×Dic197C2×D411yescharacteristic
171C4×D197—24no—
172C4×D197—24no—
173C4×D197—22no—
174C4×D197—21no—
175C4×D197—21no—
176C4×D197—22no—
177C4×D197—24no—
178C4×D197—42no—
179C4×D197C2×D412yes—
180C4×D197C2×D412yes—
181#13096802—21no—
182#13096802—21no—
183C23×C197—21no—
184C23×C197—21no—
185C23×C197C2×D411yescharacteristic
186Q8×C197—21no—
187Q8×C197C2×D412yescomplemented
188D4×C197—24no—
189D4×C197—21no—
190D4×C197—21no—
191D4×C197—21no—
192D4×C197—42no—
193D4×C197C2×D412yescomplemented
194C2×C4×C197—24no—
195C2×C4×C197—21no—
196C2×C4×C197—21no—
197C2×C4×C197—21no—
198C2×C4×C197—21no—
199C2×C4×C197C2×D411yescharacteristic
200C2×C4×C197C2411yescharacteristic
201C8×C197—21no—
202C8×C197—21no—
203C22×C197—21no—
204C22×C197—22no—
205C22×C197—21no—
206C22×C197—21no—
207C22×C197—41no—
208C22×C197—41no—
209C22×C197—41no—
210C22×C197—41no—
211C22×C197—41no—
212C22×C197C22×D411yescharacteristic
213C2×D197—21no—
214C2×D197—21no—
215C2×D197—21no—
216C2×D197—21no—
217C2×D197—22no—
218C2×D197—44no—
219C2×D197—48no—
220C2×D197—41no—
221C2×D197—41no—
222C2×D197—41no—
223C2×D197—41no—
224C2×D197—41no—
225C2×D197—84no—
226C2×D197#12012yes—
227C2×D197AGL(1,ℤ/8)12yescomplemented
228C4×C197—21no—
229C4×C197—22no—
230C4×C197—41no—
231C4×C197C22×D411yescharacteristic
232C4×C197C22×D411yescharacteristic
233Dic197—22no—
234Dic197—21no—
235Dic197—41no—
236Dic197#12012yes—
237C2×C197—41no—
238C2×C197—42no—
239C2×C197—81no—
240C2×C197C2×#12011yescharacteristic
241C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
242C2×C197#57511yescharacteristic
243D197—22no—
244D197—22no—
245D197—42no—
246D197—81no—
247C197#300611yescharacteristic, complemented, Sylow
248#3006—1971nomaximal, Sylow
249C22×#41—1971no—
250C2×AGL(1,ℤ/8)—1971no—
251C2×AGL(1,ℤ/8)—1971no—
252C2×#121—1971no—
253#449—1971no—
254#449—1971no—
255#449—1971no—
256#449D19711yescharacteristic, complemented
257#448—1971no—
258#448—1971no—
259#447—1971no—
260#447—1971no—
261#417—1971no—
262#417—1971no—
263#407—1971no—
264C2×#41—3941no—
265C2×#41—3944no—
266C2×#41—3944no—
267C2×#41—3941no—
268C2×#41—1971no—
269C2×#41—1971no—
270C2×#41—1971no—
271C2×#41C2×D19711yescharacteristic, complemented
272C22×Q8—1971no—
273C22×D4—3941no—
274C22×D4—3941no—
275C22×D4—1971no—
276C22×D4—1971no—
277C23×C4—3941no—
278C23×C4—1971no—
279AGL(1,ℤ/8)—3944no—
280AGL(1,ℤ/8)—3944no—
281C2×SD16—1971no—
282C2×SD16—1971no—
283C2×SD16—1971no—
284C2×SD16C2×D19711yescharacteristic, complemented
285C2×D8—1971no—
286C2×D8—1971no—
287C2×D8—1971no—
288C2×D8C2×D19711yescharacteristic, complemented
289C2×M16—1971no—
290C2×M16—1971no—
291#121—3941no—
292#121—3941no—
293#121—1971no—
294#121—1971no—
295#121—1971no—
296#121C2×D19711yescharacteristic, complemented
297C2×C4⋊C4—1971no—
298C2×C22⋊C4—3941no—
299#103—1971no—
300#103—1971no—
301#103—1971no—
302#103C2×D19711yescharacteristic, complemented
303#102—1971no—
304#102—1971no—
305#102—1971no—
306#102C2×D19711yescharacteristic, complemented
307C22⋊C8—1971no—
308C22⋊C8—1971no—
309C22⋊C8—1971no—
310C22⋊C8C2×D19711yescharacteristic, complemented
311C24—3941no—
312C24—3941no—
313C24—1971no—
314#41—22no—
315#41—22no—
316#41—3948no—
317#41—3942no—
318#41—3942no—
319#41—3948no—
320#41—3942no—
321#41—3942no—
322#41—1974no—
323#41—1974no—
324C2×Q8—3941no—
325C2×Q8—3942no—
326C2×Q8—3942no—
327C2×Q8—1971no—
328C2×Q8C22×D19711yescharacteristic
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—7884no—
332C2×D4—3941no—
333C2×D4—3941no—
334C2×D4—3948no—
335C2×D4—3942no—
336C2×D4—3941no—
337C2×D4—3941no—
338C2×D4—3941no—
339C2×D4—3941no—
340C2×D4—3942no—
341C2×D4—3942no—
342C2×D4—3942no—
343C2×D4—3941no—
344C2×D4—3941no—
345C2×D4—1971no—
346C2×D4—1971no—
347C2×D4C22×D19711yescharacteristic
348C2×D4C22×D19711yescharacteristic
349C22×C4—21no—
350C22×C4—3941no—
351C22×C4—3948no—
352C22×C4—3941no—
353C22×C4—3941no—
354C22×C4—3942no—
355C22×C4—3942no—
356C22×C4—3941no—
357C22×C4—3941no—
358C22×C4—1971no—
359C22×C4—1971no—
360C22×C4—1971no—
361C22×C4—1971no—
362C22×C4—1971no—
363C22×C4C22×D19711yescharacteristic
364SD16—22no—
365SD16—3942no—
366SD16—3942no—
367SD16—3942no—
368D8—22no—
369D8—3942no—
370D8—3942no—
371D8—3942no—
372M16—3942no—
373M16—3942no—
374C2×C8—1971no—
375C2×C8—1971no—
376C2×C8C22×D19711yescharacteristic
377C2×C8C22×D19711yescharacteristic
378C4⋊C4—3941no—
379C4⋊C4—1971no—
380C4⋊C4C22×D19711yescharacteristic
381C22⋊C4—21no—
382C22⋊C4—3941no—
383C22⋊C4—3941no—
384C22⋊C4—3941no—
385C23—21no—
386C23—21no—
387C23—7881no—
388C23—7882no—
389C23—7882no—
390C23—7882no—
391C23—7882no—
392C23—7882no—
393C23—3944no—
394C23—3942no—
395C23—3942no—
396C23—3941no—
397C23—3941no—
398C23—1971no—
399C23—1971no—
400C23D4×D19711yescharacteristic
401Q8—21no—
402Q8—3944no—
403Q8—3941no—
404Q8—1972no—
405Q8D4×D19712yescomplemented
406D4—24no—
407D4—21no—
408D4—21no—
409D4—21no—
410D4—42no—
411D4—7882no—
412D4—7882no—
413D4—7882no—
414D4—7882no—
415D4—3944no—
416D4—3944no—
417D4—3944no—
418D4—3944no—
419D4—3941no—
420D4—3941no—
421D4—3941no—
422D4—1972no—
423D4D4×D19712yescomplemented
424C2×C4—24no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—7882no—
430C2×C4—3944no—
431C2×C4—3944no—
432C2×C4—3942no—
433C2×C4—3942no—
434C2×C4—3944no—
435C2×C4—3941no—
436C2×C4—3941no—
437C2×C4—3944no—
438C2×C4—3942no—
439C2×C4—3941no—
440C2×C4—3941no—
441C2×C4—1971no—
442C2×C4—1971no—
443C2×C4—1972no—
444C2×C4—1972no—
445C2×C4D4×D19711yescharacteristic
446C2×C4C23×D19711yescharacteristic
447C8—21no—
448C8—21no—
449C8—3941no—
450C8—3941no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—7888no—
461C22—7884no—
462C22—7881no—
463C22—7881no—
464C22—7881no—
465C22—7881no—
466C22—7881no—
467C22—15764no—
468C22—3942no—
469C22—3941no—
470C22—3941no—
471C22—3941no—
472C22—3941no—
473C22—1972no—
474C22—1972no—
475C22C2×D4×D19711yescharacteristic
476C4—21no—
477C4—22no—
478C4—41no—
479C4—7881no—
480C4—3942no—
481C4—3941no—
482C4—1972no—
483C4C2×D4×D19711yescharacteristic
484C4C2×D4×D19711yescharacteristic
485C2—41no—
486C2—42no—
487C2—81no—
488C2—7882no—
489C2—15761no—
490C2—3942no—
491C2—3942no—
492C2#120×D19711yescharacteristic
493C2AGL(1,ℤ/8)×D19711yescharacteristic
494C2#2716187611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#449×C197C2
socleC22×C197C22×D4
hypercenter#449D197
hypocenterC197#3006
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#41×D197C2semidirect
2. C2×#20263353C2semidirect
3. C2×#20263350C2semidirect
4. #121×D197C2semidirect
5. #27161226C2semidirect
6. #27161224C2semidirect
7. #27161222C2semidirect
8. #27161219C2semidirect
9. #27161216C2semidirect
10. #27161214C2semidirect
11. #27161211C2semidirect
12. #27161079C2semidirect
13. #27161077C2semidirect
14. #27161041C2semidirect
15. #449×C197C2semidirect
18. C2×#15691358C22semidirect
23. C2×Q8×D197C22semidirect
26. C2×D4×D197C22semidirect
27. C2×D4×D197C22semidirect
32. C2×#15691347C22semidirect
33. C2×#15691346C22semidirect
34. C2×#15691345C22semidirect
35. C2×D1576C22semidirect
36. C2×#15691343C22semidirect
37. C2×#15691343C22semidirect
38. C2×#15691341C22semidirect
39. C2×#15691341C22semidirect
41. #20263300C22semidirect
45. C4⋊C4×D197C22semidirect
47. #20263251C22semidirect
48. #20263250C22semidirect
49. #20263249C22semidirect
50. #20263248C22semidirect
51. #20263247C22semidirect
52. #20263246C22semidirect
53. C22⋊#13096802C22semidirect
54. #20263238C22semidirect
55. #20263238C22semidirect
56. C2×#41×C197C22semidirect
57. C2×SD16×C197C22semidirect
58. C2×D8×C197C22semidirect
59. #121×C197C22semidirect
60. #103×C197C22semidirect
61. #102×C197C22semidirect
62. C22⋊C8×C197C22semidirect
67. #15691359D4semidirect
73. #15691357D4semidirect
152. Dic394C2×D4semidirect
158. D788C2×D4semidirect
187. Q8×C197C2×D4semidirect
193. D4×C197C2×D4semidirect
227. C2×D197AGL(1,ℤ/8)semidirect
247. C197#3006semidirect
256. #449D197semidirect
271. C2×#41C2×D197semidirect
284. C2×SD16C2×D197semidirect
288. C2×D8C2×D197semidirect
296. #121C2×D197semidirect
302. #103C2×D197semidirect
306. #102C2×D197semidirect
310. C22⋊C8C2×D197semidirect
405. Q8D4×D197semidirect
423. D4D4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1972
C2×#202633532
C2×#202633502
#121×D1972
#271612262
#271612242
#271612222
#271612192
#271612162
#271612142
#271612112
#271610792
#271610772
#271610412
#449×C1972
#3006197