Finite Group Databasebeta

← #38754192

Subgroups — #38754192

Order 25216.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263350C211yesmaximal, characteristic, complemented
2#27161723C211yesmaximal, characteristic, complemented
3D4×C22⋊D197C211yesmaximal, characteristic, complemented
4#27161467C211yesmaximal, characteristic, complemented
5#27161329C211yesmaximal, characteristic, complemented
6#27161308C211yesmaximal, characteristic, complemented
7#27161292C211yesmaximal, characteristic, complemented
8#27161272C211yesmaximal, characteristic, complemented
9#27161230C211yesmaximal, characteristic, complemented
10#27161221C211yesmaximal, characteristic, complemented
11#27161213C211yesmaximal, characteristic, complemented
12#27161211C211yesmaximal, characteristic, complemented
13#27161191C211yesmaximal, characteristic, complemented
14#27161150C211yesmaximal, characteristic, complemented
15C22×D8×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic, complemented
17C22×C22⋊D197—21no—
18C2×D4×D197C2211yescharacteristic, complemented
19#20263350—24no—
20C2×#15691347C2211yescharacteristic, complemented
21C2×#15691346C2211yescharacteristic, complemented
22C2×#15691346C2211yescharacteristic, complemented
23C2×#15691343C2211yescharacteristic, complemented
24C2×#15691343C2211yescharacteristic, complemented
25C2×#15691341C2211yescharacteristic, complemented
26#20263322C2211yescharacteristic, complemented
27#20263313C2211yescharacteristic, complemented
28#20263307—21no—
29#20263305C2211yescharacteristic, complemented
30#20263300C2211yescharacteristic
31#20263298—21no—
32#20263298—21no—
33#20263297C2211yescharacteristic, complemented
34#20263297C2211yescharacteristic, complemented
35#20263295—21no—
36#20263295—21no—
37#20263294—21no—
38D4×Dic197C2211yescharacteristic, complemented
39D4×Dic197C2211yescharacteristic, complemented
40C4×C22⋊D197C2211yescharacteristic
41C2×C22⋊Dic197—21no—
42#20263258C2211yescharacteristic, complemented
43#20263255C2211yescharacteristic, complemented
44#20263251C2211yescharacteristic, complemented
45#20263248C2211yescharacteristic, complemented
46#20263247C2211yescharacteristic, complemented
47#20263247C2211yescharacteristic, complemented
48#20263247C2211yescharacteristic, complemented
49#20263247C2211yescharacteristic, complemented
50C22⋊#13096802C2211yescharacteristic, complemented
51#20263238C2211yescharacteristic, complemented
52#20263236C2211yescharacteristic, complemented
53C22×D4×C197C2211yescharacteristic
54C22×D4×C197C2211yescharacteristic, complemented
55C2×D8×C197—24no—
56C2×D8×C197C2211yescharacteristic
57C2×D8×C197C2211yescharacteristic
58C2×D8×C197C2211yescharacteristic, complemented
59C2×D8×C197C2211yescharacteristic, complemented
60C22×C8×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62#15691357—22no—
63#15691357—41no—
64C2×Dic394C2311yescharacteristic
65C2×D788C2311yescharacteristic
66C2×C22⋊D197—21no—
67C2×C22⋊D197—21no—
68C2×C22⋊D197—21no—
69C2×C22⋊D197—21no—
70C2×C22⋊D197—21no—
71C2×C22⋊D197—21no—
72C2×C22⋊D197—21no—
73C2×C22⋊D197—44no—
74C2×C22⋊D197D411yescharacteristic
75D4×D197—22no—
76D4×D197—41no—
77C22×Dic197—21no—
78C22×Dic197—21no—
79C22×Dic197—21no—
80C2×C4×D197C2311yescharacteristic
81#15691347—22no—
82#15691346—22no—
83#15691346—41no—
84#15691343—22no—
85#15691343—41no—
86#15691341—22no—
87C2×#13096802C2311yescharacteristic
88C2×#13096802C2311yescharacteristic
89C4⋊Dic197C2311yescharacteristic
90C4⋊Dic197C2311yescharacteristic
91#15691337—21no—
92#15691337D411yescharacteristic
93C22⋊Dic197—21no—
94C22⋊Dic197—21no—
95C22⋊Dic197—21no—
96C22⋊Dic197—21no—
97C22⋊Dic197—21no—
98C22⋊Dic197—21no—
99C22⋊Dic197—21no—
100C22⋊Dic197D411yescharacteristic
101#15691335—21no—
102#15691335D411yescharacteristic
103C4×Dic197C2311yescharacteristic
104C24×C197—21no—
105C24×C197—21no—
106C2×D4×C197—22no—
107C2×D4×C197—22no—
108C2×D4×C197—21no—
109C2×D4×C197—21no—
110C2×D4×C197—41no—
111C2×D4×C197—41no—
112C2×D4×C197C2311yescharacteristic
113C2×D4×C197C2311yescharacteristic
114C2×D4×C197C2311yescharacteristic
115C2×D4×C197C2311yescharacteristic
116C22×C4×C197C2311yescharacteristic
117D8×C197—22no—
118D8×C197—22no—
119D8×C197—22no—
120D8×C197D414yescomplemented
121C2×C8×C197—22no—
122C2×C8×C197C2311yescharacteristic
123C2×C8×C197C2311yescharacteristic
124C22×D197—21no—
125C22×D197—42no—
126C22×D197—42no—
127C22×D197—81no—
128C22×D197C2×D411yescharacteristic
129Dic394—21no—
130Dic394—21no—
131D788—21no—
132D788—21no—
133C22⋊D197—22no—
134C22⋊D197—42no—
135C22⋊D197—42no—
136C22⋊D197—42no—
137C22⋊D197—42no—
138C22⋊D197—42no—
139C22⋊D197—81no—
140C22⋊D197—81no—
141C2×Dic197—21no—
142C2×Dic197—21no—
143C2×Dic197—21no—
144C2×Dic197—42no—
145C2×Dic197—42no—
146C2×Dic197—81no—
147C2×Dic197C2×D411yescharacteristic
148C2×Dic197C2×D411yescharacteristic
149C2×Dic197C2×D411yescharacteristic
150C4×D197—21no—
151C4×D197—21no—
152#13096802—21no—
153#13096802—21no—
154C23×C197—21no—
155C23×C197—21no—
156C23×C197—21no—
157C23×C197—21no—
158C23×C197—42no—
159C23×C197—42no—
160C23×C197—42no—
161C23×C197—42no—
162C23×C197—81no—
163C23×C197—81no—
164C23×C197C2×D411yescharacteristic
165D4×C197—21no—
166D4×C197—21no—
167D4×C197—21no—
168D4×C197—21no—
169D4×C197—21no—
170D4×C197—21no—
171D4×C197—42no—
172D4×C197—42no—
173D4×C197C2×D412yes—
174D4×C197C2×D412yescomplemented
175C2×C4×C197—21no—
176C2×C4×C197—21no—
177C2×C4×C197C2×D411yescharacteristic
178C2×C4×C197C2411yescharacteristic
179C8×C197—21no—
180C8×C197C2×D412yescomplemented
181C22×C197—21no—
182C22×C197—21no—
183C22×C197—22no—
184C22×C197—22no—
185C22×C197—41no—
186C22×C197—41no—
187C22×C197—41no—
188C22×C197—41no—
189C22×C197—41no—
190C22×C197—41no—
191C22×C197—41no—
192C22×C197—41no—
193C22×C197—41no—
194C22×C197—41no—
195C22×C197—84no—
196C22×C197—84no—
197C22×C197AGL(1,ℤ/8)12yescomplemented
198C22×C197C22×D411yescharacteristic
199C2×D197—21no—
200C2×D197—21no—
201C2×D197—21no—
202C2×D197—41no—
203C2×D197—41no—
204C2×D197—41no—
205C2×D197—84no—
206C4×C197—21no—
207C4×C197C22×D411yescharacteristic
208C4×C197H(5,𝔽2)11yescharacteristic
209Dic197—21no—
210Dic197—21no—
211Dic197—21no—
212Dic197—41no—
213Dic197—41no—
214Dic197—41no—
215C2×C197—22no—
216C2×C197—42no—
217C2×C197—42no—
218C2×C197—81no—
219C2×C197—81no—
220C2×C197D4211yescharacteristic
221C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C197#57611yescharacteristic
223D197—41no—
224D197—81no—
225C197#326411yescharacteristic, complemented, Sylow
226#3264—1971nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1971no—
228C22×D8D19711yescharacteristic, complemented
229#546—1971no—
230D42—1971no—
231#496—1971no—
232#469—1971no—
233#465—1971no—
234#463—1971no—
235#459—1971no—
236#450—1971no—
237#449—1971no—
238#447—1971no—
239#447—1971no—
240#442—1971no—
241#435—1971no—
242C2×#41—1971no—
243C22×D4—3941no—
244C22×D4—1971no—
245C22×D4C2×D19711yescharacteristic
246C22×D4C2×D19711yescharacteristic, complemented
247AGL(1,ℤ/8)—3944no—
248C2×SD16—1971no—
249C2×SD16—1971no—
250C2×SD16—1971no—
251C2×D8—24no—
252C2×D8—1971no—
253C2×D8—1971no—
254C2×D8C2×D19711yescharacteristic
255C2×D8C2×D19711yescharacteristic
256C2×D8C2×D19711yescharacteristic, complemented
257C2×D8C2×D19711yescharacteristic, complemented
258C2×M16—1971no—
259C22×C8C2×D19711yescharacteristic, complemented
260#127—1971no—
261#124—1971no—
262#123—3941no—
263C22⋊Q8—1971no—
264#121—3941no—
265#121—3941no—
266#121—1971no—
267#121—1971no—
268#121—1971no—
269#120—3941no—
270#120—3941no—
271#120—3941no—
272C4×D4—1971no—
273C4×D4—1971no—
274C4×D4—1971no—
275C2×C22⋊C4—3941no—
276#106—1971no—
277C4⋊C8—1971no—
278#103—1971no—
279#102—1971no—
280#102—1971no—
281#102—1971no—
282#102—1971no—
283#102—1971no—
284C22⋊C8—1971no—
285C22⋊C8—1971no—
286#97—1971no—
287C24—21no—
288C24—21no—
289C24—3941no—
290#41—7881no—
291#41—3942no—
292C2×Q8—1971no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—7884no—
300C2×D4—7881no—
301C2×D4—3941no—
302C2×D4—3941no—
303C2×D4—3941no—
304C2×D4—3942no—
305C2×D4—3941no—
306C2×D4—3941no—
307C2×D4—3941no—
308C2×D4—3941no—
309C2×D4—1971no—
310C2×D4—1971no—
311C2×D4C22×D19711yescharacteristic
312C2×D4C22×D19711yescharacteristic
313C2×D4C22×D19711yescharacteristic
314C2×D4C22×D19711yescharacteristic
315C22×C4—3941no—
316C22×C4—3941no—
317C22×C4—3941no—
318C22×C4—1971no—
319C22×C4C22×D19711yescharacteristic
320SD16—7881no—
321SD16—3942no—
322SD16—3942no—
323D8—22no—
324D8—22no—
325D8—22no—
326D8—7881no—
327D8—3942no—
328D8C22⋊D19714yescomplemented
329M16—3942no—
330C2×C8—22no—
331C2×C8—1971no—
332C2×C8—1971no—
333C2×C8C22×D19711yescharacteristic
334C2×C8C22×D19711yescharacteristic
335C4⋊C4—3941no—
336C4⋊C4—1971no—
337C4⋊C4—1971no—
338C4⋊C4—1971no—
339C22⋊C4—3941no—
340C22⋊C4—3941no—
341C22⋊C4—3941no—
342C22⋊C4—3941no—
343C22⋊C4—3941no—
344C22⋊C4—3941no—
345C22⋊C4—3941no—
346C22⋊C4—3941no—
347C22⋊C4—1971no—
348C22⋊C4—1971no—
349C42—1971no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23—7882no—
361C23—7882no—
362C23—15761no—
363C23—3941no—
364C23—1971no—
365C23D4×D19711yescharacteristic
366Q8—3941no—
367Q8—3941no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—7882no—
377D4—7882no—
378D4—7882no—
379D4—7882no—
380D4—7882no—
381D4—15761no—
382D4—15761no—
383D4—3942no—
384D4—3941no—
385D4—3941no—
386D4C2×C22⋊D19712yes—
387D4C2×C22⋊D19712yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—7882no—
391C2×C4—7882no—
392C2×C4—15761no—
393C2×C4—3941no—
394C2×C4—3941no—
395C2×C4—3941no—
396C2×C4—3941no—
397C2×C4—3941no—
398C2×C4—1971no—
399C2×C4—1971no—
400C2×C4—1971no—
401C2×C4D4×D19711yescharacteristic
402C2×C4C23×D19711yescharacteristic
403C8—21no—
404C8—3941no—
405C8—3941no—
406C8C2×C22⋊D19712yescomplemented
407C22—21no—
408C22—21no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—7881no—
424C22—7881no—
425C22—7881no—
426C22—15764no—
427C22—3941no—
428C22—3941no—
429C22—3941no—
430C22#2026335012yescomplemented
431C22C2×D4×D19711yescharacteristic
432C4—21no—
433C4—7881no—
434C4—7881no—
435C4—7881no—
436C4—3941no—
437C4—3941no—
438C4—3941no—
439C4C22×C22⋊D19711yescharacteristic
440C4#2026337111yescharacteristic
441C2—22no—
442C2—42no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—7881no—
447C2—15761no—
448C2D4×C22⋊D19711yescharacteristic
449C2C2×#2026335011yescharacteristic
450C2#2716188011yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC22×D8×C197C2
socleC22×C197C22×D4
hypercenterC22×D8D197
hypocenterC197#3264
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#20263350C2semidirect
2. #27161723C2semidirect
3. D4×C22⋊D197C2semidirect
4. #27161467C2semidirect
5. #27161329C2semidirect
6. #27161308C2semidirect
7. #27161292C2semidirect
8. #27161272C2semidirect
9. #27161230C2semidirect
10. #27161221C2semidirect
11. #27161213C2semidirect
12. #27161211C2semidirect
13. #27161191C2semidirect
14. #27161150C2semidirect
15. C22×D8×C197C2semidirect
16. C2×#15691357C22semidirect
18. C2×D4×D197C22semidirect
20. C2×#15691347C22semidirect
21. C2×#15691346C22semidirect
22. C2×#15691346C22semidirect
23. C2×#15691343C22semidirect
24. C2×#15691343C22semidirect
25. C2×#15691341C22semidirect
26. #20263322C22semidirect
27. #20263313C22semidirect
29. #20263305C22semidirect
33. #20263297C22semidirect
34. #20263297C22semidirect
38. D4×Dic197C22semidirect
39. D4×Dic197C22semidirect
42. #20263258C22semidirect
43. #20263255C22semidirect
44. #20263251C22semidirect
45. #20263248C22semidirect
46. #20263247C22semidirect
47. #20263247C22semidirect
48. #20263247C22semidirect
49. #20263247C22semidirect
50. C22⋊#13096802C22semidirect
51. #20263238C22semidirect
52. #20263236C22semidirect
54. C22×D4×C197C22semidirect
58. C2×D8×C197C22semidirect
59. C2×D8×C197C22semidirect
60. C22×C8×C197C22semidirect
120. D8×C197D4semidirect
174. D4×C197C2×D4semidirect
180. C8×C197C2×D4semidirect
197. C22×C197AGL(1,ℤ/8)semidirect
225. C197#3264semidirect
228. C22×D8D197semidirect
246. C22×D4C2×D197semidirect
256. C2×D8C2×D197semidirect
257. C2×D8C2×D197semidirect
259. C22×C8C2×D197semidirect
328. D8C22⋊D197semidirect
387. D4C2×C22⋊D197semidirect
406. C8C2×C22⋊D197semidirect
430. C22#20263350semidirect

Maximal subgroups

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

subgroupindex
C2×#202633502
#271617232
D4×C22⋊D1972
#271614672
#271613292
#271613082
#271612922
#271612722
#271612302
#271612212
#271612132
#271612112
#271611912
#271611502
C22×D8×C1972
#3264197