Finite Group Databasebeta

← #38754306

Subgroups — #38754306

Order 25216.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263350C211yesmaximal, characteristic, complemented
2#27161728C211yesmaximal, characteristic, complemented
3D4×C22⋊D197C211yesmaximal, characteristic, complemented
4#27161450C211yesmaximal, characteristic, complemented
5#27161325C211yesmaximal, characteristic, complemented
6#27161322C211yesmaximal, characteristic, complemented
7#27161282C211yesmaximal, characteristic, complemented
8#27161272C211yesmaximal, characteristic, complemented
9#27161230C211yesmaximal, characteristic, complemented
10#27161228C211yesmaximal, characteristic, complemented
11#27161224C211yesmaximal, characteristic, complemented
12#27161221C211yesmaximal, characteristic, complemented
13D8×Dic197C211yesmaximal, characteristic, complemented
14#27161155C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic, complemented
17C22×C22⋊D197—21no—
18C2×D4×D197C2211yescharacteristic
19#20263350—24no—
20C2×#15691347C2211yescharacteristic, complemented
21C2×#15691346C2211yescharacteristic
22C2×#15691346C2211yescharacteristic, complemented
23C2×#15691345C2211yescharacteristic, complemented
24C2×#15691343C2211yescharacteristic, complemented
25C2×#15691341C2211yescharacteristic, complemented
26#20263324C2211yescharacteristic, complemented
27#20263322C2211yescharacteristic
28#20263309—21no—
29#20263307—21no—
30#20263305C2211yescharacteristic, complemented
31#20263302C2211yescharacteristic, complemented
32#20263300C2211yescharacteristic
33#20263298—21no—
34#20263298—21no—
35#20263297C2211yescharacteristic
36#20263295—21no—
37#20263294—21no—
38D4×Dic197C2211yescharacteristic
39D4×Dic197C2211yescharacteristic, complemented
40C4×C22⋊D197C2211yescharacteristic
41C2×#15691337—21no—
42#20263260C2211yescharacteristic, complemented
43#20263255C2211yescharacteristic, complemented
44#20263251C2211yescharacteristic, complemented
45#20263250C2211yescharacteristic, complemented
46#20263248C2211yescharacteristic, complemented
47#20263247C2211yescharacteristic
48#20263247C2211yescharacteristic, complemented
49#20263247C2211yescharacteristic, complemented
50C22⋊#13096802C2211yescharacteristic, complemented
51#20263238C2211yescharacteristic, complemented
52C8×Dic197C2211yescharacteristic, complemented
53C2×#41×C197C2211yescharacteristic, complemented
54C22×D4×C197C2211yescharacteristic
55AGL(1,ℤ/8)×C197—24no—
56C2×SD16×C197C2211yescharacteristic
57C2×SD16×C197C2211yescharacteristic, complemented
58C2×D8×C197C2211yescharacteristic
59C2×D8×C197C2211yescharacteristic, complemented
60C2×M16×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—44no—
73C2×C22⋊D197D411yescharacteristic
74D4×D197—22no—
75D4×D197—41no—
76C22×Dic197—21no—
77C22×Dic197—21no—
78C22×Dic197—21no—
79C2×C4×D197C2311yescharacteristic
80#15691347—22no—
81#15691346—22no—
82#15691346—41no—
83#15691345—41no—
84#15691343—22no—
85#15691341—22no—
86C2×#13096802C2311yescharacteristic
87C2×#13096802C2311yescharacteristic
88C4⋊Dic197C2311yescharacteristic
89C4⋊Dic197C2311yescharacteristic
90#15691337—21no—
91#15691337—21no—
92#15691337—21no—
93#15691337—21no—
94#15691337D411yescharacteristic
95C22⋊Dic197—21no—
96C22⋊Dic197—21no—
97C22⋊Dic197—21no—
98C22⋊Dic197—21no—
99C22⋊Dic197D411yescharacteristic
100#15691335—21no—
101#15691335—21no—
102#15691335D411yescharacteristic
103C4×Dic197C2311yescharacteristic
104C24×C197—21no—
105#41×C197—22no—
106#41×C197—41no—
107C2×Q8×C197C2311yescharacteristic
108C2×D4×C197—22no—
109C2×D4×C197—21no—
110C2×D4×C197—21no—
111C2×D4×C197—41no—
112C2×D4×C197C2311yescharacteristic
113C2×D4×C197C2311yescharacteristic
114C2×D4×C197C2311yescharacteristic
115C22×C4×C197—21no—
116C22×C4×C197C2311yescharacteristic
117SD16×C197—22no—
118SD16×C197—22no—
119D8×C197—22no—
120D8×C197D414yescomplemented
121M16×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—
140C2×Dic197—21no—
141C2×Dic197—21no—
142C2×Dic197—21no—
143C2×Dic197—42no—
144C2×Dic197—42no—
145C2×Dic197—42no—
146C2×Dic197C2×D411yescharacteristic
147C2×Dic197C2×D411yescharacteristic
148C2×Dic197C2×D411yescharacteristic
149C4×D197—21no—
150C4×D197—21no—
151#13096802—21no—
152#13096802—21no—
153C23×C197—21no—
154C23×C197—21no—
155C23×C197—21no—
156C23×C197—42no—
157C23×C197—42no—
158C23×C197—81no—
159C23×C197C2×D411yescharacteristic
160Q8×C197—21no—
161Q8×C197—21no—
162D4×C197—21no—
163D4×C197—21no—
164D4×C197—21no—
165D4×C197—21no—
166D4×C197—42no—
167D4×C197—42no—
168D4×C197C2×D412yes—
169D4×C197C2×D412yescomplemented
170C2×C4×C197—21no—
171C2×C4×C197—21no—
172C2×C4×C197—21no—
173C2×C4×C197—42no—
174C2×C4×C197C2×D411yescharacteristic
175C2×C4×C197C2411yescharacteristic
176C8×C197—21no—
177C8×C197C2×D412yescomplemented
178C22×C197—21no—
179C22×C197—22no—
180C22×C197—22no—
181C22×C197—21no—
182C22×C197—21no—
183C22×C197—41no—
184C22×C197—41no—
185C22×C197—41no—
186C22×C197—41no—
187C22×C197—41no—
188C22×C197—41no—
189C22×C197—41no—
190C22×C197—84no—
191C22×C197C22×D411yescharacteristic
192C2×D197—21no—
193C2×D197—21no—
194C2×D197—21no—
195C2×D197—41no—
196C2×D197—41no—
197C2×D197—41no—
198C2×D197—84no—
199C4×C197—21no—
200C4×C197—41no—
201C4×C197C22×D411yescharacteristic
202C4×C197H(5,𝔽2)11yescharacteristic
203Dic197—21no—
204Dic197—21no—
205Dic197—41no—
206Dic197—41no—
207Dic197—41no—
208Dic197AGL(1,ℤ/8)12yescomplemented
209C2×C197—42no—
210C2×C197—41no—
211C2×C197—42no—
212C2×C197—81no—
213C2×C197D4211yescharacteristic
214C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C197#57711yescharacteristic
216D197—41no—
217D197—81no—
218C197#327211yescharacteristic, complemented, Sylow
219#3272—1971nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1971no—
221C2×AGL(1,ℤ/8)D19711yescharacteristic, complemented
222#547—1971no—
223D42—1971no—
224#492—1971no—
225#468—1971no—
226#468—1971no—
227#461—1971no—
228#459—1971no—
229#450—1971no—
230#450—1971no—
231#449—1971no—
232#449—1971no—
233C4×D8—1971no—
234#436—1971no—
235C2×#41—1971no—
236C2×#41C2×D19711yescharacteristic, complemented
237C22×D4—3941no—
238C22×D4—1971no—
239C22×D4C2×D19711yescharacteristic
240AGL(1,ℤ/8)—24no—
241AGL(1,ℤ/8)—3944no—
242C2×SD16—1971no—
243C2×SD16—1971no—
244C2×SD16—1971no—
245C2×SD16—1971no—
246C2×SD16C2×D19711yescharacteristic
247C2×SD16C2×D19711yescharacteristic, complemented
248C2×D8—1971no—
249C2×D8C2×D19711yescharacteristic
250C2×D8C2×D19711yescharacteristic, complemented
251C2×M16—1971no—
252C2×M16C2×D19711yescharacteristic, complemented
253C4⋊Q8—1971no—
254#127—1971no—
255#123—3941no—
256#123—3941no—
257C22⋊Q8—1971no—
258C22⋊Q8—1971no—
259#121—3941no—
260#121—3941no—
261#121—1971no—
262#121—1971no—
263#120—3941no—
264#120—3941no—
265C4×D4—1971no—
266C4×D4—1971no—
267C4×D4—1971no—
268C2×C4⋊C4—3941no—
269#107—1971no—
270C4⋊C8—1971no—
271#103—1971no—
272#103—1971no—
273#102—1971no—
274#102—1971no—
275#102—1971no—
276#102—1971no—
277C22⋊C8—1971no—
278C22⋊C8—1971no—
279C4×C8—1971no—
280C24—21no—
281C24—3941no—
282#41—22no—
283#41—41no—
284#41—7881no—
285#41—3942no—
286C2×Q8—1971no—
287C2×Q8C22×D19711yescharacteristic
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—7884no—
293C2×D4—7881no—
294C2×D4—3941no—
295C2×D4—3941no—
296C2×D4—3941no—
297C2×D4—3942no—
298C2×D4—3941no—
299C2×D4—3941no—
300C2×D4—3941no—
301C2×D4—1971no—
302C2×D4—1971no—
303C2×D4C22×D19711yescharacteristic
304C2×D4C22×D19711yescharacteristic
305C2×D4C22×D19711yescharacteristic
306C22×C4—21no—
307C22×C4—3941no—
308C22×C4—3941no—
309C22×C4—3941no—
310C22×C4—1971no—
311C22×C4C22×D19711yescharacteristic
312SD16—22no—
313SD16—22no—
314SD16—7881no—
315SD16—7881no—
316SD16—3942no—
317SD16—3942no—
318D8—22no—
319D8—3942no—
320D8C22⋊D19714yescomplemented
321M16—22no—
322M16—3942no—
323C2×C8—1971no—
324C2×C8—1971no—
325C2×C8C22×D19711yescharacteristic
326C2×C8C22×D19711yescharacteristic
327C4⋊C4—3941no—
328C4⋊C4—3941no—
329C4⋊C4—3941no—
330C4⋊C4—3941no—
331C4⋊C4—1971no—
332C4⋊C4—1971no—
333C4⋊C4—1971no—
334C22⋊C4—3941no—
335C22⋊C4—3941no—
336C22⋊C4—3941no—
337C22⋊C4—3941no—
338C22⋊C4—3941no—
339C22⋊C4—3941no—
340C22⋊C4—1971no—
341C22⋊C4—1971no—
342C42—1971no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—7882no—
350C23—7882no—
351C23—15761no—
352C23—3941no—
353C23—1971no—
354C23D4×D19711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—3941no—
358Q8—3941no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—7882no—
366D4—7882no—
367D4—7882no—
368D4—7882no—
369D4—7882no—
370D4—15761no—
371D4—3942no—
372D4—3941no—
373D4—3941no—
374D4C2×C22⋊D19712yes—
375D4C2×C22⋊D19712yescomplemented
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—7882no—
381C2×C4—7882no—
382C2×C4—7882no—
383C2×C4—3941no—
384C2×C4—3941no—
385C2×C4—3941no—
386C2×C4—3941no—
387C2×C4—3941no—
388C2×C4—1971no—
389C2×C4—1971no—
390C2×C4—1971no—
391C2×C4D4×D19711yescharacteristic
392C2×C4C23×D19711yescharacteristic
393C8—21no—
394C8—3941no—
395C8—3941no—
396C8C2×C22⋊D19712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—7881no—
411C22—7881no—
412C22—7881no—
413C22—15764no—
414C22—3941no—
415C22—3941no—
416C22—3941no—
417C22C2×D4×D19711yescharacteristic
418C4—21no—
419C4—41no—
420C4—7881no—
421C4—7881no—
422C4—7881no—
423C4—3941no—
424C4—3941no—
425C4—1972no—
426C4C22×C22⋊D19711yescharacteristic
427C4#2026337111yescharacteristic
428C2—41no—
429C2—42no—
430C2—42no—
431C2—81no—
432C2—7881no—
433C2—15761no—
434C2D4×C22⋊D19711yescharacteristic
435C2AGL(1,ℤ/8)×D19711yescharacteristic
436C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×AGL(1,ℤ/8)×C197C2
socleC22×C197C22×D4
hypercenterC2×AGL(1,ℤ/8)D197
hypocenterC197#3272
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20263350C2semidirect
2. #27161728C2semidirect
3. D4×C22⋊D197C2semidirect
4. #27161450C2semidirect
5. #27161325C2semidirect
6. #27161322C2semidirect
7. #27161282C2semidirect
8. #27161272C2semidirect
9. #27161230C2semidirect
10. #27161228C2semidirect
11. #27161224C2semidirect
12. #27161221C2semidirect
13. D8×Dic197C2semidirect
14. #27161155C2semidirect
15. C2×AGL(1,ℤ/8)×C197C2semidirect
16. C2×#15691357C22semidirect
20. C2×#15691347C22semidirect
22. C2×#15691346C22semidirect
23. C2×#15691345C22semidirect
24. C2×#15691343C22semidirect
25. C2×#15691341C22semidirect
26. #20263324C22semidirect
30. #20263305C22semidirect
31. #20263302C22semidirect
39. D4×Dic197C22semidirect
42. #20263260C22semidirect
43. #20263255C22semidirect
44. #20263251C22semidirect
45. #20263250C22semidirect
46. #20263248C22semidirect
48. #20263247C22semidirect
49. #20263247C22semidirect
50. C22⋊#13096802C22semidirect
51. #20263238C22semidirect
52. C8×Dic197C22semidirect
53. C2×#41×C197C22semidirect
57. C2×SD16×C197C22semidirect
59. C2×D8×C197C22semidirect
60. C2×M16×C197C22semidirect
120. D8×C197D4semidirect
169. D4×C197C2×D4semidirect
177. C8×C197C2×D4semidirect
208. Dic197AGL(1,ℤ/8)semidirect
218. C197#3272semidirect
221. C2×AGL(1,ℤ/8)D197semidirect
236. C2×#41C2×D197semidirect
247. C2×SD16C2×D197semidirect
250. C2×D8C2×D197semidirect
252. C2×M16C2×D197semidirect
320. D8C22⋊D197semidirect
375. D4C2×C22⋊D197semidirect
396. C8C2×C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633502
#271617282
D4×C22⋊D1972
#271614502
#271613252
#271613222
#271612822
#271612722
#271612302
#271612282
#271612242
#271612212
D8×Dic1972
#271611552
C2×AGL(1,ℤ/8)×C1972
#3272197