Finite Group Databasebeta

← #38752210

Subgroups — #38752210

Order 25216.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263350C211yesmaximal, characteristic, complemented
2C2×D8×D197C211yesmaximal, characteristic, complemented
3#27161642C211yesmaximal, characteristic, complemented
4#27161551C211yesmaximal, characteristic, complemented
5#27161295C211yesmaximal, characteristic, complemented
6#27161294C211yesmaximal, characteristic, complemented
7#27161293C211yesmaximal, characteristic, complemented
8#27161292C211yesmaximal, characteristic, complemented
9#27161273C211yesmaximal, characteristic, complemented
10#27161272C211yesmaximal, characteristic, complemented
11#27161271C211yesmaximal, characteristic, complemented
12#27161104C211yesmaximal, characteristic, complemented
13#27161082C211yesmaximal, characteristic, complemented
14#27161072C211yesmaximal, characteristic, complemented
15#459×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic
17C2×D4×D197C2211yescharacteristic
18C2×D4×D197C2211yescharacteristic
19#20263350—24no—
20C2×#15691347C2211yescharacteristic, complemented
21C2×#15691346C2211yescharacteristic, complemented
22C2×D1576C2211yescharacteristic
23C2×#15691343C2211yescharacteristic
24C2×#15691343C2211yescharacteristic
25C2×#15691343C2211yescharacteristic, complemented
26D8×D197—24no—
27C2×#15691341C2211yescharacteristic
28C2×C8×D197C2211yescharacteristic
29#20263322C2211yescharacteristic, complemented
30#20263314C2211yescharacteristic, complemented
31#20263313C2211yescharacteristic, complemented
32#20263298—21no—
33#20263298—21no—
34#20263294—21no—
35#20263294—21no—
36C4×D788C2211yescharacteristic, complemented
37D4×Dic197C2211yescharacteristic, complemented
38#20263288C2211yescharacteristic, complemented
39C4×C22⋊D197—21no—
40#20263286—21no—
41#20263284C2211yescharacteristic
42#20263283C2211yescharacteristic
43C2×#15691335—21no—
44C22⋊C4×D197—21no—
45C4⋊#13096802C2211yescharacteristic, complemented
46#20263255C2211yescharacteristic, complemented
47#20263255C2211yescharacteristic, complemented
48#20263251C2211yescharacteristic, complemented
49#20263249C2211yescharacteristic, complemented
50#20263248C2211yescharacteristic, complemented
51#20263247C2211yescharacteristic, complemented
52#20263247C2211yescharacteristic, complemented
53#20263246C2211yescharacteristic, complemented
54C2×D8×C197C2211yescharacteristic
55C2×D8×C197C2211yescharacteristic, complemented
56#127×C197C2211yescharacteristic, complemented
57C4×D4×C197C2211yescharacteristic, complemented
58C4⋊C8×C197C2211yescharacteristic, complemented
59#102×C197C2211yescharacteristic, complemented
60#102×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62C23×D197—21no—
63#15691357—22no—
64#15691357—22no—
65C2×Dic394C2311yescharacteristic
66C2×D788C2311yescharacteristic
67C2×D788C2311yescharacteristic
68C2×C22⋊D197—21no—
69C2×C22⋊D197—21no—
70C2×C22⋊D197—21no—
71C2×C22⋊D197—21no—
72C2×C22⋊D197—21no—
73D4×D197—22no—
74D4×D197—22no—
75D4×D197—22no—
76D4×D197D414yescomplemented
77C22×Dic197—21no—
78C22×Dic197—21no—
79C2×C4×D197—21no—
80C2×C4×D197—21no—
81C2×C4×D197C2311yescharacteristic
82#15691347—22no—
83#15691346—22no—
84D1576—22no—
85#15691343—22no—
86#15691343—22no—
87#15691343—22no—
88#15691341—22no—
89C2×#13096802C2311yescharacteristic
90C2×#13096802C2311yescharacteristic
91C8×D197—22no—
92C4⋊Dic197C2311yescharacteristic
93#15691337D411yescharacteristic
94#15691337D411yescharacteristic
95C22⋊Dic197—21no—
96C22⋊Dic197—21no—
97C22⋊Dic197—21no—
98#15691335—21no—
99#15691335—21no—
100#15691335—21no—
101#15691335—21no—
102#15691335—21no—
103#15691335D411yescharacteristic
104#15691335D411yescharacteristic
105C4×Dic197—21no—
106C4×Dic197C2311yescharacteristic
107C2×D4×C197—21no—
108C2×D4×C197—21no—
109C2×D4×C197C2311yescharacteristic
110C2×D4×C197C2311yescharacteristic
111C2×D4×C197C2311yescharacteristic
112C22×C4×C197—21no—
113D8×C197—22no—
114D8×C197—22no—
115C2×C8×C197C2311yescharacteristic
116C2×C8×C197C2311yescharacteristic
117C4⋊C4×C197C2311yescharacteristic
118C22⋊C4×C197—21no—
119C42×C197C2311yescharacteristic
120C22×D197—21no—
121C22×D197—24no—
122C22×D197—21no—
123C22×D197—42no—
124C22×D197—81no—
125C22×D197—81no—
126C22×D197—81no—
127C22×D197—81no—
128C22×D197C2×D411yescharacteristic
129Dic394—21no—
130Dic394—21no—
131D788—21no—
132D788—21no—
133D788—21no—
134D788C2×D412yes—
135C22⋊D197—24no—
136C22⋊D197—42no—
137C22⋊D197—42no—
138C22⋊D197—81no—
139C22⋊D197—81no—
140C2×Dic197—21no—
141C2×Dic197—21no—
142C2×Dic197—42no—
143C2×Dic197—81no—
144C2×Dic197C2×D411yescharacteristic
145C2×Dic197C2×D411yescharacteristic
146C2×Dic197C2×D411yescharacteristic
147C4×D197—81no—
148C4×D197—81no—
149C4×D197C2×D412yes—
150C4×D197#4112yes—
151#13096802—21no—
152#13096802—21no—
153C23×C197—21no—
154C23×C197—21no—
155C23×C197—21no—
156D4×C197—21no—
157D4×C197—21no—
158D4×C197—21no—
159D4×C197—21no—
160D4×C197—21no—
161D4×C197—81no—
162D4×C197—81no—
163D4×C197C2×D412yes—
164C2×C4×C197—21no—
165C2×C4×C197—81no—
166C2×C4×C197C2×D411yescharacteristic
167C2×C4×C197C2×D411yescharacteristic
168C2×C4×C197C2411yescharacteristic
169C8×C197—21no—
170C8×C197—21no—
171C22×C197—22no—
172C22×C197—41no—
173C22×C197—41no—
174C22×C197—41no—
175C22×C197—41no—
176C22×C197—41no—
177C22×C197—41no—
178C22×C197—41no—
179C22×C197—41no—
180C22×C197C22×D411yescharacteristic
181C2×D197—21no—
182C2×D197—21no—
183C2×D197—22no—
184C2×D197—41no—
185C2×D197—41no—
186C2×D197—41no—
187C2×D197—41no—
188C2×D197—41no—
189C2×D197—84no—
190C2×D197—84no—
191C2×D197#12112yes—
192C4×C197—21no—
193C4×C197—21no—
194C4×C197—41no—
195C4×C197C22×D411yescharacteristic
196C4×C197C2×#4111yescharacteristic
197Dic197—21no—
198Dic197—21no—
199Dic197—41no—
200Dic197—41no—
201Dic197#12112yes—
202C2×C197—42no—
203C2×C197—81no—
204C2×C197—81no—
205C2×C197C2×#12111yescharacteristic
206C2×C197#57611yescharacteristic
207C2×C197#57611yescharacteristic
208D197—22no—
209D197—42no—
210D197—81no—
211C197#304111yescharacteristic, complemented, Sylow
212#3041—1971nomaximal, Sylow
213C2×AGL(1,ℤ/8)—1971no—
214C22×D8—1971no—
215#535—1971no—
216#518—1971no—
217#463—1971no—
218#463—1971no—
219#463—1971no—
220#463—1971no—
221#459—1971no—
222#459—1971no—
223#459—1971no—
224#459D19711yescharacteristic, complemented
225#424—1971no—
226#418—1971no—
227#416—1971no—
228C2×#41—1971no—
229C22×D4—1971no—
230C22×D4—1971no—
231AGL(1,ℤ/8)—3944no—
232C2×SD16—1971no—
233C2×SD16—1971no—
234C2×D8—3944no—
235C2×D8—1971no—
236C2×D8—1971no—
237C2×D8—1971no—
238C2×D8—1971no—
239C2×D8C2×D19711yescharacteristic
240C2×D8C2×D19711yescharacteristic, complemented
241C2×M16—1971no—
242C22×C8—1971no—
243#127—1971no—
244#127C2×D19711yescharacteristic, complemented
245#124—1971no—
246#124—1971no—
247#121—3941no—
248#121—3941no—
249#120—3941no—
250#120—3941no—
251C4×D4—3941no—
252C4×D4—3941no—
253C4×D4—1971no—
254C4×D4—1971no—
255C4×D4—1971no—
256C4×D4C2×D19711yescharacteristic, complemented
257#117—1971no—
258#117—1971no—
259C2×C22⋊C4—3941no—
260C2×C22⋊C4—3941no—
261C4⋊C8—1971no—
262C4⋊C8—1971no—
263C4⋊C8—1971no—
264C4⋊C8C2×D19711yescharacteristic, complemented
265#103—1971no—
266#103—1971no—
267#102—1971no—
268#102—1971no—
269#102—1971no—
270#102—1971no—
271#102C2×D19711yescharacteristic, complemented
272#102C2×D19711yescharacteristic, complemented
273C24—3941no—
274C24—3941no—
275#41—3942no—
276#41—3942no—
277C2×Q8—1971no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—3941no—
281C2×D4—3942no—
282C2×D4—3942no—
283C2×D4—3942no—
284C2×D4—3941no—
285C2×D4—3941no—
286C2×D4—3941no—
287C2×D4—3941no—
288C2×D4—1974no—
289C2×D4—1971no—
290C2×D4—1971no—
291C2×D4C22×D19711yescharacteristic
292C2×D4C22×D19711yescharacteristic
293C2×D4C22×D19711yescharacteristic
294C22×C4—21no—
295C22×C4—3941no—
296C22×C4—3941no—
297C22×C4—3941no—
298C22×C4—3941no—
299C22×C4—1971no—
300SD16—3942no—
301SD16—3942no—
302D8—22no—
303D8—22no—
304D8—3942no—
305D8—3942no—
306D8—3942no—
307D8—3942no—
308M16—3942no—
309C2×C8—3942no—
310C2×C8—1971no—
311C2×C8—1971no—
312C2×C8C22×D19711yescharacteristic
313C2×C8C22×D19711yescharacteristic
314C4⋊C4—1971no—
315C4⋊C4—1971no—
316C4⋊C4—1971no—
317C4⋊C4C22×D19711yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—3941no—
320C22⋊C4—3941no—
321C22⋊C4—3941no—
322C22⋊C4—3941no—
323C22⋊C4—3941no—
324C22⋊C4—3941no—
325C22⋊C4—3941no—
326C22⋊C4—3941no—
327C22⋊C4—1971no—
328C22⋊C4—1971no—
329C42—3941no—
330C42—1971no—
331C42C22×D19711yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—7882no—
336C23—15761no—
337C23—15761no—
338C23—15761no—
339C23—15761no—
340C23—3944no—
341C23—3941no—
342C23—3941no—
343C23—1971no—
344Q8—3941no—
345Q8—3941no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—81no—
352D4—81no—
353D4—7882no—
354D4—7882no—
355D4—15761no—
356D4—15761no—
357D4—3944no—
358D4—3941no—
359D4—3941no—
360D4—3941no—
361D4—1972no—
362D4D4×D19712yes—
363C2×C4—21no—
364C2×C4—81no—
365C2×C4—7882no—
366C2×C4—15761no—
367C2×C4—15761no—
368C2×C4—15761no—
369C2×C4—3941no—
370C2×C4—3941no—
371C2×C4—1971no—
372C2×C4—1971no—
373C2×C4—1971no—
374C2×C4—1972no—
375C2×C4—1972no—
376C2×C4D4×D19711yescharacteristic
377C2×C4D4×D19711yescharacteristic
378C2×C4C23×D19711yescharacteristic
379C8—21no—
380C8—21no—
381C8—3941no—
382C8—3941no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—7881no—
393C22—7881no—
394C22—7881no—
395C22—7881no—
396C22—7881no—
397C22—15764no—
398C22—15764no—
399C22—3942no—
400C22—3941no—
401C22—3941no—
402C22—1972no—
403C22C2×D4×D19711yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—7881no—
408C4—7881no—
409C4—3941no—
410C4—3941no—
411C4—1972no—
412C4C2×D4×D19711yescharacteristic
413C4#41×D19711yescharacteristic
414C2—42no—
415C2—81no—
416C2—81no—
417C2—7882no—
418C2—15761no—
419C2—3942no—
420C2#121×D19711yescharacteristic
421C2#2716188011yescharacteristic
422C2#2716188311yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#459×C197C2
socleC22×C197C22×D4
hypercenter#459D197
hypocenterC197#3041
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#20263350C2semidirect
2. C2×D8×D197C2semidirect
3. #27161642C2semidirect
4. #27161551C2semidirect
5. #27161295C2semidirect
6. #27161294C2semidirect
7. #27161293C2semidirect
8. #27161292C2semidirect
9. #27161273C2semidirect
10. #27161272C2semidirect
11. #27161271C2semidirect
12. #27161104C2semidirect
13. #27161082C2semidirect
14. #27161072C2semidirect
15. #459×C197C2semidirect
20. C2×#15691347C22semidirect
21. C2×#15691346C22semidirect
25. C2×#15691343C22semidirect
29. #20263322C22semidirect
30. #20263314C22semidirect
31. #20263313C22semidirect
36. C4×D788C22semidirect
37. D4×Dic197C22semidirect
38. #20263288C22semidirect
45. C4⋊#13096802C22semidirect
46. #20263255C22semidirect
47. #20263255C22semidirect
48. #20263251C22semidirect
49. #20263249C22semidirect
50. #20263248C22semidirect
51. #20263247C22semidirect
52. #20263247C22semidirect
53. #20263246C22semidirect
55. C2×D8×C197C22semidirect
56. #127×C197C22semidirect
57. C4×D4×C197C22semidirect
58. C4⋊C8×C197C22semidirect
59. #102×C197C22semidirect
60. #102×C197C22semidirect
76. D4×D197D4semidirect
211. C197#3041semidirect
224. #459D197semidirect
240. C2×D8C2×D197semidirect
244. #127C2×D197semidirect
256. C4×D4C2×D197semidirect
264. C4⋊C8C2×D197semidirect
271. #102C2×D197semidirect
272. #102C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633502
C2×D8×D1972
#271616422
#271615512
#271612952
#271612942
#271612932
#271612922
#271612732
#271612722
#271612712
#271611042
#271610822
#271610722
#459×C1972
#3041197