Finite Group Databasebeta

← #38752015

Subgroups — #38752015

Order 25216.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263371C211yesmaximal, characteristic, complemented
2C2×#20263346C211yesmaximal, characteristic, complemented
3#27161657C211yesmaximal, characteristic, complemented
4#27161238C212yesmaximal, complemented
5#27161228C212yesmaximal, complemented
6#27161221C212yesmaximal, complemented
7#27161211C212yesmaximal, complemented
8#27161081C211yesmaximal, characteristic, complemented
9C2×#20263247C211yesmaximal, characteristic, complemented
10#27161047C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C197C211yesmaximal, characteristic, complemented
12#20263371—24no—
13#20263371—24no—
14C2×#15691358C2211yescharacteristic
15C2×#15691358C2211yescharacteristic
16C2×#15691357—21no—
17C2×#15691357C2212yes—
18C22×C22⋊D197—22no—
19C2×D4×D197—21no—
20C2×D4×D197C2212yes—
21#20263346—24no—
22C2×#15691348C2211yescharacteristic, complemented
23C2×#15691346C2211yescharacteristic, complemented
24C2×#15691345C2211yescharacteristic, complemented
25C2×#15691343C2211yescharacteristic, complemented
26C22×#13096802C2211yescharacteristic, complemented
27#20263305—21no—
28#20263302C2212yescomplemented
29#20263300—21no—
30#20263298—22no—
31#20263297C2212yescomplemented
32C2×C4⋊Dic197C2211yescharacteristic, complemented
33#20263251C2212yescomplemented
34#20263248C2212yescomplemented
35#20263247C2212yescomplemented
36#20263247C2212yescomplemented
37#20263238C2212yescomplemented
38#20263238C2212yescomplemented
39C2×#41×C197C2211yescharacteristic, complemented
40C22×D4×C197C2211yescharacteristic
41AGL(1,ℤ/8)×C197—24no—
42C2×SD16×C197C2212yescomplemented
43C2×D8×C197C2212yescomplemented
44C2×M16×C197C2211yescharacteristic, complemented
45C23×D197—22no—
46#15691358—22no—
47#15691358—22no—
48#15691358—22no—
49#15691358D414yescomplemented
50#15691357—28no—
51#15691357—24no—
52#15691357—24no—
53C2×Dic394D411yescharacteristic
54C2×Dic394C2311yescharacteristic
55C2×D788D411yescharacteristic
56C2×D788C2311yescharacteristic
57C2×C22⋊D197—216no—
58C2×C22⋊D197—22no—
59C2×C22⋊D197—22no—
60C2×C22⋊D197—22no—
61C2×C22⋊D197—21no—
62C2×C22⋊D197D412yes—
63D4×D197—28no—
64D4×D197—24no—
65D4×D197—24no—
66C22×Dic197—22no—
67C22×Dic197—21no—
68C2×C4×D197—21no—
69C2×C4×D197C2312yes—
70#15691348—22no—
71#15691346—22no—
72#15691345—22no—
73#15691343—22no—
74C2×#13096802—22no—
75C2×#13096802C2311yescharacteristic
76C2×#13096802C2311yescharacteristic
77C4⋊Dic197—21no—
78C4⋊Dic197C2312yes—
79#15691337—22no—
80C22⋊Dic197—22no—
81#15691335—22no—
82C24×C197—21no—
83#41×C197—22no—
84#41×C197—22no—
85C2×Q8×C197C2311yescharacteristic
86C2×D4×C197—21no—
87C2×D4×C197—21no—
88C2×D4×C197—22no—
89C2×D4×C197D414yescomplemented
90C2×D4×C197C2311yescharacteristic
91C2×D4×C197C2312yes—
92C22×C4×C197—21no—
93C22×C4×C197C2311yescharacteristic
94SD16×C197—24no—
95D8×C197—24no—
96M16×C197—22no—
97C2×C8×C197C2312yes—
98C22×D197—21no—
99C22×D197—28no—
100C22×D197—82no—
101C22×D197—82no—
102C22×D197C2×D412yes—
103Dic394—22no—
104Dic394—21no—
105Dic394C2×D412yes—
106D788—22no—
107D788—21no—
108D788C2×D412yes—
109C22⋊D197—28no—
110C22⋊D197—28no—
111C22⋊D197—24no—
112C22⋊D197—24no—
113C22⋊D197—82no—
114C2×Dic197—28no—
115C2×Dic197—21no—
116C2×Dic197—22no—
117C2×Dic197—81no—
118C2×Dic197C2×D412yes—
119C4×D197—24no—
120C4×D197—22no—
121C4×D197—22no—
122#13096802—21no—
123#13096802—21no—
124C23×C197—22no—
125C23×C197—24no—
126C23×C197—21no—
127C23×C197—81no—
128C23×C197—81no—
129C23×C197C2×D411yescharacteristic
130Q8×C197—21no—
131Q8×C197—21no—
132D4×C197—24no—
133D4×C197—22no—
134D4×C197—21no—
135D4×C197—21no—
136D4×C197—42no—
137D4×C197C2×D414yes—
138C2×C4×C197—21no—
139C2×C4×C197—42no—
140C2×C4×C197C2×D411yescharacteristic
141C2×C4×C197C2×D412yes—
142C2×C4×C197C2×D412yes—
143C2×C4×C197C2411yescharacteristic
144C8×C197—22no—
145C22×C197—21no—
146C22×C197—21no—
147C22×C197—24no—
148C22×C197—42no—
149C22×C197—42no—
150C22×C197—41no—
151C22×C197—41no—
152C22×C197—41no—
153C22×C197—84no—
154C22×C197#12012yes—
155C22×C197C22×D411yescharacteristic
156C2×D197—22no—
157C2×D197—22no—
158C2×D197—22no—
159C2×D197—22no—
160C2×D197—41no—
161C2×D197—41no—
162C2×D197—88no—
163C4×C197—41no—
164C4×C197#12012yes—
165C4×C197C22×D411yescharacteristic
166C4×C197C22×D411yescharacteristic
167Dic197—22no—
168Dic197—22no—
169Dic197—42no—
170C2×C197—22no—
171C2×C197—44no—
172C2×C197—81no—
173C2×C197C2×#12011yescharacteristic
174C2×C197#57611yescharacteristic
175C2×C197#57711yescharacteristic
176D197—42no—
177D197—42no—
178C197#301211yescharacteristic, complemented, Sylow
179#3012—1971nomaximal, Sylow
180C2×H(5,𝔽2)—1971no—
181C2×AGL(1,ℤ/8)D19711yescharacteristic, complemented
182C2×#135—1971no—
183#537—1971no—
184#452—1972no—
185#450—1972no—
186#449—1972no—
187#447—1972no—
188#417—1971no—
189C2×#102—1971no—
190#408—1971no—
191H(5,𝔽2)—3944no—
192H(5,𝔽2)—3944no—
193C2×#41—3941no—
194C2×#41—1972no—
195C2×#41—1971no—
196C2×#41—1971no—
197C2×#41C2×D19711yescharacteristic, complemented
198C22×D4—3941no—
199C22×D4—3942no—
200C22×D4—1972no—
201C22×D4C2×D19711yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—3944no—
204C2×Q16—1971no—
205C2×SD16—1971no—
206C2×SD16—1971no—
207C2×SD16C2×D19712yescomplemented
208C2×D8—1971no—
209C2×D8C2×D19712yescomplemented
210C2×M16C2×D19711yescharacteristic, complemented
211C22×C8—1971no—
212C22⋊Q8—3941no—
213C22⋊Q8—1972no—
214#121—3941no—
215#121—3942no—
216#121—1972no—
217C2×C4⋊C4—1971no—
218#103—1972no—
219#102—1972no—
220#102—1972no—
221#102—1972no—
222C22⋊C8—1972no—
223C22⋊C8—1972no—
224C24—21no—
225C24—3942no—
226#41—22no—
227#41—22no—
228#41—3948no—
229#41—3944no—
230#41—3944no—
231#41—3942no—
232#41—3942no—
233#41—3942no—
234#41—1974no—
235C2×Q8—1971no—
236C2×Q8—1971no—
237C2×Q8C22×D19711yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—3948no—
242C2×D4—3944no—
243C2×D4—3942no—
244C2×D4—39416no—
245C2×D4—3942no—
246C2×D4—3941no—
247C2×D4—3944no—
248C2×D4—3942no—
249C2×D4—1972no—
250C2×D4—1971no—
251C2×D4—1971no—
252C2×D4C22⋊D19714yescomplemented
253C2×D4C22×D19711yescharacteristic
254C2×D4C22×D19712yes—
255C22×C4—21no—
256C22×C4—3941no—
257C22×C4—3942no—
258C22×C4—3941no—
259C22×C4—1972no—
260C22×C4C22×D19711yescharacteristic
261Q16—3942no—
262SD16—24no—
263SD16—3942no—
264SD16—3942no—
265D8—24no—
266D8—3942no—
267M16—22no—
268C2×C8—3942no—
269C2×C8—1971no—
270C2×C8—1971no—
271C2×C8C22×D19712yes—
272C4⋊C4—3942no—
273C4⋊C4—3941no—
274C4⋊C4—1972no—
275C22⋊C4—3942no—
276C22⋊C4—3942no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—15762no—
283C23—15762no—
284C23—3948no—
285C23—3941no—
286C23—1972no—
287C23D4×D19711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—3942no—
291Q8—3941no—
292Q8—1972no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—15762no—
299D4—3948no—
300D4—3948no—
301D4—3944no—
302D4—3944no—
303D4—3942no—
304D4—3941no—
305D4—1972no—
306D4C2×C22⋊D19714yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—15761no—
310C2×C4—3942no—
311C2×C4—3948no—
312C2×C4—3941no—
313C2×C4—3944no—
314C2×C4—3942no—
315C2×C4—3942no—
316C2×C4—1972no—
317C2×C4D4×D19711yescharacteristic
318C2×C4D4×D19712yes—
319C2×C4C2×C22⋊D19712yes—
320C2×C4C23×D19711yescharacteristic
321C8—22no—
322C8—3941no—
323C8—3941no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—7881no—
334C22—7881no—
335C22—15768no—
336C22—3942no—
337C22—3942no—
338C22—3942no—
339C22—3942no—
340C22#2026329412yes—
341C22C2×D4×D19711yescharacteristic
342C4—41no—
343C4—7882no—
344C4—3942no—
345C4—3942no—
346C4#2026329412yes—
347C4C2×D4×D19711yescharacteristic
348C4C22×C22⋊D19711yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—7882no—
353C2—7882no—
354C2C2×#2026329411yescharacteristic
355C2#2716188311yescharacteristic
356C2#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#3012
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×#20263371C2semidirect
2. C2×#20263346C2semidirect
3. #27161657C2semidirect
4. #27161238C2semidirect
5. #27161228C2semidirect
6. #27161221C2semidirect
7. #27161211C2semidirect
8. #27161081C2semidirect
9. C2×#20263247C2semidirect
10. #27161047C2semidirect
11. C2×AGL(1,ℤ/8)×C197C2semidirect
22. C2×#15691348C22semidirect
23. C2×#15691346C22semidirect
24. C2×#15691345C22semidirect
25. C2×#15691343C22semidirect
26. C22×#13096802C22semidirect
28. #20263302C22semidirect
31. #20263297C22semidirect
32. C2×C4⋊Dic197C22semidirect
33. #20263251C22semidirect
34. #20263248C22semidirect
35. #20263247C22semidirect
36. #20263247C22semidirect
37. #20263238C22semidirect
38. #20263238C22semidirect
39. C2×#41×C197C22semidirect
42. C2×SD16×C197C22semidirect
43. C2×D8×C197C22semidirect
44. C2×M16×C197C22semidirect
49. #15691358D4semidirect
89. C2×D4×C197D4semidirect
178. C197#3012semidirect
181. C2×AGL(1,ℤ/8)D197semidirect
197. C2×#41C2×D197semidirect
207. C2×SD16C2×D197semidirect
209. C2×D8C2×D197semidirect
210. C2×M16C2×D197semidirect
252. C2×D4C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633712
C2×#202633462
#271616572
#271612382
#271612282
#271612212
#271612112
#271610812
C2×#202632472
#271610472
C2×AGL(1,ℤ/8)×C1972
#3012197