Finite Group Databasebeta

← #38752173

Subgroups — #38752173

Order 25216.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×#20263352C211yesmaximal, characteristic, complemented
2#27161625C211yesmaximal, characteristic, complemented
3C2×D4×Dic197C211yesmaximal, characteristic, complemented
4#27161296C212yesmaximal, complemented
5#27161292C212yesmaximal, complemented
6#27161282C212yesmaximal, complemented
7#27161272C212yesmaximal, complemented
8#27161102C211yesmaximal, characteristic, complemented
9#27161081C211yesmaximal, characteristic, complemented
10#27161080C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2211yescharacteristic
13C23×Dic197—21no—
14#20263352—24no—
15C2×#15691346C2212yes—
16C2×#15691343C2212yes—
17C2×#15691342C2211yescharacteristic
18#20263324C2211yescharacteristic, complemented
19#20263322C2211yescharacteristic, complemented
20#20263315C2211yescharacteristic, complemented
21#20263313C2211yescharacteristic, complemented
22#20263298—21no—
23#20263298—21no—
24D4×Dic197—21no—
25D4×Dic197—21no—
26D4×Dic197C2212yes—
27D4×Dic197C2212yes—
28C4×C22⋊D197—21no—
29C4×C22⋊D197—21no—
30C2×C4⋊Dic197C2211yescharacteristic
31C2×C22⋊Dic197—21no—
32C2×C4×Dic197C2211yescharacteristic
33#20263255C2212yescomplemented
34#20263255C2212yescomplemented
35#20263251C2212yescomplemented
36#20263250C2212yescomplemented
37#20263248C2212yescomplemented
38#20263247C2212yescomplemented
39C2×#41×C197C2211yescharacteristic
40C22×D4×C197C2211yescharacteristic
41AGL(1,ℤ/8)×C197—24no—
42C2×SD16×C197C2212yes—
43C2×D8×C197C2212yes—
44C2×M16×C197C2211yescharacteristic
45#15691358—22no—
46#15691358—22no—
47C2×Dic394C2311yescharacteristic
48C2×D788C2311yescharacteristic
49C2×C22⋊D197—21no—
50C2×C22⋊D197—21no—
51C2×C22⋊D197—21no—
52C22×Dic197—22no—
53C22×Dic197—22no—
54C22×Dic197—21no—
55C22×Dic197—44no—
56C22×Dic197D411yescharacteristic
57C22×Dic197D411yescharacteristic
58C2×C4×D197—21no—
59#15691346—24no—
60#15691343—24no—
61#15691342—22no—
62C2×#13096802C2312yes—
63C4⋊Dic197—21no—
64C4⋊Dic197C2312yes—
65#15691337—21no—
66#15691337—21no—
67C22⋊Dic197—22no—
68C22⋊Dic197—22no—
69C22⋊Dic197—21no—
70C22⋊Dic197—21no—
71#15691335—21no—
72#15691335—21no—
73C4×Dic197D411yescharacteristic
74C4×Dic197D411yescharacteristic
75C4×Dic197C2311yescharacteristic
76C4×Dic197C2311yescharacteristic
77C24×C197—21no—
78#41×C197—22no—
79#41×C197—22no—
80C2×Q8×C197C2311yescharacteristic
81C2×D4×C197—21no—
82C2×D4×C197—21no—
83C2×D4×C197—22no—
84C2×D4×C197D414yescomplemented
85C2×D4×C197C2311yescharacteristic
86C2×D4×C197C2312yes—
87C22×C4×C197—21no—
88C22×C4×C197C2311yescharacteristic
89SD16×C197—24no—
90D8×C197—24no—
91M16×C197—22no—
92C2×C8×C197C2312yes—
93C22×D197—21no—
94Dic394—21no—
95Dic394—21no—
96D788—21no—
97D788—21no—
98C22⋊D197—42no—
99C22⋊D197—42no—
100C22⋊D197—42no—
101C2×Dic197—22no—
102C2×Dic197—21no—
103C2×Dic197—22no—
104C2×Dic197—41no—
105C2×Dic197—44no—
106C2×Dic197—42no—
107C2×Dic197—44no—
108C2×Dic197C2×D411yescharacteristic
109C2×Dic197C2×D411yescharacteristic
110C2×Dic197C2×D411yescharacteristic
111C2×Dic197C2×D411yescharacteristic
112C4×D197—42no—
113#13096802—22no—
114C23×C197—24no—
115C23×C197—22no—
116C23×C197—21no—
117C23×C197—42no—
118C23×C197—42no—
119C23×C197C2×D411yescharacteristic
120Q8×C197—21no—
121Q8×C197—21no—
122D4×C197—24no—
123D4×C197—21no—
124D4×C197—22no—
125D4×C197—21no—
126D4×C197—42no—
127D4×C197C2×D414yes—
128C2×C4×C197—21no—
129C2×C4×C197—42no—
130C2×C4×C197C2×D411yescharacteristic
131C2×C4×C197C2×D412yes—
132C2×C4×C197#4112yes—
133C2×C4×C197C2411yescharacteristic
134C8×C197—22no—
135C22×C197—21no—
136C22×C197—21no—
137C22×C197—24no—
138C22×C197—48no—
139C22×C197—42no—
140C22×C197—42no—
141C22×C197—41no—
142C22×C197—41no—
143C22×C197—41no—
144C22×C197#12112yes—
145C22×C197C22×D411yescharacteristic
146C2×D197—41no—
147C2×D197—41no—
148C2×D197—41no—
149C4×C197—41no—
150C4×C197#12112yes—
151C4×C197C22×D411yescharacteristic
152C4×C197C2×#4111yescharacteristic
153Dic197—21no—
154Dic197—21no—
155Dic197—21no—
156Dic197—41no—
157Dic197—42no—
158Dic197AGL(1,ℤ/8)12yescomplemented
159C2×C197—22no—
160C2×C197—44no—
161C2×C197—81no—
162C2×C197C2×#12111yescharacteristic
163C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
164C2×C197#57511yescharacteristic
165D197—81no—
166C197#303611yescharacteristic, complemented, Sylow
167#3036—1971nomaximal, Sylow
168C2×AGL(1,ℤ/8)—1971no—
169C2×AGL(1,ℤ/8)D19711yescharacteristic, complemented
170#532—1971no—
171C2×C4×D4—1971no—
172#463—1972no—
173#463—1972no—
174#461—1972no—
175#459—1972no—
176#423—1971no—
177#417—1971no—
178#417—1971no—
179C2×#41—1971no—
180C2×#41C2×D19711yescharacteristic
181C22×D4C2×D19711yescharacteristic
182C23×C4—3941no—
183AGL(1,ℤ/8)—24no—
184AGL(1,ℤ/8)—3944no—
185C2×SD16—1972no—
186C2×SD16C2×D19712yes—
187C2×D8—1972no—
188C2×D8C2×D19712yes—
189C2×M16—1971no—
190C2×M16C2×D19711yescharacteristic
191C4⋊Q8—1971no—
192#127—1971no—
193#124—1971no—
194#124—1971no—
195#121—3941no—
196#121—3941no—
197C4×D4—3941no—
198C4×D4—3941no—
199C4×D4—3941no—
200C4×D4—3941no—
201C4×D4—1972no—
202C4×D4—1972no—
203C2×C4⋊C4—1971no—
204C2×C22⋊C4—3941no—
205C2×C42—1971no—
206C4⋊C8—1972no—
207C4⋊C8—1972no—
208#103—1972no—
209#103—1972no—
210#102—1972no—
211#102—1972no—
212C24—21no—
213#41—22no—
214#41—22no—
215#41—3942no—
216#41—3942no—
217C2×Q8—1971no—
218C2×Q8C22×D19711yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—3941no—
223C2×D4—3941no—
224C2×D4—3941no—
225C2×D4—1971no—
226C2×D4C22⋊D19714yescomplemented
227C2×D4C22×D19711yescharacteristic
228C2×D4C22×D19712yes—
229C22×C4—21no—
230C22×C4—7884no—
231C22×C4—3941no—
232C22×C4—3942no—
233C22×C4—3942no—
234C22×C4—3941no—
235C22×C4—1971no—
236C22×C4—1971no—
237C22×C4C22×D19711yescharacteristic
238SD16—24no—
239SD16—3944no—
240D8—24no—
241D8—3944no—
242M16—22no—
243M16—3942no—
244C2×C8—1972no—
245C2×C8C22×D19712yes—
246C4⋊C4—3941no—
247C4⋊C4—3941no—
248C4⋊C4—3941no—
249C4⋊C4—1972no—
250C22⋊C4—3941no—
251C22⋊C4—3941no—
252C22⋊C4—3942no—
253C22⋊C4—3942no—
254C22⋊C4—3941no—
255C22⋊C4—3941no—
256C42—1971no—
257C42—1971no—
258C42—1971no—
259C42—1971no—
260C23—24no—
261C23—22no—
262C23—21no—
263C23—42no—
264C23—42no—
265C23—3941no—
266C23D4×D19711yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—3941no—
270Q8—3941no—
271D4—24no—
272D4—21no—
273D4—22no—
274D4—21no—
275D4—42no—
276D4—7882no—
277D4—7882no—
278D4—7882no—
279D4—3941no—
280D4—3941no—
281D4C2×C22⋊D19714yes—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—7882no—
285C2×C4—7884no—
286C2×C4—7884no—
287C2×C4—7882no—
288C2×C4—7881no—
289C2×C4—3941no—
290C2×C4—3942no—
291C2×C4—3942no—
292C2×C4—1971no—
293C2×C4—1971no—
294C2×C4—1971no—
295C2×C4—1971no—
296C2×C4D4×D19711yescharacteristic
297C2×C4C2×C22⋊D19712yes—
298C2×C4#1569135712yes—
299C2×C4C23×D19711yescharacteristic
300C8—22no—
301C8—3942no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—48no—
306C22—42no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—7881no—
312C22—7881no—
313C22—7881no—
314C22#2026329812yes—
315C22C2×D4×D19711yescharacteristic
316C4—41no—
317C4—7882no—
318C4—7881no—
319C4—3941no—
320C4—3941no—
321C4—3941no—
322C4—1972no—
323C4#2026329812yes—
324C4C22×C22⋊D19711yescharacteristic
325C4C2×#1569135711yescharacteristic
326C2—22no—
327C2—44no—
328C2—81no—
329C2—15761no—
330C2C2×#2026329811yescharacteristic
331C2AGL(1,ℤ/8)×D19711yescharacteristic
332C2#2716187511yescharacteristic
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#3036
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×#20263352C2semidirect
2. #27161625C2semidirect
3. C2×D4×Dic197C2semidirect
4. #27161296C2semidirect
5. #27161292C2semidirect
6. #27161282C2semidirect
7. #27161272C2semidirect
8. #27161102C2semidirect
9. #27161081C2semidirect
10. #27161080C2semidirect
11. C2×AGL(1,ℤ/8)×C197C2semidirect
18. #20263324C22semidirect
19. #20263322C22semidirect
20. #20263315C22semidirect
21. #20263313C22semidirect
33. #20263255C22semidirect
34. #20263255C22semidirect
35. #20263251C22semidirect
36. #20263250C22semidirect
37. #20263248C22semidirect
38. #20263247C22semidirect
84. C2×D4×C197D4semidirect
158. Dic197AGL(1,ℤ/8)semidirect
166. C197#3036semidirect
169. C2×AGL(1,ℤ/8)D197semidirect
226. C2×D4C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633522
#271616252
C2×D4×Dic1972
#271612962
#271612922
#271612822
#271612722
#271611022
#271610812
#271610802
C2×AGL(1,ℤ/8)×C1972
#3036197