Finite Group Databasebeta

← #27151002

Subgroups — #27151002

Order 12416.

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

normal subgroups only

This group has 784 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×#15690856C211yesmaximal, characteristic, complemented
2C2×#15690836C211yesmaximal, characteristic, complemented
3C22×#13095496C211yesmaximal, characteristic, complemented
4#20260878C214yesmaximal, complemented
5#20260868C214yesmaximal, complemented
6#20260739C211yesmaximal, characteristic, complemented
7#20260729C211yesmaximal, characteristic, complemented
8#20260702C211yesmaximal, characteristic, complemented
9#535×C97C211yesmaximal, characteristic, complemented
10#15690856—24no—
11#15690856—24no—
12C2×#13095512—21no—
13C2×#13095511C2211yescharacteristic
14C2×#13095511C2212yescomplemented
15C22×D388C2211yescharacteristic
16C22×D388C2212yescomplemented
17C2×D4×D97—22no—
18C2×D4×D97—21no—
19#15690836—24no—
20C2×#13095499C2212yescomplemented
21C2×#13095496—24no—
22C2×#13095496C2212yescomplemented
23C2×#13095496C2214yescomplemented
24C2×#13095495C2211yescharacteristic, complemented
25C22×#11715895C2211yescharacteristic, complemented
26#15690733C2212yescomplemented
27#15690730C2212yescomplemented
28#15690730C2214yescomplemented
29C22⋊#11715895C2214yescomplemented
30C2×#41×C97C2211yescharacteristic
31C22×D4×C97C2211yescharacteristic, complemented
32#127×C97—21no—
33#124×C97—21no—
34#121×C97C2214yescomplemented
35#120×C97—22no—
36#117×C97C2211yescharacteristic, complemented
37C23×D97—21no—
38C23×D97—22no—
39#13095512—28no—
40#13095511—22no—
41#13095511—24no—
42#13095511—24no—
43#13095511D414yescomplemented
44C2×Dic194C2311yescharacteristic
45C2×D388—21no—
46C2×D388—22no—
47C2×D388—22no—
48C2×D388—24no—
49C2×D388—24no—
50C2×D388D414yescomplemented
51C2×D388C2312yes—
52C2×D388C2311yescharacteristic
53C2×C22⋊D97—21no—
54C2×C22⋊D97—22no—
55D4×D97—216no—
56D4×D97—28no—
57C2×C4×D97—21no—
58C2×C4×D97—22no—
59#13095499—24no—
60#13095496—28no—
61#13095496—24no—
62#13095495—22no—
63C2×#11715895—22no—
64C2×#11715895C2312yes—
65C2×#11715895C2312yes—
66C24×C97—21no—
67#41×C97—22no—
68#41×C97—22no—
69C2×Q8×C97D411yescharacteristic
70C2×D4×C97—24no—
71C2×D4×C97—22no—
72C2×D4×C97—22no—
73C2×D4×C97—21no—
74C2×D4×C97D412yes—
75C2×D4×C97D411yescharacteristic
76C2×D4×C97C2312yes—
77C22×C4×C97C2312yes—
78C22×C4×C97C2311yescharacteristic
79C4⋊C4×C97C2312yes—
80C22⋊C4×C97—21no—
81C22⋊C4×C97—24no—
82C42×C97—21no—
83C23×C97—22no—
84C23×C97—42no—
85C23×C97—82no—
86C23×C97C2×D412yes—
87C23×C97C2×D411yescharacteristic
88C22×D97—21no—
89C22×D97—22no—
90C22×D97—28no—
91C22×D97—24no—
92C22×D97—82no—
93C22×D97—84no—
94Q8×C97—22no—
95D4×C97—24no—
96D4×C97—22no—
97D4×C97—22no—
98D4×C97—22no—
99D4×C97—42no—
100D4×C97—84no—
101C2×C4×C97—22no—
102C2×C4×C97—22no—
103C2×C4×C97—22no—
104C2×C4×C97C2×D412yes—
105C2×C4×C97C2×D412yes—
106C2×C4×C97C2×D412yes—
107C2×C4×C97C2×D411yescharacteristic
108C2×C4×C97C2411yescharacteristic
109C22⋊D97—28no—
110C22⋊D97—24no—
111C2×Dic97—21no—
112D388—28no—
113D388—24no—
114D388—21no—
115D388—22no—
116D388C2×D412yes—
117D388C2×D414yes—
118C4×D97—28no—
119C4×D97—24no—
120Dic194—21no—
121Dic194C2×D412yes—
122#11715895—22no—
123#11715895—22no—
124C22×C97—22no—
125C22×C97—22no—
126C22×C97—24no—
127C22×C97—42no—
128C22×C97—44no—
129C22×C97—84no—
130C22×C97#12012yes—
131C22×C97C22×D411yescharacteristic
132C2×D97—24no—
133C2×D97—22no—
134C2×D97—44no—
135C2×D97—42no—
136C2×D97—84no—
137C2×D97—88no—
138C4×C97—22no—
139C4×C97—42no—
140C4×C97#12012yes—
141C4×C97C22×D411yescharacteristic
142C4×C97C22×D411yescharacteristic
143Dic97—22no—
144C2×C97—22no—
145C2×C97—42no—
146C2×C97—82no—
147C2×C97C2×#12011yescharacteristic
148C2×C97#57612yes—
149D97—44no—
150D97—42no—
151#3008—971nomaximal, Sylow
152C97#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)—971no—
154C2×AGL(1,ℤ/8)—971no—
155C22×D8—971no—
156#535D9711yescharacteristic, complemented
157#449—974no—
158#447—974no—
159#418—971no—
160#416—971no—
161#408—971no—
162H(5,𝔽2)—1944no—
163H(5,𝔽2)—1944no—
164C2×#41—971no—
165C2×#41—972no—
166C2×#41—1941no—
167C2×#41C2×D9711yescharacteristic
168C22×D4—971no—
169C22×D4—972no—
170C22×D4—1942no—
171C22×D4—1941no—
172C22×D4C2×D9711yescharacteristic, complemented
173AGL(1,ℤ/8)—1944no—
174C2×SD16—972no—
175C2×D8—974no—
176C2×D8—972no—
177C2×D8—1944no—
178C2×M16—971no—
179C22×C8—971no—
180#127—21no—
181#124—21no—
182#121C2×D9714yescomplemented
183#120—22no—
184#117C2×D9711yescharacteristic, complemented
185#103—972no—
186#102—972no—
187#102—974no—
188C22⋊C8—974no—
189C24—21no—
190C24—1942no—
191C24—1941no—
192#41—22no—
193#41—22no—
194#41—974no—
195#41—1944no—
196#41—1944no—
197#41—1948no—
198#41—1942no—
199C2×Q8—971no—
200C2×Q8C22⋊D9711yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—974no—
206C2×D4—972no—
207C2×D4—971no—
208C2×D4—1944no—
209C2×D4—1944no—
210C2×D4—19416no—
211C2×D4—1948no—
212C2×D4—1942no—
213C2×D4—1941no—
214C2×D4—1942no—
215C2×D4—1942no—
216C2×D4—1941no—
217C2×D4C22⋊D9712yes—
218C2×D4C22⋊D9711yescharacteristic
219C2×D4C22×D9712yes—
220C22×C4—1942no—
221C22×C4—1941no—
222C22×C4C22×D9712yes—
223C22×C4C22×D9711yescharacteristic
224SD16—1944no—
225D8—1948no—
226D8—1944no—
227M16—1942no—
228C2×C8—972no—
229C2×C8—972no—
230C2×C8—1942no—
231C4⋊C4C22×D9712yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—7762no—
239C23—7764no—
240C23—1948no—
241C23—1944no—
242C23—1942no—
243C23—1941no—
244C23C2×C22⋊D9712yes—
245C23C2×C22⋊D9711yescharacteristic
246Q8—22no—
247Q8—972no—
248Q8—1941no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—974no—
256D4—972no—
257D4—1948no—
258D4—1948no—
259D4—1944no—
260D4—1944no—
261D4—1941no—
262D4—1942no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—1948no—
267C2×C4—1941no—
268C2×C4—1944no—
269C2×C4D4×D9712yes—
270C2×C4D4×D9712yes—
271C2×C4C2×C22⋊D9712yes—
272C2×C4C2×C22⋊D9711yescharacteristic
273C2×C4C23×D9711yescharacteristic
274C8—1942no—
275C8—1942no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—7764no—
283C22—7768no—
284C22—3884no—
285C22—3882no—
286C22—1944no—
287C22—1942no—
288C22#1569077812yes—
289C22C22×C22⋊D9711yescharacteristic
290C4—22no—
291C4—42no—
292C4—1942no—
293C4#1569077812yes—
294C4C2×D4×D9711yescharacteristic
295C4C2×D4×D9711yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—3884no—
300C2—3882no—
301C2C2×#1569077811yescharacteristic
302C2#2026153812yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#535×C97C2
socleC22×C97C22×D4
hypercenter#535D97
hypocenterC97#3008
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#15690856C2semidirect
2. C2×#15690836C2semidirect
3. C22×#13095496C2semidirect
4. #20260878C2semidirect
5. #20260868C2semidirect
6. #20260739C2semidirect
7. #20260729C2semidirect
8. #20260702C2semidirect
9. #535×C97C2semidirect
14. C2×#13095511C22semidirect
16. C22×D388C22semidirect
20. C2×#13095499C22semidirect
22. C2×#13095496C22semidirect
23. C2×#13095496C22semidirect
24. C2×#13095495C22semidirect
25. C22×#11715895C22semidirect
26. #15690733C22semidirect
27. #15690730C22semidirect
28. #15690730C22semidirect
29. C22⋊#11715895C22semidirect
31. C22×D4×C97C22semidirect
34. #121×C97C22semidirect
36. #117×C97C22semidirect
43. #13095511D4semidirect
50. C2×D388D4semidirect
152. C97#3008semidirect
156. #535D97semidirect
172. C22×D4C2×D97semidirect
182. #121C2×D97semidirect
184. #117C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908562
C2×#156908362
C22×#130954962
#202608782
#202608682
#202607392
#202607292
#202607022
#535×C972
#300897