Finite Group Databasebeta

← #27151004

Subgroups — #27151004

Order 12416.

784 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15690835C211yesmaximal, characteristic, complemented
2C22×#13095496C211yesmaximal, characteristic, complemented
3#20261299C211yesmaximal, characteristic, complemented
4#20260880C214yesmaximal, complemented
5#20260869C214yesmaximal, complemented
6#20260740C211yesmaximal, characteristic, complemented
7#20260731C211yesmaximal, characteristic, complemented
8#20260702C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C97C211yesmaximal, characteristic, complemented
10C22×D388C2211yescharacteristic, complemented
11#15690835—24no—
12C2×#13095498C2212yescomplemented
13C2×#13095496—24no—
14C2×#13095496C2212yescomplemented
15C2×#13095496C2214yescomplemented
16C2×#13095495C2211yescharacteristic, complemented
17C22×#11715895C2211yescharacteristic, complemented
18#15690806—21no—
19#15690799—21no—
20#15690784C2214yescomplemented
21#15690778—22no—
22#15690769C2211yescharacteristic, complemented
23#15690734C2212yescomplemented
24#15690731C2212yescomplemented
25#15690731C2214yescomplemented
26C22⋊#11715895C2214yescomplemented
27H(5,𝔽2)×C97—24no—
28H(5,𝔽2)×C97—24no—
29C2×#41×C97—21no—
30C2×#41×C97C2211yescharacteristic
31C2×#41×C97C2211yescharacteristic
32C2×#41×C97C2212yescomplemented
33C22×D4×C97—21no—
34C22×D4×C97—22no—
35C22×D4×C97C2211yescharacteristic
36C22×D4×C97C2212yescomplemented
37C23×D97—21no—
38C2×D388—22no—
39C2×D388—22no—
40C2×D388—21no—
41C2×D388C2312yes—
42C2×C22⋊D97—24no—
43#13095498—24no—
44#13095496—28no—
45#13095496—24no—
46#13095495—22no—
47C2×#11715895—22no—
48C2×#11715895C2312yes—
49C2×#11715895C2312yes—
50C4⋊Dic97C2312yes—
51C22⋊Dic97—21no—
52#13095488—24no—
53C4×Dic97—21no—
54C24×C97—21no—
55C24×C97—22no—
56#41×C97—28no—
57#41×C97—24no—
58#41×C97—24no—
59#41×C97—22no—
60#41×C97—22no—
61#41×C97—22no—
62#41×C97D414yescomplemented
63C2×Q8×C97D411yescharacteristic
64C2×Q8×C97C2311yescharacteristic
65C2×D4×C97—216no—
66C2×D4×C97—28no—
67C2×D4×C97—21no—
68C2×D4×C97—22no—
69C2×D4×C97—24no—
70C2×D4×C97—24no—
71C2×D4×C97—21no—
72C2×D4×C97—22no—
73C2×D4×C97—22no—
74C2×D4×C97D412yes—
75C2×D4×C97D411yescharacteristic
76C2×D4×C97D414yescomplemented
77C2×D4×C97C2312yes—
78C2×D4×C97C2311yescharacteristic
79C22×C4×C97—21no—
80C22×C4×C97—22no—
81C22×C4×C97C2312yes—
82C22×C4×C97C2311yescharacteristic
83C23×C97—28no—
84C23×C97—21no—
85C23×C97—22no—
86C23×C97—24no—
87C23×C97—82no—
88C23×C97—84no—
89C23×C97C2×D412yes—
90C23×C97C2×D411yescharacteristic
91C22×D97—22no—
92C22×D97—42no—
93C22×D97—82no—
94Q8×C97—22no—
95Q8×C97—21no—
96Q8×C97C2×D412yes—
97D4×C97—28no—
98D4×C97—28no—
99D4×C97—24no—
100D4×C97—24no—
101D4×C97—22no—
102D4×C97—24no—
103D4×C97—22no—
104D4×C97—21no—
105D4×C97C2×D412yes—
106D4×C97C2×D414yes—
107C2×C4×C97—28no—
108C2×C4×C97—21no—
109C2×C4×C97—24no—
110C2×C4×C97—22no—
111C2×C4×C97—22no—
112C2×C4×C97C2×D412yes—
113C2×C4×C97C2×D411yescharacteristic
114C2×C4×C97C2×D412yes—
115C2×C4×C97C2×D412yes—
116C2×C4×C97C2411yescharacteristic
117C22⋊D97—84no—
118C2×Dic97—22no—
119D388—22no—
120D388—22no—
121D388—42no—
122#11715895—22no—
123#11715895—22no—
124C22×C97—22no—
125C22×C97—22no—
126C22×C97—24no—
127C22×C97—22no—
128C22×C97—24no—
129C22×C97—44no—
130C22×C97—42no—
131C22×C97—84no—
132C22×C97—88no—
133C22×C97#12012yes—
134C22×C97C22×D411yescharacteristic
135C2×D97—42no—
136C2×D97—44no—
137C2×D97—84no—
138C4×C97—22no—
139C4×C97—22no—
140C4×C97#12012yes—
141C4×C97C22×D411yescharacteristic
142C4×C97C22×D411yescharacteristic
143Dic97—42no—
144C2×C97—22no—
145C2×C97—42no—
146C2×C97—44no—
147C2×C97—42no—
148C2×C97C2×#12011yescharacteristic
149C2×C97#57612yes—
150D97—82no—
151#3008—971nomaximal, Sylow
152C97#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)D9711yescharacteristic, complemented
154C2×AGL(1,ℤ/8)—971no—
155C22×D8—971no—
156#535—971no—
157#449—974no—
158#447—974no—
159#418—971no—
160#416—971no—
161#408—971no—
162H(5,𝔽2)—24no—
163H(5,𝔽2)—24no—
164C2×#41—21no—
165C2×#41C2×D9711yescharacteristic
166C2×#41C2×D9711yescharacteristic
167C2×#41C2×D9712yescomplemented
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—971no—
171C22×D4C2×D9711yescharacteristic
172C22×D4C2×D9712yescomplemented
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—1941no—
181#124—1941no—
182#121—974no—
183#120—1942no—
184#117—971no—
185#103—972no—
186#102—972no—
187#102—974no—
188C22⋊C8—974no—
189C24—21no—
190C24—22no—
191C24—1941no—
192#41—28no—
193#41—24no—
194#41—24no—
195#41—22no—
196#41—22no—
197#41—22no—
198#41C22⋊D9714yescomplemented
199C2×Q8C22⋊D9711yescharacteristic
200C2×Q8C22×D9711yescharacteristic
201C2×D4—216no—
202C2×D4—28no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—972no—
211C2×D4—1944no—
212C2×D4—1941no—
213C2×D4—1942no—
214C2×D4—1942no—
215C2×D4C22⋊D9712yes—
216C2×D4C22⋊D9711yescharacteristic
217C2×D4C22⋊D9714yescomplemented
218C2×D4C22×D9712yes—
219C2×D4C22×D9711yescharacteristic
220C22×C4—21no—
221C22×C4—22no—
222C22×C4C22×D9712yes—
223C22×C4C22×D9711yescharacteristic
224SD16—1944no—
225D8—1948no—
226D8—1944no—
227M16—1942no—
228C2×C8—972no—
229C2×C8—972no—
230C2×C8—1942no—
231C4⋊C4—972no—
232C22⋊C4—1941no—
233C22⋊C4—1944no—
234C42—1941no—
235C23—28no—
236C23—21no—
237C23—22no—
238C23—24no—
239C23—82no—
240C23—84no—
241C23—7762no—
242C23—3882no—
243C23—1942no—
244C23C2×C22⋊D9712yes—
245C23C2×C22⋊D9711yescharacteristic
246Q8—22no—
247Q8—21no—
248Q8C2×C22⋊D9712yes—
249D4—28no—
250D4—28no—
251D4—24no—
252D4—24no—
253D4—22no—
254D4—24no—
255D4—22no—
256D4—21no—
257D4—7764no—
258D4—3882no—
259D4—1942no—
260D4—1942no—
261D4C2×C22⋊D9712yes—
262D4C2×C22⋊D9714yes—
263C2×C4—28no—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—1942no—
269C2×C4C2×C22⋊D9712yes—
270C2×C4C2×C22⋊D9711yescharacteristic
271C2×C4C2×C22⋊D9712yes—
272C2×C4C2×C22⋊D9712yes—
273C2×C4C23×D9711yescharacteristic
274C8—1942no—
275C8—1942no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—24no—
281C22—44no—
282C22—42no—
283C22—84no—
284C22—88no—
285C22—7764no—
286C22—3882no—
287C22—3884no—
288C22#1569077912yes—
289C22C22×C22⋊D9711yescharacteristic
290C4—22no—
291C4—22no—
292C4—3882no—
293C4#1569077912yes—
294C4C22×C22⋊D9711yescharacteristic
295C4C22×C22⋊D9711yescharacteristic
296C2—22no—
297C2—42no—
298C2—44no—
299C2—42no—
300C2—7762no—
301C2C2×#1569077911yescharacteristic
302C2#2026153812yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×H(5,𝔽2)×C97C2
socleC22×C97C22×D4
hypercenterC2×H(5,𝔽2)D97
hypocenterC97#3008
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. C22×#13095496C2semidirect
3. #20261299C2semidirect
4. #20260880C2semidirect
5. #20260869C2semidirect
6. #20260740C2semidirect
7. #20260731C2semidirect
8. #20260702C2semidirect
9. C2×H(5,𝔽2)×C97C2semidirect
10. C22×D388C22semidirect
12. C2×#13095498C22semidirect
14. C2×#13095496C22semidirect
15. C2×#13095496C22semidirect
16. C2×#13095495C22semidirect
17. C22×#11715895C22semidirect
20. #15690784C22semidirect
22. #15690769C22semidirect
23. #15690734C22semidirect
24. #15690731C22semidirect
25. #15690731C22semidirect
26. C22⋊#11715895C22semidirect
32. C2×#41×C97C22semidirect
36. C22×D4×C97C22semidirect
62. #41×C97D4semidirect
76. C2×D4×C97D4semidirect
152. C97#3008semidirect
153. C2×H(5,𝔽2)D97semidirect
167. C2×#41C2×D97semidirect
172. C22×D4C2×D97semidirect
198. #41C22⋊D97semidirect
217. C2×D4C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
C22×#130954962
#202612992
#202608802
#202608692
#202607402
#202607312
#202607022
C2×H(5,𝔽2)×C972
#300897