Finite Group Databasebeta

← #27151011

Subgroups — #27151011

Order 12416.

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

normal subgroups only

This group has 752 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
2C2×#15690841C211yesmaximal, characteristic, complemented
3C22×#13095498C211yesmaximal, characteristic, complemented
4#20260895C214yesmaximal, complemented
5#20260885C214yesmaximal, complemented
6#20260739C211yesmaximal, characteristic, complemented
7#20260729C211yesmaximal, characteristic, complemented
8#20260702C211yesmaximal, characteristic, complemented
9#536×C97C211yesmaximal, characteristic, complemented
10#15690856—24no—
11#15690856—24no—
12C2×#13095512—21no—
13C2×#13095511C2212yes—
14C2×#13095511C2211yescharacteristic
15C22×D388C2212yes—
16C22×D388C2211yescharacteristic
17C2×D4×D97—22no—
18C2×D4×D97—21no—
19#15690841—24no—
20C2×#13095501C2212yescomplemented
21C2×#13095498—24no—
22C2×#13095498C2212yescomplemented
23C2×#13095498C2214yescomplemented
24C2×#13095495C2211yescharacteristic, complemented
25C22×#11715895C2211yescharacteristic, complemented
26#15690733C2212yescomplemented
27#15690730C2212yescomplemented
28#15690730C2214yescomplemented
29C22⋊#11715895C2214yescomplemented
30C2×#41×C97C2211yescharacteristic
31C22×Q8×C97C2211yescharacteristic, complemented
32C4⋊Q8×C97—21no—
33#124×C97—21no—
34#123×C97—22no—
35C22⋊Q8×C97C2214yescomplemented
36#117×C97C2211yescharacteristic, complemented
37C23×D97—22no—
38C23×D97—21no—
39#13095512—28no—
40#13095511—22no—
41#13095511—24no—
42#13095511—24no—
43#13095511D414yes—
44C2×Dic194C2311yescharacteristic
45C2×D388—22no—
46C2×D388—21no—
47C2×D388—22no—
48C2×D388—24no—
49C2×D388—24no—
50C2×D388D414yes—
51C2×D388C2311yescharacteristic
52C2×D388C2312yes—
53C2×C22⋊D97—22no—
54C2×C22⋊D97—21no—
55D4×D97—216no—
56D4×D97—28no—
57C2×C4×D97—22no—
58C2×C4×D97—21no—
59#13095501—24no—
60#13095498—28no—
61#13095498—24no—
62#13095495—22no—
63C2×#11715895—22no—
64C2×#11715895C2312yes—
65C2×#11715895C2312yes—
66#41×C97—22no—
67#41×C97—22no—
68C2×Q8×C97—21no—
69C2×Q8×C97—22no—
70C2×Q8×C97—22no—
71C2×Q8×C97D411yescharacteristic
72C2×Q8×C97C2312yes—
73C2×D4×C97D412yes—
74C2×D4×C97D411yescharacteristic
75C22×C4×C97—21no—
76C22×C4×C97C2312yes—
77C22×C4×C97C2311yescharacteristic
78C4⋊C4×C97—24no—
79C4⋊C4×C97C2312yes—
80C22⋊C4×C97—21no—
81C22⋊C4×C97—24no—
82C42×C97—21no—
83C23×C97C2×D412yes—
84C23×C97C2×D411yescharacteristic
85C22×D97—22no—
86C22×D97—21no—
87C22×D97—28no—
88C22×D97—24no—
89C22×D97—82no—
90C22×D97—84no—
91Q8×C97—22no—
92Q8×C97—22no—
93Q8×C97—22no—
94Q8×C97—42no—
95D4×C97—24no—
96D4×C97—22no—
97C2×C4×C97—22no—
98C2×C4×C97—22no—
99C2×C4×C97—22no—
100C2×C4×C97—22no—
101C2×C4×C97—42no—
102C2×C4×C97C2×D412yes—
103C2×C4×C97C2×D412yes—
104C2×C4×C97C2×D412yes—
105C2×C4×C97C2×D411yescharacteristic
106C2×C4×C97C2411yescharacteristic
107C22⋊D97—28no—
108C22⋊D97—24no—
109C2×Dic97—21no—
110D388—28no—
111D388—24no—
112D388—21no—
113D388—22no—
114D388C2×D412yes—
115D388C2×D414yes—
116C4×D97—28no—
117C4×D97—24no—
118Dic194—21no—
119Dic194C2×D412yes—
120#11715895—22no—
121#11715895—22no—
122C22×C97—22no—
123C22×C97—22no—
124C22×C97—24no—
125C22×C97#12012yes—
126C22×C97C22×D411yescharacteristic
127C2×D97—24no—
128C2×D97—22no—
129C2×D97—44no—
130C2×D97—42no—
131C2×D97—84no—
132C2×D97—88no—
133C4×C97—22no—
134C4×C97—42no—
135C4×C97—42no—
136C4×C97#12012yes—
137C4×C97C22×D411yescharacteristic
138C4×C97C22×D411yescharacteristic
139Dic97—22no—
140C2×C97—22no—
141C2×C97—42no—
142C2×C97C2×#12011yescharacteristic
143C2×C97#57712yes—
144D97—44no—
145D97—42no—
146#3009—971nomaximal, Sylow
147C97#300911yescharacteristic, complemented, Sylow
148C2×H(5,𝔽2)—971no—
149C2×#137—971no—
150C22×SD16—971no—
151#536D9711yescharacteristic, complemented
152#452—974no—
153#450—974no—
154#418—971no—
155#416—971no—
156#408—971no—
157H(5,𝔽2)—1944no—
158H(5,𝔽2)—1944no—
159C2×#41—971no—
160C2×#41—972no—
161C2×#41—1941no—
162C2×#41C2×D9711yescharacteristic
163C22×Q8C2×D9711yescharacteristic, complemented
164C22×D4—971no—
165C22×D4—972no—
166C22×D4—1942no—
167C22×D4—1941no—
168#137—1944no—
169C2×Q16—972no—
170C2×SD16—974no—
171C2×SD16—972no—
172C2×SD16—1944no—
173C2×M16—971no—
174C22×C8—971no—
175C4⋊Q8—21no—
176#124—21no—
177#123—22no—
178C22⋊Q8C2×D9714yescomplemented
179#117C2×D9711yescharacteristic, complemented
180#103—972no—
181#102—972no—
182#102—974no—
183C22⋊C8—974no—
184C24—1942no—
185C24—1941no—
186#41—22no—
187#41—22no—
188#41—974no—
189#41—1944no—
190#41—1944no—
191#41—1948no—
192#41—1942no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—971no—
197C2×Q8C22⋊D9711yescharacteristic
198C2×Q8C22×D9712yes—
199C2×D4—974no—
200C2×D4—972no—
201C2×D4—971no—
202C2×D4—1944no—
203C2×D4—1944no—
204C2×D4—19416no—
205C2×D4—1948no—
206C2×D4—1942no—
207C2×D4—1941no—
208C2×D4—1942no—
209C2×D4—1942no—
210C2×D4—1941no—
211C2×D4C22⋊D9712yes—
212C2×D4C22⋊D9711yescharacteristic
213C22×C4—21no—
214C22×C4—1942no—
215C22×C4—1941no—
216C22×C4C22×D9712yes—
217C22×C4C22×D9711yescharacteristic
218Q16—1944no—
219SD16—1948no—
220SD16—1944no—
221M16—1942no—
222C2×C8—972no—
223C2×C8—972no—
224C2×C8—1942no—
225C4⋊C4—24no—
226C4⋊C4C22×D9712yes—
227C22⋊C4—21no—
228C22⋊C4—24no—
229C42—21no—
230C23—7762no—
231C23—7764no—
232C23—1948no—
233C23—1944no—
234C23—1942no—
235C23—1941no—
236C23C2×C22⋊D9712yes—
237C23C2×C22⋊D9711yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—972no—
243Q8—1941no—
244D4—24no—
245D4—22no—
246D4—974no—
247D4—972no—
248D4—1948no—
249D4—1948no—
250D4—1944no—
251D4—1944no—
252D4—1941no—
253D4—1942no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—1948no—
260C2×C4—1941no—
261C2×C4—1944no—
262C2×C4D4×D9712yes—
263C2×C4D4×D9712yes—
264C2×C4C2×C22⋊D9712yes—
265C2×C4C2×C22⋊D9711yescharacteristic
266C2×C4C23×D9711yescharacteristic
267C8—1942no—
268C8—1942no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—7764no—
273C22—7768no—
274C22—3884no—
275C22—3882no—
276C22—1944no—
277C22—1942no—
278C22#1569077812yes—
279C22C22×C22⋊D9711yescharacteristic
280C4—22no—
281C4—42no—
282C4—42no—
283C4—1942no—
284C4#1569077812yes—
285C4C2×D4×D9711yescharacteristic
286C4C2×D4×D9711yescharacteristic
287C2—22no—
288C2—42no—
289C2—3884no—
290C2—3882no—
291C2C2×#1569077811yescharacteristic
292C2#2026154312yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#536×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690856C2
hypercenter#536D97
hypocenterC97#3009
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#15690841C2semidirect
3. C22×#13095498C2semidirect
4. #20260895C2semidirect
5. #20260885C2semidirect
6. #20260739C2semidirect
7. #20260729C2semidirect
8. #20260702C2semidirect
9. #536×C97C2semidirect
20. C2×#13095501C22semidirect
22. C2×#13095498C22semidirect
23. C2×#13095498C22semidirect
24. C2×#13095495C22semidirect
25. C22×#11715895C22semidirect
26. #15690733C22semidirect
27. #15690730C22semidirect
28. #15690730C22semidirect
29. C22⋊#11715895C22semidirect
31. C22×Q8×C97C22semidirect
35. C22⋊Q8×C97C22semidirect
36. #117×C97C22semidirect
147. C97#3009semidirect
151. #536D97semidirect
163. C22×Q8C2×D97semidirect
178. C22⋊Q8C2×D97semidirect
179. #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×#156908412
C22×#130954982
#202608952
#202608852
#202607392
#202607292
#202607022
#536×C972
#300997