Finite Group Databasebeta

← #27150996

Subgroups — #27150996

Order 12416.

754 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 169 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×#15690840C212yesmaximal, complemented
2C2×#15690789C211yesmaximal, characteristic, complemented
3#20260897C214yesmaximal, complemented
4#20260891C212yesmaximal, complemented
5#20260886C212yesmaximal, complemented
6#20260736C212yesmaximal, complemented
7#20260699C211yesmaximal, characteristic, complemented
8C22×#41×C97C211yesmaximal, characteristic
9C22×Dic194C2211yescharacteristic, complemented
10#15690840—28no—
11C2×#13095501C2214yescomplemented
12C2×#13095499C2214yescomplemented
13C2×#13095495C2212yescomplemented
14#15690789—21no—
15#15690789—21no—
16#15690789C2212yes—
17#15690789C2212yescomplemented
18C2×C4⋊Dic97C2211yescharacteristic, complemented
19C2×#13095490—21no—
20C2×C22⋊Dic97—21no—
21#15690734C2214yescomplemented
22#15690731C2214yescomplemented
23C22⋊#11715895C2212yes—
24C22⋊#11715895C2212yescomplemented
25C2×#41×C97—28no—
26C2×#41×C97—21no—
27C2×#41×C97—21no—
28C2×#41×C97C2212yes—
29C2×#41×C97C2212yes—
30C22×Q8×C97C2211yescharacteristic
31C22×D4×C97—21no—
32C22×D4×C97C2211yescharacteristic
33C23×C4×C97—21no—
34C23×C4×C97C2211yescharacteristic
35C2×Dic194—24no—
36C2×Dic194—21no—
37C2×Dic194C2312yes—
38C22×Dic97—21no—
39C22×Dic97—21no—
40#13095501—28no—
41#13095499—28no—
42#13095495—24no—
43C2×#11715895C2314yes—
44C4⋊Dic97—21no—
45C4⋊Dic97C2312yes—
46#13095490—22no—
47#13095490—22no—
48C22⋊Dic97—22no—
49C22⋊Dic97—22no—
50C24×C97—21no—
51C24×C97D411yescharacteristic
52#41×C97—28no—
53#41×C97—28no—
54#41×C97—24no—
55#41×C97—24no—
56#41×C97—24no—
57#41×C97D418yes—
58C2×Q8×C97—24no—
59C2×Q8×C97—21no—
60C2×Q8×C97C2312yes—
61C2×D4×C97—28no—
62C2×D4×C97—22no—
63C2×D4×C97—22no—
64C2×D4×C97—24no—
65C2×D4×C97—21no—
66C2×D4×C97C2312yes—
67C22×C4×C97—28no—
68C22×C4×C97—21no—
69C22×C4×C97—22no—
70C22×C4×C97—22no—
71C22×C4×C97—24no—
72C22×C4×C97D411yescharacteristic
73C22×C4×C97D411yescharacteristic
74C22×C4×C97D411yescharacteristic
75C22×C4×C97C2311yescharacteristic
76C22×C4×C97C2311yescharacteristic
77C22×C4×C97C2311yescharacteristic
78C23×C97—24no—
79C23×C97—22no—
80C23×C97—22no—
81C23×C97—22no—
82C23×C97—41no—
83C23×C97—42no—
84C23×C97—42no—
85C23×C97C2×D411yescharacteristic
86C23×C97C2×D411yescharacteristic
87C23×C97C2×D411yescharacteristic
88Q8×C97—24no—
89Q8×C97—22no—
90Q8×C97C2×D414yes—
91D4×C97—28no—
92D4×C97—24no—
93D4×C97—28no—
94D4×C97—22no—
95D4×C97C2×D414yes—
96C2×C4×C97—24no—
97C2×C4×C97—28no—
98C2×C4×C97—22no—
99C2×C4×C97—22no—
100C2×C4×C97—22no—
101C2×C4×C97—28no—
102C2×C4×C97—22no—
103C2×C4×C97—22no—
104C2×C4×C97—22no—
105C2×C4×C97C2×D411yescharacteristic
106C2×C4×C97C2×D411yescharacteristic
107C2×C4×C97C2×D411yescharacteristic
108C2×C4×C97C2×D414yes—
109C2×C4×C97C2411yescharacteristic
110C2×Dic97—22no—
111C2×Dic97—22no—
112C2×Dic97—42no—
113C2×Dic97—42no—
114Dic194—24no—
115Dic194—42no—
116#11715895—24no—
117C22×C97—21no—
118C22×C97—21no—
119C22×C97—21no—
120C22×C97—21no—
121C22×C97—21no—
122C22×C97—24no—
123C22×C97—21no—
124C22×C97—21no—
125C22×C97—44no—
126C22×C97—48no—
127C22×C97—42no—
128C22×C97—42no—
129C22×C97#12012yes—
130C22×C97#13712yescomplemented
131C22×C97C22×D411yescharacteristic
132C4×C97—21no—
133C4×C97—24no—
134C4×C97—21no—
135C4×C97#12012yes—
136C4×C97C22×D412yes—
137Dic97—42no—
138Dic97—42no—
139C2×C97—22no—
140C2×C97—22no—
141C2×C97—41no—
142C2×C97—44no—
143C2×C97C2×#12011yescharacteristic
144C2×C97C2×#13711yescharacteristic
145C2×C97#57511yescharacteristic
146#3007—971nomaximal, Sylow
147C97#300711yescharacteristic, complemented, Sylow
148C22×#41D9711yescharacteristic
149C2×#137—972no—
150C2×C22⋊Q8—971no—
151#452—974no—
152#451—972no—
153#450—972no—
154#417—972no—
155#407—971no—
156C2×#41—28no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41C2×D9712yes—
160C2×#41C2×D9712yes—
161C22×Q8—971no—
162C22×Q8C2×D9711yescharacteristic
163C22×D4—21no—
164C22×D4C2×D9711yescharacteristic
165C23×C4—21no—
166C23×C4C2×D9711yescharacteristic
167#137—1948no—
168C2×Q16—974no—
169C2×SD16—974no—
170C2×M16—972no—
171C22⋊Q8—972no—
172C22⋊Q8—972no—
173C22⋊Q8—1941no—
174C22⋊Q8—1941no—
175C2×C4⋊C4—971no—
176C2×C4⋊C4—1941no—
177C2×C22⋊C4—1941no—
178#103—974no—
179#102—974no—
180C22⋊C8—972no—
181C22⋊C8—972no—
182C24—21no—
183C24C22⋊D9711yescharacteristic
184#41—28no—
185#41—28no—
186#41—24no—
187#41—24no—
188#41—24no—
189#41C22⋊D9718yes—
190C2×Q8—24no—
191C2×Q8—21no—
192C2×Q8—972no—
193C2×Q8—1944no—
194C2×Q8—1941no—
195C2×Q8C22×D9712yes—
196C2×D4—28no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—24no—
200C2×D4—21no—
201C2×D4C22×D9712yes—
202C22×C4—28no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—1941no—
208C22×C4—1941no—
209C22×C4C22⋊D9711yescharacteristic
210C22×C4C22⋊D9711yescharacteristic
211C22×C4C22⋊D9711yescharacteristic
212C22×C4C22×D9711yescharacteristic
213C22×C4C22×D9711yescharacteristic
214C22×C4C22×D9711yescharacteristic
215Q16—1948no—
216SD16—1948no—
217M16—1944no—
218C2×C8—974no—
219C4⋊C4—972no—
220C4⋊C4—1942no—
221C4⋊C4—1942no—
222C4⋊C4—1941no—
223C22⋊C4—1942no—
224C22⋊C4—1942no—
225C23—24no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—41no—
230C23—42no—
231C23—42no—
232C23C2×C22⋊D9711yescharacteristic
233C23C2×C22⋊D9711yescharacteristic
234C23C2×C22⋊D9711yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—3882no—
238Q8—1944no—
239Q8C2×C22⋊D9714yes—
240D4—28no—
241D4—24no—
242D4—28no—
243D4—22no—
244D4C2×C22⋊D9714yes—
245C2×C4—24no—
246C2×C4—28no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—28no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—3882no—
255C2×C4—3882no—
256C2×C4—1942no—
257C2×C4—1942no—
258C2×C4C2×C22⋊D9711yescharacteristic
259C2×C4C2×C22⋊D9711yescharacteristic
260C2×C4C2×C22⋊D9711yescharacteristic
261C2×C4C2×C22⋊D9714yes—
262C2×C4C23×D9711yescharacteristic
263C8—1944no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—24no—
270C22—21no—
271C22—21no—
272C22—44no—
273C22—48no—
274C22—42no—
275C22—42no—
276C22#1569077912yes—
277C22#1569084012yescomplemented
278C22C22×C22⋊D9711yescharacteristic
279C4—21no—
280C4—24no—
281C4—21no—
282C4—3882no—
283C4—3882no—
284C4#1569077912yes—
285C4C22×C22⋊D9712yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—44no—
290C2C2×#1569077911yescharacteristic
291C2C2×#1569084011yescharacteristic
292C2#2026153311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting, prime radicalC22×#41×C97C2
socleC22×C97C22×D4
hypercenterC22×#41D97
hypocenterC97#3007
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#15690840C2semidirect
2. C2×#15690789C2semidirect
3. #20260897C2semidirect
4. #20260891C2semidirect
5. #20260886C2semidirect
6. #20260736C2semidirect
7. #20260699C2semidirect
9. C22×Dic194C22semidirect
11. C2×#13095501C22semidirect
12. C2×#13095499C22semidirect
13. C2×#13095495C22semidirect
17. #15690789C22semidirect
18. C2×C4⋊Dic97C22semidirect
21. #15690734C22semidirect
22. #15690731C22semidirect
24. C22⋊#11715895C22semidirect
130. C22×C97#137semidirect
147. C97#3007semidirect
277. C22#15690840semidirect

Maximal subgroups

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

subgroupindex
C2×#156908402
C2×#156907892
#202608972
#202608912
#202608862
#202607362
#202606992
C22×#41×C972
#300797