Finite Group Databasebeta

← #27150978

Subgroups — #27150978

Order 12416.

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

normal subgroups only

This group has 758 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×#15690841C211yesmaximal, characteristic, complemented
2C2×#15690836C211yesmaximal, characteristic, complemented
3#20261246C211yesmaximal, characteristic, complemented
4#20260897C214yesmaximal, complemented
5#20260880C214yesmaximal, complemented
6#20260744C211yesmaximal, characteristic, complemented
7#20260740C211yesmaximal, characteristic, complemented
8#20260699C211yesmaximal, characteristic, complemented
9C22×#41×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2211yescharacteristic
11#15690841—24no—
12#15690836—24no—
13C2×#13095501C2212yescomplemented
14C2×#13095499C2212yescomplemented
15C2×#13095498C2212yescomplemented
16C2×#13095496C2212yescomplemented
17C2×#13095495C2211yescharacteristic
18C2×#13095495C2211yescharacteristic
19#15690793—21no—
20#15690789C2212yescomplemented
21#15690784C2212yescomplemented
22#15690779—21no—
23C4×C22⋊D97—22no—
24#15690769C2211yescharacteristic
25#15690734C2214yescomplemented
26#15690731C2214yescomplemented
27C22⋊#11715895C2214yescomplemented
28C2×#41×C97—24no—
29C2×#41×C97—24no—
30C2×#41×C97—22no—
31C2×#41×C97C2214yes—
32C22×Q8×C97C2211yescharacteristic
33C22×D4×C97—21no—
34C22×D4×C97C2211yescharacteristic
35C23×C4×C97—21no—
36C23×C4×C97C2211yescharacteristic
37#13095511—22no—
38#13095511—22no—
39C2×Dic194C2311yescharacteristic
40C2×D388C2311yescharacteristic
41C2×C22⋊D97—22no—
42C2×C22⋊D97—21no—
43C2×C4×D97—21no—
44#13095501—24no—
45#13095499—24no—
46#13095498—24no—
47#13095496—24no—
48#13095495—22no—
49#13095495—22no—
50C2×#11715895C2312yes—
51C2×#11715895C2312yes—
52C4⋊Dic97C2312yes—
53#13095490—22no—
54C22⋊Dic97—21no—
55C22⋊Dic97—22no—
56#13095488—22no—
57C4×Dic97—21no—
58C24×C97—21no—
59C24×C97D411yescharacteristic
60#41×C97—216no—
61#41×C97—28no—
62#41×C97—28no—
63C2×Q8×C97—22no—
64C2×Q8×C97—21no—
65C2×Q8×C97D414yescomplemented
66C2×Q8×C97C2312yes—
67C2×D4×C97—28no—
68C2×D4×C97—24no—
69C2×D4×C97—21no—
70C2×D4×C97—22no—
71C2×D4×C97D414yescomplemented
72C2×D4×C97C2312yes—
73C22×C4×C97—28no—
74C22×C4×C97—21no—
75C22×C4×C97—24no—
76C22×C4×C97—22no—
77C22×C4×C97—22no—
78C22×C4×C97D412yes—
79C22×C4×C97D411yescharacteristic
80C22×C4×C97C2312yes—
81C22×C4×C97C2311yescharacteristic
82C23×C97—22no—
83C23×C97—24no—
84C23×C97—22no—
85C23×C97—42no—
86C23×C97—44no—
87C23×C97C2×D412yes—
88C23×C97C2×D411yescharacteristic
89C22×D97—21no—
90Q8×C97—24no—
91Q8×C97—22no—
92Q8×C97C2×D414yes—
93D4×C97—216no—
94D4×C97—24no—
95D4×C97—22no—
96D4×C97C2×D414yes—
97C2×C4×C97—216no—
98C2×C4×C97—22no—
99C2×C4×C97—24no—
100C2×C4×C97—22no—
101C2×C4×C97—24no—
102C2×C4×C97—22no—
103C2×C4×C97—22no—
104C2×C4×C97C2×D412yes—
105C2×C4×C97C2×D411yescharacteristic
106C2×C4×C97C2×D412yes—
107C2×C4×C97C2×D412yes—
108C2×C4×C97C2411yescharacteristic
109C22⋊D97—44no—
110C22⋊D97—42no—
111C2×Dic97—22no—
112C2×Dic97—21no—
113D388—21no—
114D388—21no—
115C4×D97—42no—
116Dic194—21no—
117Dic194—21no—
118#11715895—22no—
119#11715895—22no—
120C22×C97—22no—
121C22×C97—22no—
122C22×C97—24no—
123C22×C97—22no—
124C22×C97—22no—
125C22×C97—44no—
126C22×C97—48no—
127C22×C97—44no—
128C22×C97#12012yes—
129C22×C97C22×D411yescharacteristic
130C2×D97—41no—
131C2×D97—42no—
132C4×C97—22no—
133C4×C97—24no—
134C4×C97#12012yes—
135C4×C97C22×D411yescharacteristic
136C4×C97C22×D411yescharacteristic
137Dic97—41no—
138Dic97—42no—
139C2×C97—22no—
140C2×C97—42no—
141C2×C97—44no—
142C2×C97C2×#12011yescharacteristic
143C2×C97#57512yes—
144D97—81no—
145#3005—971nomaximal, Sylow
146C97#300511yescharacteristic, complemented, Sylow
147C22×#41D9711yescharacteristic, complemented
148C2×#137—971no—
149C2×AGL(1,ℤ/8)—971no—
150#525—971no—
151#452—974no—
152#449—974no—
153#419—971no—
154#418—971no—
155#407—971no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—971no—
160C2×#41C2×D9714yes—
161C22×Q8C2×D9711yescharacteristic
162C22×D4—21no—
163C22×D4C2×D9711yescharacteristic
164C23×C4—21no—
165C23×C4C2×D9711yescharacteristic
166#137—1944no—
167AGL(1,ℤ/8)—1944no—
168C2×Q16—972no—
169C2×SD16—972no—
170C2×SD16—972no—
171C2×D8—972no—
172C2×M16—971no—
173C2×M16—971no—
174#123—1941no—
175C22⋊Q8—972no—
176#121—972no—
177#120—1941no—
178C4×D4—1942no—
179#117—971no—
180#103—974no—
181#102—974no—
182C22⋊C8—974no—
183C24—21no—
184C24C22⋊D9711yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188#41—1942no—
189#41—1942no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—971no—
193C2×Q8C22⋊D9714yescomplemented
194C2×Q8C22×D9712yes—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—971no—
200C2×D4—1942no—
201C2×D4—1941no—
202C2×D4C22⋊D9714yescomplemented
203C2×D4C22×D9712yes—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—1941no—
210C22×C4C22⋊D9712yes—
211C22×C4C22⋊D9711yescharacteristic
212C22×C4C22×D9712yes—
213C22×C4C22×D9711yescharacteristic
214Q16—1944no—
215SD16—1944no—
216SD16—1944no—
217D8—1944no—
218M16—1942no—
219M16—1942no—
220C2×C8—972no—
221C2×C8—972no—
222C4⋊C4—972no—
223C4⋊C4—1942no—
224C22⋊C4—1941no—
225C22⋊C4—1942no—
226C22⋊C4—1942no—
227C42—1941no—
228C23—22no—
229C23—24no—
230C23—22no—
231C23—42no—
232C23—44no—
233C23—1941no—
234C23C2×C22⋊D9712yes—
235C23C2×C22⋊D9711yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—1941no—
239Q8—1941no—
240Q8C2×C22⋊D9714yes—
241D4—216no—
242D4—24no—
243D4—22no—
244D4—3884no—
245D4—3882no—
246D4—1941no—
247D4—1941no—
248D4C2×C22⋊D9714yes—
249C2×C4—216no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—3882no—
257C2×C4—1942no—
258C2×C4—1941no—
259C2×C4C2×C22⋊D9712yes—
260C2×C4C2×C22⋊D9711yescharacteristic
261C2×C4C2×C22⋊D9712yes—
262C2×C4C2×C22⋊D9712yes—
263C2×C4C23×D9711yescharacteristic
264C8—1942no—
265C8—1942no—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—3881no—
275C22—3882no—
276C22#1569077912yes—
277C22C22×C22⋊D9711yescharacteristic
278C4—22no—
279C4—24no—
280C4—3881no—
281C4—3882no—
282C4#1569077912yes—
283C4C22×C22⋊D9711yescharacteristic
284C4C22×C22⋊D9711yescharacteristic
285C2—22no—
286C2—42no—
287C2—44no—
288C2—7761no—
289C2C2×#1569077911yescharacteristic
290C2#2026153312yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×#41×C97C2
socleC22×C97C22×D4
hypercenterC22×#41D97
hypocenterC97#3005
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×#15690841C2semidirect
2. C2×#15690836C2semidirect
3. #20261246C2semidirect
4. #20260897C2semidirect
5. #20260880C2semidirect
6. #20260744C2semidirect
7. #20260740C2semidirect
8. #20260699C2semidirect
9. C22×#41×C97C2semidirect
13. C2×#13095501C22semidirect
14. C2×#13095499C22semidirect
15. C2×#13095498C22semidirect
16. C2×#13095496C22semidirect
20. #15690789C22semidirect
21. #15690784C22semidirect
25. #15690734C22semidirect
26. #15690731C22semidirect
27. C22⋊#11715895C22semidirect
65. C2×Q8×C97D4semidirect
71. C2×D4×C97D4semidirect
146. C97#3005semidirect
147. C22×#41D97semidirect
193. C2×Q8C22⋊D97semidirect
202. C2×D4C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
C2×#156908362
#202612462
#202608972
#202608802
#202607442
#202607402
#202606992
C22×#41×C972
#300597