Finite Group Databasebeta

← #27150968

Subgroups — #27150968

Order 12416.

738 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×Q8×D97C211yesmaximal, characteristic, complemented
2C2×#15690840C211yesmaximal, characteristic, complemented
3#20261242C211yesmaximal, characteristic
4#20260889C214yesmaximal, complemented
5#20260872C214yesmaximal, complemented
6#20260745C211yesmaximal, characteristic, complemented
7#20260744C211yesmaximal, characteristic, complemented
8#20260698C211yesmaximal, characteristic, complemented
9C2×#137×C97C211yesmaximal, characteristic, complemented
10C22×Dic194—21no—
11C22×Dic194C2211yescharacteristic
12C2×Q8×D97—24no—
13C2×Q8×D97—22no—
14C2×Q8×D97—24no—
15C2×Q8×D97C2214yes—
16C22×C4×D97—21no—
17C22×C4×D97C2211yescharacteristic
18#15690840—24no—
19C2×#13095501C2212yescomplemented
20C2×#13095499C2212yescomplemented
21C2×#13095495C2211yescharacteristic
22#15690791—21no—
23#15690786C2212yes—
24#15690781C2212yes—
25#15690778—21no—
26C4×C22⋊D97—22no—
27#15690769C2211yescharacteristic
28#15690735C2214yescomplemented
29#15690734C2214yescomplemented
30#15690722C2214yes—
31C2×#41×C97C2211yescharacteristic
32C22×Q8×C97C2211yescharacteristic
33#137×C97—24no—
34C2×Q16×C97C2212yescomplemented
35C2×SD16×C97C2212yescomplemented
36C2×M16×C97C2211yescharacteristic
37C23×D97D411yescharacteristic
38C2×Dic194—28no—
39C2×Dic194—21no—
40C2×Dic194—22no—
41C2×Dic194—24no—
42C2×Dic194D414yescomplemented
43C2×Dic194C2312yes—
44Q8×D97—216no—
45Q8×D97—28no—
46Q8×D97—28no—
47C2×C22⋊D97—22no—
48C22×Dic97—21no—
49C22×Dic97D411yescharacteristic
50C2×C4×D97—28no—
51C2×C4×D97—24no—
52C2×C4×D97—22no—
53C2×C4×D97—22no—
54C2×C4×D97D412yes—
55C2×C4×D97C2312yes—
56#13095501—24no—
57#13095499—24no—
58#13095495—22no—
59C2×#11715895C2312yes—
60C4⋊Dic97C2312yes—
61#13095490—22no—
62C22⋊Dic97—21no—
63C22⋊Dic97—22no—
64#13095488—22no—
65C4×Dic97—21no—
66#41×C97—22no—
67#41×C97—22no—
68C2×Q8×C97—22no—
69C2×Q8×C97—21no—
70C2×Q8×C97D414yescomplemented
71C2×Q8×C97C2311yescharacteristic
72C2×Q8×C97C2312yes—
73C2×D4×C97—21no—
74C2×D4×C97C2311yescharacteristic
75C22×C4×C97—21no—
76C22×C4×C97—21no—
77C22×C4×C97C2311yescharacteristic
78Q16×C97—24no—
79SD16×C97—24no—
80M16×C97—22no—
81C2×C8×C97C2312yes—
82C23×C97—21no—
83C23×C97C2×D411yescharacteristic
84C22×D97—22no—
85C22×D97—24no—
86C22×D97C2×D412yes—
87Q8×C97—24no—
88Q8×C97—22no—
89Q8×C97—21no—
90Q8×C97—21no—
91Q8×C97C2×D414yes—
92D4×C97—21no—
93D4×C97—21no—
94D4×C97—42no—
95C2×C4×C97—24no—
96C2×C4×C97—22no—
97C2×C4×C97—21no—
98C2×C4×C97—42no—
99C2×C4×C97C2×D411yescharacteristic
100C2×C4×C97C2×D412yes—
101C2×C4×C97C2×D412yes—
102C2×C4×C97C2411yescharacteristic
103C22⋊D97—44no—
104C2×Dic97—24no—
105C2×Dic97—22no—
106C2×Dic97—22no—
107C2×Dic97—22no—
108C2×Dic97C2×D412yes—
109C4×D97—216no—
110C4×D97—24no—
111C4×D97—22no—
112C4×D97—22no—
113Dic194—216no—
114Dic194—24no—
115Dic194—22no—
116Dic194C2×D414yes—
117C8×C97—22no—
118#11715895—22no—
119C22×C97—22no—
120C22×C97—41no—
121C22×C97—42no—
122C22×C97#12012yes—
123C22×C97C22×D411yescharacteristic
124C2×D97—28no—
125C2×D97—22no—
126C2×D97—22no—
127C2×D97#13714yescomplemented
128C4×C97—24no—
129C4×C97—41no—
130C4×C97#12012yes—
131C4×C97C22×D411yescharacteristic
132C4×C97C22×D411yescharacteristic
133Dic97—22no—
134Dic97—24no—
135Dic97—42no—
136C2×C97—22no—
137C2×C97—81no—
138C2×C97C2×#12011yescharacteristic
139C2×C97C2×#13712yes—
140D97—24no—
141#3004—971nomaximal, Sylow
142C97#300411yescharacteristic, complemented, Sylow
143C23×Q8—971no—
144C2×#137—971no—
145C2×#137D9711yescharacteristic, complemented
146#525—971no—
147#451—974no—
148#448—974no—
149#419—971no—
150#419—971no—
151#407—971no—
152C2×#41C2×D9711yescharacteristic
153C22×Q8—974no—
154C22×Q8—971no—
155C22×Q8—1941no—
156C22×Q8—1942no—
157C22×Q8—1944no—
158C22×Q8—1944no—
159C22×Q8C2×D9711yescharacteristic
160C23×C4—971no—
161C23×C4—1941no—
162#137—24no—
163#137—1944no—
164C2×Q16—972no—
165C2×Q16C2×D9712yescomplemented
166C2×SD16—972no—
167C2×SD16C2×D9712yescomplemented
168C2×M16—971no—
169C2×M16C2×D9711yescharacteristic
170#123—1941no—
171C22⋊Q8—972no—
172#121—972no—
173#120—1941no—
174C4×D4—1942no—
175#117—971no—
176#103—974no—
177#103—974no—
178C22⋊C8—974no—
179C24—971no—
180#41—22no—
181#41—22no—
182C2×Q8—22no—
183C2×Q8—21no—
184C2×Q8—974no—
185C2×Q8—972no—
186C2×Q8—1944no—
187C2×Q8—1948no—
188C2×Q8—1948no—
189C2×Q8—19416no—
190C2×Q8—1948no—
191C2×Q8—1941no—
192C2×Q8—1942no—
193C2×Q8C22⋊D9714yescomplemented
194C2×Q8C22×D9711yescharacteristic
195C2×Q8C22×D9712yes—
196C2×D4—21no—
197C2×D4—1942no—
198C2×D4C22×D9711yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—971no—
202C22×C4—972no—
203C22×C4—972no—
204C22×C4—1944no—
205C22×C4—1948no—
206C22×C4—1941no—
207C22×C4—1942no—
208C22×C4—1942no—
209C22×C4C22×D9711yescharacteristic
210Q16—24no—
211Q16—1944no—
212SD16—24no—
213SD16—1944no—
214M16—22no—
215M16—1942no—
216C2×C8—972no—
217C2×C8C22×D9712yes—
218C4⋊C4—972no—
219C4⋊C4—1942no—
220C22⋊C4—1942no—
221C22⋊C4—1942no—
222C22⋊C4—1941no—
223C42—1941no—
224C23—21no—
225C23—972no—
226C23—1944no—
227C23—1942no—
228C23D4×D9711yescharacteristic
229Q8—24no—
230Q8—22no—
231Q8—21no—
232Q8—21no—
233Q8—974no—
234Q8—19416no—
235Q8—1944no—
236Q8—1942no—
237Q8C2×C22⋊D9714yes—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—3884no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—42no—
246C2×C4—972no—
247C2×C4—19416no—
248C2×C4—1942no—
249C2×C4—1942no—
250C2×C4—1944no—
251C2×C4—1942no—
252C2×C4—1944no—
253C2×C4—1942no—
254C2×C4—1942no—
255C2×C4D4×D9711yescharacteristic
256C2×C4D4×D9712yes—
257C2×C4C2×C22⋊D9712yes—
258C2×C4C23×D9711yescharacteristic
259C8—22no—
260C8—1942no—
261C22—22no—
262C22—41no—
263C22—42no—
264C22—974no—
265C22—1948no—
266C22—1942no—
267C22—1942no—
268C22#1569077812yes—
269C22C2×D4×D9711yescharacteristic
270C4—24no—
271C4—41no—
272C4—3882no—
273C4—1942no—
274C4—1944no—
275C4#1569077812yes—
276C4C2×D4×D9711yescharacteristic
277C4C22×C22⋊D9711yescharacteristic
278C2—22no—
279C2—81no—
280C2—1944no—
281C2C2×#1569077811yescharacteristic
282C2#137×D9712yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#137×C97C2
socleC22×C97C22×D4
prime radical#20261242C2
hypercenterC2×#137D97
hypocenterC97#3004
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D97C2semidirect
2. C2×#15690840C2semidirect
4. #20260889C2semidirect
5. #20260872C2semidirect
6. #20260745C2semidirect
7. #20260744C2semidirect
8. #20260698C2semidirect
9. C2×#137×C97C2semidirect
19. C2×#13095501C22semidirect
20. C2×#13095499C22semidirect
28. #15690735C22semidirect
29. #15690734C22semidirect
34. C2×Q16×C97C22semidirect
35. C2×SD16×C97C22semidirect
42. C2×Dic194D4semidirect
70. C2×Q8×C97D4semidirect
127. C2×D97#137semidirect
142. C97#3004semidirect
145. C2×#137D97semidirect
165. C2×Q16C2×D97semidirect
167. C2×SD16C2×D97semidirect
193. C2×Q8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D972
C2×#156908402
#202612422
#202608892
#202608722
#202607452
#202607442
#202606982
C2×#137×C972
#300497