Finite Group Databasebeta

← #27145096

Subgroups — #27145096

Order 12416.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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×#15690788C212yesmaximal, complemented
3C2×#15690772C211yesmaximal, characteristic
4#20260655C211yesmaximal, characteristic, complemented
5#20260648C211yesmaximal, characteristic, complemented
6#20260646C211yesmaximal, characteristic, complemented
7#20260639C212yesmaximal, complemented
8#20260613C212yesmaximal
9#20260606C212yesmaximal, complemented
10#20260594C211yesmaximal, characteristic, complemented
11#395×C97C211yesmaximal, characteristic, complemented
12C22×Dic194C2211yescharacteristic
13C22×Dic194C2212yes—
14C2×Q8×D97—28no—
15C2×Q8×D97—24no—
16C22×D388C2211yescharacteristic
17C22×C4×D97C2212yes—
18C22×C4×D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20#15690788—28no—
21#15690772—24no—
22C2×#13095490C2212yes—
23C2×#13095490C2212yes—
24C2×#13095488C2212yes—
25C2×#13095488C2212yes—
26C2×#13095488C2212yes—
27C2×#13095488C2211yescharacteristic
28C2×#13095488C2211yescharacteristic
29C2×C4×Dic97C2211yescharacteristic
30#15690717C2212yes—
31#15690717C2212yes—
32#15690717C2212yes—
33#15690717C2211yescharacteristic
34C22×Q8×C97C2211yescharacteristic
35C2×C4⋊C4×C97C2211yescharacteristic
36C2×C4⋊C4×C97C2212yes—
37#95×C97C2212yes—
38#95×C97C2211yescharacteristic
39C23×D97C2311yescharacteristic
40C23×D97C2311yescharacteristic
41C2×Dic194—22no—
42C2×Dic194—24no—
43C2×Dic194—24no—
44C2×Dic194—24no—
45C2×Dic194D418yescomplemented
46Q8×D97—232no—
47C2×D388—22no—
48C2×D388—22no—
49C2×D388Q814yescomplemented
50C22×Dic97C2311yescharacteristic
51C22×Dic97C2312yes—
52C22×Dic97C2312yes—
53C22×Dic97C2311yescharacteristic
54C2×C4×D97—24no—
55C2×C4×D97—24no—
56C2×C4×D97—24no—
57C2×C4×D97—24no—
58C2×C4×D97—22no—
59C2×C4×D97—24no—
60C2×C4×D97D414yes—
61#13095490—24no—
62#13095490—24no—
63#13095488—22no—
64#13095488—24no—
65#13095488—22no—
66#13095488—24no—
67#13095488—24no—
68C4×Dic97—22no—
69C2×Q8×C97—24no—
70C2×Q8×C97—22no—
71C22×C4×C97C2312yes—
72C22×C4×C97C2311yescharacteristic
73C22×C4×C97C2312yes—
74C22×C4×C97C2311yescharacteristic
75C22×C4×C97C2311yescharacteristic
76C4⋊C4×C97—22no—
77C4⋊C4×C97—24no—
78C23×C97C2411yescharacteristic
79C22×D97—22no—
80C22×D97—22no—
81C22×D97—22no—
82C22×D97—21no—
83C22×D97—21no—
84C22×D97—22no—
85C22×D97C2×D412yes—
86C22×D97C2×Q812yes—
87C22×D97C2×Q812yescomplemented
88C22×D97#4112yes—
89Q8×C97—28no—
90C2×C4×C97—21no—
91C2×C4×C97—22no—
92C2×C4×C97—22no—
93C2×C4×C97—22no—
94C2×C4×C97—21no—
95C2×C4×C97—22no—
96C2×C4×C97—21no—
97C2×C4×C97—22no—
98C2×C4×C97—22no—
99C2×C4×C97—22no—
100C2×C4×C97C2×D414yes—
101C2×C4×C97C2×D412yes—
102C2×C4×C97C2×Q812yescomplemented
103C2×Dic97—22no—
104C2×Dic97—22no—
105C2×Dic97—21no—
106C2×Dic97—22no—
107C2×Dic97—22no—
108C2×Dic97—22no—
109C2×Dic97—22no—
110C2×Dic97C2×D414yes—
111C2×Dic97C2×D412yes—
112C2×Dic97C2×D414yes—
113D388—28no—
114C4×D97—216no—
115C4×D97—28no—
116C4×D97—44no—
117Dic194—28no—
118Dic194—216no—
119C22×C97C22×D411yescharacteristic
120C22×C97C22×D412yes—
121C22×C97C22×Q811yescharacteristic
122C22×C97C2×#4111yescharacteristic
123C22×C972−1+412yes—
124C2×D97—22no—
125C2×D97—22no—
126C2×D97—24no—
127C2×D97—24no—
128C2×D97—22no—
129C2×D97—24no—
130C2×D97—41no—
131C2×D97—41no—
132C2×D97—42no—
133C2×D97C22⋊Q814yescomplemented
134C4×C97—22no—
135C4×C97—24no—
136C4×C97—22no—
137C4×C97—41no—
138C4×C97—42no—
139Dic97—24no—
140Dic97—41no—
141Dic97—42no—
142Dic97#12014yes—
143C2×C97C2×#12011yescharacteristic
144C2×C97C2×C22⋊Q811yescharacteristic
145C2×C97#53611yescharacteristic
146C2×C97#54812yes—
147C2×C97D4×Q812yes—
148D97—24no—
149D97—42no—
150#2409—971nomaximal, Sylow
151C97#240911yescharacteristic, complemented, Sylow
152C23×Q8—971no—
153C2×C22⋊Q8—972no—
154C2×C4×D4—971no—
155#395—971no—
156#395D9711yescharacteristic, complemented
157#393—971no—
158#393—971no—
159#391—972no—
160#386—972no—
161#385—972no—
162#380—971no—
163C22×Q8—972no—
164C22×Q8—971no—
165C22×Q8—1944no—
166C22×Q8—1948no—
167C22×Q8C2×D9711yescharacteristic
168C22×D4—971no—
169C23×C4—971no—
170C23×C4—971no—
171C23×C4—972no—
172C22⋊Q8—1948no—
173C4×D4—1944no—
174C2×C4⋊C4—972no—
175C2×C4⋊C4—972no—
176C2×C4⋊C4C2×D9711yescharacteristic
177C2×C4⋊C4C2×D9712yes—
178C2×C22⋊C4—971no—
179C2×C22⋊C4—971no—
180C2×C22⋊C4—972no—
181C2×C22⋊C4—972no—
182C2×C22⋊C4—972no—
183C2×C42—971no—
184#95—971no—
185#95—972no—
186#95—972no—
187#95—972no—
188#95C2×D9711yescharacteristic
189#95C2×D9712yes—
190C24—971no—
191C24—971no—
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—978no—
195C2×Q8—1944no—
196C2×Q8—1944no—
197C2×Q8—1944no—
198C2×Q8—19432no—
199C2×Q8—1942no—
200C2×D4—974no—
201C2×D4—1942no—
202C2×D4—1942no—
203C22×C4—971no—
204C22×C4—972no—
205C22×C4—974no—
206C22×C4—971no—
207C22×C4—972no—
208C22×C4—1942no—
209C22×C4—1944no—
210C22×C4—1944no—
211C22×C4—1944no—
212C22×C4—1944no—
213C22×C4—1944no—
214C22×C4C22×D9712yes—
215C22×C4C22×D9711yescharacteristic
216C22×C4C22×D9711yescharacteristic
217C22×C4C22×D9712yes—
218C22×C4C22×D9711yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—1944no—
222C4⋊C4—1944no—
223C22⋊C4—1944no—
224C22⋊C4—1944no—
225C22⋊C4—1942no—
226C22⋊C4—1942no—
227C22⋊C4—1944no—
228C42—1942no—
229C23—972no—
230C23—972no—
231C23—972no—
232C23—972no—
233C23—1942no—
234C23—1941no—
235C23—1941no—
236C23—1942no—
237C23—1942no—
238C23—1942no—
239C23C23×D9711yescharacteristic
240Q8—28no—
241Q8—19416no—
242Q8—1948no—
243D4—1948no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—972no—
255C2×C4—974no—
256C2×C4—974no—
257C2×C4—3884no—
258C2×C4—1948no—
259C2×C4—19416no—
260C2×C4—1941no—
261C2×C4—1942no—
262C2×C4—1942no—
263C2×C4—1942no—
264C2×C4—1942no—
265C2×C4—1942no—
266C2×C4—1942no—
267C2×C4D4×D9712yes—
268C2×C4D4×D9714yes—
269C2×C4Q8×D9712yescomplemented
270C22—974no—
271C22—3882no—
272C22—3881no—
273C22—3881no—
274C22—1944no—
275C22—1942no—
276C22—1944no—
277C22—1944no—
278C22—1942no—
279C22—1942no—
280C22C2×D4×D9712yes—
281C22C2×D4×D9711yescharacteristic
282C22C2×Q8×D9711yescharacteristic
283C22#41×D9711yescharacteristic
284C22#1569085812yes—
285C4—22no—
286C4—24no—
287C4—22no—
288C4—42no—
289C4—41no—
290C4—974no—
291C4—3882no—
292C4—3881no—
293C4—1944no—
294C2—3882no—
295C2—1944no—
296C2#120×D9711yescharacteristic
297C2C22⋊Q8×D9711yescharacteristic
298C2#2026131011yescharacteristic
299C2#2026139512yes—
300C2#2026140012yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#395×C97C2
prime radicalC22×D388C22
hypercenter#395D97
hypocenterC97#2409
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D97C2semidirect
2. C2×#15690788C2semidirect
4. #20260655C2semidirect
5. #20260648C2semidirect
6. #20260646C2semidirect
7. #20260639C2semidirect
9. #20260606C2semidirect
10. #20260594C2semidirect
11. #395×C97C2semidirect
45. C2×Dic194D4semidirect
49. C2×D388Q8semidirect
87. C22×D97C2×Q8semidirect
102. C2×C4×C97C2×Q8semidirect
133. C2×D97C22⋊Q8semidirect
151. C97#2409semidirect
156. #395D97semidirect
269. C2×C4Q8×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×#156907882
C2×#156907722
#202606552
#202606482
#202606462
#202606392
#202606132
#202606062
#202605942
#395×C972
#240997