Finite Group Databasebeta

← #27145094

Subgroups — #27145094

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×#15690787C212yesmaximal, complemented
3C2×#15690770C211yesmaximal, characteristic
4#20260656C211yesmaximal, characteristic, complemented
5#20260654C211yesmaximal, characteristic, complemented
6#20260647C211yesmaximal, characteristic, complemented
7#20260639C212yesmaximal, complemented
8#20260611C212yesmaximal
9#20260609C212yesmaximal, complemented
10#20260594C211yesmaximal, characteristic, complemented
11#393×C97C211yesmaximal, characteristic, complemented
12C22×Dic194C2211yescharacteristic
13C22×Dic194C2212yes—
14C2×Q8×D97—28no—
15C2×Q8×D97—24no—
16C22×C22⋊D97C2211yescharacteristic
17C23×Dic97C2211yescharacteristic
18C22×C4×D97C2212yes—
19C22×C4×D97C2211yescharacteristic
20#15690787—28no—
21#15690770—24no—
22C2×C4⋊Dic97C2212yes—
23C2×#13095490C2212yes—
24C2×#13095490C2212yes—
25C2×#13095490C2211yescharacteristic
26C2×C22⋊Dic97C2212yes—
27C2×#13095488C2211yescharacteristic
28C2×#13095488C2212yes—
29C2×C4×Dic97C2211yescharacteristic
30#15690717C2212yes—
31#15690717C2212yes—
32#15690717C2212yes—
33#15690717C2211yescharacteristic
34C22×Q8×C97C2211yescharacteristic
35C2×C22⋊C4×C97C2212yes—
36C2×C22⋊C4×C97C2211yescharacteristic
37#95×C97C2212yes—
38#95×C97C2211yescharacteristic
39C23×D97C2311yescharacteristic
40C2×Dic194—22no—
41C2×Dic194—24no—
42C2×Dic194—24no—
43C2×Dic194—24no—
44C2×Dic194D418yescomplemented
45Q8×D97—232no—
46C2×C22⋊D97—22no—
47C2×C22⋊D97—22no—
48C2×C22⋊D97Q814yescomplemented
49C22×Dic97—22no—
50C22×Dic97—24no—
51C22×Dic97C2312yes—
52C22×Dic97C2311yescharacteristic
53C22×Dic97C2312yes—
54C22×Dic97C2311yescharacteristic
55C22×Dic97C2311yescharacteristic
56C2×C4×D97—24no—
57C2×C4×D97—24no—
58C2×C4×D97—24no—
59C2×C4×D97—24no—
60C2×C4×D97D414yes—
61C4⋊Dic97—24no—
62#13095490—22no—
63#13095490—24no—
64#13095490—24no—
65C22⋊Dic97—24no—
66#13095488—22no—
67#13095488—24no—
68C4×Dic97—22no—
69C24×C97C2311yescharacteristic
70C2×Q8×C97—24no—
71C2×Q8×C97—22no—
72C22×C4×C97C2311yescharacteristic
73C22×C4×C97C2312yes—
74C22×C4×C97C2312yes—
75C22×C4×C97C2311yescharacteristic
76C22⋊C4×C97—22no—
77C22⋊C4×C97—24no—
78C23×C97—22no—
79C23×C97—21no—
80C23×C97—21no—
81C23×C97—22no—
82C23×C97C2×Q812yescomplemented
83C23×C97C2411yescharacteristic
84C22×D97—22no—
85C22×D97—22no—
86C22×D97C2×D412yes—
87C22×D97C2×Q812yes—
88C22×D97#4112yes—
89Q8×C97—28no—
90C2×C4×C97—22no—
91C2×C4×C97—22no—
92C2×C4×C97—22no—
93C2×C4×C97—21no—
94C2×C4×C97—22no—
95C2×C4×C97—22no—
96C2×C4×C97—22no—
97C2×C4×C97—22no—
98C2×C4×C97C2×D414yes—
99C2×C4×C97C2×D412yes—
100C22⋊D97—28no—
101C2×Dic97—21no—
102C2×Dic97—22no—
103C2×Dic97—22no—
104C2×Dic97—21no—
105C2×Dic97—21no—
106C2×Dic97—22no—
107C2×Dic97—22no—
108C2×Dic97—22no—
109C2×Dic97—22no—
110C2×Dic97—44no—
111C2×Dic97C2×D414yes—
112C2×Dic97C2×D412yes—
113C2×Dic97C2×D414yes—
114C2×Dic97C2×Q812yescomplemented
115C4×D97—216no—
116C4×D97—28no—
117Dic194—28no—
118Dic194—216no—
119C22×C97—22no—
120C22×C97—24no—
121C22×C97—41no—
122C22×C97—41no—
123C22×C97—42no—
124C22×C97C22×D411yescharacteristic
125C22×C97C22×D412yes—
126C22×C97C22×Q811yescharacteristic
127C22×C97C2×#4111yescharacteristic
128C22×C972−1+412yes—
129C2×D97—22no—
130C2×D97—22no—
131C2×D97—24no—
132C2×D97—24no—
133C2×D97C22⋊Q814yescomplemented
134C4×C97—22no—
135C4×C97—24no—
136C4×C97—41no—
137C4×C97—42no—
138Dic97—24no—
139Dic97—22no—
140Dic97—41no—
141Dic97—42no—
142Dic97#12014yes—
143C2×C97—42no—
144C2×C97C2×#12011yescharacteristic
145C2×C97C2×C22⋊Q811yescharacteristic
146C2×C97#53611yescharacteristic
147C2×C97#54812yes—
148C2×C97D4×Q812yes—
149D97—24no—
150#2409—971nomaximal, Sylow
151C97#240911yescharacteristic, complemented, Sylow
152C23×Q8—971no—
153C2×C22⋊Q8—972no—
154C2×C4×D4—971no—
155#395—971no—
156#395—971no—
157#393—971no—
158#393D9711yescharacteristic, complemented
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—971no—
175C2×C4⋊C4—972no—
176C2×C4⋊C4—972no—
177C2×C4⋊C4—972no—
178C2×C22⋊C4—972no—
179C2×C22⋊C4—971no—
180C2×C22⋊C4—972no—
181C2×C22⋊C4C2×D9711yescharacteristic
182C2×C22⋊C4C2×D9712yes—
183C2×C42—971no—
184#95—971no—
185#95—972no—
186#95—972no—
187#95—972no—
188#95C2×D9711yescharacteristic
189#95C2×D9712yes—
190C24—971no—
191C24C22×D9711yescharacteristic
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—971no—
205C22×C4—972no—
206C22×C4—974no—
207C22×C4—972no—
208C22×C4—971no—
209C22×C4—1944no—
210C22×C4—1944no—
211C22×C4—1944no—
212C22×C4—1942no—
213C22×C4—1944no—
214C22×C4—1944no—
215C22×C4C22×D9711yescharacteristic
216C22×C4C22×D9712yes—
217C22×C4C22×D9711yescharacteristic
218C22×C4C22×D9712yes—
219C4⋊C4—1944no—
220C4⋊C4—1944no—
221C4⋊C4—1942no—
222C4⋊C4—1944no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—1944no—
226C22⋊C4—1942no—
227C22⋊C4—1944no—
228C42—1942no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—972no—
234C23—972no—
235C23—972no—
236C23—1942no—
237C23—1942no—
238C23Q8×D9712yescomplemented
239C23C23×D9711yescharacteristic
240Q8—28no—
241Q8—19416no—
242Q8—1948no—
243D4—1948no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—972no—
253C2×C4—974no—
254C2×C4—974no—
255C2×C4—972no—
256C2×C4—3884no—
257C2×C4—1948no—
258C2×C4—19416no—
259C2×C4—1942no—
260C2×C4—1941no—
261C2×C4—1942no—
262C2×C4—1942no—
263C2×C4—1942no—
264C2×C4—1941no—
265C2×C4—1942no—
266C2×C4—1942no—
267C2×C4—1941no—
268C2×C4D4×D9712yes—
269C2×C4D4×D9714yes—
270C22—24no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—974no—
276C22—1944no—
277C22—1944no—
278C22—1942no—
279C22—1942no—
280C22C2×D4×D9712yes—
281C22C2×D4×D9711yescharacteristic
282C22C2×Q8×D9711yescharacteristic
283C22#41×D9711yescharacteristic
284C22#1569085712yes—
285C4—22no—
286C4—24no—
287C4—42no—
288C4—41no—
289C4—974no—
290C4—3882no—
291C4—3881no—
292C4—1944no—
293C4—1942no—
294C2—42no—
295C2—1944no—
296C2#120×D9711yescharacteristic
297C2C22⋊Q8×D9711yescharacteristic
298C2#2026130511yescharacteristic
299C2#2026139312yes—
300C2#2026139812yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#393×C97C2
prime radicalC22×C22⋊D97C22
hypercenter#393D97
hypocenterC97#2409
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D97C2semidirect
2. C2×#15690787C2semidirect
4. #20260656C2semidirect
5. #20260654C2semidirect
6. #20260647C2semidirect
7. #20260639C2semidirect
9. #20260609C2semidirect
10. #20260594C2semidirect
11. #393×C97C2semidirect
44. C2×Dic194D4semidirect
48. C2×C22⋊D97Q8semidirect
82. C23×C97C2×Q8semidirect
114. C2×Dic97C2×Q8semidirect
133. C2×D97C22⋊Q8semidirect
151. C97#2409semidirect
158. #393D97semidirect
238. C23Q8×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×#156907872
C2×#156907702
#202606562
#202606542
#202606472
#202606392
#202606112
#202606092
#202605942
#393×C972
#240997