Finite Group Databasebeta

← #27144586

Subgroups — #27144586

Order 12416.

792 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13095488C211yesmaximal, characteristic
2C2×C22⋊C4×D97C211yesmaximal, characteristic
3#20260626C212yesmaximal, complemented
4#20260625C212yesmaximal, complemented
5#20260606C212yesmaximal, complemented
6#20260598C211yesmaximal, characteristic, complemented
7#20260593C211yesmaximal, characteristic, complemented
8#20260590C211yesmaximal, characteristic
9C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
10C4×#13095488C212yesmaximal, complemented
11C2×#117×C97C211yesmaximal, characteristic, complemented
12C24×D97C2211yescharacteristic
13C23×Dic97C2211yescharacteristic
14C22×C4×D97C2212yes—
15C2×#13095490C2212yescomplemented
16C2×C22⋊Dic97C2211yescharacteristic
17C2×C22⋊Dic97C2211yescharacteristic
18C2×#13095488C418yescomplemented
19C2×#13095488C2212yes—
20C2×#13095488C2212yes—
21C2×#13095488C2212yes—
22C2×#13095488C2211yescharacteristic
23C2×#13095488C2211yescharacteristic
24C2×#13095488C2211yescharacteristic
25C2×#13095488C2211yescharacteristic
26C22⋊C4×D97—24no—
27C2×C4×Dic97C2212yescomplemented
28#15690717C2212yescomplemented
29#15690717C2212yescomplemented
30#15690717C2212yescomplemented
31#15690717C2212yescomplemented
32C23×C4×C97C2211yescharacteristic
33#117×C97—24no—
34C2×C4⋊C4×C97C2212yescomplemented
35C2×C22⋊C4×C97C2211yescharacteristic
36C2×C22⋊C4×C97C2211yescharacteristic
37C2×C42×C97C2212yescomplemented
38C23×D97—22no—
39C23×D97—22no—
40C23×D97—24no—
41C23×D97—42no—
42C23×D97C2×C414yes—
43C23×D97C2311yescharacteristic
44C23×D97C2311yescharacteristic
45C22×Dic97—42no—
46C22×Dic97C2×C414yescomplemented
47C22×Dic97C2312yes—
48C22×Dic97C2312yes—
49C22×Dic97C2311yescharacteristic
50C22×Dic97C2311yescharacteristic
51C2×C4×D97—24no—
52C2×C4×D97—24no—
53C2×C4×D97—42no—
54#13095490—42no—
55C22⋊Dic97—22no—
56C22⋊Dic97—22no—
57#13095488—24no—
58#13095488—24no—
59#13095488—42no—
60#13095488C2×C414yes—
61#13095488C2×C414yes—
62#13095488C2×C414yescomplemented
63#13095488C2×C414yescomplemented
64C4×Dic97—42no—
65C24×C97C2311yescharacteristic
66C22×C4×C97—22no—
67C22×C4×C97—22no—
68C22×C4×C97C2×C414yescomplemented
69C22×C4×C97C2312yes—
70C22×C4×C97C2311yescharacteristic
71C22×C4×C97C2311yescharacteristic
72C22×C4×C97C2312yes—
73C4⋊C4×C97—24no—
74C22⋊C4×C97—22no—
75C22⋊C4×C97D414yes—
76C42×C97—24no—
77C23×C97—22no—
78C23×C97—22no—
79C23×C97C22×C412yes—
80C23×C97C2×D412yes—
81C23×C97#4112yes—
82C23×C97C2411yescharacteristic
83C22×D97—21no—
84C22×D97—21no—
85C22×D97—22no—
86C22×D97—28no—
87C22×D97—21no—
88C22×D97—22no—
89C22×D97—21no—
90C22×D97—22no—
91C22×D97—28no—
92C22×D97—44no—
93C22×D97—48no—
94C22×D97—48no—
95C22×D97C22×C412yes—
96C22×D97C22×C412yes—
97C22×D97#4114yes—
98C2×C4×C97—28no—
99C2×C4×C97—21no—
100C2×C4×C97—21no—
101C2×C4×C97—21no—
102C2×C4×C97—21no—
103C2×C4×C97—22no—
104C2×C4×C97—22no—
105C2×C4×C97—22no—
106C2×C4×C97—22no—
107C2×C4×C97—22no—
108C2×C4×C97C22×C412yes—
109C2×C4×C97C22×C412yes—
110C2×C4×C97C2×D414yes—
111C2×Dic97—22no—
112C2×Dic97—22no—
113C2×Dic97—22no—
114C2×Dic97—22no—
115C2×Dic97—22no—
116C2×Dic97—22no—
117C2×Dic97—22no—
118C2×Dic97—22no—
119C2×Dic97—44no—
120C2×Dic97C22×C412yes—
121C2×Dic97C22×C412yes—
122C4×D97—48no—
123C22×C97—22no—
124C22×C97—22no—
125C22×C97—24no—
126C22×C97—24no—
127C22×C97C4×D414yes—
128C22×C97C23×C411yescharacteristic
129C22×C97C22×D411yescharacteristic
130C22×C97C2×#4112yes—
131C22×C97C2×#4111yescharacteristic
132C22×C97H(5,𝔽2)12yes—
133C2×D97—22no—
134C2×D97—22no—
135C2×D97—24no—
136C2×D97—24no—
137C2×D97—22no—
138C2×D97—22no—
139C2×D97—22no—
140C2×D97—416no—
141C2×D97—42no—
142C2×D97—42no—
143C2×D97#11714yescomplemented
144C4×C97—22no—
145C4×C97—22no—
146C4×C97—24no—
147C4×C97—42no—
148Dic97—22no—
149Dic97—22no—
150Dic97—42no—
151Dic97—42no—
152C2×C97—24no—
153C2×C97C2×#11711yescharacteristic
154C2×C97C2×C4×D411yescharacteristic
155C2×C97#51811yescharacteristic
156C2×C97#54612yes—
157C2×C97#55112yes—
158D97—24no—
159D97—42no—
160#2342—971nomaximal, Sylow
161C97#234211yescharacteristic, complemented, Sylow
162C2×#117D9711yescharacteristic, complemented
163C22×C22⋊C4—971no—
164C22×C22⋊C4—971no—
165#388—972no—
166#388—972no—
167#385—972no—
168#381—971no—
169#380—971no—
170#379—971no—
171C4×C22⋊C4—971no—
172C4×C22⋊C4—972no—
173C25—971no—
174C23×C4—971no—
175C23×C4—972no—
176C23×C4C2×D9711yescharacteristic
177#117—24no—
178C2×C4⋊C4—972no—
179C2×C4⋊C4C2×D9712yescomplemented
180C2×C22⋊C4—978no—
181C2×C22⋊C4—971no—
182C2×C22⋊C4—971no—
183C2×C22⋊C4—971no—
184C2×C22⋊C4—971no—
185C2×C22⋊C4—972no—
186C2×C22⋊C4—971no—
187C2×C22⋊C4—971no—
188C2×C22⋊C4—972no—
189C2×C22⋊C4—972no—
190C2×C22⋊C4—1944no—
191C2×C22⋊C4C2×D9711yescharacteristic
192C2×C22⋊C4C2×D9711yescharacteristic
193C2×C42—972no—
194C2×C42C2×D9712yescomplemented
195#95—972no—
196#95—972no—
197#95—972no—
198#95—972no—
199C24—971no—
200C24—974no—
201C24—971no—
202C24—3882no—
203C24—1944no—
204C24—1942no—
205C24—1942no—
206C24C22×D9711yescharacteristic
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—974no—
210C22×C4—971no—
211C22×C4—971no—
212C22×C4—972no—
213C22×C4—972no—
214C22×C4—3882no—
215C22×C4—3882no—
216C22×C4—1944no—
217C22×C4—1944no—
218C22×C4C4×D9714yescomplemented
219C22×C4C22×D9712yes—
220C22×C4C22×D9711yescharacteristic
221C22×C4C22×D9711yescharacteristic
222C22×C4C22×D9712yes—
223C4⋊C4—24no—
224C4⋊C4—3882no—
225C22⋊C4—22no—
226C22⋊C4—974no—
227C22⋊C4—974no—
228C22⋊C4—974no—
229C22⋊C4—974no—
230C22⋊C4—3882no—
231C22⋊C4—1944no—
232C22⋊C4—1942no—
233C22⋊C4—1942no—
234C22⋊C4—1944no—
235C22⋊C4C22⋊D9714yes—
236C42—24no—
237C42—3882no—
238C23—22no—
239C23—22no—
240C23—972no—
241C23—974no—
242C23—972no—
243C23—3888no—
244C23—3888no—
245C23—3884no—
246C23—1948no—
247C23—1948no—
248C23—1941no—
249C23—1942no—
250C23—1941no—
251C23—1942no—
252C23—1941no—
253C23—1941no—
254C23—1942no—
255C23C2×C4×D9712yes—
256C23C2×C22⋊D9712yes—
257C23#1309551112yes—
258C23C23×D9711yescharacteristic
259C2×C4—28no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—972no—
270C2×C4—972no—
271C2×C4—3888no—
272C2×C4—3884no—
273C2×C4—1942no—
274C2×C4—1942no—
275C2×C4—1942no—
276C2×C4—1942no—
277C2×C4—1942no—
278C2×C4—1942no—
279C2×C4—1942no—
280C2×C4—1942no—
281C2×C4C2×C4×D9712yes—
282C2×C4C2×C4×D9712yes—
283C2×C4C2×C22⋊D9714yes—
284C22—24no—
285C22—22no—
286C22—22no—
287C22—24no—
288C22—974no—
289C22—38816no—
290C22—3882no—
291C22—3882no—
292C22—1942no—
293C22—1942no—
294C22—1942no—
295C22—1944no—
296C22—1942no—
297C22—1942no—
298C22—1944no—
299C22C4×C22⋊D9714yes—
300C22C22×C4×D9711yescharacteristic
301C22C22×C22⋊D9711yescharacteristic
302C22#41×D9712yes—
303C22C2×#1309551111yescharacteristic
304C22#1569085612yes—
305C4—22no—
306C4—22no—
307C4—24no—
308C4—42no—
309C4—3882no—
310C4—3882no—
311C4—1942no—
312C4—1942no—
313C2—24no—
314C2—3882no—
315C2—1944no—
316C2#117×D9711yescharacteristic
317C2C2×C4×C22⋊D9711yescharacteristic
318C2#2026120911yescharacteristic
319C2#2026137312yes—
320C2#2026140912yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
FittingC2×#117×C97C2
socle, cosocleC23×C97C24
prime radicalC24×D97C22
hypercenterC2×#117D97
hypocenterC97#2342
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #20260626C2semidirect
4. #20260625C2semidirect
5. #20260606C2semidirect
6. #20260598C2semidirect
7. #20260593C2semidirect
9. C22⋊C4×Dic97C2semidirect
10. C4×#13095488C2semidirect
11. C2×#117×C97C2semidirect
15. C2×#13095490C22semidirect
18. C2×#13095488C4semidirect
27. C2×C4×Dic97C22semidirect
28. #15690717C22semidirect
29. #15690717C22semidirect
30. #15690717C22semidirect
31. #15690717C22semidirect
34. C2×C4⋊C4×C97C22semidirect
37. C2×C42×C97C22semidirect
46. C22×Dic97C2×C4semidirect
62. #13095488C2×C4semidirect
63. #13095488C2×C4semidirect
68. C22×C4×C97C2×C4semidirect
143. C2×D97#117semidirect
161. C97#2342semidirect
162. C2×#117D97semidirect
179. C2×C4⋊C4C2×D97semidirect
194. C2×C42C2×D97semidirect
218. C22×C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130954882
C2×C22⋊C4×D972
#202606262
#202606252
#202606062
#202605982
#202605932
#202605902
C22⋊C4×Dic972
C4×#130954882
C2×#117×C972
#234297