Finite Group Databasebeta

← #27144347

Subgroups — #27144347

Order 12416.

816 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13095488C211yesmaximal, characteristic
2C2×C22⋊C4×D97C211yesmaximal, characteristic
3C2×C22⋊C4×D97C211yesmaximal, characteristic
4#20260610C211yesmaximal, characteristic, complemented
5#20260609C212yesmaximal, complemented
6#20260608C211yesmaximal, characteristic, complemented
7#20260607C211yesmaximal, characteristic, complemented
8#20260606C212yesmaximal, complemented
9#20260594C212yesmaximal, complemented
10#20260592C212yesmaximal, complemented
11#385×C97C211yesmaximal, characteristic, complemented
12C24×D97C2211yescharacteristic
13C23×Dic97C2211yescharacteristic
14C22×C4×D97C2211yescharacteristic
15C22×C4×D97C2211yescharacteristic
16C22×C4×D97C2212yes—
17C2×C4⋊Dic97C2212yescomplemented
18C2×#13095490C2212yescomplemented
19C2×#13095490C2212yescomplemented
20C2×C22⋊Dic97C2211yescharacteristic
21C2×C22⋊Dic97C2211yescharacteristic
22C2×#13095488—24no—
23C2×#13095488C2212yes—
24C2×#13095488C2211yescharacteristic
25C2×#13095488C2211yescharacteristic
26C2×#13095488C2212yes—
27C2×#13095488C2212yes—
28C22⋊C4×D97—24no—
29C22⋊C4×D97C418yescomplemented
30#15690717C2212yescomplemented
31#15690717C2212yescomplemented
32#15690717C2212yescomplemented
33C23×C4×C97C2211yescharacteristic
34C2×C4⋊C4×C97C2212yescomplemented
35C2×C22⋊C4×C97C2211yescharacteristic
36C2×C22⋊C4×C97C2211yescharacteristic
37#95×C97C2212yescomplemented
38C23×D97—42no—
39C23×D97—42no—
40C23×D97C2×C414yes—
41C23×D97D414yes—
42C23×D97Q814yes—
43C23×D97C2311yescharacteristic
44C23×D97C2311yescharacteristic
45C22×Dic97—22no—
46C22×Dic97—42no—
47C22×Dic97C2312yes—
48C22×Dic97C2311yescharacteristic
49C22×Dic97C2311yescharacteristic
50C22×Dic97C2312yes—
51C2×C4×D97—22no—
52C2×C4×D97—22no—
53C2×C4×D97—42no—
54C2×C4×D97—42no—
55C2×C4×D97—42no—
56C2×C4×D97—42no—
57C2×C4×D97C2×C418yescomplemented
58C4⋊Dic97—42no—
59#13095490—42no—
60#13095490—42no—
61C22⋊Dic97—22no—
62C22⋊Dic97C2×C414yescomplemented
63#13095488—22no—
64#13095488—22no—
65#13095488—24no—
66#13095488—24no—
67#13095488C2×C418yescomplemented
68C24×C97C2311yescharacteristic
69C22×C4×C97—22no—
70C22×C4×C97—42no—
71C22×C4×C97C2312yes—
72C22×C4×C97C2311yescharacteristic
73C22×C4×C97C2311yescharacteristic
74C22×C4×C97C2312yes—
75C4⋊C4×C97—42no—
76C22⋊C4×C97—22no—
77C22⋊C4×C97C2×C414yescomplemented
78C23×C97—22no—
79C23×C97—22no—
80C23×C97C22×C412yes—
81C23×C97C2×D412yes—
82C23×C97C2×Q812yes—
83C23×C97C2411yescharacteristic
84C22×D97—22no—
85C22×D97—22no—
86C22×D97—22no—
87C22×D97—22no—
88C22×D97—48no—
89C22×D97—44no—
90C22×D97—44no—
91C22×D97—48no—
92C22×D97C4⋊C4116yescomplemented
93C22×D97C22×C412yes—
94C22×D97C22×C412yes—
95C22×D97C2×D412yes—
96C22×D97C2×D412yes—
97C22×D97C2×Q812yes—
98C22×D97C2×Q812yes—
99C2×C4×C97—22no—
100C2×C4×C97—22no—
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×C97—44no—
109C2×C4×C97C22×C414yes—
110C2×Dic97—22no—
111C2×Dic97—22no—
112C2×Dic97—22no—
113C2×Dic97—21no—
114C2×Dic97—21no—
115C2×Dic97—22no—
116C2×Dic97—22no—
117C2×Dic97—22no—
118C2×Dic97—22no—
119C2×Dic97—44no—
120C2×Dic97C22×C414yes—
121C4×D97—48no—
122C4×D97—44no—
123C4×D97—44no—
124C22×C97—24no—
125C22×C97—22no—
126C22×C97—22no—
127C22×C97—24no—
128C22×C97C2×C4⋊C414yes—
129C22×C97C23×C411yescharacteristic
130C22×C97C22×D411yescharacteristic
131C22×C97C22×Q811yescharacteristic
132C22×C97H(5,𝔽2)12yes—
133C22×C97H(5,𝔽2)12yes—
134C2×D97—24no—
135C2×D97—24no—
136C2×D97—22no—
137C2×D97—22no—
138C2×D97—22no—
139C2×D97—22no—
140C2×D97—24no—
141C2×D97—24no—
142C2×D97—416no—
143C2×D97C2×C4⋊C414yes—
144C2×D97C2×C4⋊C414yes—
145C4×C97—24no—
146C4×C97—41no—
147C4×C97—41no—
148C4×C97—42no—
149Dic97—24no—
150Dic97—41no—
151Dic97—41no—
152Dic97—42no—
153C2×C97—24no—
154C2×C97C22×C4⋊C411yescharacteristic
155C2×C97#51812yes—
156C2×C97#53411yescharacteristic
157C2×C97#53411yescharacteristic
158C2×C97#54312yes—
159D97—24no—
160D97—24no—
161#2315—971nomaximal, Sylow
162C97#231511yescharacteristic, complemented, Sylow
163C22×C22⋊C4—971no—
164C22×C22⋊C4—971no—
165C22×C22⋊C4—971no—
166#385—971no—
167#385—972no—
168#385—971no—
169#385—971no—
170#385—972no—
171#385D9711yescharacteristic, complemented
172#380—972no—
173#380—972no—
174C25—971no—
175C23×C4—972no—
176C23×C4—971no—
177C23×C4—971no—
178C23×C4—971no—
179C23×C4C2×D9711yescharacteristic
180C2×C4⋊C4—972no—
181C2×C4⋊C4—972no—
182C2×C4⋊C4—972no—
183C2×C4⋊C4C2×D9712yescomplemented
184C2×C22⋊C4—971no—
185C2×C22⋊C4—972no—
186C2×C22⋊C4—978no—
187C2×C22⋊C4—971no—
188C2×C22⋊C4—971no—
189C2×C22⋊C4—972no—
190C2×C22⋊C4—971no—
191C2×C22⋊C4—972no—
192C2×C22⋊C4—1944no—
193C2×C22⋊C4—1944no—
194C2×C22⋊C4C2×D9711yescharacteristic
195C2×C22⋊C4C2×D9711yescharacteristic
196#95—972no—
197#95—972no—
198#95—972no—
199#95C2×D9712yescomplemented
200C24—974no—
201C24—974no—
202C24—974no—
203C24—971no—
204C24—971no—
205C24—3882no—
206C24—3882no—
207C24C22×D9711yescharacteristic
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—972no—
211C22×C4—978no—
212C22×C4—971no—
213C22×C4—972no—
214C22×C4—971no—
215C22×C4—3882no—
216C22×C4—3882no—
217C22×C4—3882no—
218C22×C4—3882no—
219C22×C4—3882no—
220C22×C4—1942no—
221C22×C4—1942no—
222C22×C4—1942no—
223C22×C4C22×D9711yescharacteristic
224C22×C4C22×D9712yes—
225C22×C4C22×D9711yescharacteristic
226C22×C4C22×D9712yes—
227C4⋊C4—42no—
228C4⋊C4—3882no—
229C4⋊C4—3882no—
230C4⋊C4—3882no—
231C22⋊C4—22no—
232C22⋊C4—974no—
233C22⋊C4—978no—
234C22⋊C4—1942no—
235C22⋊C4—1942no—
236C22⋊C4—1944no—
237C22⋊C4—1944no—
238C22⋊C4—1942no—
239C22⋊C4C4×D9714yescomplemented
240C23—22no—
241C23—22no—
242C23—9716no—
243C23—972no—
244C23—972no—
245C23—972no—
246C23—972no—
247C23—972no—
248C23—972no—
249C23—3888no—
250C23—3884no—
251C23—3884no—
252C23—3888no—
253C23—1942no—
254C23—1942no—
255C23—1942no—
256C23—1942no—
257C23C2×C4×D9712yes—
258C23D4×D9712yes—
259C23Q8×D9712yes—
260C23C23×D9711yescharacteristic
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—44no—
271C2×C4—974no—
272C2×C4—3888no—
273C2×C4—3884no—
274C2×C4—3884no—
275C2×C4—3884no—
276C2×C4—1941no—
277C2×C4—1942no—
278C2×C4—1941no—
279C2×C4—1942no—
280C2×C4—1942no—
281C2×C4—1942no—
282C2×C4—1942no—
283C2×C4—1942no—
284C2×C4—1942no—
285C2×C4C2×C4×D9714yes—
286C22—24no—
287C22—22no—
288C22—22no—
289C22—24no—
290C22—974no—
291C22—974no—
292C22—38816no—
293C22—1944no—
294C22—1944no—
295C22—1942no—
296C22—1942no—
297C22—1942no—
298C22—1942no—
299C22—1944no—
300C22—1944no—
301C22C4⋊C4×D9714yes—
302C22C22×C4×D9711yescharacteristic
303C22C2×D4×D9711yescharacteristic
304C22C2×Q8×D9711yescharacteristic
305C22#1569085512yes—
306C22#1569085612yes—
307C4—24no—
308C4—42no—
309C4—41no—
310C4—41no—
311C4—3881no—
312C4—3881no—
313C4—3882no—
314C4—1944no—
315C2—24no—
316C2—1944no—
317C2—1944no—
318C2C2×C4⋊C4×D9711yescharacteristic
319C2#2026120712yes—
320C2#2026129511yescharacteristic
321C2#2026129611yescharacteristic
322C2#2026136012yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#385×C97C2
socle, cosocleC23×C97C24
prime radicalC24×D97C22
hypercenter#385D97
hypocenterC97#2315
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. #20260610C2semidirect
5. #20260609C2semidirect
6. #20260608C2semidirect
7. #20260607C2semidirect
8. #20260606C2semidirect
9. #20260594C2semidirect
10. #20260592C2semidirect
11. #385×C97C2semidirect
17. C2×C4⋊Dic97C22semidirect
18. C2×#13095490C22semidirect
19. C2×#13095490C22semidirect
29. C22⋊C4×D97C4semidirect
30. #15690717C22semidirect
31. #15690717C22semidirect
32. #15690717C22semidirect
34. C2×C4⋊C4×C97C22semidirect
37. #95×C97C22semidirect
57. C2×C4×D97C2×C4semidirect
62. C22⋊Dic97C2×C4semidirect
67. #13095488C2×C4semidirect
77. C22⋊C4×C97C2×C4semidirect
92. C22×D97C4⋊C4semidirect
162. C97#2315semidirect
171. #385D97semidirect
183. C2×C4⋊C4C2×D97semidirect
199. #95C2×D97semidirect
239. 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
#202606102
#202606092
#202606082
#202606072
#202606062
#202605942
#202605922
#385×C972
#231597