Finite Group Databasebeta

← #27144427

Subgroups — #27144427

Order 12416.

668 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690785C211yesmaximal, characteristic
2C2×#15690772C211yesmaximal, characteristic, complemented
3C2×#15690767C211yesmaximal, characteristic, complemented
4#20260637C211yesmaximal, characteristic, complemented
5#20260636C212yesmaximal, complemented
6#20260624C212yesmaximal, complemented
7#20260613C212yesmaximal, complemented
8#20260606C212yesmaximal, complemented
9#20260592C211yesmaximal, characteristic, complemented
10C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
11#389×C97C211yesmaximal, characteristic, complemented
12C22×D388C2211yescharacteristic
13C22×D388C2212yes—
14C22×C4×D97C2211yescharacteristic
15C22×C4×D97C2212yes—
16#15690785C418yescomplemented
17#15690772—24no—
18#15690767—24no—
19C2×C4⋊Dic97C2211yescharacteristic
20C2×#13095490C2211yescharacteristic
21C2×#13095490C2211yescharacteristic
22C2×#13095488C2211yescharacteristic
23C2×#13095488C2212yes—
24C2×#13095488C2211yescharacteristic
25C2×#13095488C2212yes—
26C2×#13095488C2212yes—
27C2×#13095488C2212yes—
28C2×#13095488C2212yes—
29C2×C4×Dic97C2212yes—
30C2×C4×Dic97C2211yescharacteristic
31#15690717C2212yes—
32#15690717C2212yes—
33C2×C4⋊C4×C97C2211yescharacteristic
34C2×C4⋊C4×C97C2212yes—
35C2×C4⋊C4×C97C2211yescharacteristic
36C2×C42×C97C2211yescharacteristic
37#95×C97C2212yes—
38C23×D97C2311yescharacteristic
39C23×D97C2312yes—
40C2×D388—22no—
41C2×D388—22no—
42C2×D388—42no—
43C2×D388—42no—
44C2×D388C2×C414yes—
45C2×D388C2×C418yescomplemented
46C22×Dic97C2312yes—
47C22×Dic97C2311yescharacteristic
48C22×Dic97C2311yescharacteristic
49C22×Dic97C2311yescharacteristic
50C2×C4×D97—22no—
51C2×C4×D97—22no—
52C2×C4×D97—24no—
53C2×C4×D97—42no—
54C2×C4×D97—42no—
55C2×C4×D97C2×C414yescomplemented
56C4⋊Dic97—22no—
57#13095490—22no—
58#13095490—22no—
59#13095488—22no—
60#13095488—22no—
61#13095488—24no—
62#13095488—42no—
63#13095488—42no—
64#13095488—42no—
65#13095488C2×C418yescomplemented
66C4×Dic97—42no—
67C4×Dic97D414yescomplemented
68C22×C4×C97C2311yescharacteristic
69C22×C4×C97C2311yescharacteristic
70C22×C4×C97C2312yes—
71C22×C4×C97C2312yes—
72C22×C4×C97C2311yescharacteristic
73C4⋊C4×C97—22no—
74C4⋊C4×C97—42no—
75C4⋊C4×C97C2×C414yescomplemented
76C42×C97—22no—
77C23×C97C2411yescharacteristic
78C22×D97—21no—
79C22×D97—22no—
80C22×D97—21no—
81C22×D97—22no—
82C22×D97—21no—
83C22×D97—22no—
84C22×D97—22no—
85C22×D97—22no—
86C22×D97—21no—
87C22×D97—21no—
88C22×D97—21no—
89C22×D97—22no—
90C22×D97C22×C414yes—
91C22×D97C22×C412yes—
92C2×C4×C97—22no—
93C2×C4×C97—22no—
94C2×C4×C97—21no—
95C2×C4×C97—21no—
96C2×C4×C97—22no—
97C2×C4×C97—21no—
98C2×C4×C97—22no—
99C2×C4×C97—21no—
100C2×C4×C97—21no—
101C2×C4×C97—22no—
102C2×C4×C97—21no—
103C2×C4×C97C22×C414yes—
104C2×C4×C97C22×C412yes—
105C2×C4×C97C2×D412yes—
106C2×C4×C97#4112yes—
107C2×Dic97—21no—
108C2×Dic97—21no—
109C2×Dic97—21no—
110C2×Dic97—21no—
111C2×Dic97—21no—
112C2×Dic97—21no—
113C2×Dic97—22no—
114C2×Dic97—22no—
115C2×Dic97—22no—
116C2×Dic97C22×C412yes—
117C2×Dic97C2×D412yes—
118C2×Dic97C2×D412yes—
119D388—28no—
120D388—48no—
121C4×D97—28no—
122C4×D97—48no—
123C22×C97C23×C411yescharacteristic
124C22×C97C22×D411yescharacteristic
125C22×C97C2×#4111yescharacteristic
126C22×C97H(5,𝔽2)11yescharacteristic
127C22×C97H(5,𝔽2)11yescharacteristic
128C22×C97H(5,𝔽2)11yescharacteristic
129C22×C972−1+411yescharacteristic
130C2×D97—22no—
131C2×D97—24no—
132C2×D97—22no—
133C2×D97—24no—
134C2×D97—22no—
135C2×D97—24no—
136C2×D97—42no—
137C2×D97—41no—
138C2×D97—41no—
139C2×D97—42no—
140C2×D97—41no—
141C2×D97—42no—
142C2×D97—41no—
143C2×D97—42no—
144C4×C97—24no—
145C4×C97—42no—
146C4×C97—41no—
147C4×C97—41no—
148C4×C97C4×D414yes—
149Dic97—22no—
150Dic97—22no—
151Dic97—22no—
152Dic97—42no—
153C2×C97C2×C4×D411yescharacteristic
154C2×C97#51811yescharacteristic
155C2×C97#52011yescharacteristic
156C2×C97#53511yescharacteristic
157C2×C97#53711yescharacteristic
158C2×C97#54011yescharacteristic
159C2×C97#54211yescharacteristic
160D97—42no—
161D97—44no—
162#2324—971nomaximal, Sylow
163C97#232411yescharacteristic, complemented, Sylow
164C2×#121—971no—
165C2×C4×D4—971no—
166C2×#117—971no—
167#390—972no—
168#390—971no—
169#389D9711yescharacteristic, complemented
170#388—972no—
171#386—972no—
172#385—972no—
173#380—971no—
174C4×C4⋊C4—971no—
175C22×D4—972no—
176C22×D4—971no—
177C23×C4—971no—
178C23×C4—972no—
179#121—978no—
180C4×D4—1944no—
181#117—1944no—
182C2×C4⋊C4—971no—
183C2×C4⋊C4—971no—
184C2×C4⋊C4—971no—
185C2×C4⋊C4C2×D9711yescharacteristic
186C2×C4⋊C4C2×D9712yes—
187C2×C4⋊C4C2×D9711yescharacteristic
188C2×C22⋊C4—972no—
189C2×C22⋊C4—972no—
190C2×C22⋊C4—972no—
191C2×C22⋊C4—972no—
192C2×C22⋊C4—971no—
193C2×C22⋊C4—971no—
194C2×C22⋊C4—972no—
195C2×C42—971no—
196C2×C42—972no—
197C2×C42C2×D9711yescharacteristic
198#95—972no—
199#95—972no—
200#95C2×D9712yes—
201C24—972no—
202C24—971no—
203C2×D4—978no—
204C2×D4—974no—
205C2×D4—3882no—
206C2×D4—3882no—
207C2×D4—1942no—
208C2×D4—1942no—
209C22×C4—971no—
210C22×C4—971no—
211C22×C4—971no—
212C22×C4—974no—
213C22×C4—972no—
214C22×C4—3882no—
215C22×C4—3882no—
216C22×C4—1944no—
217C22×C4—1942no—
218C22×C4—1942no—
219C22×C4C22×D9712yes—
220C22×C4C22×D9711yescharacteristic
221C22×C4C22×D9711yescharacteristic
222C22×C4C22×D9712yes—
223C22×C4C22×D9711yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—42no—
226C4⋊C4—1942no—
227C4⋊C4—1942no—
228C4⋊C4—1942no—
229C4⋊C4C4×D9714yescomplemented
230C22⋊C4—978no—
231C22⋊C4—3882no—
232C22⋊C4—3882no—
233C22⋊C4—3882no—
234C22⋊C4—1942no—
235C22⋊C4—1942no—
236C22⋊C4—1944no—
237C42—22no—
238C42—974no—
239C42—3882no—
240C23—974no—
241C23—972no—
242C23—1941no—
243C23—1942no—
244C23—1941no—
245C23—1942no—
246C23—1942no—
247C23—1941no—
248C23—1941no—
249C23—1942no—
250C23—1942no—
251C23—1941no—
252C23—1941no—
253C23—1942no—
254C23C23×D9711yescharacteristic
255D4—3888no—
256D4—1948no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—972no—
269C2×C4—972no—
270C2×C4—972no—
271C2×C4—3888no—
272C2×C4—1942no—
273C2×C4—1942no—
274C2×C4—1942no—
275C2×C4—1941no—
276C2×C4—1941no—
277C2×C4—1941no—
278C2×C4—1941no—
279C2×C4—1941no—
280C2×C4—1941no—
281C2×C4—1948no—
282C2×C4C2×C4×D9714yes—
283C2×C4C2×C4×D9712yes—
284C2×C4D4×D9712yes—
285C2×C4#1309551212yes—
286C22—3882no—
287C22—3881no—
288C22—3881no—
289C22—3882no—
290C22—3881no—
291C22—3882no—
292C22—3881no—
293C22—3882no—
294C22—1942no—
295C22—1944no—
296C22—1942no—
297C22—1944no—
298C22—1944no—
299C22—1942no—
300C22C22×C4×D9711yescharacteristic
301C22C2×D4×D9711yescharacteristic
302C22C2×#1309551211yescharacteristic
303C22#1569085511yescharacteristic
304C22#1569085611yescharacteristic
305C22#1569085611yescharacteristic
306C22#1569085811yescharacteristic
307C4—24no—
308C4—42no—
309C4—41no—
310C4—41no—
311C4—3882no—
312C4—1942no—
313C4—1942no—
314C4—1942no—
315C4#1569077214yes—
316C2—3884no—
317C2—3882no—
318C2C2×#1569077211yescharacteristic
319C2#2026120711yescharacteristic
320C2#2026121711yescharacteristic
321C2#2026130111yescharacteristic
322C2#2026131511yescharacteristic
323C2#2026134111yescharacteristic
324C2#2026135911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#389×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690785C2
hypercenter#389D97
hypocenterC97#2324
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C2×#15690772C2semidirect
3. C2×#15690767C2semidirect
4. #20260637C2semidirect
5. #20260636C2semidirect
6. #20260624C2semidirect
7. #20260613C2semidirect
8. #20260606C2semidirect
9. #20260592C2semidirect
10. C4⋊C4×Dic97C2semidirect
11. #389×C97C2semidirect
16. #15690785C4semidirect
45. C2×D388C2×C4semidirect
55. C2×C4×D97C2×C4semidirect
65. #13095488C2×C4semidirect
67. C4×Dic97D4semidirect
75. C4⋊C4×C97C2×C4semidirect
163. C97#2324semidirect
169. #389D97semidirect
229. C4⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907852
C2×#156907722
C2×#156907672
#202606372
#202606362
#202606242
#202606132
#202606062
#202605922
C4⋊C4×Dic972
#389×C972
#232497