Finite Group Databasebeta

← #27144432

Subgroups — #27144432

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×#15690781C211yesmaximal, characteristic
2C2×D4×Dic97C211yesmaximal, characteristic, complemented
3C2×#15690767C211yesmaximal, characteristic, complemented
4#20260635C212yesmaximal, complemented
5#20260633C211yesmaximal, characteristic, complemented
6#20260623C212yesmaximal, complemented
7#20260612C212yesmaximal, complemented
8#20260607C212yesmaximal, complemented
9#20260592C211yesmaximal, characteristic, complemented
10C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
11#390×C97C211yesmaximal, characteristic, complemented
12C22×C22⋊D97C2212yes—
13C23×Dic97C2212yes—
14C22×C4×D97C2211yescharacteristic
15#15690781C418yescomplemented
16D4×Dic97—24no—
17#15690767—24no—
18C2×C4⋊Dic97C2211yescharacteristic
19C2×C4⋊Dic97C2211yescharacteristic
20C2×#13095490C2211yescharacteristic
21C2×#13095490C2211yescharacteristic
22C2×#13095490C2212yes—
23C2×C22⋊Dic97C2212yes—
24C2×C22⋊Dic97C2212yes—
25C2×#13095488C2211yescharacteristic
26C2×#13095488C2211yescharacteristic
27C2×#13095488C2212yes—
28C2×C4×Dic97C2212yes—
29C2×C4×Dic97C2211yescharacteristic
30#15690717C2212yes—
31#15690717C2212yes—
32#15690717C2212yes—
33C22×D4×C97C2211yescharacteristic
34C2×C4⋊C4×C97C2211yescharacteristic
35C2×C22⋊C4×C97C2212yes—
36C2×C22⋊C4×C97C2212yes—
37C2×C42×C97C2211yescharacteristic
38C23×D97C2311yescharacteristic
39C2×C22⋊D97—42no—
40C2×C22⋊D97—42no—
41C2×C22⋊D97C2×C418yescomplemented
42C22×Dic97—24no—
43C22×Dic97—42no—
44C22×Dic97—42no—
45C22×Dic97C2312yes—
46C22×Dic97C2312yes—
47C22×Dic97C2311yescharacteristic
48C22×Dic97C2311yescharacteristic
49C22×Dic97C2311yescharacteristic
50C2×C4×D97—22no—
51C2×C4×D97—22no—
52C2×C4×D97C2×C414yescomplemented
53C4⋊Dic97—22no—
54C4⋊Dic97C2×C414yescomplemented
55#13095490—22no—
56#13095490—22no—
57#13095490—42no—
58C22⋊Dic97—24no—
59C22⋊Dic97C2×C418yescomplemented
60#13095488—22no—
61#13095488—22no—
62#13095488—42no—
63C4×Dic97—42no—
64C4×Dic97D414yescomplemented
65C24×C97C2312yes—
66C2×D4×C97—22no—
67C2×D4×C97—22no—
68C2×D4×C97C2×C414yes—
69C22×C4×C97C2312yes—
70C22×C4×C97C2311yescharacteristic
71C22×C4×C97C2311yescharacteristic
72C22×C4×C97C2311yescharacteristic
73C4⋊C4×C97—22no—
74C22⋊C4×C97—42no—
75C22⋊C4×C97—42no—
76C42×C97—22no—
77C23×C97—22no—
78C23×C97—22no—
79C23×C97—22no—
80C23×C97—22no—
81C23×C97—22no—
82C23×C97—22no—
83C23×C97C22×C414yes—
84C23×C97C2411yescharacteristic
85C22×D97—21no—
86C22×D97—21no—
87C22×D97—21no—
88C22×D97—21no—
89C22×D97—21no—
90C22×D97—21no—
91C22×D97C22×C412yes—
92D4×C97—28no—
93C2×C4×C97—21no—
94C2×C4×C97—21no—
95C2×C4×C97—22no—
96C2×C4×C97—21no—
97C2×C4×C97—21no—
98C2×C4×C97—22no—
99C2×C4×C97—22no—
100C2×C4×C97—21no—
101C2×C4×C97—21no—
102C2×C4×C97C22×C412yes—
103C2×C4×C97C2×D412yes—
104C2×C4×C97#4112yes—
105C22⋊D97—48no—
106C2×Dic97—21no—
107C2×Dic97—21no—
108C2×Dic97—21no—
109C2×Dic97—22no—
110C2×Dic97—22no—
111C2×Dic97—21no—
112C2×Dic97—21no—
113C2×Dic97—21no—
114C2×Dic97—22no—
115C2×Dic97—22no—
116C2×Dic97—22no—
117C2×Dic97—48no—
118C2×Dic97C22×C412yes—
119C2×Dic97C22×C414yes—
120C2×Dic97C2×D412yes—
121C2×Dic97C2×D412yes—
122C4×D97—28no—
123C22×C97—24no—
124C22×C97—24no—
125C22×C97—24no—
126C22×C97—42no—
127C22×C97—42no—
128C22×C97—42no—
129C22×C97—42no—
130C22×C97C23×C411yescharacteristic
131C22×C97C22×D411yescharacteristic
132C22×C97C2×#4111yescharacteristic
133C22×C97H(5,𝔽2)11yescharacteristic
134C22×C97H(5,𝔽2)11yescharacteristic
135C22×C97H(5,𝔽2)11yescharacteristic
136C22×C972−1+411yescharacteristic
137C2×D97—22no—
138C2×D97—22no—
139C2×D97—22no—
140C2×D97—41no—
141C2×D97—41no—
142C2×D97—41no—
143C2×D97—41no—
144C4×C97—41no—
145C4×C97—41no—
146C4×C97—42no—
147C4×C97C4×D414yes—
148Dic97—22no—
149Dic97—24no—
150Dic97—22no—
151Dic97—22no—
152Dic97—42no—
153C2×C97—44no—
154C2×C97C2×C4×D411yescharacteristic
155C2×C97#51811yescharacteristic
156C2×C97#52011yescharacteristic
157C2×C97#53511yescharacteristic
158C2×C97#53711yescharacteristic
159C2×C97#54011yescharacteristic
160C2×C97#54211yescharacteristic
161D97—42no—
162#2324—971nomaximal, Sylow
163C97#232411yescharacteristic, complemented, Sylow
164C2×#121—971no—
165C2×C4×D4—971no—
166C2×#117—971no—
167#390—972no—
168#390D9711yescharacteristic, complemented
169#389—971no—
170#388—972no—
171#386—972no—
172#385—972no—
173#380—971no—
174C4×C4⋊C4—971no—
175C22×D4—972no—
176C22×D4C2×D9711yescharacteristic
177C23×C4—971no—
178C23×C4—972no—
179#121—978no—
180C4×D4—1944no—
181#117—1944no—
182C2×C4⋊C4—971no—
183C2×C4⋊C4—972no—
184C2×C4⋊C4—971no—
185C2×C4⋊C4—971no—
186C2×C4⋊C4—971no—
187C2×C4⋊C4C2×D9711yescharacteristic
188C2×C22⋊C4—972no—
189C2×C22⋊C4—972no—
190C2×C22⋊C4—972no—
191C2×C22⋊C4—971no—
192C2×C22⋊C4—971no—
193C2×C22⋊C4C2×D9712yes—
194C2×C22⋊C4C2×D9712yes—
195C2×C42—971no—
196C2×C42—972no—
197C2×C42C2×D9711yescharacteristic
198#95—972no—
199#95—972no—
200#95—972no—
201C24—971no—
202C24C22×D9712yes—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—978no—
206C2×D4—3882no—
207C2×D4—3882no—
208C2×D4C4×D9714yes—
209C22×C4—972no—
210C22×C4—971no—
211C22×C4—971no—
212C22×C4—971no—
213C22×C4—974no—
214C22×C4—972no—
215C22×C4—3882no—
216C22×C4—3882no—
217C22×C4—1944no—
218C22×C4—1942no—
219C22×C4—1942no—
220C22×C4C22×D9712yes—
221C22×C4C22×D9711yescharacteristic
222C22×C4C22×D9711yescharacteristic
223C22×C4C22×D9711yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—974no—
226C4⋊C4—3882no—
227C4⋊C4—1942no—
228C4⋊C4—1942no—
229C4⋊C4—1942no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—978no—
233C22⋊C4—3882no—
234C22⋊C4—1942no—
235C22⋊C4—1942no—
236C22⋊C4—1944no—
237C42—22no—
238C42—974no—
239C42—3882no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—972no—
247C23—1941no—
248C23—1941no—
249C23—1941no—
250C23—1941no—
251C23—1941no—
252C23—1941no—
253C23C2×C4×D9714yes—
254C23C23×D9711yescharacteristic
255D4—28no—
256D4—3888no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—972no—
267C2×C4—974no—
268C2×C4—972no—
269C2×C4—972no—
270C2×C4—3888no—
271C2×C4—1942no—
272C2×C4—1942no—
273C2×C4—1942no—
274C2×C4—1941no—
275C2×C4—1941no—
276C2×C4—1941no—
277C2×C4—1942no—
278C2×C4—1941no—
279C2×C4—1942no—
280C2×C4—1941no—
281C2×C4—1941no—
282C2×C4—1948no—
283C2×C4C2×C4×D9712yes—
284C2×C4D4×D9712yes—
285C2×C4#1309551012yes—
286C22—24no—
287C22—24no—
288C22—24no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—3881no—
294C22—3881no—
295C22—3881no—
296C22—3881no—
297C22—1942no—
298C22—1942no—
299C22—1942no—
300C22C22×C4×D9711yescharacteristic
301C22C2×D4×D9711yescharacteristic
302C22C2×#1309551011yescharacteristic
303C22#1569085511yescharacteristic
304C22#1569085511yescharacteristic
305C22#1569085611yescharacteristic
306C22#1569085711yescharacteristic
307C4—41no—
308C4—41no—
309C4—42no—
310C4—3882no—
311C4—1942no—
312C4—1944no—
313C4—1942no—
314C4—1942no—
315C4#1569077014yes—
316C2—44no—
317C2—3882no—
318C2C2×#1569077011yescharacteristic
319C2#2026120711yescharacteristic
320C2#2026121411yescharacteristic
321C2#2026130111yescharacteristic
322C2#2026131411yescharacteristic
323C2#2026133911yescharacteristic
324C2#2026135411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#390×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690781C2
hypercenter#390D97
hypocenterC97#2324
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic97C2semidirect
3. C2×#15690767C2semidirect
4. #20260635C2semidirect
5. #20260633C2semidirect
6. #20260623C2semidirect
7. #20260612C2semidirect
8. #20260607C2semidirect
9. #20260592C2semidirect
10. C4⋊C4×Dic97C2semidirect
11. #390×C97C2semidirect
15. #15690781C4semidirect
41. C2×C22⋊D97C2×C4semidirect
52. C2×C4×D97C2×C4semidirect
54. C4⋊Dic97C2×C4semidirect
59. C22⋊Dic97C2×C4semidirect
64. C4×Dic97D4semidirect
163. C97#2324semidirect
168. #390D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907812
C2×D4×Dic972
C2×#156907672
#202606352
#202606332
#202606232
#202606122
#202606072
#202605922
C4⋊C4×Dic972
#390×C972
#232497