Finite Group Databasebeta

← #27144751

Subgroups — #27144751

Order 12416.

788 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690785C211yesmaximal, characteristic
2C2×C4×D388C211yesmaximal, characteristic, complemented
3C2×#15690772C212yesmaximal, complemented
4C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
5#20260636C211yesmaximal, characteristic, complemented
6#20260624C212yesmaximal, complemented
7#20260613C212yesmaximal, complemented
8#20260606C212yesmaximal, complemented
9C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
10C4×#13095488C211yesmaximal, characteristic, complemented
11#389×C97C211yesmaximal, characteristic, complemented
12C22×D388C2212yes—
13C22×D388C2211yescharacteristic
14C22×C4×D97C2212yes—
15C22×C4×D97C2212yes—
16C22×C4×D97C2211yescharacteristic
17#15690785C418yescomplemented
18C4×D388—24no—
19#15690772—28no—
20C2×C4⋊Dic97C2211yescharacteristic
21C2×#13095490C2212yes—
22C4⋊C4×D97—24no—
23C2×#13095488C2211yescharacteristic
24C2×#13095488C2212yes—
25C2×#13095488C2212yes—
26C2×#13095488C2212yes—
27C2×#13095488C2212yes—
28C2×#13095488C2211yescharacteristic
29C2×C4×Dic97C2212yes—
30C2×C4×Dic97C2211yescharacteristic
31#15690717C2212yes—
32#15690717C2211yescharacteristic
33#15690717C2211yescharacteristic
34C2×C4⋊C4×C97C2211yescharacteristic
35C2×C4⋊C4×C97C2212yes—
36C2×C4⋊C4×C97C2211yescharacteristic
37C2×C42×C97C2211yescharacteristic
38#95×C97C2212yes—
39C23×D97C2312yes—
40C23×D97C2311yescharacteristic
41C2×D388—24no—
42C2×D388—24no—
43C2×D388—22no—
44C2×D388—22no—
45C2×D388C2×C418yescomplemented
46C2×D388C2×C414yescomplemented
47C22×Dic97C2311yescharacteristic
48C22×Dic97C2311yescharacteristic
49C22×Dic97C2311yescharacteristic
50C22×Dic97C2312yes—
51C2×C4×D97—22no—
52C2×C4×D97—22no—
53C2×C4×D97—24no—
54C2×C4×D97—24no—
55C2×C4×D97—24no—
56C2×C4×D97—24no—
57C2×C4×D97—42no—
58C2×C4×D97—42no—
59C2×C4×D97C2×C414yescomplemented
60C4⋊Dic97D414yescomplemented
61#13095490—24no—
62#13095488—22no—
63#13095488—24no—
64#13095488—24no—
65#13095488—42no—
66#13095488C2×C418yescomplemented
67#13095488D414yescomplemented
68C4×Dic97—22no—
69C4×Dic97—24no—
70C22×C4×C97C2311yescharacteristic
71C22×C4×C97C2312yes—
72C22×C4×C97C2312yes—
73C22×C4×C97C2311yescharacteristic
74C22×C4×C97C2311yescharacteristic
75C4⋊C4×C97—22no—
76C4⋊C4×C97—24no—
77C4⋊C4×C97C2×C414yescomplemented
78C42×C97—22no—
79C23×C97C2411yescharacteristic
80C22×D97—21no—
81C22×D97—21no—
82C22×D97—21no—
83C22×D97—21no—
84C22×D97—22no—
85C22×D97—22no—
86C22×D97—22no—
87C22×D97—22no—
88C22×D97—22no—
89C22×D97—22no—
90C22×D97C22×C414yes—
91C22×D97C22×C412yes—
92C22×D97C2×D412yes—
93C22×D97#4112yes—
94C2×C4×C97—21no—
95C2×C4×C97—22no—
96C2×C4×C97—22no—
97C2×C4×C97—21no—
98C2×C4×C97—21no—
99C2×C4×C97—22no—
100C2×C4×C97—22no—
101C2×C4×C97—21no—
102C2×C4×C97—22no—
103C2×C4×C97—21no—
104C2×C4×C97C22×C414yes—
105C2×C4×C97C22×C412yes—
106C2×C4×C97C2×D412yes—
107C2×C4×C97C2×D412yes—
108C2×C4×C97#4112yes—
109C2×Dic97—22no—
110C2×Dic97—21no—
111C2×Dic97—21no—
112C2×Dic97—21no—
113C2×Dic97—22no—
114C2×Dic97—21no—
115C2×Dic97—21no—
116C2×Dic97—21no—
117C2×Dic97—21no—
118C2×Dic97C22×C412yes—
119C2×Dic97C2×D412yes—
120C2×Dic97C2×D414yes—
121D388—216no—
122D388—28no—
123C4×D97—28no—
124C4×D97—48no—
125C4×D97—48no—
126C22×C97C23×C411yescharacteristic
127C22×C97C22×D411yescharacteristic
128C22×C97C22×D411yescharacteristic
129C22×C97C2×#4111yescharacteristic
130C22×C97C2×#4111yescharacteristic
131C22×C97H(5,𝔽2)11yescharacteristic
132C22×C972−1+411yescharacteristic
133C2×D97—22no—
134C2×D97—22no—
135C2×D97—22no—
136C2×D97—22no—
137C2×D97—22no—
138C2×D97—22no—
139C2×D97—24no—
140C2×D97—24no—
141C2×D97—24no—
142C2×D97—42no—
143C2×D97—42no—
144C2×D97—42no—
145C2×D97—42no—
146C2×D97C4×D414yescomplemented
147C4×C97—24no—
148C4×C97—22no—
149C4×C97—42no—
150C4×C97—41no—
151C4×C97C4×D414yescomplemented
152Dic97—22no—
153Dic97—24no—
154Dic97—22no—
155Dic97—41no—
156C2×C97C2×C4×D411yescharacteristic
157C2×C97C2×C4×D411yescharacteristic
158C2×C97#52011yescharacteristic
159C2×C97D4211yescharacteristic
160C2×C97#54711yescharacteristic
161C2×C97#54811yescharacteristic
162C2×C97#55311yescharacteristic
163D97—24no—
164D97—44no—
165#2361—971nomaximal, Sylow
166C97#236111yescharacteristic, complemented, Sylow
167C2×#121—971no—
168C2×C4×D4—972no—
169C2×C4×D4—971no—
170C22×C4⋊C4—971no—
171#390—971no—
172#389D9711yescharacteristic, complemented
173#388—972no—
174#386—972no—
175#385—972no—
176C4×C4⋊C4—971no—
177C4×C22⋊C4—971no—
178C22×D4—972no—
179C22×D4—971no—
180C23×C4—971no—
181C23×C4—972no—
182C23×C4—972no—
183#121—978no—
184C4×D4—1944no—
185C4×D4—1948no—
186C2×C4⋊C4—971no—
187C2×C4⋊C4—972no—
188C2×C4⋊C4—1944no—
189C2×C4⋊C4C2×D9711yescharacteristic
190C2×C4⋊C4C2×D9712yes—
191C2×C4⋊C4C2×D9711yescharacteristic
192C2×C22⋊C4—971no—
193C2×C22⋊C4—972no—
194C2×C22⋊C4—972no—
195C2×C22⋊C4—972no—
196C2×C22⋊C4—972no—
197C2×C22⋊C4—971no—
198C2×C42—972no—
199C2×C42—971no—
200C2×C42C2×D9711yescharacteristic
201#95—971no—
202#95—971no—
203#95—972no—
204#95C2×D9712yes—
205C24—971no—
206C24—972no—
207C2×D4—978no—
208C2×D4—974no—
209C2×D4—1944no—
210C2×D4—1944no—
211C2×D4—1942no—
212C2×D4—1942no—
213C22×C4—971no—
214C22×C4—971no—
215C22×C4—972no—
216C22×C4—974no—
217C22×C4—971no—
218C22×C4—3882no—
219C22×C4—3882no—
220C22×C4—1944no—
221C22×C4—1944no—
222C22×C4—1944no—
223C22×C4—1944no—
224C22×C4—1942no—
225C22×C4—1942no—
226C22×C4C22×D9711yescharacteristic
227C22×C4C22×D9712yes—
228C22×C4C22×D9712yes—
229C22×C4C22×D9711yescharacteristic
230C22×C4C22×D9711yescharacteristic
231C4⋊C4—22no—
232C4⋊C4—24no—
233C4⋊C4—974no—
234C4⋊C4—1944no—
235C4⋊C4C4×D9714yescomplemented
236C22⋊C4—978no—
237C22⋊C4—974no—
238C22⋊C4—3882no—
239C22⋊C4—1944no—
240C22⋊C4—1944no—
241C22⋊C4—1942no—
242C42—22no—
243C42—1944no—
244C42—1942no—
245C23—972no—
246C23—972no—
247C23—972no—
248C23—974no—
249C23—1942no—
250C23—1942no—
251C23—1942no—
252C23—1942no—
253C23—1942no—
254C23—1942no—
255C23—1941no—
256C23—1941no—
257C23—1941no—
258C23—1941no—
259C23C23×D9711yescharacteristic
260D4—19416no—
261D4—1948no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—972no—
273C2×C4—972no—
274C2×C4—974no—
275C2×C4—3888no—
276C2×C4—3888no—
277C2×C4—1941no—
278C2×C4—1941no—
279C2×C4—1941no—
280C2×C4—1942no—
281C2×C4—1941no—
282C2×C4—1941no—
283C2×C4—1941no—
284C2×C4—1942no—
285C2×C4—1941no—
286C2×C4—1948no—
287C2×C4C2×C4×D9714yes—
288C2×C4C2×C4×D9712yes—
289C2×C4D4×D9712yes—
290C2×C4D4×D9712yes—
291C2×C4#1309551212yes—
292C22—974no—
293C22—3882no—
294C22—3882no—
295C22—3882no—
296C22—3882no—
297C22—1944no—
298C22—1944no—
299C22—1944no—
300C22—1942no—
301C22—1942no—
302C22—1942no—
303C22—1942no—
304C22—1942no—
305C22—1942no—
306C22C22×C4×D9711yescharacteristic
307C22C2×D4×D9711yescharacteristic
308C22C2×D4×D9711yescharacteristic
309C22#41×D9711yescharacteristic
310C22C2×#1309551211yescharacteristic
311C22#1569085611yescharacteristic
312C22#1569085811yescharacteristic
313C4—24no—
314C4—22no—
315C4—41no—
316C4—42no—
317C4—3881no—
318C4—1942no—
319C4—1942no—
320C4—1944no—
321C4#1569077214yescomplemented
322C2—3884no—
323C2—1944no—
324C2C4×D4×D9711yescharacteristic
325C2C2×#1569077211yescharacteristic
326C2#2026121711yescharacteristic
327C2#2026136911yescharacteristic
328C2#2026138811yescharacteristic
329C2#2026139511yescharacteristic
330C2#2026143811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#389×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690785C2
hypercenter#389D97
hypocenterC97#2361
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C2×C4×D388C2semidirect
3. C2×#15690772C2semidirect
4. C2×C4⋊C4×D97C2semidirect
5. #20260636C2semidirect
6. #20260624C2semidirect
7. #20260613C2semidirect
8. #20260606C2semidirect
9. C4⋊C4×Dic97C2semidirect
10. C4×#13095488C2semidirect
11. #389×C97C2semidirect
17. #15690785C4semidirect
45. C2×D388C2×C4semidirect
46. C2×D388C2×C4semidirect
59. C2×C4×D97C2×C4semidirect
60. C4⋊Dic97D4semidirect
66. #13095488C2×C4semidirect
67. #13095488D4semidirect
77. C4⋊C4×C97C2×C4semidirect
146. C2×D97C4×D4semidirect
151. C4×C97C4×D4semidirect
166. C97#2361semidirect
172. #389D97semidirect
235. C4⋊C4C4×D97semidirect
321. C4#15690772semidirect

Maximal subgroups

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

subgroupindex
C2×#156907852
C2×C4×D3882
C2×#156907722
C2×C4⋊C4×D972
#202606362
#202606242
#202606132
#202606062
C4⋊C4×Dic972
C4×#130954882
#389×C972
#236197