Finite Group Databasebeta

← #385×D97 #27144144

Subgroups — #385×D97 #27144144

Order 12416.

1232 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D97C211yesmaximal, characteristic
2C2×C4⋊C4×D97C212yesmaximal, complemented
3C2×C22⋊C4×D97C211yesmaximal, characteristic
4C2×C22⋊C4×D97C211yesmaximal, characteristic
5#20260610C211yesmaximal, characteristic, complemented
6#20260609C212yesmaximal, complemented
7#20260608C211yesmaximal, characteristic, complemented
8#20260607C211yesmaximal, characteristic, complemented
9#20260606C212yesmaximal, complemented
10#95×D97C212yesmaximal, complemented
11#385×C97C211yesmaximal, characteristic, complemented
12C24×D97C2211yescharacteristic
13C23×Dic97C2211yescharacteristic
14C22×C4×D97—24no—
15C22×C4×D97—28no—
16C22×C4×D97C2212yes—
17C22×C4×D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C22×C4×D97C2212yes—
20C22×C4×D97C2212yes—
21C2×C4⋊Dic97C2212yescomplemented
22C2×#13095490C2212yescomplemented
23C2×#13095490C2212yescomplemented
24C4⋊C4×D97—28no—
25C2×C22⋊Dic97C2211yescharacteristic
26C2×C22⋊Dic97C2211yescharacteristic
27C2×#13095488C2211yescharacteristic
28C2×#13095488C2211yescharacteristic
29C2×#13095488C2212yes—
30C22⋊C4×D97—24no—
31C22⋊C4×D97C418yescomplemented
32#15690717C2212yescomplemented
33#15690717C2212yescomplemented
34#15690717C2212yescomplemented
35C23×C4×C97C2211yescharacteristic
36C2×C4⋊C4×C97C2212yescomplemented
37C2×C22⋊C4×C97C2211yescharacteristic
38C2×C22⋊C4×C97C2211yescharacteristic
39#95×C97C2212yescomplemented
40C23×D97—24no—
41C23×D97—24no—
42C23×D97C2×C414yes—
43C23×D97D414yes—
44C23×D97Q814yes—
45C23×D97C2311yescharacteristic
46C23×D97C2311yescharacteristic
47C22×Dic97—22no—
48C22×Dic97—24no—
49C22×Dic97C2311yescharacteristic
50C22×Dic97C2311yescharacteristic
51C22×Dic97C2312yes—
52C22×Dic97C2312yes—
53C2×C4×D97—232no—
54C2×C4×D97—24no—
55C2×C4×D97—24no—
56C2×C4×D97—24no—
57C2×C4×D97—24no—
58C2×C4×D97—24no—
59C2×C4×D97—24no—
60C2×C4×D97—24no—
61C2×C4×D97—22no—
62C2×C4×D97—22no—
63C2×C4×D97—24no—
64C2×C4×D97—24no—
65C2×C4×D97C2×C418yescomplemented
66C2×C4×D97D418yes—
67C4⋊Dic97—24no—
68#13095490—24no—
69#13095490—24no—
70C22⋊Dic97—22no—
71C22⋊Dic97C2×C414yescomplemented
72#13095488—22no—
73#13095488—22no—
74#13095488C2×C418yescomplemented
75C24×C97C2311yescharacteristic
76C22×C4×C97—22no—
77C22×C4×C97—24no—
78C22×C4×C97C2311yescharacteristic
79C22×C4×C97C2311yescharacteristic
80C22×C4×C97C2312yes—
81C22×C4×C97C2312yes—
82C4⋊C4×C97—24no—
83C22⋊C4×C97—22no—
84C22⋊C4×C97C2×C414yescomplemented
85C23×C97—22no—
86C23×C97—22no—
87C23×C97C22×C412yes—
88C23×C97C2×D412yes—
89C23×C97C2×Q812yes—
90C23×C97C2411yescharacteristic
91C22×D97—22no—
92C22×D97—22no—
93C22×D97—216no—
94C22×D97—28no—
95C22×D97—216no—
96C22×D97—28no—
97C22×D97C4⋊C4116yescomplemented
98C22×D97C22×C412yes—
99C22×D97C22×C412yes—
100C22×D97C2×D412yes—
101C22×D97C2×D412yes—
102C22×D97C2×D414yes—
103C22×D97C2×Q812yes—
104C22×D97C2×Q812yes—
105C22×D97#4114yes—
106C2×C4×C97—28no—
107C2×C4×C97—22no—
108C2×C4×C97—22no—
109C2×C4×C97—22no—
110C2×C4×C97—22no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—22no—
114C2×C4×C97—22no—
115C2×C4×C97C22×C414yescomplemented
116C2×C4×C97C2×D414yes—
117C2×Dic97—28no—
118C2×Dic97—22no—
119C2×Dic97—22no—
120C2×Dic97—22no—
121C2×Dic97—22no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97—22no—
125C2×Dic97—22no—
126C2×Dic97C22×C414yescomplemented
127C2×Dic97C2×D414yes—
128C4×D97—216no—
129C4×D97—216no—
130C4×D97—216no—
131C4×D97—44no—
132C4×D97—44no—
133C22×C97—24no—
134C22×C97—22no—
135C22×C97—24no—
136C22×C97—22no—
137C22×C97C2×C4⋊C414yescomplemented
138C22×C97C23×C411yescharacteristic
139C22×C97C22×D411yescharacteristic
140C22×C97C22×D412yes—
141C22×C97C22×Q811yescharacteristic
142C22×C97C2×#4112yes—
143C2×D97—24no—
144C2×D97—22no—
145C2×D97—24no—
146C2×D97—22no—
147C2×D97—232no—
148C2×D97C2×C4⋊C414yes—
149C2×D97C2×C4⋊C414yescomplemented
150C2×D97C4×D418yes—
151C2×D97#12014yes—
152C2×D97C22⋊Q818yes—
153C2×D97#12314yes—
154C4×C97—24no—
155C4×C97—24no—
156C4×C97—41no—
157C4×C97—41no—
158Dic97—24no—
159Dic97—24no—
160Dic97—41no—
161Dic97—41no—
162C2×C97—24no—
163C2×C97C22×C4⋊C411yescharacteristic
164C2×C97C2×C4×D412yes—
165C2×C97C2×#12011yescharacteristic
166C2×C97C2×C22⋊Q812yes—
167C2×C97C2×#12311yescharacteristic
168D97—24no—
169D97#38518yesdirect product factor
170C2×#385—971nomaximal, Sylow
171C97C2×#38511yescharacteristic, complemented, Sylow
172C24×C4—971no—
173C22×C4⋊C4—972no—
174C22×C22⋊C4—971no—
175C22×C22⋊C4—971no—
176#385—971no—
177#385—972no—
178#385—971no—
179#385—971no—
180#385—972no—
181#385D9711yescharacteristic, direct product factor
182C2×#95—972no—
183C25—971no—
184C23×C4—972no—
185C23×C4—972no—
186C23×C4—971no—
187C23×C4—971no—
188C23×C4—971no—
189C23×C4—972no—
190C23×C4—1948no—
191C23×C4—1944no—
192C23×C4C2×D9711yescharacteristic
193C2×C4⋊C4—972no—
194C2×C4⋊C4—972no—
195C2×C4⋊C4—972no—
196C2×C4⋊C4—1948no—
197C2×C4⋊C4C2×D9712yescomplemented
198C2×C22⋊C4—971no—
199C2×C22⋊C4—972no—
200C2×C22⋊C4—978no—
201C2×C22⋊C4—971no—
202C2×C22⋊C4—971no—
203C2×C22⋊C4—971no—
204C2×C22⋊C4—1944no—
205C2×C22⋊C4C2×D9711yescharacteristic
206C2×C22⋊C4C2×D9711yescharacteristic
207#95—972no—
208#95—972no—
209#95—972no—
210#95C2×D9712yescomplemented
211C24—971no—
212C24—974no—
213C24—974no—
214C24—974no—
215C24—971no—
216C24—1944no—
217C24—1944no—
218C24C22×D9711yescharacteristic
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—972no—
222C22×C4—972no—
223C22×C4—978no—
224C22×C4—978no—
225C22×C4—971no—
226C22×C4—971no—
227C22×C4—1944no—
228C22×C4—1942no—
229C22×C4—1942no—
230C22×C4—1944no—
231C22×C4—1944no—
232C22×C4—1944no—
233C22×C4—1944no—
234C22×C4—1944no—
235C22×C4—1944no—
236C22×C4—1944no—
237C22×C4—19432no—
238C22×C4—1944no—
239C22×C4—1942no—
240C22×C4—1944no—
241C22×C4C22×D9711yescharacteristic
242C22×C4C22×D9711yescharacteristic
243C22×C4C22×D9712yes—
244C22×C4C22×D9712yes—
245C4⋊C4—24no—
246C4⋊C4—1944no—
247C4⋊C4—1944no—
248C4⋊C4—1944no—
249C22⋊C4—22no—
250C22⋊C4—974no—
251C22⋊C4—978no—
252C22⋊C4—1942no—
253C22⋊C4—1942no—
254C22⋊C4—1942no—
255C22⋊C4C4×D9714yescomplemented
256C23—22no—
257C23—22no—
258C23—972no—
259C23—974no—
260C23—972no—
261C23—972no—
262C23—974no—
263C23—9716no—
264C23—972no—
265C23—972no—
266C23—972no—
267C23—1948no—
268C23—19416no—
269C23—19416no—
270C23—1948no—
271C23—1942no—
272C23—1942no—
273C23C2×C4×D9712yes—
274C23D4×D9712yes—
275C23Q8×D9712yes—
276C23C23×D9711yescharacteristic
277C2×C4—28no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—974no—
287C2×C4—974no—
288C2×C4—3884no—
289C2×C4—3884no—
290C2×C4—19416no—
291C2×C4—19416no—
292C2×C4—19416no—
293C2×C4—1941no—
294C2×C4—1941no—
295C2×C4—1942no—
296C2×C4—1942no—
297C2×C4—1948no—
298C2×C4—1942no—
299C2×C4—1942no—
300C2×C4—1942no—
301C2×C4—1942no—
302C2×C4C2×C4×D9714yescomplemented
303C2×C4D4×D9714yes—
304C22—24no—
305C22—22no—
306C22—24no—
307C22—22no—
308C22—974no—
309C22—978no—
310C22—974no—
311C22—978no—
312C22—974no—
313C22—974no—
314C22—19432no—
315C22—1944no—
316C22—1942no—
317C22—1944no—
318C22—1942no—
319C22C4⋊C4×D9714yescomplemented
320C22C22×C4×D9711yescharacteristic
321C22C2×D4×D9711yescharacteristic
322C22C2×D4×D9712yes—
323C22C2×Q8×D9711yescharacteristic
324C22#41×D9712yes—
325C4—24no—
326C4—24no—
327C4—41no—
328C4—41no—
329C4—3881no—
330C4—3881no—
331C4—1944no—
332C4—1944no—
333C2—24no—
334C2—978no—
335C2—1944no—
336C2C2×C4⋊C4×D9711yescharacteristic
337C2C4×D4×D9712yes—
338C2#120×D9711yescharacteristic
339C2C22⋊Q8×D9712yes—
340C2#123×D9711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#385×C97C2
socle, cosocleC23×C97C24
prime radicalC24×D97C22
hypercenter#385D97
hypocenterC97C2×#385
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D97C2semidirect
5. #20260610C2semidirect
6. #20260609C2semidirect
7. #20260608C2semidirect
8. #20260607C2semidirect
9. #20260606C2semidirect
10. #95×D97C2semidirect
11. #385×C97C2semidirect
21. C2×C4⋊Dic97C22semidirect
22. C2×#13095490C22semidirect
23. C2×#13095490C22semidirect
31. C22⋊C4×D97C4semidirect
32. #15690717C22semidirect
33. #15690717C22semidirect
34. #15690717C22semidirect
36. C2×C4⋊C4×C97C22semidirect
39. #95×C97C22semidirect
65. C2×C4×D97C2×C4semidirect
71. C22⋊Dic97C2×C4semidirect
74. #13095488C2×C4semidirect
84. C22⋊C4×C97C2×C4semidirect
97. C22×D97C4⋊C4semidirect
115. C2×C4×C97C22×C4semidirect
126. C2×Dic97C22×C4semidirect
137. C22×C97C2×C4⋊C4semidirect
149. C2×D97C2×C4⋊C4semidirect
169. D97#385direct (N × H)
171. C97C2×#385semidirect
181. #385D97direct (N × H)
197. C2×C4⋊C4C2×D97semidirect
210. #95C2×D97semidirect
255. C22⋊C4C4×D97semidirect
302. C2×C4C2×C4×D97semidirect
319. C22C4⋊C4×D97semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D972
C2×C4⋊C4×D972
C2×C22⋊C4×D972
#202606102
#202606092
#202606082
#202606072
#202606062
#95×D972
#385×C972
C2×#38597