Finite Group Databasebeta

← #387×D97 #27144156

Subgroups — #387×D97 #27144156

Order 12416.

712 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D97C211yesmaximal, characteristic
2C2×C4⋊C4×D97C211yesmaximal, characteristic
3C2×C42×D97C211yesmaximal, characteristic
4#20260622C211yesmaximal, characteristic, complemented
5#20260621C211yesmaximal, characteristic, complemented
6#20260620C211yesmaximal, characteristic, complemented
7#20260619C211yesmaximal, characteristic, complemented
8#20260618C211yesmaximal, characteristic, complemented
9#20260617C211yesmaximal, characteristic, complemented
10#20260616C211yesmaximal, characteristic, complemented
11#95×D97C211yesmaximal, characteristic
12#95×D97C211yesmaximal, characteristic
13#95×D97C211yesmaximal, characteristic
14#95×D97C211yesmaximal, characteristic
15#387×C97C211yesmaximal, characteristic, complemented
16C22×C4×D97C2211yescharacteristic
17C22×C4×D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21C22×C4×D97C2211yescharacteristic
22C22×C4×D97C2211yescharacteristic
23C2×C4⋊Dic97C2211yescharacteristic
24C2×C4⋊Dic97C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×#13095490C2211yescharacteristic
28C2×#13095490C2211yescharacteristic
29C4⋊C4×D97—24no—
30C4⋊C4×D97C418yescomplemented
31C2×C4×Dic97C2211yescharacteristic
32C2×C4×Dic97C2211yescharacteristic
33C2×C4×Dic97C2211yescharacteristic
34C42×D97—24no—
35#15690717C2211yescharacteristic
36#15690717C2211yescharacteristic
37#15690717C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46#15690717C2211yescharacteristic
47C2×C4⋊C4×C97C2211yescharacteristic
48C2×C4⋊C4×C97C2211yescharacteristic
49C2×C42×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52#95×C97C2211yescharacteristic
53#95×C97C2211yescharacteristic
54C23×D97C2311yescharacteristic
55C22×Dic97C2311yescharacteristic
56C22×Dic97C2311yescharacteristic
57C22×Dic97C2311yescharacteristic
58C22×Dic97C2311yescharacteristic
59C22×Dic97C2311yescharacteristic
60C22×Dic97C2311yescharacteristic
61C22×Dic97C2311yescharacteristic
62C2×C4×D97—22no—
63C2×C4×D97—22no—
64C2×C4×D97—22no—
65C2×C4×D97—22no—
66C2×C4×D97—22no—
67C2×C4×D97—22no—
68C2×C4×D97—22no—
69C2×C4×D97—22no—
70C2×C4×D97—22no—
71C2×C4×D97—22no—
72C2×C4×D97—22no—
73C2×C4×D97—22no—
74C2×C4×D97—22no—
75C2×C4×D97—22no—
76C2×C4×D97—22no—
77C2×C4×D97—22no—
78C2×C4×D97C2×C414yes—
79C2×C4×D97C2×C414yes—
80C2×C4×D97C2×C414yes—
81C2×C4×D97D414yes—
82C2×C4×D97Q814yes—
83C4⋊Dic97—22no—
84C4⋊Dic97C2×C414yescomplemented
85#13095490—22no—
86#13095490—22no—
87#13095490C2×C414yescomplemented
88#13095490C2×C414yescomplemented
89C4×Dic97—22no—
90C4×Dic97—22no—
91C4×Dic97—22no—
92C22×C4×C97C2311yescharacteristic
93C22×C4×C97C2311yescharacteristic
94C22×C4×C97C2311yescharacteristic
95C22×C4×C97C2311yescharacteristic
96C22×C4×C97C2311yescharacteristic
97C22×C4×C97C2311yescharacteristic
98C22×C4×C97C2311yescharacteristic
99C4⋊C4×C97—22no—
100C4⋊C4×C97C2×C414yescomplemented
101C42×C97—22no—
102C23×C97C2411yescharacteristic
103C22×D97C22×C412yes—
104C22×D97C2×D412yes—
105C22×D97C2×Q812yes—
106C22×D97#4112yes—
107C22×D97#4112yes—
108C22×D97#4112yes—
109C22×D97#4112yes—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97—21no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97C22×C412yes—
127C2×C4×C97C22×C412yes—
128C2×C4×C97C22×C412yes—
129C2×C4×C97C2×D412yes—
130C2×C4×C97C2×Q812yes—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—21no—
139C2×Dic97—21no—
140C2×Dic97—21no—
141C2×Dic97—21no—
142C2×Dic97—21no—
143C2×Dic97—21no—
144C2×Dic97—21no—
145C2×Dic97—21no—
146C2×Dic97—21no—
147C2×Dic97C22×C412yes—
148C2×Dic97C22×C412yes—
149C2×Dic97C22×C412yes—
150C2×Dic97C2×D412yes—
151C2×Dic97C2×Q812yes—
152C4×D97—28no—
153C4×D97—28no—
154C4×D97—28no—
155C4×D97—28no—
156C4×D97—28no—
157C4×D97—44no—
158C4×D97—44no—
159C22×C97C23×C411yescharacteristic
160C22×C97C22×D411yescharacteristic
161C22×C97C22×Q811yescharacteristic
162C22×C97C2×#4111yescharacteristic
163C22×C97C2×#4111yescharacteristic
164C22×C97C2×#4111yescharacteristic
165C22×C97C2×#4111yescharacteristic
166C2×D97#11714yes—
167C2×D97C4×D414yes—
168C2×D97C4×Q814yes—
169C2×D97C22⋊Q814yes—
170C2×D97#12314yes—
171C2×D97#12514yes—
172C2×D97#12614yes—
173C4×C97—22no—
174C4×C97—22no—
175C4×C97—22no—
176C4×C97—22no—
177C4×C97—22no—
178C4×C97—41no—
179C4×C97—41no—
180Dic97—22no—
181Dic97—22no—
182Dic97—22no—
183Dic97—22no—
184Dic97—22no—
185Dic97—41no—
186Dic97—41no—
187C2×C97C2×#11711yescharacteristic
188C2×C97C2×C4×D411yescharacteristic
189C2×C97C2×C4×Q811yescharacteristic
190C2×C97C2×C22⋊Q811yescharacteristic
191C2×C97C2×#12311yescharacteristic
192C2×C97C2×#12511yescharacteristic
193C2×C97C2×#12611yescharacteristic
194D97#38718yesdirect product factor
195C2×#387—971nomaximal, Sylow
196C97C2×#38711yescharacteristic, complemented, Sylow
197C22×C4⋊C4—971no—
198C22×C4⋊C4—971no—
199C22×C42—971no—
200#387—971no—
201#387—971no—
202#387—971no—
203#387—971no—
204#387—971no—
205#387—971no—
206#387—971no—
207#387D9711yescharacteristic, direct product factor
208C2×#95—971no—
209C2×#95—971no—
210C2×#95—971no—
211C2×#95—971no—
212C23×C4—971no—
213C23×C4—971no—
214C23×C4—971no—
215C23×C4—971no—
216C23×C4—971no—
217C23×C4—971no—
218C23×C4—971no—
219C2×C4⋊C4—971no—
220C2×C4⋊C4—971no—
221C2×C4⋊C4—971no—
222C2×C4⋊C4—978no—
223C2×C4⋊C4—971no—
224C2×C4⋊C4—971no—
225C2×C4⋊C4—971no—
226C2×C4⋊C4—1944no—
227C2×C4⋊C4C2×D9711yescharacteristic
228C2×C4⋊C4C2×D9711yescharacteristic
229C2×C42—971no—
230C2×C42—971no—
231C2×C42—971no—
232C2×C42—1944no—
233C2×C42C2×D9711yescharacteristic
234#95—971no—
235#95—971no—
236#95—971no—
237#95—971no—
238#95—971no—
239#95—971no—
240#95—971no—
241#95—971no—
242#95—971no—
243#95—971no—
244#95—971no—
245#95—971no—
246#95C2×D9711yescharacteristic
247#95C2×D9711yescharacteristic
248#95C2×D9711yescharacteristic
249#95C2×D9711yescharacteristic
250C24—971no—
251C22×C4—971no—
252C22×C4—971no—
253C22×C4—971no—
254C22×C4—971no—
255C22×C4—971no—
256C22×C4—974no—
257C22×C4—974no—
258C22×C4—974no—
259C22×C4—974no—
260C22×C4—974no—
261C22×C4—971no—
262C22×C4—971no—
263C22×C4—1942no—
264C22×C4—1942no—
265C22×C4—1942no—
266C22×C4—1942no—
267C22×C4—1942no—
268C22×C4—1942no—
269C22×C4—1942no—
270C22×C4—1942no—
271C22×C4—1942no—
272C22×C4—1942no—
273C22×C4—1942no—
274C22×C4—1942no—
275C22×C4—1942no—
276C22×C4—1942no—
277C22×C4—1942no—
278C22×C4—1942no—
279C22×C4C22×D9711yescharacteristic
280C22×C4C22×D9711yescharacteristic
281C22×C4C22×D9711yescharacteristic
282C22×C4C22×D9711yescharacteristic
283C22×C4C22×D9711yescharacteristic
284C22×C4C22×D9711yescharacteristic
285C22×C4C22×D9711yescharacteristic
286C4⋊C4—22no—
287C4⋊C4—974no—
288C4⋊C4—974no—
289C4⋊C4—974no—
290C4⋊C4—1942no—
291C4⋊C4—1942no—
292C4⋊C4—1942no—
293C4⋊C4C4×D9714yescomplemented
294C42—22no—
295C42—1942no—
296C42—1942no—
297C42—1942no—
298C23—972no—
299C23—972no—
300C23—972no—
301C23—972no—
302C23—972no—
303C23—972no—
304C23—972no—
305C23C23×D9711yescharacteristic
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—972no—
323C2×C4—972no—
324C2×C4—972no—
325C2×C4—972no—
326C2×C4—972no—
327C2×C4—3884no—
328C2×C4—3884no—
329C2×C4—1948no—
330C2×C4—1948no—
331C2×C4—1948no—
332C2×C4—1948no—
333C2×C4—1948no—
334C2×C4—1941no—
335C2×C4—1941no—
336C2×C4—1941no—
337C2×C4—1941no—
338C2×C4—1941no—
339C2×C4—1941no—
340C2×C4—1941no—
341C2×C4—1941no—
342C2×C4—1941no—
343C2×C4—1941no—
344C2×C4—1941no—
345C2×C4—1941no—
346C2×C4—1941no—
347C2×C4—1941no—
348C2×C4—1941no—
349C2×C4—1941no—
350C2×C4C2×C4×D9712yes—
351C2×C4C2×C4×D9712yes—
352C2×C4C2×C4×D9712yes—
353C2×C4D4×D9712yes—
354C2×C4Q8×D9712yes—
355C22—974no—
356C22—974no—
357C22—974no—
358C22—974no—
359C22—974no—
360C22—974no—
361C22—974no—
362C22C22×C4×D9711yescharacteristic
363C22C2×D4×D9711yescharacteristic
364C22C2×Q8×D9711yescharacteristic
365C22#41×D9711yescharacteristic
366C22#41×D9711yescharacteristic
367C22#41×D9711yescharacteristic
368C22#41×D9711yescharacteristic
369C4—22no—
370C4—22no—
371C4—22no—
372C4—22no—
373C4—22no—
374C4—41no—
375C4—41no—
376C4—3881no—
377C4—3881no—
378C4—1942no—
379C4—1942no—
380C4—1942no—
381C4—1942no—
382C4—1942no—
383C2—978no—
384C2#117×D9711yescharacteristic
385C2C4×D4×D9711yescharacteristic
386C2C4×Q8×D9711yescharacteristic
387C2C22⋊Q8×D9711yescharacteristic
388C2#123×D9711yescharacteristic
389C2#125×D9711yescharacteristic
390C2#126×D9711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#387×C97C2
socle, cosocleC23×C97C24
prime radicalC23×D97C23
hypercenter#387D97
hypocenterC97C2×#387
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
4. #20260622C2semidirect
5. #20260621C2semidirect
6. #20260620C2semidirect
7. #20260619C2semidirect
8. #20260618C2semidirect
9. #20260617C2semidirect
10. #20260616C2semidirect
15. #387×C97C2semidirect
30. C4⋊C4×D97C4semidirect
84. C4⋊Dic97C2×C4semidirect
87. #13095490C2×C4semidirect
88. #13095490C2×C4semidirect
100. C4⋊C4×C97C2×C4semidirect
194. D97#387direct (N × H)
196. C97C2×#387semidirect
207. #387D97direct (N × H)
293. C4⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D972
C2×C42×D972
#202606222
#202606212
#202606202
#202606192
#202606182
#202606172
#202606162
#95×D972
#387×C972
C2×#38797