Finite Group Databasebeta

← #417×D97 #27150244

Subgroups — #417×D97 #27150244

Order 12416.

816 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D97C211yesmaximal, characteristic
2C2×M16×D97C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
4#20260737C211yesmaximal, characteristic, complemented
5#20260736C211yesmaximal, characteristic, complemented
6#20260735C212yesmaximal, complemented
7#20260734C211yesmaximal, characteristic, complemented
8#20260733C212yesmaximal, complemented
9#103×D97C212yesmaximal, complemented
10#102×D97C212yesmaximal, complemented
11#417×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2211yescharacteristic
13C2×#13095511—21no—
14C2×#13095511C2211yescharacteristic
15C2×#13095510—21no—
16C2×#13095510C2211yescharacteristic
17#41×D97—24no—
18#41×D97C418yescomplemented
19C2×Q8×D97C2211yescharacteristic
20C2×D4×D97—21no—
21C2×D4×D97C2211yescharacteristic
22C22×C4×D97—21no—
23C22×C4×D97—21no—
24C22×C4×D97C2211yescharacteristic
25C2×#13095495C2211yescharacteristic, complemented
26C2×#13095494C2212yescomplemented
27M16×D97—24no—
28C2×C8×D97C2212yes—
29C2×C4⋊Dic97C2211yescharacteristic, complemented
30C2×#13095490—21no—
31C4⋊C4×D97—21no—
32C4⋊C4×D97—21no—
33C4⋊C4×D97C2212yes—
34C4⋊C4×D97C2212yescomplemented
35#15690735C2212yescomplemented
36#15690734C2212yescomplemented
37#15690733C2212yescomplemented
38#15690732C2212yescomplemented
39#15690731C2212yescomplemented
40#15690730C2212yescomplemented
41C2×#41×C97C2211yescharacteristic
42C2×M16×C97C2211yescharacteristic, complemented
43C2×C4⋊C4×C97C2211yescharacteristic, complemented
44#103×C97C2212yescomplemented
45#102×C97C2212yescomplemented
46C23×D97—21no—
47C23×D97D411yescharacteristic
48#13095512—22no—
49#13095512C2×C414yescomplemented
50#13095511—22no—
51#13095511—28no—
52#13095511C2×C414yescomplemented
53#13095510—28no—
54#13095510—22no—
55#13095510C2×C414yescomplemented
56C2×Dic194—21no—
57C2×Dic194C2311yescharacteristic
58Q8×D97—22no—
59Q8×D97C2×C414yescomplemented
60C2×D388—21no—
61C2×D388C2311yescharacteristic
62C2×C22⋊D97—21no—
63C2×C22⋊D97—21no—
64C2×C22⋊D97—21no—
65D4×D97—28no—
66D4×D97—22no—
67D4×D97C2×C414yescomplemented
68C22×Dic97—21no—
69C22×Dic97—21no—
70C22×Dic97D411yescharacteristic
71C2×C4×D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74C2×C4×D97—28no—
75C2×C4×D97—22no—
76C2×C4×D97—22no—
77C2×C4×D97—44no—
78C2×C4×D97C2×C414yes—
79C2×C4×D97D411yescharacteristic
80C2×C4×D97D411yescharacteristic
81C2×C4×D97D414yes—
82C2×C4×D97C2311yescharacteristic
83C2×C4×D97C2311yescharacteristic
84#13095495—22no—
85#13095494—24no—
86C2×#11715895C2312yes—
87C8×D97—24no—
88C4⋊Dic97—21no—
89C4⋊Dic97C2312yes—
90#13095490—22no—
91#13095490—22no—
92#41×C97—22no—
93#41×C97C2×C414yescomplemented
94C2×Q8×C97C2311yescharacteristic
95C2×D4×C97—21no—
96C2×D4×C97C2311yescharacteristic
97C22×C4×C97—21no—
98C22×C4×C97—21no—
99C22×C4×C97C2311yescharacteristic
100M16×C97—22no—
101C2×C8×C97C2312yes—
102C4⋊C4×C97—21no—
103C4⋊C4×C97C2312yes—
104C23×C97—21no—
105C23×C97C2×D411yescharacteristic
106C22×D97—22no—
107C22×D97—22no—
108C22×D97—21no—
109C22×D97—24no—
110C22×D97—42no—
111C22×D97—42no—
112C22×D97C22⋊C414yes—
113C22×D97C2×D411yescharacteristic
114C22×D97C2×D411yescharacteristic
115Q8×C97—21no—
116Q8×C97C22×C412yescomplemented
117D4×C97—24no—
118D4×C97—21no—
119D4×C97C22×C412yescomplemented
120C2×C4×C97—21no—
121C2×C4×C97—24no—
122C2×C4×C97—22no—
123C2×C4×C97—42no—
124C2×C4×C97C22×C412yes—
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2×D412yes—
127C2×C4×C97C2411yescharacteristic
128C22⋊D97—24no—
129C22⋊D97—24no—
130C22⋊D97—24no—
131C2×Dic97—21no—
132C2×Dic97—24no—
133C2×Dic97—22no—
134C2×Dic97—42no—
135C2×Dic97C22⋊C414yes—
136C2×Dic97C2×D411yescharacteristic
137C2×Dic97C2×D411yescharacteristic
138D388—24no—
139D388—21no—
140D388C22×C412yescomplemented
141C4×D97—24no—
142C4×D97—24no—
143C4×D97—24no—
144C4×D97—44no—
145C4×D97—44no—
146C4×D97C22⋊C414yes—
147C4×D97C22⋊C414yes—
148C4×D97C22×C412yes—
149C4×D97C22×C412yes—
150C4×D97C2×D412yes—
151C4×D97C2×D412yes—
152Dic194—21no—
153Dic194—24no—
154Dic194C22×C412yescomplemented
155C8×C97—22no—
156#11715895—22no—
157C22×C97—21no—
158C22×C97—21no—
159C22×C97—22no—
160C22×C97—41no—
161C22×C97—41no—
162C22×C97C2×C22⋊C412yes—
163C22×C97C22×D411yescharacteristic
164C2×D97—21no—
165C2×D97—21no—
166C2×D97—28no—
167C2×D97—22no—
168C2×D97—41no—
169C2×D97—41no—
170C2×D97—48no—
171C2×D97C2×C22⋊C412yes—
172C2×D97C2×C22⋊C412yes—
173C2×D97AGL(1,ℤ/8)12yes—
174C2×D97#13712yes—
175C4×C97—22no—
176C4×C97—42no—
177C4×C97C2×C22⋊C412yes—
178C4×C97C23×C411yescharacteristic
179C4×C97C22×D411yescharacteristic
180Dic97—22no—
181Dic97—42no—
182Dic97C2×C22⋊C412yes—
183Dic97C2×C22⋊C412yes—
184C2×C97—22no—
185C2×C97—42no—
186C2×C97C22×C22⋊C411yescharacteristic
187C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C97C2×#13711yescharacteristic
189D97—22no—
190D97—42no—
191D97#41714yesdirect product factor
192C2×#417—971nomaximal, Sylow
193C97C2×#41711yescharacteristic, complemented, Sylow
194C22×#41—971no—
195C22×M16—971no—
196C22×C4⋊C4—971no—
197#417—971no—
198#417—971no—
199#417—971no—
200#417—972no—
201#417—972no—
202#417D9711yescharacteristic, direct product factor
203C2×#103—972no—
204C2×#102—972no—
205C2×#41—978no—
206C2×#41—971no—
207C2×#41—971no—
208C2×#41—971no—
209C2×#41—1941no—
210C2×#41—1941no—
211C2×#41—1944no—
212C2×#41C2×D9711yescharacteristic
213C22×Q8—971no—
214C22×D4—971no—
215C22×D4—1941no—
216C23×C4—971no—
217C23×C4—1941no—
218C23×C4—1941no—
219C2×M16—971no—
220C2×M16—972no—
221C2×M16—1944no—
222C2×M16C2×D9711yescharacteristic, complemented
223C22×C8—972no—
224C2×C4⋊C4—971no—
225C2×C4⋊C4—972no—
226C2×C4⋊C4—972no—
227C2×C4⋊C4—1941no—
228C2×C4⋊C4—1941no—
229C2×C4⋊C4—1941no—
230C2×C4⋊C4C2×D9711yescharacteristic, complemented
231#103—972no—
232#103—972no—
233#103—972no—
234#103C2×D9712yescomplemented
235#102—972no—
236#102—972no—
237#102—972no—
238#102C2×D9712yescomplemented
239C24—971no—
240C24—1941no—
241#41—22no—
242#41—974no—
243#41—974no—
244#41—974no—
245#41—1948no—
246#41—1948no—
247#41—1942no—
248#41—1942no—
249#41—1942no—
250#41C4×D9714yescomplemented
251C2×Q8—974no—
252C2×Q8—971no—
253C2×Q8—1941no—
254C2×Q8—1942no—
255C2×Q8C22×D9711yescharacteristic
256C2×D4—21no—
257C2×D4—974no—
258C2×D4—971no—
259C2×D4—1941no—
260C2×D4—1942no—
261C2×D4—1941no—
262C2×D4—1948no—
263C2×D4—1941no—
264C2×D4—1941no—
265C2×D4C22×D9711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—971no—
269C22×C4—974no—
270C22×C4—971no—
271C22×C4—971no—
272C22×C4—974no—
273C22×C4—971no—
274C22×C4—971no—
275C22×C4—3884no—
276C22×C4—1941no—
277C22×C4—1948no—
278C22×C4—1942no—
279C22×C4—1942no—
280C22×C4—1941no—
281C22×C4—1941no—
282C22×C4—1941no—
283C22×C4—1941no—
284C22×C4C22×D9711yescharacteristic
285M16—22no—
286M16—1944no—
287M16—1942no—
288C2×C8—972no—
289C2×C8—1944no—
290C2×C8C22×D9712yes—
291C4⋊C4—21no—
292C4⋊C4—972no—
293C4⋊C4—1942no—
294C4⋊C4—1942no—
295C4⋊C4—1941no—
296C4⋊C4C22×D9712yes—
297C23—21no—
298C23—971no—
299C23—974no—
300C23—971no—
301C23—3882no—
302C23—3882no—
303C23—1944no—
304C23—1942no—
305C23—1942no—
306C23—1941no—
307C23D4×D9711yescharacteristic
308Q8—21no—
309Q8—972no—
310Q8—1944no—
311Q8—1941no—
312Q8C2×C4×D9712yescomplemented
313D4—24no—
314D4—21no—
315D4—972no—
316D4—1944no—
317D4—1944no—
318D4—1944no—
319D4—1944no—
320D4—1941no—
321D4C2×C4×D9712yescomplemented
322C2×C4—21no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—42no—
326C2×C4—974no—
327C2×C4—971no—
328C2×C4—971no—
329C2×C4—974no—
330C2×C4—974no—
331C2×C4—972no—
332C2×C4—972no—
333C2×C4—972no—
334C2×C4—972no—
335C2×C4—3884no—
336C2×C4—3884no—
337C2×C4—3882no—
338C2×C4—1944no—
339C2×C4—1944no—
340C2×C4—1944no—
341C2×C4—1942no—
342C2×C4—1941no—
343C2×C4—1944no—
344C2×C4C2×C4×D9712yes—
345C2×C4D4×D9711yescharacteristic
346C2×C4D4×D9712yes—
347C2×C4C23×D9711yescharacteristic
348C8—22no—
349C8—1942no—
350C22—21no—
351C22—21no—
352C22—22no—
353C22—41no—
354C22—41no—
355C22—972no—
356C22—972no—
357C22—972no—
358C22—972no—
359C22—3888no—
360C22—3881no—
361C22—3881no—
362C22—1948no—
363C22—1942no—
364C22—1941no—
365C22—1941no—
366C22C22⋊C4×D9712yes—
367C22C2×D4×D9711yescharacteristic
368C4—22no—
369C4—42no—
370C4—972no—
371C4—972no—
372C4—3882no—
373C4—1942no—
374C4C22⋊C4×D9712yes—
375C4C22×C4×D9711yescharacteristic
376C4C2×D4×D9711yescharacteristic
377C2—22no—
378C2—42no—
379C2—974no—
380C2—3882no—
381C2—1942no—
382C2C2×C22⋊C4×D9711yescharacteristic
383C2AGL(1,ℤ/8)×D9711yescharacteristic
384C2#137×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
Fitting#417×C97C2
socleC22×C97C22×D4
prime radicalC2×#41×D97C2
cosocleC2×C4×C97C24
hypercenter#417D97
hypocenterC97C2×#417
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×M16×D97C2semidirect
3. C2×C4⋊C4×D97C2semidirect
4. #20260737C2semidirect
5. #20260736C2semidirect
6. #20260735C2semidirect
7. #20260734C2semidirect
8. #20260733C2semidirect
9. #103×D97C2semidirect
10. #102×D97C2semidirect
11. #417×C97C2semidirect
18. #41×D97C4semidirect
25. C2×#13095495C22semidirect
26. C2×#13095494C22semidirect
29. C2×C4⋊Dic97C22semidirect
34. C4⋊C4×D97C22semidirect
35. #15690735C22semidirect
36. #15690734C22semidirect
37. #15690733C22semidirect
38. #15690732C22semidirect
39. #15690731C22semidirect
40. #15690730C22semidirect
42. C2×M16×C97C22semidirect
43. C2×C4⋊C4×C97C22semidirect
44. #103×C97C22semidirect
45. #102×C97C22semidirect
49. #13095512C2×C4semidirect
52. #13095511C2×C4semidirect
55. #13095510C2×C4semidirect
59. Q8×D97C2×C4semidirect
67. D4×D97C2×C4semidirect
93. #41×C97C2×C4semidirect
116. Q8×C97C22×C4semidirect
119. D4×C97C22×C4semidirect
140. D388C22×C4semidirect
154. Dic194C22×C4semidirect
191. D97#417direct (N × H)
193. C97C2×#417semidirect
202. #417D97direct (N × H)
222. C2×M16C2×D97semidirect
230. C2×C4⋊C4C2×D97semidirect
234. #103C2×D97semidirect
238. #102C2×D97semidirect
250. #41C4×D97semidirect
312. Q8C2×C4×D97semidirect
321. D4C2×C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D972
C2×M16×D972
C2×C4⋊C4×D972
#202607372
#202607362
#202607352
#202607342
#202607332
#103×D972
#102×D972
#417×C972
C2×#41797