Finite Group Databasebeta

← #447×D97 #27150909

Subgroups — #447×D97 #27150909

Order 12416.

1048 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D97C211yesmaximal, characteristic, complemented
2C2×D8×D97C212yesmaximal, complemented
3#121×D97C211yesmaximal, characteristic, complemented
4#20260871C211yesmaximal, characteristic, complemented
5#20260870C212yesmaximal, complemented
6#20260869C211yesmaximal, characteristic, complemented
7#20260868C211yesmaximal, characteristic, complemented
8#20260867C212yesmaximal, complemented
9#102×D97C212yesmaximal, complemented
10C22⋊C8×D97C211yesmaximal, characteristic, complemented
11#447×C97C211yesmaximal, characteristic, complemented
12C24×D97—21no—
13C22×D388C2211yescharacteristic, complemented
14C22×C22⋊D97—21no—
15C2×D4×D97—28no—
16C2×D4×D97—21no—
17C2×D4×D97—21no—
18C2×D4×D97—21no—
19C2×D4×D97C2212yes—
20C2×D4×D97C2212yescomplemented
21C2×D4×D97C2211yescharacteristic, complemented
22C22×C4×D97C2211yescharacteristic
23C2×D776C2212yescomplemented
24C2×#13095496C2212yescomplemented
25C2×#13095496C2212yescomplemented
26D8×D97—28no—
27C2×C8×D97C2212yescomplemented
28#15690785C2211yescharacteristic, complemented
29#15690784C2211yescharacteristic, complemented
30#15690783—21no—
31#15690782—21no—
32#15690781C2211yescharacteristic, complemented
33C4⋊C4×D97C2211yescharacteristic, complemented
34C22⋊C4×D97—21no—
35#15690732C2212yescomplemented
36#15690731C2212yescomplemented
37#15690730C2212yescomplemented
38C22⋊#11715895C2211yescharacteristic, complemented
39#15690722C2212yescomplemented
40C22×D4×C97C2211yescharacteristic, complemented
41C2×D8×C97C2212yescomplemented
42#121×C97C2211yescharacteristic, complemented
43#102×C97C2212yescomplemented
44C22⋊C8×C97C2211yescharacteristic, complemented
45C23×D97—28no—
46C23×D97—21no—
47C23×D97—22no—
48C23×D97—22no—
49C23×D97—21no—
50C23×D97—44no—
51C23×D97—44no—
52C23×D97D411yescharacteristic
53C2×D388—21no—
54C2×D388—24no—
55C2×D388—21no—
56C2×D388C2312yes—
57C2×D388C2311yescharacteristic, complemented
58C2×C22⋊D97—28no—
59C2×C22⋊D97—21no—
60C2×C22⋊D97—21no—
61C2×C22⋊D97—22no—
62C2×C22⋊D97—22no—
63C2×C22⋊D97—21no—
64D4×D97—28no—
65D4×D97—28no—
66D4×D97—24no—
67D4×D97—24no—
68D4×D97—24no—
69D4×D97—24no—
70D4×D97—44no—
71D4×D97D418yescomplemented
72C22×Dic97D411yescharacteristic
73C2×C4×D97—24no—
74C2×C4×D97—21no—
75C2×C4×D97D411yescharacteristic
76C2×C4×D97D411yescharacteristic
77C2×C4×D97C2311yescharacteristic
78C2×C4×D97C2311yescharacteristic
79D776—24no—
80#13095496—24no—
81#13095496—24no—
82C2×#11715895C2312yescomplemented
83C8×D97—24no—
84C4⋊Dic97C2311yescharacteristic, complemented
85#13095490—21no—
86C22⋊Dic97—21no—
87#13095488—21no—
88#13095488—21no—
89C24×C97—21no—
90C2×D4×C97—24no—
91C2×D4×C97—21no—
92C2×D4×C97—21no—
93C2×D4×C97C2312yes—
94C2×D4×C97C2311yescharacteristic, complemented
95C22×C4×C97C2311yescharacteristic
96D8×C97—24no—
97C2×C8×C97C2312yescomplemented
98C4⋊C4×C97C2311yescharacteristic, complemented
99C22⋊C4×C97—21no—
100C23×C97—24no—
101C23×C97—22no—
102C23×C97—21no—
103C23×C97—42no—
104C23×C97—42no—
105C23×C97C2×D411yescharacteristic
106C22×D97—22no—
107C22×D97—22no—
108C22×D97—28no—
109C22×D97—24no—
110C22×D97—22no—
111C22×D97—28no—
112C22×D97—21no—
113C22×D97—44no—
114C22×D97—432no—
115C22×D97—42no—
116C22×D97—42no—
117C22×D97—44no—
118C22×D97—44no—
119C22×D97—44no—
120C22×D97—42no—
121C22×D97—42no—
122C22×D97—42no—
123C22×D97D814yescomplemented
124C22×D97C2×D411yescharacteristic
125C22×D97C2×D411yescharacteristic
126D4×C97—24no—
127D4×C97—22no—
128D4×C97—22no—
129D4×C97—42no—
130D4×C97C2×D414yescomplemented
131C2×C4×C97—22no—
132C2×C4×C97—21no—
133C2×C4×C97C2×D411yescharacteristic
134C2×C4×C97C2411yescharacteristic
135C22⋊D97—28no—
136C22⋊D97—28no—
137C22⋊D97—42no—
138C22⋊D97—42no—
139C22⋊D97—42no—
140C2×Dic97—22no—
141C2×Dic97—21no—
142C2×Dic97C2×D411yescharacteristic
143C2×Dic97C2×D411yescharacteristic
144D388—24no—
145D388—22no—
146D388—22no—
147D388—42no—
148D388C2×D414yescomplemented
149C4×D97—22no—
150C4×D97—22no—
151C4×D97—22no—
152C4×D97—42no—
153C4×D97C2×D414yes—
154C8×C97—22no—
155#11715895—22no—
156C22×C97—21no—
157C22×C97—21no—
158C22×C97—24no—
159C22×C97—48no—
160C22×C97—42no—
161C22×C97—42no—
162C22×C97—41no—
163C22×C97—41no—
164C22×C97—41no—
165C22×C97C2×D812yescomplemented
166C22×C97C22×D411yescharacteristic
167C2×D97—21no—
168C2×D97—21no—
169C2×D97—28no—
170C2×D97—24no—
171C2×D97—416no—
172C2×D97—48no—
173C2×D97—42no—
174C2×D97—42no—
175C2×D97—416no—
176C2×D97—41no—
177C2×D97—41no—
178C2×D97—41no—
179C2×D97—84no—
180C2×D97#12012yes—
181C2×D97C2×D812yes—
182C2×D97C2×D812yescomplemented
183C2×D97AGL(1,ℤ/8)12yes—
184C4×C97—21no—
185C4×C97—41no—
186C4×C97C22×D412yes—
187Dic97—21no—
188Dic97—41no—
189Dic97#12012yes—
190C2×C97—22no—
191C2×C97—44no—
192C2×C97—81no—
193C2×C97C2×#12011yescharacteristic
194C2×C97C22×D811yescharacteristic
195C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
196D97—22no—
197D97—44no—
198D97—81no—
199D97#44714yesdirect product factor
200C2×#447—971nomaximal, Sylow
201C97C2×#44711yescharacteristic, complemented, Sylow
202C23×D4—971no—
203C22×D8—972no—
204C2×#121—971no—
205#447—971no—
206#447—972no—
207#447—971no—
208#447—972no—
209#447—971no—
210#447D9711yescharacteristic, direct product factor
211C2×#102—972no—
212C2×C22⋊C8—971no—
213C25—1941no—
214C22×D4—972no—
215C22×D4—972no—
216C22×D4—971no—
217C22×D4—971no—
218C22×D4—1941no—
219C22×D4—1941no—
220C22×D4—1941no—
221C22×D4—1948no—
222C22×D4—1941no—
223C22×D4C2×D9711yescharacteristic, complemented
224C23×C4—971no—
225C2×D8—972no—
226C2×D8—972no—
227C2×D8—972no—
228C2×D8—1948no—
229C2×D8C2×D9712yescomplemented
230C22×C8—972no—
231#121—971no—
232#121—971no—
233#121—971no—
234#121—1941no—
235#121—1941no—
236#121C2×D9711yescharacteristic, complemented
237C2×C4⋊C4—971no—
238C2×C22⋊C4—1941no—
239#102—972no—
240#102—972no—
241#102—972no—
242#102C2×D9712yescomplemented
243C22⋊C8—971no—
244C22⋊C8—972no—
245C22⋊C8C2×D9711yescharacteristic, complemented
246C24—21no—
247C24—971no—
248C24—3884no—
249C24—3884no—
250C24—1941no—
251C24—1942no—
252C24—1942no—
253C24—1948no—
254C24—1941no—
255C2×D4—24no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—978no—
259C2×D4—972no—
260C2×D4—971no—
261C2×D4—3884no—
262C2×D4—1944no—
263C2×D4—1941no—
264C2×D4—1944no—
265C2×D4—1944no—
266C2×D4—1941no—
267C2×D4—1942no—
268C2×D4—1942no—
269C2×D4—1944no—
270C2×D4—1948no—
271C2×D4—1948no—
272C2×D4—1941no—
273C2×D4—1941no—
274C2×D4—1948no—
275C2×D4—1944no—
276C2×D4—1941no—
277C2×D4C22×D9712yes—
278C2×D4C22×D9711yescharacteristic, complemented
279C22×C4—971no—
280C22×C4—971no—
281C22×C4—971no—
282C22×C4—971no—
283C22×C4—971no—
284C22×C4—1944no—
285C22×C4—1941no—
286C22×C4C22×D9711yescharacteristic
287D8—24no—
288D8—1944no—
289D8—1944no—
290D8—1944no—
291C2×C8—972no—
292C2×C8—1944no—
293C2×C8C22×D9712yescomplemented
294C4⋊C4—971no—
295C4⋊C4—1941no—
296C4⋊C4C22×D9711yescharacteristic, complemented
297C22⋊C4—21no—
298C22⋊C4—1941no—
299C22⋊C4—1941no—
300C22⋊C4—1941no—
301C23—24no—
302C23—22no—
303C23—21no—
304C23—42no—
305C23—42no—
306C23—971no—
307C23—974no—
308C23—971no—
309C23—3882no—
310C23—3882no—
311C23—3882no—
312C23—3882no—
313C23—3884no—
314C23—3884no—
315C23—3884no—
316C23—38832no—
317C23—3884no—
318C23—3882no—
319C23—1948no—
320C23—1942no—
321C23—1942no—
322C23—1944no—
323C23—1948no—
324C23—1941no—
325C23—1942no—
326C23D4×D9711yescharacteristic
327D4—24no—
328D4—22no—
329D4—22no—
330D4—42no—
331D4—974no—
332D4—3882no—
333D4—3882no—
334D4—3882no—
335D4—3882no—
336D4—1944no—
337D4—1948no—
338D4—1948no—
339D4—1942no—
340D4—1942no—
341D4D4×D9714yescomplemented
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—971no—
345C2×C4—974no—
346C2×C4—971no—
347C2×C4—3882no—
348C2×C4—1942no—
349C2×C4—1942no—
350C2×C4—1942no—
351C2×C4—1941no—
352C2×C4—1942no—
353C2×C4D4×D9711yescharacteristic
354C2×C4C23×D9711yescharacteristic
355C8—22no—
356C8—1942no—
357C22—21no—
358C22—21no—
359C22—24no—
360C22—48no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—7764no—
367C22—972no—
368C22—972no—
369C22—972no—
370C22—972no—
371C22—3888no—
372C22—38816no—
373C22—38816no—
374C22—3881no—
375C22—3881no—
376C22—3881no—
377C22—3882no—
378C22—3882no—
379C22—1948no—
380C22—1944no—
381C22—1941no—
382C22—1941no—
383C22D8×D9712yescomplemented
384C22C2×D4×D9711yescharacteristic
385C4—21no—
386C4—41no—
387C4—972no—
388C4—3881no—
389C4—1941no—
390C4C2×D4×D9712yes—
391C2—22no—
392C2—44no—
393C2—81no—
394C2—7761no—
395C2—974no—
396C2—3884no—
397C2—1942no—
398C2#120×D9711yescharacteristic
399C2C2×D8×D9711yescharacteristic
400C2AGL(1,ℤ/8)×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#447×C97C2
socleC22×C97C22×D4
hypercenter#447D97
hypocenterC97C2×#447
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×D4×D97C2semidirect
2. C2×D8×D97C2semidirect
3. #121×D97C2semidirect
4. #20260871C2semidirect
5. #20260870C2semidirect
6. #20260869C2semidirect
7. #20260868C2semidirect
8. #20260867C2semidirect
9. #102×D97C2semidirect
10. C22⋊C8×D97C2semidirect
11. #447×C97C2semidirect
13. C22×D388C22semidirect
20. C2×D4×D97C22semidirect
21. C2×D4×D97C22semidirect
23. C2×D776C22semidirect
24. C2×#13095496C22semidirect
25. C2×#13095496C22semidirect
27. C2×C8×D97C22semidirect
28. #15690785C22semidirect
29. #15690784C22semidirect
32. #15690781C22semidirect
33. C4⋊C4×D97C22semidirect
35. #15690732C22semidirect
36. #15690731C22semidirect
37. #15690730C22semidirect
38. C22⋊#11715895C22semidirect
39. #15690722C22semidirect
40. C22×D4×C97C22semidirect
41. C2×D8×C97C22semidirect
42. #121×C97C22semidirect
43. #102×C97C22semidirect
44. C22⋊C8×C97C22semidirect
57. C2×D388C23semidirect
71. D4×D97D4semidirect
82. C2×#11715895C23semidirect
84. C4⋊Dic97C23semidirect
94. C2×D4×C97C23semidirect
97. C2×C8×C97C23semidirect
98. C4⋊C4×C97C23semidirect
123. C22×D97D8semidirect
130. D4×C97C2×D4semidirect
148. D388C2×D4semidirect
165. C22×C97C2×D8semidirect
182. C2×D97C2×D8semidirect
199. D97#447direct (N × H)
201. C97C2×#447semidirect
210. #447D97direct (N × H)
223. C22×D4C2×D97semidirect
229. C2×D8C2×D97semidirect
236. #121C2×D97semidirect
242. #102C2×D97semidirect
245. C22⋊C8C2×D97semidirect
278. C2×D4C22×D97semidirect
293. C2×C8C22×D97semidirect
296. C4⋊C4C22×D97semidirect
341. D4D4×D97semidirect
383. C22D8×D97semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D972
C2×D8×D972
#121×D972
#202608712
#202608702
#202608692
#202608682
#202608672
#102×D972
C22⋊C8×D972
#447×C972
C2×#44797