Finite Group Databasebeta

← #482×D97 #27151616

Subgroups — #482×D97 #27151616

Order 12416.

468 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
1#121×D97C211yesmaximal, characteristic
2#117×D97C211yesmaximal, characteristic, complemented
3#20261062C211yesmaximal, characteristic, complemented
4#20261061C211yesmaximal, characteristic, complemented
5#20261060C211yesmaximal, characteristic, complemented
6#20261059C211yesmaximal, characteristic, complemented
7#20261058C211yesmaximal, characteristic, complemented
8#20261057C211yesmaximal, characteristic, complemented
9#20261056C211yesmaximal, characteristic, complemented
10#107×D97C211yesmaximal, characteristic, complemented
11#106×D97C211yesmaximal, characteristic, complemented
12#102×D97C211yesmaximal, characteristic, complemented
13#102×D97C211yesmaximal, characteristic, complemented
14C22⋊C8×D97C211yesmaximal, characteristic, complemented
15#482×C97C211yesmaximal, characteristic, complemented
16C2×D4×D97—21no—
17C2×D4×D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20C2×C8×D97C2211yescharacteristic
21#15690785C2211yescharacteristic
22#15690784C2211yescharacteristic
23#15690783—21no—
24#15690782—21no—
25#15690781C2211yescharacteristic
26#15690769C2211yescharacteristic, complemented
27#15690768C2211yescharacteristic, complemented
28#15690768C2211yescharacteristic, complemented
29#15690767—21no—
30#15690766—21no—
31C4⋊C4×D97C2211yescharacteristic
32C4⋊C4×D97C2211yescharacteristic
33C4⋊C4×D97C2211yescharacteristic
34C22⋊C4×D97—21no—
35C22⋊C4×D97—21no—
36C42×D97—21no—
37#15690744C2211yescharacteristic, complemented
38#15690743C2211yescharacteristic, complemented
39#15690743C2211yescharacteristic, complemented
40#15690742C2211yescharacteristic, complemented
41#15690741C2211yescharacteristic, complemented
42#15690741C2211yescharacteristic, complemented
43#15690732C2211yescharacteristic, complemented
44#15690732C2211yescharacteristic, complemented
45#15690731C2211yescharacteristic, complemented
46#15690731C2211yescharacteristic, complemented
47#15690730C2211yescharacteristic, complemented
48#15690730C2211yescharacteristic, complemented
49C22⋊#11715895C2211yescharacteristic, complemented
50#15690722C2211yescharacteristic, complemented
51#15690722C2211yescharacteristic, complemented
52#121×C97C2211yescharacteristic
53#117×C97C2211yescharacteristic, complemented
54#107×C97C2211yescharacteristic, complemented
55#106×C97C2211yescharacteristic, complemented
56#102×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic, complemented
58C22⋊C8×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60C23×D97D411yescharacteristic
61C2×D388—21no—
62C2×D388C2311yescharacteristic
63C2×C22⋊D97—21no—
64C2×C22⋊D97—21no—
65C2×C22⋊D97—21no—
66D4×D97—22no—
67D4×D97—22no—
68D4×D97—44no—
69C22×Dic97D411yescharacteristic
70C2×C4×D97—22no—
71C2×C4×D97—22no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74C2×C4×D97—21no—
75C2×C4×D97D411yescharacteristic
76C2×C4×D97D411yescharacteristic
77C2×C4×D97C2311yescharacteristic
78C2×C4×D97C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C2×#11715895C2311yescharacteristic
81C8×D97—22no—
82C8×D97—22no—
83C4⋊Dic97C2311yescharacteristic
84C4⋊Dic97C2311yescharacteristic
85C4⋊Dic97C2311yescharacteristic
86#13095490—21no—
87#13095490—21no—
88#13095490—21no—
89C22⋊Dic97—21no—
90C22⋊Dic97—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95C4×Dic97—21no—
96C4×Dic97—21no—
97C4×Dic97—21no—
98C2×D4×C97—21no—
99C2×D4×C97C2311yescharacteristic
100C22×C4×C97C2311yescharacteristic
101C2×C8×C97C2311yescharacteristic
102C2×C8×C97C2311yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C4⋊C4×C97C2311yescharacteristic
105C4⋊C4×C97C2311yescharacteristic
106C22⋊C4×C97—21no—
107C22⋊C4×C97—21no—
108C42×C97—21no—
109C23×C97—21no—
110C23×C97C2×D411yescharacteristic
111C22×D97—22no—
112C22×D97—22no—
113C22×D97—22no—
114C22×D97—21no—
115C22×D97—42no—
116C22×D97—42no—
117C22×D97—42no—
118C22×D97C2×D411yescharacteristic
119C22×D97C2×D411yescharacteristic
120D4×C97—21no—
121D4×C97—21no—
122D4×C97—42no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2411yescharacteristic
130C22⋊D97—42no—
131C22⋊D97—42no—
132C22⋊D97—42no—
133C2×Dic97—22no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97C2×D411yescharacteristic
138C2×Dic97C2×D411yescharacteristic
139D388—21no—
140D388—21no—
141D388—42no—
142C4×D97—22no—
143C4×D97—21no—
144C4×D97—21no—
145C4×D97—22no—
146C4×D97—42no—
147C4×D97—42no—
148C4×D97—42no—
149C4×D97#4112yes—
150C4×D97#4112yes—
151C8×C97—21no—
152C8×C97—21no—
153#11715895—21no—
154#11715895—21no—
155C22×C97—21no—
156C22×C97—21no—
157C22×C97—21no—
158C22×C97—41no—
159C22×C97—41no—
160C22×C97—41no—
161C22×C97C22×D411yescharacteristic
162C2×D97—21no—
163C2×D97—21no—
164C2×D97—21no—
165C2×D97—44no—
166C2×D97—41no—
167C2×D97—41no—
168C2×D97—41no—
169C2×D97—84no—
170C2×D97#12312yes—
171C2×D97#13512yes—
172C2×D97AGL(1,ℤ/8)12yes—
173C4×C97—21no—
174C4×C97—41no—
175C4×C97—41no—
176C4×C97—41no—
177C4×C97C2×#4111yescharacteristic
178C4×C97C2×#4111yescharacteristic
179Dic97—21no—
180Dic97—41no—
181Dic97—41no—
182Dic97—41no—
183Dic97#12312yes—
184C2×C97—41no—
185C2×C97—81no—
186C2×C97C2×#12311yescharacteristic
187C2×C97C2×#13511yescharacteristic
188C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
189D97—41no—
190D97—81no—
191D97#48214yesdirect product factor
192C2×#482—971nomaximal, Sylow
193C97C2×#48211yescharacteristic, complemented, Sylow
194C2×#121—971no—
195C2×#117—971no—
196#482—971no—
197#482—971no—
198#482—971no—
199#482—971no—
200#482—971no—
201#482—971no—
202#482—971no—
203#482D9711yescharacteristic, direct product factor
204C2×#107—971no—
205C2×#106—971no—
206C2×#102—971no—
207C2×#102—971no—
208C2×C22⋊C8—971no—
209C22×D4—971no—
210C22×D4—1941no—
211C23×C4—971no—
212C22×C8—971no—
213C22×C8—971no—
214#121—971no—
215#121—971no—
216#121—971no—
217#121—1941no—
218#121—1941no—
219#121C2×D9711yescharacteristic
220#117—971no—
221#117—971no—
222#117—971no—
223#117—1941no—
224#117—1941no—
225#117C2×D9711yescharacteristic, complemented
226C2×C4⋊C4—971no—
227C2×C4⋊C4—971no—
228C2×C4⋊C4—971no—
229C2×C22⋊C4—1941no—
230C2×C22⋊C4—1941no—
231C2×C42—1941no—
232#107—971no—
233#107—971no—
234#107—971no—
235#107C2×D9711yescharacteristic, complemented
236#106—971no—
237#106—971no—
238#106—971no—
239#106C2×D9711yescharacteristic, complemented
240#102—971no—
241#102—971no—
242#102—971no—
243#102—971no—
244#102—971no—
245#102—971no—
246#102C2×D9711yescharacteristic, complemented
247#102C2×D9711yescharacteristic, complemented
248C22⋊C8—971no—
249C22⋊C8—971no—
250C22⋊C8—971no—
251C22⋊C8C2×D9711yescharacteristic, complemented
252C24—971no—
253C24—1941no—
254C2×D4—21no—
255C2×D4—971no—
256C2×D4—3884no—
257C2×D4—1941no—
258C2×D4—1942no—
259C2×D4—1942no—
260C2×D4—1941no—
261C2×D4—1941no—
262C2×D4—1941no—
263C2×D4C22×D9711yescharacteristic
264C22×C4—971no—
265C22×C4—971no—
266C22×C4—971no—
267C22×C4—971no—
268C22×C4—971no—
269C22×C4—1942no—
270C22×C4—1942no—
271C22×C4—1941no—
272C22×C4—1941no—
273C22×C4—1941no—
274C22×C4C22×D9711yescharacteristic
275C2×C8—971no—
276C2×C8—971no—
277C2×C8—1942no—
278C2×C8—1942no—
279C2×C8C22×D9711yescharacteristic
280C2×C8C22×D9711yescharacteristic
281C4⋊C4—971no—
282C4⋊C4—971no—
283C4⋊C4—971no—
284C4⋊C4—1941no—
285C4⋊C4—1941no—
286C4⋊C4—1941no—
287C4⋊C4C22×D9711yescharacteristic
288C4⋊C4C22×D9711yescharacteristic
289C4⋊C4C22×D9711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—1941no—
293C22⋊C4—1941no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C22⋊C4—1941no—
298C42—21no—
299C42—1941no—
300C42—1941no—
301C42—1941no—
302C23—21no—
303C23—971no—
304C23—971no—
305C23—3882no—
306C23—3882no—
307C23—3882no—
308C23—1942no—
309C23—1942no—
310C23—1942no—
311C23—1941no—
312C23D4×D9711yescharacteristic
313D4—21no—
314D4—21no—
315D4—42no—
316D4—3882no—
317D4—3882no—
318D4—3882no—
319D4—3882no—
320D4—1941no—
321D4—1941no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—971no—
328C2×C4—972no—
329C2×C4—972no—
330C2×C4—971no—
331C2×C4—3882no—
332C2×C4—3882no—
333C2×C4—3882no—
334C2×C4—1942no—
335C2×C4—1942no—
336C2×C4—1942no—
337C2×C4—1941no—
338C2×C4—1941no—
339C2×C4—1941no—
340C2×C4—1941no—
341C2×C4—1941no—
342C2×C4D4×D9711yescharacteristic
343C2×C4C23×D9711yescharacteristic
344C8—21no—
345C8—21no—
346C8—1941no—
347C8—1941no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—7764no—
355C22—972no—
356C22—972no—
357C22—972no—
358C22—3884no—
359C22—3881no—
360C22—3881no—
361C22—3881no—
362C22—1941no—
363C22—1941no—
364C22—1941no—
365C22C2×D4×D9711yescharacteristic
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—972no—
371C4—3881no—
372C4—3881no—
373C4—3881no—
374C4—1941no—
375C4#41×D9711yescharacteristic
376C4#41×D9711yescharacteristic
377C2—41no—
378C2—81no—
379C2—7761no—
380C2—974no—
381C2—3881no—
382C2#123×D9711yescharacteristic
383C2#135×D9711yescharacteristic
384C2AGL(1,ℤ/8)×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#482×C97C2
socleC22×C97C22×D4
prime radical#121×D97C2
hypercenter#482D97
hypocenterC97C2×#482
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #117×D97C2semidirect
3. #20261062C2semidirect
4. #20261061C2semidirect
5. #20261060C2semidirect
6. #20261059C2semidirect
7. #20261058C2semidirect
8. #20261057C2semidirect
9. #20261056C2semidirect
10. #107×D97C2semidirect
11. #106×D97C2semidirect
12. #102×D97C2semidirect
13. #102×D97C2semidirect
14. C22⋊C8×D97C2semidirect
15. #482×C97C2semidirect
26. #15690769C22semidirect
27. #15690768C22semidirect
28. #15690768C22semidirect
37. #15690744C22semidirect
38. #15690743C22semidirect
39. #15690743C22semidirect
40. #15690742C22semidirect
41. #15690741C22semidirect
42. #15690741C22semidirect
43. #15690732C22semidirect
44. #15690732C22semidirect
45. #15690731C22semidirect
46. #15690731C22semidirect
47. #15690730C22semidirect
48. #15690730C22semidirect
49. C22⋊#11715895C22semidirect
50. #15690722C22semidirect
51. #15690722C22semidirect
53. #117×C97C22semidirect
54. #107×C97C22semidirect
55. #106×C97C22semidirect
56. #102×C97C22semidirect
57. #102×C97C22semidirect
58. C22⋊C8×C97C22semidirect
191. D97#482direct (N × H)
193. C97C2×#482semidirect
203. #482D97direct (N × H)
225. #117C2×D97semidirect
235. #107C2×D97semidirect
239. #106C2×D97semidirect
246. #102C2×D97semidirect
247. #102C2×D97semidirect
251. C22⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#121×D972
#117×D972
#202610622
#202610612
#202610602
#202610592
#202610582
#202610572
#202610562
#107×D972
#106×D972
#102×D972
C22⋊C8×D972
#482×C972
C2×#48297