Finite Group Databasebeta

← #464×D97 #27151176

Subgroups — #464×D97 #27151176

Order 12416.

500 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q16×D97C211yesmaximal, characteristic, complemented
2C2×SD16×D97C211yesmaximal, characteristic
3#124×D97C211yesmaximal, characteristic
4C4×Q8×D97C211yesmaximal, characteristic, complemented
5#20260960C211yesmaximal, characteristic, complemented
6#20260959C211yesmaximal, characteristic, complemented
7#20260958C211yesmaximal, characteristic, complemented
8#20260957C211yesmaximal, characteristic, complemented
9#20260956C211yesmaximal, characteristic, complemented
10#20260955C211yesmaximal, characteristic, complemented
11#20260954C211yesmaximal, characteristic, complemented
12C4⋊C8×D97C211yesmaximal, characteristic, complemented
13#103×D97C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic
15#464×C97C211yesmaximal, characteristic, complemented
16C2×Q8×D97C2211yescharacteristic
17C2×Q8×D97C2211yescharacteristic
18C2×D4×D97C2211yescharacteristic
19C2×Dic388C2211yescharacteristic, complemented
20C2×#13095501C2211yescharacteristic, complemented
21C2×#13095501C2211yescharacteristic, complemented
22Q16×D97—24no—
23C2×#13095500C2211yescharacteristic
24C2×#13095499C2211yescharacteristic
25C2×#13095498C2211yescharacteristic
26SD16×D97—24no—
27C2×C8×D97C2211yescharacteristic
28C2×C8×D97C2211yescharacteristic
29#15690799C2211yescharacteristic
30#15690798C2211yescharacteristic
31#15690797C2211yescharacteristic
32#15690796—21no—
33#15690796—21no—
34Q8×Dic97C2211yescharacteristic, complemented
35C4×Dic194—21no—
36C4×Dic194C2211yescharacteristic, complemented
37#15690775—21no—
38#15690775C2211yescharacteristic, complemented
39C4⋊C4×D97—21no—
40C4⋊C4×D97C2211yescharacteristic
41C22⋊C4×D97—21no—
42C22⋊C4×D97—21no—
43C42×D97—21no—
44C42×D97C2211yescharacteristic
45C4⋊#11715895C2211yescharacteristic, complemented
46#15690739C2211yescharacteristic, complemented
47#15690739C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690734C2211yescharacteristic, complemented
50#15690733C2211yescharacteristic, complemented
51#15690732C2211yescharacteristic
52#15690731C2211yescharacteristic
53#15690730C2211yescharacteristic
54C2×Q16×C97C2211yescharacteristic, complemented
55C2×SD16×C97C2211yescharacteristic
56#124×C97C2211yescharacteristic
57C4×Q8×C97C2211yescharacteristic, complemented
58C4⋊C8×C97C2211yescharacteristic, complemented
59#103×C97C2211yescharacteristic, complemented
60#102×C97C2211yescharacteristic
61C23×D97—21no—
62C2×Dic194—21no—
63C2×Dic194—21no—
64C2×Dic194C2311yescharacteristic
65C2×Dic194C2311yescharacteristic
66Q8×D97—22no—
67Q8×D97—22no—
68Q8×D97—22no—
69Q8×D97D414yes—
70C2×D388C2311yescharacteristic
71C2×C22⋊D97—21no—
72D4×D97—22no—
73D4×D97—22no—
74C2×C4×D97—21no—
75C2×C4×D97—21no—
76C2×C4×D97—21no—
77C2×C4×D97D411yescharacteristic
78C2×C4×D97D411yescharacteristic
79C2×C4×D97C2311yescharacteristic
80Dic388—22no—
81#13095501—22no—
82#13095501—22no—
83#13095500—22no—
84#13095499—22no—
85#13095498—22no—
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C8×D97—22no—
89C8×D97—22no—
90C4⋊Dic97—21no—
91C4⋊Dic97C2311yescharacteristic
92#13095490—21no—
93#13095490—21no—
94#13095490—21no—
95C22⋊Dic97—21no—
96C22⋊Dic97—21no—
97#13095488—21no—
98#13095488—21no—
99#13095488—21no—
100#13095488—21no—
101C4×Dic97—21no—
102C4×Dic97—21no—
103C4×Dic97—21no—
104C4×Dic97D411yescharacteristic
105C4×Dic97D411yescharacteristic
106C4×Dic97C2311yescharacteristic
107C2×Q8×C97C2311yescharacteristic
108C2×Q8×C97C2311yescharacteristic
109C2×D4×C97C2311yescharacteristic
110Q16×C97—22no—
111SD16×C97—22no—
112C2×C8×C97C2311yescharacteristic
113C2×C8×C97C2311yescharacteristic
114C4⋊C4×C97—21no—
115C4⋊C4×C97C2311yescharacteristic
116C22⋊C4×C97—21no—
117C22⋊C4×C97—21no—
118C42×C97—21no—
119C42×C97C2311yescharacteristic
120C23×C97—21no—
121C22×D97—21no—
122C22×D97—42no—
123C22×D97—42no—
124C22×D97—42no—
125C22×D97C2×D411yescharacteristic
126Q8×C97—21no—
127Q8×C97—21no—
128Q8×C97—21no—
129Q8×C97C2×D412yes—
130D4×C97—21no—
131D4×C97—21no—
132C2×C4×C97—21no—
133C2×C4×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97C2×D411yescharacteristic
136C2×C4×C97C2×D411yescharacteristic
137C2×C4×C97C2411yescharacteristic
138C22⋊D97—42no—
139C2×Dic97—21no—
140C2×Dic97—21no—
141C2×Dic97—21no—
142C2×Dic97C2×D411yescharacteristic
143C2×Dic97C2×D411yescharacteristic
144C2×Dic97C2×D411yescharacteristic
145D388—21no—
146D388—21no—
147C4×D97—24no—
148C4×D97—22no—
149C4×D97—22no—
150C4×D97—42no—
151C4×D97—42no—
152C4×D97C2×D412yes—
153C4×D97#4112yes—
154Dic194—21no—
155Dic194—24no—
156Dic194—21no—
157Dic194—21no—
158Dic194—42no—
159Dic194C2×D412yes—
160C8×C97—21no—
161C8×C97—21no—
162#11715895—21no—
163#11715895—21no—
164C22×C97—41no—
165C22×C97—41no—
166C22×C97—41no—
167C22×C97C22×D411yescharacteristic
168C2×D97—41no—
169C2×D97—41no—
170C2×D97—41no—
171C2×D97—84no—
172C2×D97#12112yes—
173C2×D97#13512yes—
174C2×D97#13712yes—
175C4×C97—22no—
176C4×C97—21no—
177C4×C97—21no—
178C4×C97—41no—
179C4×C97—41no—
180C4×C97C22×D411yescharacteristic
181C4×C97C2×#4111yescharacteristic
182Dic97—22no—
183Dic97—21no—
184Dic97—21no—
185Dic97—41no—
186Dic97—41no—
187Dic97#12112yes—
188C2×C97—81no—
189C2×C97C2×#12111yescharacteristic
190C2×C97C2×#13511yescharacteristic
191C2×C97C2×#13711yescharacteristic
192D97—81no—
193D97#46414yesdirect product factor
194C2×#464—971nomaximal, Sylow
195C97C2×#46411yescharacteristic, complemented, Sylow
196C22×Q16—971no—
197C22×SD16—971no—
198C2×#124—971no—
199C2×C4×Q8—971no—
200#464—971no—
201#464—971no—
202#464—971no—
203#464—971no—
204#464—971no—
205#464—971no—
206#464—971no—
207#464D9711yescharacteristic, direct product factor
208C2×C4⋊C8—971no—
209C2×#103—971no—
210C2×#102—971no—
211C22×Q8—971no—
212C22×Q8—971no—
213C22×D4—971no—
214C2×Q16—971no—
215C2×Q16—971no—
216C2×Q16—971no—
217C2×Q16—1944no—
218C2×Q16C2×D9711yescharacteristic, complemented
219C2×SD16—971no—
220C2×SD16—971no—
221C2×SD16—971no—
222C2×SD16—1944no—
223C2×SD16C2×D9711yescharacteristic
224C22×C8—971no—
225C22×C8—971no—
226#124—971no—
227#124—971no—
228#124—971no—
229#124—1941no—
230#124—1941no—
231#124C2×D9711yescharacteristic
232C4×Q8—971no—
233C4×Q8—971no—
234C4×Q8—971no—
235C4×Q8—1941no—
236C4×Q8—1941no—
237C4×Q8C2×D9711yescharacteristic, complemented
238C2×C4⋊C4—971no—
239C2×C4⋊C4—1941no—
240C2×C22⋊C4—1941no—
241C2×C22⋊C4—1941no—
242C2×C42—971no—
243C2×C42—1941no—
244C4⋊C8—971no—
245C4⋊C8—971no—
246C4⋊C8—971no—
247C4⋊C8C2×D9711yescharacteristic, complemented
248#103—971no—
249#103—971no—
250#103—971no—
251#103C2×D9711yescharacteristic, complemented
252#102—971no—
253#102—971no—
254#102—971no—
255#102C2×D9711yescharacteristic
256C24—1941no—
257C2×Q8—974no—
258C2×Q8—971no—
259C2×Q8—971no—
260C2×Q8—1941no—
261C2×Q8—1941no—
262C2×Q8—1942no—
263C2×Q8—1942no—
264C2×Q8—1942no—
265C2×Q8C22×D9711yescharacteristic
266C2×Q8C22×D9711yescharacteristic
267C2×D4—971no—
268C2×D4—1942no—
269C2×D4—1942no—
270C2×D4—1941no—
271C2×D4C22×D9711yescharacteristic
272C22×C4—971no—
273C22×C4—971no—
274C22×C4—971no—
275C22×C4—1941no—
276C22×C4—1941no—
277C22×C4—1941no—
278Q16—22no—
279Q16—1942no—
280Q16—1942no—
281Q16—1942no—
282SD16—22no—
283SD16—1942no—
284SD16—1942no—
285SD16—1942no—
286C2×C8—971no—
287C2×C8—971no—
288C2×C8—1942no—
289C2×C8—1942no—
290C2×C8C22×D9711yescharacteristic
291C2×C8C22×D9711yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—971no—
294C4⋊C4—1941no—
295C4⋊C4—1941no—
296C4⋊C4—1941no—
297C4⋊C4—1941no—
298C4⋊C4C22×D9711yescharacteristic
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—1941no—
302C22⋊C4—1941no—
303C22⋊C4—1941no—
304C22⋊C4—1941no—
305C22⋊C4—1941no—
306C22⋊C4—1941no—
307C42—21no—
308C42—971no—
309C42—971no—
310C42—971no—
311C42—1941no—
312C42—1941no—
313C42—1941no—
314C42C22×D9711yescharacteristic
315C23—21no—
316C23—971no—
317C23—3882no—
318C23—3882no—
319C23—3882no—
320C23—1941no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—972no—
325Q8—3882no—
326Q8—1944no—
327Q8—1941no—
328Q8—1941no—
329Q8—1941no—
330Q8D4×D9712yes—
331D4—21no—
332D4—21no—
333D4—3882no—
334D4—1941no—
335D4—1941no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—971no—
340C2×C4—972no—
341C2×C4—972no—
342C2×C4—971no—
343C2×C4—971no—
344C2×C4—3882no—
345C2×C4—3882no—
346C2×C4—1942no—
347C2×C4—1942no—
348C2×C4—1944no—
349C2×C4—1941no—
350C2×C4—1941no—
351C2×C4—1941no—
352C2×C4D4×D9711yescharacteristic
353C2×C4D4×D9711yescharacteristic
354C2×C4C23×D9711yescharacteristic
355C8—21no—
356C8—21no—
357C8—1941no—
358C8—1941no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—7764no—
363C22—972no—
364C22—972no—
365C22—972no—
366C22—3881no—
367C22—3881no—
368C22—3881no—
369C22C2×D4×D9711yescharacteristic
370C4—22no—
371C4—21no—
372C4—21no—
373C4—41no—
374C4—41no—
375C4—972no—
376C4—3881no—
377C4—3881no—
378C4—1942no—
379C4—1941no—
380C4—1941no—
381C4C2×D4×D9711yescharacteristic
382C4#41×D9711yescharacteristic
383C2—81no—
384C2—7761no—
385C2—974no—
386C2#121×D9711yescharacteristic
387C2#135×D9711yescharacteristic
388C2#137×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#464×C97C2
socleC22×C97C22×D4
prime radicalC2×D4×D97C22
hypercenter#464D97
hypocenterC97C2×#464
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D97C2semidirect
4. C4×Q8×D97C2semidirect
5. #20260960C2semidirect
6. #20260959C2semidirect
7. #20260958C2semidirect
8. #20260957C2semidirect
9. #20260956C2semidirect
10. #20260955C2semidirect
11. #20260954C2semidirect
12. C4⋊C8×D97C2semidirect
13. #103×D97C2semidirect
15. #464×C97C2semidirect
19. C2×Dic388C22semidirect
20. C2×#13095501C22semidirect
21. C2×#13095501C22semidirect
34. Q8×Dic97C22semidirect
36. C4×Dic194C22semidirect
38. #15690775C22semidirect
45. C4⋊#11715895C22semidirect
46. #15690739C22semidirect
47. #15690739C22semidirect
48. #15690735C22semidirect
49. #15690734C22semidirect
50. #15690733C22semidirect
54. C2×Q16×C97C22semidirect
57. C4×Q8×C97C22semidirect
58. C4⋊C8×C97C22semidirect
59. #103×C97C22semidirect
193. D97#464direct (N × H)
195. C97C2×#464semidirect
207. #464D97direct (N × H)
218. C2×Q16C2×D97semidirect
237. C4×Q8C2×D97semidirect
247. C4⋊C8C2×D97semidirect
251. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D972
C2×SD16×D972
#124×D972
C4×Q8×D972
#202609602
#202609592
#202609582
#202609572
#202609562
#202609552
#202609542
C4⋊C8×D972
#103×D972
#102×D972
#464×C972
C2×#46497