Finite Group Databasebeta

← #27153243

Subgroups — #27153243

Order 12416.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690842C211yesmaximal, characteristic, complemented
2C22×#13095500C211yesmaximal, characteristic, complemented
3#20261398C211yesmaximal, characteristic, complemented
4#20261376C211yesmaximal, characteristic
5#20261128C211yesmaximal, characteristic, complemented
6#20260982C211yesmaximal, characteristic, complemented
7#20260960C211yesmaximal, characteristic, complemented
8#20260939C211yesmaximal, characteristic, complemented
9#20260900C211yesmaximal, characteristic, complemented
10#20260890C211yesmaximal, characteristic, complemented
11#20260885C211yesmaximal, characteristic, complemented
12#20260875C211yesmaximal, characteristic, complemented
13#20260834C211yesmaximal, characteristic, complemented
14#20260806C211yesmaximal, characteristic, complemented
15#467×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C22×Dic194C2211yescharacteristic
18C2×Q8×D97C2211yescharacteristic, complemented
19C22×D388C2211yescharacteristic
20#15690842—24no—
21C2×#13095501C2211yescharacteristic, complemented
22C2×#13095501C2211yescharacteristic, complemented
23C2×#13095500—24no—
24C2×#13095500C2211yescharacteristic
25C2×#13095500C2211yescharacteristic
26C2×#13095500C2211yescharacteristic
27C2×#13095500C2211yescharacteristic
28C2×#13095498C2211yescharacteristic
29C2×#13095498C2211yescharacteristic
30C2×#13095494C2211yescharacteristic, complemented
31#15690809—21no—
32#15690808C2211yescharacteristic, complemented
33#15690799C2211yescharacteristic
34#15690795—21no—
35#15690790C2211yescharacteristic, complemented
36#15690788—21no—
37#15690787—21no—
38#15690785—21no—
39#15690785C2211yescharacteristic
40#15690780—21no—
41#15690775C2211yescharacteristic, complemented
42#15690772C2211yescharacteristic
43C4×C22⋊D97C2211yescharacteristic
44#15690770—21no—
45C22⋊C4×D97—21no—
46#15690739C2211yescharacteristic, complemented
47#15690733C2211yescharacteristic, complemented
48#15690733C2211yescharacteristic, complemented
49#15690730C2211yescharacteristic
50#15690730C2211yescharacteristic
51C22⋊#11715895C2211yescharacteristic
52#15690722C2211yescharacteristic, complemented
53#15690720C2211yescharacteristic, complemented
54C2×Q16×C97C2211yescharacteristic, complemented
55C22×C8×C97C2211yescharacteristic
56C22⋊Q8×C97C2211yescharacteristic
57C22⋊Q8×C97C2211yescharacteristic
58#107×C97C2211yescharacteristic, complemented
59#103×C97C2211yescharacteristic, complemented
60#103×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62#13095512—22no—
63#13095512—41no—
64C2×Dic194—22no—
65C2×Dic194—21no—
66C2×Dic194—21no—
67C2×Dic194—41no—
68C2×Dic194C2311yescharacteristic
69C2×Dic194C2311yescharacteristic
70Q8×D97—22no—
71Q8×D97—41no—
72C2×D388—22no—
73C2×D388—21no—
74C2×D388—21no—
75C2×D388—21no—
76C2×D388—41no—
77C2×D388C2311yescharacteristic
78C2×D388C2311yescharacteristic
79C2×C22⋊D97D411yescharacteristic
80C22×Dic97—21no—
81C2×C4×D97—21no—
82C2×C4×D97—21no—
83C2×C4×D97—21no—
84C2×C4×D97C2311yescharacteristic
85#13095501—22no—
86#13095501—41no—
87#13095500—22no—
88#13095500—22no—
89#13095500—22no—
90#13095500D414yescomplemented
91#13095498—22no—
92#13095498—41no—
93#13095494—22no—
94C2×#11715895C2311yescharacteristic
95C2×#11715895C2311yescharacteristic
96C4⋊Dic97—21no—
97#13095490—21no—
98#13095490—21no—
99#13095490—21no—
100#13095490D411yescharacteristic
101C22⋊Dic97D411yescharacteristic
102#13095488—21no—
103#13095488—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488—21no—
107#13095488D411yescharacteristic
108C4×Dic97—21no—
109C4×Dic97C2311yescharacteristic
110C2×Q8×C97C2311yescharacteristic
111C2×Q8×C97C2311yescharacteristic
112C22×C4×C97C2311yescharacteristic
113Q16×C97—22no—
114C2×C8×C97—22no—
115C2×C8×C97C2311yescharacteristic
116C2×C8×C97C2311yescharacteristic
117C4⋊C4×C97—21no—
118C4⋊C4×C97—21no—
119C4⋊C4×C97C2311yescharacteristic
120C4⋊C4×C97C2311yescharacteristic
121C22⋊C4×C97—21no—
122C22⋊C4×C97—21no—
123C23×C97C2×D411yescharacteristic
124C22×D97—21no—
125C22×D97—21no—
126C22×D97—42no—
127C22×D97—42no—
128C22×D97—81no—
129C22×D97C2×D411yescharacteristic
130Q8×C97—21no—
131Q8×C97—21no—
132Q8×C97—21no—
133Q8×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97—21no—
136C2×C4×C97—21no—
137C2×C4×C97—21no—
138C2×C4×C97—21no—
139C2×C4×C97—21no—
140C2×C4×C97C2×D411yescharacteristic
141C2×C4×C97C2411yescharacteristic
142C22⋊D97—22no—
143C2×Dic97—21no—
144C2×Dic97—21no—
145C2×Dic97—42no—
146C2×Dic97C2×D411yescharacteristic
147C2×Dic97C2×D411yescharacteristic
148C2×Dic97C2×D411yescharacteristic
149D388—21no—
150D388—21no—
151D388—21no—
152D388—42no—
153D388—42no—
154D388—81no—
155D388C2×D412yes—
156C4×D97—21no—
157C4×D97—21no—
158C4×D97—42no—
159C4×D97—42no—
160C4×D97—81no—
161Dic194—21no—
162Dic194—21no—
163Dic194—21no—
164Dic194—42no—
165Dic194—42no—
166Dic194C2×D412yes—
167C8×C97—21no—
168C8×C97C2×D412yes—
169#11715895—21no—
170#11715895—21no—
171C22×C97—21no—
172C22×C97—21no—
173C22×C97#13712yescomplemented
174C22×C97C22×D411yescharacteristic
175C2×D97—21no—
176C2×D97—22no—
177C2×D97—21no—
178C2×D97—21no—
179C2×D97—41no—
180C2×D97—41no—
181C2×D97—41no—
182C2×D97—41no—
183C2×D97—41no—
184C2×D97—84no—
185C4×C97—21no—
186C4×C97—41no—
187C4×C97—41no—
188C4×C97—41no—
189C4×C97—41no—
190C4×C97C22×D411yescharacteristic
191C4×C97H(5,𝔽2)11yescharacteristic
192Dic97—22no—
193Dic97—21no—
194Dic97—21no—
195Dic97—21no—
196Dic97—41no—
197C2×C97—22no—
198C2×C97D4211yescharacteristic
199C2×C97C2×#13711yescharacteristic
200C2×C97#57711yescharacteristic
201D97—41no—
202D97—42no—
203D97—81no—
204#3266—971nomaximal, Sylow
205C97#326611yescharacteristic, complemented, Sylow
206C2×#137—971no—
207C22×SD16—971no—
208D4×Q8—971no—
209#546—971no—
210#497—971no—
211#468—971no—
212#467D9711yescharacteristic, complemented
213#464—971no—
214#461—971no—
215#452—971no—
216#451—971no—
217#450—971no—
218#448—971no—
219#440—971no—
220#435—971no—
221C2×#41—971no—
222C22×Q8—971no—
223C22×Q8—971no—
224C22×D4—971no—
225#137—1944no—
226C2×Q16—971no—
227C2×Q16—971no—
228C2×Q16C2×D9711yescharacteristic, complemented
229C2×SD16—971no—
230C2×SD16—971no—
231C2×SD16—971no—
232C2×SD16—971no—
233C2×SD16—971no—
234C2×SD16—971no—
235C2×SD16—1944no—
236C2×M16—971no—
237C22×C8C2×D9711yescharacteristic
238C4⋊Q8—971no—
239C4⋊Q8—1941no—
240#124—971no—
241#123—1941no—
242C22⋊Q8—971no—
243C22⋊Q8—1941no—
244C22⋊Q8—1941no—
245C22⋊Q8C2×D9711yescharacteristic
246C22⋊Q8C2×D9711yescharacteristic
247#121—971no—
248#121—1941no—
249#120—1941no—
250C4×Q8—971no—
251C4×D4—971no—
252C4×D4—971no—
253C4×D4—1941no—
254C2×C22⋊C4—1941no—
255#107C2×D9711yescharacteristic, complemented
256C4⋊C8—971no—
257#103—971no—
258#103—971no—
259#103C2×D9711yescharacteristic, complemented
260#103C2×D9711yescharacteristic, complemented
261#102—971no—
262#102—971no—
263C22⋊C8—971no—
264C22⋊C8—971no—
265#97—971no—
266C24—1941no—
267#41—3881no—
268#41—1942no—
269C2×Q8—971no—
270C2×Q8—971no—
271C2×Q8—3881no—
272C2×Q8—3881no—
273C2×Q8—1941no—
274C2×Q8—1942no—
275C2×Q8—1942no—
276C2×Q8—1941no—
277C2×Q8C22×D9711yescharacteristic
278C2×Q8C22×D9711yescharacteristic
279C2×D4—971no—
280C2×D4—971no—
281C2×D4—971no—
282C2×D4—3881no—
283C2×D4—1941no—
284C2×D4—1942no—
285C2×D4—1941no—
286C2×D4—1941no—
287C22×C4—971no—
288C22×C4—1941no—
289C22×C4—1941no—
290C22×C4—1941no—
291C22×C4—1941no—
292C22×C4C22×D9711yescharacteristic
293Q16—22no—
294Q16—3881no—
295Q16—1942no—
296SD16—974no—
297SD16—3881no—
298SD16—1942no—
299SD16—1942no—
300SD16—1942no—
301SD16—1942no—
302M16—1942no—
303C2×C8—22no—
304C2×C8—971no—
305C2×C8—971no—
306C2×C8C22×D9711yescharacteristic
307C2×C8C22×D9711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—971no—
311C4⋊C4—1941no—
312C4⋊C4—1941no—
313C4⋊C4—1941no—
314C4⋊C4—1941no—
315C4⋊C4C22×D9711yescharacteristic
316C4⋊C4C22×D9711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—971no—
320C22⋊C4—971no—
321C22⋊C4—1941no—
322C22⋊C4—1941no—
323C22⋊C4—1941no—
324C22⋊C4—1941no—
325C22⋊C4—1941no—
326C42—971no—
327C42—1941no—
328C23—7761no—
329C23—971no—
330C23—3882no—
331C23—3882no—
332C23—1941no—
333C23—1941no—
334C23D4×D9711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—972no—
340Q8—3882no—
341Q8—3882no—
342Q8—1941no—
343Q8—1941no—
344Q8—1941no—
345D4—7761no—
346D4—972no—
347D4—3882no—
348D4—3882no—
349D4—1942no—
350D4—1941no—
351D4—1941no—
352D4—1941no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—7761no—
360C2×C4—971no—
361C2×C4—971no—
362C2×C4—971no—
363C2×C4—3882no—
364C2×C4—3882no—
365C2×C4—3882no—
366C2×C4—1941no—
367C2×C4—1941no—
368C2×C4—1941no—
369C2×C4—1941no—
370C2×C4D4×D9711yescharacteristic
371C2×C4C23×D9711yescharacteristic
372C8—21no—
373C8—1941no—
374C8—1941no—
375C8D4×D9712yes—
376C22—21no—
377C22—21no—
378C22—7764no—
379C22—3881no—
380C22—3881no—
381C22—3881no—
382C22—3881no—
383C22—3881no—
384C22—1941no—
385C22—1941no—
386C22—1941no—
387C22—1942no—
388C22#1569084212yescomplemented
389C22C2×D4×D9711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—3881no—
396C4—1941no—
397C4—1941no—
398C4—1941no—
399C4—1942no—
400C4C2×D4×D9711yescharacteristic
401C4#1569085611yescharacteristic
402C2—22no—
403C2—7761no—
404C2—3882no—
405C2—3881no—
406C2#2026136911yescharacteristic
407C2C2×#1569084211yescharacteristic
408C2#2026154211yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#467×C97C2
socleC22×C97C22×D4
prime radical#20261376C2
hypercenter#467D97
hypocenterC97#3266
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#15690842C2semidirect
2. C22×#13095500C2semidirect
3. #20261398C2semidirect
5. #20261128C2semidirect
6. #20260982C2semidirect
7. #20260960C2semidirect
8. #20260939C2semidirect
9. #20260900C2semidirect
10. #20260890C2semidirect
11. #20260885C2semidirect
12. #20260875C2semidirect
13. #20260834C2semidirect
14. #20260806C2semidirect
15. #467×C97C2semidirect
18. C2×Q8×D97C22semidirect
21. C2×#13095501C22semidirect
22. C2×#13095501C22semidirect
30. C2×#13095494C22semidirect
32. #15690808C22semidirect
35. #15690790C22semidirect
41. #15690775C22semidirect
46. #15690739C22semidirect
47. #15690733C22semidirect
48. #15690733C22semidirect
52. #15690722C22semidirect
53. #15690720C22semidirect
54. C2×Q16×C97C22semidirect
58. #107×C97C22semidirect
59. #103×C97C22semidirect
60. #103×C97C22semidirect
90. #13095500D4semidirect
173. C22×C97#137semidirect
205. C97#3266semidirect
212. #467D97semidirect
228. C2×Q16C2×D97semidirect
255. #107C2×D97semidirect
259. #103C2×D97semidirect
260. #103C2×D97semidirect
388. C22#15690842semidirect

Maximal subgroups

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

subgroupindex
C2×#156908422
C22×#130955002
#202613982
#202613762
#202611282
#202609822
#202609602
#202609392
#202609002
#202608902
#202608852
#202608752
#202608342
#202608062
#467×C972
#326697