Finite Group Databasebeta

← #27152197

Subgroups — #27152197

Order 12416.

572 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×#15690842C211yesmaximal, characteristic, complemented
2C2×#15690839C211yesmaximal, characteristic, complemented
3C2×#15690837C211yesmaximal, characteristic, complemented
4C2×#15690834C211yesmaximal, characteristic, complemented
5#20261284C211yesmaximal, characteristic, complemented
6#124×D97C211yesmaximal, characteristic, complemented
7#20261128C211yesmaximal, characteristic, complemented
8#20261126C211yesmaximal, characteristic, complemented
9#20261123C211yesmaximal, characteristic, complemented
10#20261121C211yesmaximal, characteristic, complemented
11#20261119C211yesmaximal, characteristic, complemented
12#20261117C211yesmaximal, characteristic, complemented
13#20261117C211yesmaximal, characteristic, complemented
14C4×#13095494C211yesmaximal, characteristic, complemented
15#495×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095510C2211yescharacteristic
18C2×Q8×D97C2211yescharacteristic
19C2×D4×D97C2211yescharacteristic
20#15690842—24no—
21#15690839—24no—
22#15690837—24no—
23#15690834—24no—
24C2×Dic388C2211yescharacteristic, complemented
25C2×#13095501C2211yescharacteristic, complemented
26C2×#13095501C2211yescharacteristic, complemented
27C2×#13095500C2211yescharacteristic
28C2×#13095500C2211yescharacteristic, complemented
29C2×#13095499C2211yescharacteristic
30C2×#13095499C2211yescharacteristic, complemented
31C2×#13095498C2211yescharacteristic
32C2×#13095498C2211yescharacteristic, complemented
33C2×D776C2211yescharacteristic
34C2×#13095496C2211yescharacteristic
35C2×#13095496C2211yescharacteristic
36C2×#13095494C2211yescharacteristic
37C2×#13095494C2211yescharacteristic
38#15690808C2211yescharacteristic, complemented
39#15690806C2211yescharacteristic, complemented
40#15690799C2211yescharacteristic
41#15690798C2211yescharacteristic
42#15690798C2211yescharacteristic, complemented
43#15690797C2211yescharacteristic
44#15690796—22no—
45#15690783—22no—
46#15690770—22no—
47C22⋊C4×D97—22no—
48C42×D97C2211yescharacteristic
49#15690720C2211yescharacteristic, complemented
50#15690720C2211yescharacteristic, complemented
51C4×#11715895C2211yescharacteristic, complemented
52C2×Q16×C97C2211yescharacteristic, complemented
53C2×SD16×C97C2211yescharacteristic
54C2×SD16×C97C2211yescharacteristic, complemented
55C2×D8×C97C2211yescharacteristic
56#124×C97C2211yescharacteristic
57#124×C97C2211yescharacteristic, complemented
58C4×C8×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60#13095512—22no—
61#13095512—22no—
62#13095510—22no—
63#13095510—22no—
64C2×Dic194—21no—
65C2×Dic194C2311yescharacteristic
66C2×Dic194C2311yescharacteristic
67Q8×D97—22no—
68Q8×D97—22no—
69C2×D388—21no—
70C2×D388C2311yescharacteristic
71C2×D388C2311yescharacteristic
72C2×C22⋊D97—21no—
73C2×C22⋊D97—22no—
74C2×C22⋊D97—21no—
75D4×D97—22no—
76D4×D97—22no—
77C22×Dic97—21no—
78C2×C4×D97—21no—
79C2×C4×D97—21no—
80C2×C4×D97D412yes—
81C2×C4×D97C2311yescharacteristic
82Dic388—22no—
83#13095501—22no—
84#13095501—22no—
85#13095500—22no—
86#13095500—22no—
87#13095499—22no—
88#13095499—22no—
89#13095498—22no—
90#13095498—22no—
91D776—22no—
92#13095496—22no—
93#13095496—22no—
94#13095494D414yescomplemented
95#13095494D414yescomplemented
96C2×#11715895C2311yescharacteristic
97C2×#11715895C2311yescharacteristic
98#13095490—22no—
99C22⋊Dic97—22no—
100#13095488—22no—
101#13095488—22no—
102#13095488—22no—
103C4×Dic97D412yes—
104C4×Dic97C2311yescharacteristic
105C2×Q8×C97C2311yescharacteristic
106C2×Q8×C97C2311yescharacteristic
107C2×D4×C97C2311yescharacteristic
108C2×D4×C97C2311yescharacteristic
109Q16×C97—22no—
110SD16×C97—22no—
111SD16×C97—22no—
112D8×C97—22no—
113C2×C8×C97C2311yescharacteristic
114C2×C8×C97C2311yescharacteristic
115C22⋊C4×C97—22no—
116C22⋊C4×C97—22no—
117C42×C97C2311yescharacteristic
118C23×C97—21no—
119C23×C97—21no—
120C22×D97—21no—
121C22×D97—21no—
122C22×D97—44no—
123C22×D97—42no—
124C22×D97C2×D411yescharacteristic
125Q8×C97—21no—
126Q8×C97—21no—
127Q8×C97—21no—
128Q8×C97—21no—
129D4×C97—21no—
130D4×C97—21no—
131D4×C97—21no—
132D4×C97—21no—
133C2×C4×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97C2×D412yes—
136C2×C4×C97C2411yescharacteristic
137C22⋊D97—44no—
138C22⋊D97—42no—
139C22⋊D97—42no—
140C2×Dic97—21no—
141C2×Dic97—21no—
142C2×Dic97—42no—
143C2×Dic97C2×D412yes—
144C2×Dic97C2×D411yescharacteristic
145D388—21no—
146D388—21no—
147D388—21no—
148D388—21no—
149D388—42no—
150C4×D97—42no—
151C4×D97—42no—
152C4×D97—42no—
153C4×D97C2×D412yes—
154C4×D97C2×D412yes—
155Dic194—21no—
156Dic194—21no—
157Dic194—21no—
158Dic194—21no—
159Dic194—42no—
160C8×C97C2×D412yes—
161C8×C97C2×D412yes—
162#11715895C2×D412yes—
163#11715895C2×D412yes—
164C22×C97—41no—
165C22×C97—41no—
166C22×C97—42no—
167C22×C97—42no—
168C22×C97C22×D411yescharacteristic
169C2×D97—22no—
170C2×D97—41no—
171C2×D97—41no—
172C2×D97—42no—
173C2×D97—42no—
174C2×D97—84no—
175C2×D97#12712yes—
176C4×C97—22no—
177C4×C97—41no—
178C4×C97—41no—
179C4×C97C22×D411yescharacteristic
180C4×C97C22×D411yescharacteristic
181Dic97—22no—
182Dic97—41no—
183Dic97—41no—
184Dic97#12712yes—
185C2×C97—81no—
186C2×C97—81no—
187C2×C97C2×#12711yescharacteristic
188C2×C97#57512yes—
189D97—22no—
190D97—81no—
191D97—81no—
192#3153—971nomaximal, Sylow
193C97#315311yescharacteristic, complemented, Sylow
194C2×#137—971no—
195C2×#137—971no—
196C2×AGL(1,ℤ/8)—971no—
197C2×AGL(1,ℤ/8)—971no—
198#532—971no—
199C2×#124—971no—
200#497—971no—
201#497—971no—
202#496—971no—
203#496—971no—
204#495—971no—
205#495—971no—
206#495—971no—
207#495D9711yescharacteristic, complemented
208C4×M16—971no—
209C2×#41—971no—
210C2×#41—971no—
211C22×Q8—971no—
212C22×D4—971no—
213#137—1944no—
214#137—1944no—
215AGL(1,ℤ/8)—1944no—
216AGL(1,ℤ/8)—1944no—
217C2×Q16—971no—
218C2×Q16—971no—
219C2×Q16—971no—
220C2×Q16C2×D9711yescharacteristic, complemented
221C2×SD16—971no—
222C2×SD16—971no—
223C2×SD16—971no—
224C2×SD16—971no—
225C2×SD16—971no—
226C2×SD16—971no—
227C2×SD16C2×D9711yescharacteristic
228C2×SD16C2×D9711yescharacteristic, complemented
229C2×D8—971no—
230C2×D8—971no—
231C2×D8—971no—
232C2×D8C2×D9711yescharacteristic
233C2×M16—971no—
234C2×M16—971no—
235C4⋊Q8—971no—
236#127—971no—
237#124—971no—
238#124—971no—
239#124—971no—
240#124—971no—
241#124—1942no—
242#124C2×D9711yescharacteristic
243#124C2×D9711yescharacteristic, complemented
244#121—1942no—
245C4×D4—1942no—
246C2×C22⋊C4—1942no—
247C2×C42—971no—
248#97—971no—
249#97—971no—
250C4×C8—971no—
251C4×C8C2×D9711yescharacteristic, complemented
252C24—1941no—
253#41—1942no—
254#41—1942no—
255#41—1942no—
256#41—1942no—
257C2×Q8—971no—
258C2×Q8—971no—
259C2×Q8—1941no—
260C2×Q8—1942no—
261C2×Q8—1942no—
262C2×Q8C22×D9711yescharacteristic
263C2×Q8C22×D9711yescharacteristic
264C2×D4—971no—
265C2×D4—971no—
266C2×D4—1941no—
267C2×D4—1942no—
268C2×D4—1942no—
269C2×D4—1942no—
270C2×D4—1941no—
271C2×D4—1941no—
272C2×D4C22×D9711yescharacteristic
273C2×D4C22×D9711yescharacteristic
274C22×C4—971no—
275C22×C4—972no—
276C22×C4—1941no—
277C22×C4—1941no—
278C22×C4—1941no—
279Q16—22no—
280Q16—1942no—
281Q16—1942no—
282Q16—1942no—
283SD16—22no—
284SD16—22no—
285SD16—1942no—
286SD16—1942no—
287SD16—1942no—
288SD16—1942no—
289SD16—1942no—
290SD16—1942no—
291D8—22no—
292D8—1942no—
293D8—1942no—
294D8—1942no—
295M16—974no—
296M16—974no—
297C2×C8—971no—
298C2×C8—971no—
299C2×C8C22×D9711yescharacteristic
300C2×C8C22×D9711yescharacteristic
301C4⋊C4—1942no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—1942no—
305C22⋊C4—1942no—
306C22⋊C4—1942no—
307C22⋊C4—1942no—
308C42—972no—
309C42—971no—
310C42C22×D9711yescharacteristic
311C23—21no—
312C23—21no—
313C23—971no—
314C23—3882no—
315C23—3884no—
316C23—1941no—
317C23—1941no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—3882no—
323Q8—1941no—
324Q8—1941no—
325Q8—1941no—
326Q8—1941no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—3882no—
332D4—3882no—
333D4—3882no—
334D4—3884no—
335D4—1941no—
336D4—1941no—
337D4—1941no—
338D4—1941no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—971no—
342C2×C4—972no—
343C2×C4—972no—
344C2×C4—972no—
345C2×C4—3882no—
346C2×C4—3882no—
347C2×C4—3882no—
348C2×C4—3882no—
349C2×C4—1941no—
350C2×C4—1941no—
351C2×C4D4×D9712yes—
352C2×C4C23×D9711yescharacteristic
353C8—972no—
354C8—972no—
355C8D4×D9712yes—
356C8D4×D9712yes—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—42no—
361C22—7764no—
362C22—972no—
363C22—3881no—
364C22—3881no—
365C22—3882no—
366C22—3882no—
367C22—1942no—
368C22C2×D4×D9711yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—972no—
373C4—3881no—
374C4—3881no—
375C4—1942no—
376C4C2×D4×D9711yescharacteristic
377C4C2×D4×D9711yescharacteristic
378C2—81no—
379C2—81no—
380C2—7761no—
381C2—7761no—
382C2—1942no—
383C2#127×D9711yescharacteristic
384C2#2026153212yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#495×C97C2
socleC22×C97C22×D4
hypercenter#495D97
hypocenterC97#3153
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#15690842C2semidirect
2. C2×#15690839C2semidirect
3. C2×#15690837C2semidirect
4. C2×#15690834C2semidirect
5. #20261284C2semidirect
6. #124×D97C2semidirect
7. #20261128C2semidirect
8. #20261126C2semidirect
9. #20261123C2semidirect
10. #20261121C2semidirect
11. #20261119C2semidirect
12. #20261117C2semidirect
13. #20261117C2semidirect
14. C4×#13095494C2semidirect
15. #495×C97C2semidirect
24. C2×Dic388C22semidirect
25. C2×#13095501C22semidirect
26. C2×#13095501C22semidirect
28. C2×#13095500C22semidirect
30. C2×#13095499C22semidirect
32. C2×#13095498C22semidirect
38. #15690808C22semidirect
39. #15690806C22semidirect
42. #15690798C22semidirect
49. #15690720C22semidirect
50. #15690720C22semidirect
51. C4×#11715895C22semidirect
52. C2×Q16×C97C22semidirect
54. C2×SD16×C97C22semidirect
57. #124×C97C22semidirect
58. C4×C8×C97C22semidirect
94. #13095494D4semidirect
95. #13095494D4semidirect
193. C97#3153semidirect
207. #495D97semidirect
220. C2×Q16C2×D97semidirect
228. C2×SD16C2×D97semidirect
243. #124C2×D97semidirect
251. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908422
C2×#156908392
C2×#156908372
C2×#156908342
#202612842
#124×D972
#202611282
#202611262
#202611232
#202611212
#202611192
#202611172
C4×#130954942
#495×C972
#315397