Finite Group Databasebeta

← #27152211

Subgroups — #27152211

Order 12416.

564 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×#15690837C211yesmaximal, characteristic, complemented
3C2×#15690833C211yesmaximal, characteristic, complemented
4C2×SD16×D97C211yesmaximal, characteristic, complemented
5#20261310C211yesmaximal, characteristic, complemented
6#20261303C211yesmaximal, characteristic, complemented
7#20261128C211yesmaximal, characteristic, complemented
8#20261124C211yesmaximal, characteristic, complemented
9#20261121C211yesmaximal, characteristic, complemented
10#20261120C211yesmaximal, characteristic, complemented
11#20261117C211yesmaximal, characteristic, complemented
12#20261109C211yesmaximal, characteristic, complemented
13#20261106C211yesmaximal, characteristic, complemented
14#20260692C211yesmaximal, characteristic, complemented
15#497×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095512C2211yescharacteristic
18C2×Q8×D97C2211yescharacteristic
19C2×D4×D97C2211yescharacteristic
20#15690842—24no—
21#15690837—24no—
22#15690833—24no—
23C2×#13095501C2211yescharacteristic, complemented
24C2×#13095500C2211yescharacteristic
25C2×#13095500C2211yescharacteristic, complemented
26C2×#13095499C2211yescharacteristic
27C2×#13095498C2211yescharacteristic
28C2×#13095498C2211yescharacteristic
29C2×#13095498C2211yescharacteristic, complemented
30C2×#13095498C2211yescharacteristic, complemented
31C2×#13095498C2211yescharacteristic, complemented
32SD16×D97—24no—
33C2×D776C2211yescharacteristic
34C2×D776C2211yescharacteristic, complemented
35C2×#13095496C2211yescharacteristic
36C2×#13095494C2211yescharacteristic
37C2×C8×D97C2211yescharacteristic
38#15690808C2211yescharacteristic, complemented
39#15690807C2211yescharacteristic
40#15690806C2211yescharacteristic
41#15690799C2211yescharacteristic
42#15690799C2211yescharacteristic, complemented
43#15690798C2211yescharacteristic, complemented
44#15690795—22no—
45#15690788—22no—
46#15690783—22no—
47#15690780—22no—
48#15690767C2211yescharacteristic
49#15690720C2211yescharacteristic, complemented
50C4×#11715895C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C2×Q16×C97C2211yescharacteristic, complemented
53C2×Q16×C97C2211yescharacteristic, complemented
54C2×SD16×C97C2211yescharacteristic
55C2×SD16×C97C2211yescharacteristic
56C4⋊Q8×C97C2211yescharacteristic, complemented
57#124×C97C2211yescharacteristic
58#97×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60#13095512—22no—
61#13095512—22no—
62#13095512—22no—
63#13095512—22no—
64C2×Dic194—21no—
65C2×Dic194C2311yescharacteristic
66Q8×D97—22no—
67Q8×D97—22no—
68C2×D388—21no—
69C2×D388—21no—
70C2×D388—22no—
71C2×D388C2311yescharacteristic
72C2×D388C2311yescharacteristic
73C2×D388C2311yescharacteristic
74C2×C22⋊D97—21no—
75C2×C22⋊D97—22no—
76D4×D97—22no—
77D4×D97—22no—
78C2×C4×D97—21no—
79C2×C4×D97—21no—
80C2×C4×D97—21no—
81C2×C4×D97C2311yescharacteristic
82#13095501—22no—
83#13095500—22no—
84#13095500—22no—
85#13095499—22no—
86#13095498—22no—
87#13095498—22no—
88#13095498—22no—
89#13095498—22no—
90#13095498—22no—
91D776—22no—
92D776—22no—
93#13095496—22no—
94#13095494D414yescomplemented
95C2×#11715895C2311yescharacteristic
96C2×#11715895C2311yescharacteristic
97C8×D97D414yescomplemented
98#13095490—22no—
99#13095490D412yes—
100#13095488—22no—
101#13095488—22no—
102#13095488—22no—
103#13095488D412yes—
104C4×Dic97C2311yescharacteristic
105C2×Q8×C97C2311yescharacteristic
106C2×Q8×C97C2311yescharacteristic
107C2×Q8×C97C2311yescharacteristic
108C2×D4×C97C2311yescharacteristic
109Q16×C97—22no—
110Q16×C97—22no—
111SD16×C97—22no—
112SD16×C97—22no—
113C2×C8×C97C2311yescharacteristic
114C2×C8×C97C2311yescharacteristic
115C4⋊C4×C97—22no—
116C22⋊C4×C97—22no—
117C42×C97C2311yescharacteristic
118C23×C97—21no—
119C22×D97—21no—
120C22×D97—21no—
121C22×D97—21no—
122C22×D97—42no—
123C22×D97—82no—
124C22×D97C2×D411yescharacteristic
125Q8×C97—21no—
126Q8×C97—21no—
127Q8×C97—21no—
128Q8×C97—21no—
129Q8×C97—21no—
130Q8×C97—21no—
131D4×C97—21no—
132D4×C97—21no—
133C2×C4×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97—21no—
136C2×C4×C97C2×D412yes—
137C2×C4×C97C2411yescharacteristic
138C22⋊D97—42no—
139C22⋊D97—82no—
140C2×Dic97—21no—
141C2×Dic97C2×D412yes—
142C2×Dic97C2×D411yescharacteristic
143D388—21no—
144D388—21no—
145D388—21no—
146D388—21no—
147D388—21no—
148D388—21no—
149D388—42no—
150D388—42no—
151D388—82no—
152C4×D97—42no—
153C4×D97—42no—
154C4×D97—42no—
155C4×D97C2×D412yes—
156C4×D97C2×D412yes—
157Dic194—21no—
158Dic194—21no—
159Dic194—42no—
160C8×C97C2×D412yes—
161C8×C97C2×D412yes—
162#11715895C2×D412yes—
163#11715895C2×D412yes—
164C22×C97—41no—
165C22×C97—42no—
166C22×C97C22×D411yescharacteristic
167C2×D97—22no—
168C2×D97—41no—
169C2×D97—41no—
170C2×D97—41no—
171C2×D97—42no—
172C2×D97—42no—
173C2×D97—42no—
174C2×D97—84no—
175C2×D97#12712yes—
176C4×C97—22no—
177C4×C97—41no—
178C4×C97—41no—
179C4×C97—41no—
180C4×C97C22×D411yescharacteristic
181C4×C97C22×D411yescharacteristic
182Dic97—22no—
183Dic97—41no—
184Dic97#12712yes—
185C2×C97—81no—
186C2×C97C2×#12711yescharacteristic
187C2×C97#57712yes—
188D97—22no—
189D97—81no—
190D97—81no—
191D97—81no—
192#3154—971nomaximal, Sylow
193C97#315411yescharacteristic, complemented, Sylow
194C2×#137—971no—
195C2×AGL(1,ℤ/8)—971no—
196C2×#135—971no—
197C22×SD16—971no—
198#536—971no—
199#535—971no—
200#497—971no—
201#497D9711yescharacteristic, complemented
202#496—971no—
203#496—971no—
204#495—971no—
205#495—971no—
206#492—971no—
207#492—971no—
208#405—971no—
209C2×#41—971no—
210C2×#41—971no—
211C22×Q8—971no—
212C22×D4—971no—
213#137—1944no—
214AGL(1,ℤ/8)—1944no—
215#135—1944no—
216C2×Q16—971no—
217C2×Q16C2×D9711yescharacteristic, complemented
218C2×Q16C2×D9711yescharacteristic, complemented
219C2×SD16—971no—
220C2×SD16—971no—
221C2×SD16—971no—
222C2×SD16—971no—
223C2×SD16—971no—
224C2×SD16—971no—
225C2×SD16—971no—
226C2×SD16—971no—
227C2×SD16—1944no—
228C2×SD16C2×D9711yescharacteristic
229C2×SD16C2×D9711yescharacteristic
230C2×D8—971no—
231C2×D8—971no—
232C2×D8—971no—
233C2×M16—971no—
234C22×C8—971no—
235C4⋊Q8—971no—
236C4⋊Q8C2×D9711yescharacteristic, complemented
237#127—971no—
238#127—971no—
239#124—971no—
240#124—971no—
241#124—971no—
242#124C2×D9711yescharacteristic
243#123—1942no—
244C22⋊Q8—1942no—
245#121—1942no—
246#120—1942no—
247#117—971no—
248#97—971no—
249#97C2×D9711yescharacteristic, complemented
250C4×C8—971no—
251C4×C8—971no—
252C24—1941no—
253#41—1942no—
254#41—1942no—
255#41—1942no—
256#41—1942no—
257C2×Q8—971no—
258C2×Q8—1941no—
259C2×Q8—1942no—
260C2×Q8—1942no—
261C2×Q8C22×D9711yescharacteristic
262C2×Q8C22×D9711yescharacteristic
263C2×Q8C22×D9711yescharacteristic
264C2×D4—971no—
265C2×D4—971no—
266C2×D4—971no—
267C2×D4—1941no—
268C2×D4—1941no—
269C2×D4—1942no—
270C2×D4—1942no—
271C2×D4—1942no—
272C2×D4—1941no—
273C2×D4—1942no—
274C2×D4C22×D9711yescharacteristic
275C22×C4—971no—
276C22×C4—1941no—
277C22×C4—1941no—
278C22×C4—1941no—
279Q16—22no—
280Q16—22no—
281Q16—1942no—
282SD16—22no—
283SD16—22no—
284SD16—1942no—
285SD16—1942no—
286SD16—1942no—
287SD16—1942no—
288SD16—1942no—
289SD16—1942no—
290SD16—1942no—
291SD16—1942no—
292D8—1942no—
293D8—1942no—
294D8—1942no—
295M16—974no—
296C2×C8—971no—
297C2×C8—971no—
298C2×C8—974no—
299C2×C8C22×D9711yescharacteristic
300C2×C8C22×D9711yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—972no—
303C4⋊C4—1942no—
304C22⋊C4—22no—
305C22⋊C4—972no—
306C22⋊C4—1942no—
307C22⋊C4—1942no—
308C22⋊C4—1942no—
309C42—971no—
310C42C22×D9711yescharacteristic
311C23—21no—
312C23—7762no—
313C23—971no—
314C23—3882no—
315C23—1941no—
316C23—1941no—
317C23—1941no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—3882no—
325Q8—1941no—
326Q8—1941no—
327D4—21no—
328D4—21no—
329D4—7762no—
330D4—7762no—
331D4—3882no—
332D4—3882no—
333D4—3882no—
334D4—1941no—
335D4—1941no—
336D4—1941no—
337D4—1941no—
338D4—1941no—
339D4—1941no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—971no—
344C2×C4—972no—
345C2×C4—972no—
346C2×C4—972no—
347C2×C4—3882no—
348C2×C4—3882no—
349C2×C4—3882no—
350C2×C4—1941no—
351C2×C4D4×D9712yes—
352C2×C4C23×D9711yescharacteristic
353C8—972no—
354C8—972no—
355C8D4×D9712yes—
356C8D4×D9712yes—
357C22—41no—
358C22—42no—
359C22—7764no—
360C22—972no—
361C22—3881no—
362C22—3881no—
363C22—3881no—
364C22—3882no—
365C22—3882no—
366C22—3882no—
367C22—1942no—
368C22C2×D4×D9711yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—972no—
374C4—3881no—
375C4—1942no—
376C4C2×D4×D9711yescharacteristic
377C4C2×D4×D9711yescharacteristic
378C2—81no—
379C2—7761no—
380C2—7761no—
381C2—7761no—
382C2—1942no—
383C2#127×D9711yescharacteristic
384C2#2026154412yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#497×C97C2
socleC22×C97C22×D4
hypercenter#497D97
hypocenterC97#3154
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#15690842C2semidirect
2. C2×#15690837C2semidirect
3. C2×#15690833C2semidirect
4. C2×SD16×D97C2semidirect
5. #20261310C2semidirect
6. #20261303C2semidirect
7. #20261128C2semidirect
8. #20261124C2semidirect
9. #20261121C2semidirect
10. #20261120C2semidirect
11. #20261117C2semidirect
12. #20261109C2semidirect
13. #20261106C2semidirect
14. #20260692C2semidirect
15. #497×C97C2semidirect
23. C2×#13095501C22semidirect
25. C2×#13095500C22semidirect
29. C2×#13095498C22semidirect
30. C2×#13095498C22semidirect
31. C2×#13095498C22semidirect
34. C2×D776C22semidirect
38. #15690808C22semidirect
42. #15690799C22semidirect
43. #15690798C22semidirect
49. #15690720C22semidirect
50. C4×#11715895C22semidirect
51. C8×Dic97C22semidirect
52. C2×Q16×C97C22semidirect
53. C2×Q16×C97C22semidirect
56. C4⋊Q8×C97C22semidirect
58. #97×C97C22semidirect
94. #13095494D4semidirect
97. C8×D97D4semidirect
193. C97#3154semidirect
201. #497D97semidirect
217. C2×Q16C2×D97semidirect
218. C2×Q16C2×D97semidirect
236. C4⋊Q8C2×D97semidirect
249. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908422
C2×#156908372
C2×#156908332
C2×SD16×D972
#202613102
#202613032
#202611282
#202611242
#202611212
#202611202
#202611172
#202611092
#202611062
#202606922
#497×C972
#315497