Finite Group Databasebeta

← #27153266

Subgroups — #27153266

Order 12416.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690838C211yesmaximal, characteristic, complemented
2C2×#15690832C211yesmaximal, characteristic, complemented
3#20261390C211yesmaximal, characteristic, complemented
4#20261381C211yesmaximal, characteristic, complemented
5#20261129C211yesmaximal, characteristic, complemented
6#20260993C211yesmaximal, characteristic, complemented
7#20260964C211yesmaximal, characteristic, complemented
8#20260951C211yesmaximal, characteristic, complemented
9#20260941C211yesmaximal, characteristic, complemented
10#20260899C211yesmaximal, characteristic, complemented
11#20260899C211yesmaximal, characteristic, complemented
12#20260888C211yesmaximal, characteristic, complemented
13#20260882C211yesmaximal, characteristic, complemented
14#20260848C211yesmaximal, characteristic, complemented
15#435×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C2×#13095511C2211yescharacteristic
18C2×#13095511C2211yescharacteristic, complemented
19C22×D388C2211yescharacteristic
20#15690838—24no—
21#15690832—24no—
22C2×Dic388C2211yescharacteristic, complemented
23C2×Dic388C2211yescharacteristic, complemented
24C2×#13095500C2211yescharacteristic
25C2×#13095500C2211yescharacteristic
26C2×#13095500C2211yescharacteristic
27C2×#13095500C2211yescharacteristic
28C2×#13095500C2211yescharacteristic, complemented
29C2×D776C2211yescharacteristic
30C2×D776C2211yescharacteristic
31#15690810C2211yescharacteristic, complemented
32#15690798C2211yescharacteristic
33#15690795—21no—
34#15690792—21no—
35#15690792—21no—
36#15690789C2211yescharacteristic
37#15690789C2211yescharacteristic
38#15690789C2211yescharacteristic, complemented
39#15690784C2211yescharacteristic
40#15690783—21no—
41#15690781—21no—
42#15690778—21no—
43C4×D388C2211yescharacteristic
44C4×D388C2211yescharacteristic, complemented
45C4⋊C4×D97—21no—
46C22⋊C4×D97—21no—
47#15690742C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690735C2211yescharacteristic, complemented
50#15690735C2211yescharacteristic, complemented
51#15690732C2211yescharacteristic
52#15690732C2211yescharacteristic
53#15690732C2211yescharacteristic, complemented
54C2×M16×C97C2211yescharacteristic
55C22×C8×C97C2211yescharacteristic, complemented
56C4×D4×C97C2211yescharacteristic
57C4⋊C8×C97C2211yescharacteristic, complemented
58C22⋊C8×C97C2211yescharacteristic
59C22⋊C8×C97C2211yescharacteristic, complemented
60#97×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62#13095511—22no—
63#13095511—22no—
64#13095511—22no—
65#13095511—41no—
66#13095511—41no—
67#13095511—41no—
68C2×Dic194C2311yescharacteristic
69C2×Dic194C2311yescharacteristic
70C2×Dic194C2311yescharacteristic
71C2×D388—22no—
72C2×D388—21no—
73C2×D388—41no—
74C2×D388C2311yescharacteristic
75C2×D388C2311yescharacteristic
76C2×D388C2311yescharacteristic
77C2×C22⋊D97—21no—
78C2×C22⋊D97—21no—
79C2×C22⋊D97—21no—
80C2×C22⋊D97—21no—
81C2×C4×D97—21no—
82C2×C4×D97—21no—
83C2×C4×D97—21no—
84C2×C4×D97—21no—
85C2×C4×D97—21no—
86Dic388—22no—
87Dic388—41no—
88#13095500—22no—
89#13095500—22no—
90#13095500—22no—
91#13095500—22no—
92#13095500—41no—
93D776—22no—
94D776D414yescomplemented
95C4⋊Dic97—21no—
96C4⋊Dic97—21no—
97C4⋊Dic97C2311yescharacteristic
98C4⋊Dic97C2311yescharacteristic
99#13095490—21no—
100#13095490—21no—
101#13095490—21no—
102C22⋊Dic97—21no—
103C22⋊Dic97—21no—
104C22⋊Dic97—21no—
105#13095488—21no—
106#13095488—21no—
107#13095488—21no—
108#13095488—21no—
109#13095488—21no—
110C2×D4×C97D411yescharacteristic
111C22×C4×C97C2311yescharacteristic
112C22×C4×C97C2311yescharacteristic
113M16×C97—22no—
114C2×C8×C97—22no—
115C2×C8×C97C2311yescharacteristic
116C2×C8×C97C2311yescharacteristic
117C2×C8×C97C2311yescharacteristic
118C2×C8×C97C2311yescharacteristic
119C4⋊C4×C97D411yescharacteristic
120C22⋊C4×C97D411yescharacteristic
121C22⋊C4×C97D411yescharacteristic
122C42×C97C2311yescharacteristic
123C23×C97C2×D411yescharacteristic
124C23×C97C2×D411yescharacteristic
125C22×D97—21no—
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—42no—
129C22×D97—42no—
130C22×D97—81no—
131D4×C97—41no—
132C2×C4×C97—21no—
133C2×C4×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97—21no—
136C2×C4×C97C2×D411yescharacteristic
137C2×C4×C97C2×D411yescharacteristic
138C2×C4×C97C2×D411yescharacteristic
139C2×C4×C97C2×D411yescharacteristic
140C2×C4×C97C2411yescharacteristic
141C22⋊D97—42no—
142C22⋊D97—42no—
143C22⋊D97—42no—
144C22⋊D97—81no—
145C2×Dic97—21no—
146C2×Dic97—21no—
147C2×Dic97—21no—
148C2×Dic97—21no—
149C2×Dic97—21no—
150D388—21no—
151D388—21no—
152D388—21no—
153D388—21no—
154D388—42no—
155D388C2×D412yes—
156D388C2×D412yes—
157C4×D97—42no—
158C4×D97—42no—
159C4×D97—42no—
160C4×D97—42no—
161C4×D97—81no—
162Dic194—21no—
163Dic194—21no—
164Dic194—21no—
165Dic194—21no—
166Dic194—21no—
167Dic194—21no—
168C8×C97—21no—
169C8×C97—21no—
170C8×C97—21no—
171C8×C97C2×D412yes—
172C22×C97—21no—
173C22×C97—21no—
174C22×C97—21no—
175C22×C97—21no—
176C22×C97—21no—
177C22×C97—21no—
178C22×C97C22×D411yescharacteristic
179C2×D97—22no—
180C2×D97—22no—
181C2×D97—41no—
182C2×D97—41no—
183C2×D97—41no—
184C2×D97—41no—
185C2×D97—41no—
186C2×D97—41no—
187C2×D97—41no—
188C2×D97—84no—
189C4×C97—21no—
190C4×C97—21no—
191C4×C97—21no—
192C4×C97—21no—
193C4×C97C22×D411yescharacteristic
194C4×C97H(5,𝔽2)11yescharacteristic
195Dic97—41no—
196Dic97—41no—
197Dic97—41no—
198Dic97—41no—
199Dic97—41no—
200C2×C97—41no—
201C2×C97—41no—
202C2×C97D4211yescharacteristic
203C2×C97#57511yescharacteristic
204C2×C97#57711yescharacteristic
205D97—42no—
206D97—42no—
207D97—81no—
208#3267—971nomaximal, Sylow
209C97#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—971no—
211C2×#135—971no—
212#547—971no—
213#546—971no—
214#497—971no—
215#470—971no—
216#465—971no—
217#463—971no—
218#461—971no—
219#452—971no—
220#452—971no—
221#450—971no—
222#449—971no—
223#442—971no—
224#435D9711yescharacteristic, complemented
225C2×#41—971no—
226C2×#41—971no—
227C2×#41—971no—
228C22×D4—971no—
229AGL(1,ℤ/8)—1944no—
230#135—1944no—
231C2×Q16—971no—
232C2×Q16—971no—
233C2×SD16—971no—
234C2×SD16—971no—
235C2×SD16—971no—
236C2×SD16—971no—
237C2×SD16—971no—
238C2×D8—971no—
239C2×D8—971no—
240C2×M16C2×D9711yescharacteristic
241C22×C8C2×D9711yescharacteristic, complemented
242C4⋊Q8—971no—
243#124—971no—
244#123—1941no—
245#123—1941no—
246#123—1941no—
247C22⋊Q8—971no—
248C22⋊Q8—971no—
249C22⋊Q8—971no—
250#121—971no—
251#121—1941no—
252#121—1941no—
253#120—1941no—
254C4×D4—971no—
255C4×D4—971no—
256C4×D4C2×D9711yescharacteristic
257C2×C4⋊C4—1941no—
258C2×C22⋊C4—1941no—
259#106—971no—
260C4⋊C8C2×D9711yescharacteristic, complemented
261#103—971no—
262#103—971no—
263#103—971no—
264#102—971no—
265#102—971no—
266#102—971no—
267C22⋊C8C2×D9711yescharacteristic
268C22⋊C8C2×D9711yescharacteristic, complemented
269#97C2×D9711yescharacteristic, complemented
270C24—1941no—
271#41—3881no—
272#41—3881no—
273#41—3881no—
274#41—1942no—
275#41—1942no—
276#41—1942no—
277C2×Q8—971no—
278C2×Q8—971no—
279C2×Q8—971no—
280C2×D4—971no—
281C2×D4—971no—
282C2×D4—971no—
283C2×D4—3881no—
284C2×D4—1942no—
285C2×D4—1941no—
286C2×D4—1941no—
287C2×D4—1941no—
288C2×D4—1941no—
289C2×D4—1941no—
290C2×D4D38811yescharacteristic
291C22×C4—1941no—
292C22×C4—1941no—
293C22×C4—1941no—
294C22×C4—1941no—
295C22×C4—1941no—
296C22×C4C22×D9711yescharacteristic
297C22×C4C22×D9711yescharacteristic
298Q16—3881no—
299Q16—1942no—
300SD16—3881no—
301SD16—1942no—
302SD16—1942no—
303SD16—1942no—
304SD16—1942no—
305D8—974no—
306D8—1942no—
307M16—22no—
308C2×C8—22no—
309C2×C8C22×D9711yescharacteristic
310C2×C8C22×D9711yescharacteristic
311C2×C8C22×D9711yescharacteristic
312C2×C8C22×D9711yescharacteristic
313C4⋊C4—971no—
314C4⋊C4—971no—
315C4⋊C4—1941no—
316C4⋊C4—1941no—
317C4⋊C4—1941no—
318C4⋊C4—1941no—
319C4⋊C4—1941no—
320C4⋊C4D38811yescharacteristic
321C22⋊C4—1941no—
322C22⋊C4—1941no—
323C22⋊C4—1941no—
324C22⋊C4—1941no—
325C22⋊C4—1941no—
326C22⋊C4—1941no—
327C22⋊C4—1941no—
328C22⋊C4—1941no—
329C22⋊C4D38811yescharacteristic
330C22⋊C4D38811yescharacteristic
331C42C22×D9711yescharacteristic
332C23—7761no—
333C23—3882no—
334C23—3882no—
335C23—1941no—
336C23—1941no—
337C23—1941no—
338C23C2×D38811yescharacteristic
339C23C2×D38811yescharacteristic
340Q8—1941no—
341Q8—1941no—
342Q8—1941no—
343Q8—1941no—
344Q8—1941no—
345Q8—1941no—
346D4—41no—
347D4—7761no—
348D4—972no—
349D4—972no—
350D4—3882no—
351D4—3882no—
352D4—3882no—
353D4—3882no—
354D4—1941no—
355D4—1941no—
356D4—1941no—
357D4—1941no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—7761no—
363C2×C4—3882no—
364C2×C4—3882no—
365C2×C4—3882no—
366C2×C4—3882no—
367C2×C4—1941no—
368C2×C4—1941no—
369C2×C4—1941no—
370C2×C4—1941no—
371C2×C4—1941no—
372C2×C4C2×D38811yescharacteristic
373C2×C4C2×D38811yescharacteristic
374C2×C4C2×D38811yescharacteristic
375C2×C4C2×D38811yescharacteristic
376C2×C4C23×D9711yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8D4×D9712yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—7764no—
388C22—3881no—
389C22—3881no—
390C22—3881no—
391C22—3881no—
392C22—3881no—
393C22—3881no—
394C22—3881no—
395C22—1942no—
396C22—1942no—
397C22C22×D38811yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—3881no—
403C4—3881no—
404C4—3881no—
405C4—3881no—
406C4—3881no—
407C4C2×D4×D9711yescharacteristic
408C4#1569085611yescharacteristic
409C2—41no—
410C2—41no—
411C2—7761no—
412C2—3882no—
413C2—3882no—
414C2D4×D38811yescharacteristic
415C2#2026153511yescharacteristic
416C2#2026154711yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#435×C97C2
socleC22×C97C22×D4
hypercenter#435D97
hypocenterC97#3267
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#15690838C2semidirect
2. C2×#15690832C2semidirect
3. #20261390C2semidirect
4. #20261381C2semidirect
5. #20261129C2semidirect
6. #20260993C2semidirect
7. #20260964C2semidirect
8. #20260951C2semidirect
9. #20260941C2semidirect
10. #20260899C2semidirect
11. #20260899C2semidirect
12. #20260888C2semidirect
13. #20260882C2semidirect
14. #20260848C2semidirect
15. #435×C97C2semidirect
18. C2×#13095511C22semidirect
22. C2×Dic388C22semidirect
23. C2×Dic388C22semidirect
28. C2×#13095500C22semidirect
31. #15690810C22semidirect
38. #15690789C22semidirect
44. C4×D388C22semidirect
47. #15690742C22semidirect
48. #15690735C22semidirect
49. #15690735C22semidirect
50. #15690735C22semidirect
53. #15690732C22semidirect
55. C22×C8×C97C22semidirect
57. C4⋊C8×C97C22semidirect
59. C22⋊C8×C97C22semidirect
60. #97×C97C22semidirect
94. D776D4semidirect
209. C97#3267semidirect
224. #435D97semidirect
241. C22×C8C2×D97semidirect
260. C4⋊C8C2×D97semidirect
268. C22⋊C8C2×D97semidirect
269. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908382
C2×#156908322
#202613902
#202613812
#202611292
#202609932
#202609642
#202609512
#202609412
#202608992
#202608882
#202608822
#202608482
#435×C972
#326797