Finite Group Databasebeta

← #27153221

Subgroups — #27153221

Order 12416.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690838C211yesmaximal, characteristic, complemented
2C22×D776C211yesmaximal, characteristic, complemented
3#20261381C211yesmaximal, characteristic, complemented
4D4×D388C211yesmaximal, characteristic, complemented
5#20261124C211yesmaximal, characteristic, complemented
6#20260988C211yesmaximal, characteristic, complemented
7#20260964C211yesmaximal, characteristic, complemented
8#20260951C211yesmaximal, characteristic, complemented
9#20260931C211yesmaximal, characteristic, complemented
10#20260888C211yesmaximal, characteristic, complemented
11#20260882C211yesmaximal, characteristic, complemented
12#20260871C211yesmaximal, characteristic, complemented
13#20260871C211yesmaximal, characteristic, complemented
14#20260848C211yesmaximal, characteristic, complemented
15#435×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic, complemented
17C22×D388C2211yescharacteristic
18C22×D388C2211yescharacteristic, complemented
19C22×D388C2211yescharacteristic, complemented
20C2×D4×D97—21no—
21#15690838—24no—
22C2×#13095500C2211yescharacteristic, complemented
23C2×#13095500C2211yescharacteristic, complemented
24C2×#13095500C2211yescharacteristic, complemented
25C2×D776—24no—
26C2×D776C2211yescharacteristic
27C2×D776C2211yescharacteristic
28C2×D776C2211yescharacteristic, complemented
29C2×D776C2211yescharacteristic, complemented
30C2×D776C2211yescharacteristic, complemented
31C2×D776C2211yescharacteristic, complemented
32#15690807C2211yescharacteristic, complemented
33#15690798C2211yescharacteristic, complemented
34#15690795—21no—
35#15690789C2211yescharacteristic, complemented
36#15690785—21no—
37#15690784C2211yescharacteristic
38#15690784C2211yescharacteristic, complemented
39#15690784C2211yescharacteristic, complemented
40#15690783—21no—
41#15690780—21no—
42#15690780—21no—
43#15690778—21no—
44C4×D388C2211yescharacteristic, complemented
45C4×D388C2211yescharacteristic, complemented
46C22⋊C4×D97—21no—
47#15690742C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690732C2211yescharacteristic, complemented
50#15690732C2211yescharacteristic, complemented
51#15690732C2211yescharacteristic, complemented
52#15690732C2211yescharacteristic, complemented
53#15690732C2211yescharacteristic, complemented
54C2×M16×C97C2211yescharacteristic, complemented
55C22×C8×C97C2211yescharacteristic, complemented
56C4×D4×C97C2211yescharacteristic
57C4⋊C8×C97C2211yescharacteristic, complemented
58C22⋊C8×C97C2211yescharacteristic, complemented
59C22⋊C8×C97C2211yescharacteristic, complemented
60#97×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97—21no—
63C23×D97—21no—
64#13095511—22no—
65#13095511—41no—
66C2×Dic194C2311yescharacteristic
67C2×D388—22no—
68C2×D388—22no—
69C2×D388—22no—
70C2×D388—21no—
71C2×D388—21no—
72C2×D388—21no—
73C2×D388—21no—
74C2×D388—21no—
75C2×D388—41no—
76C2×D388—41no—
77C2×D388—41no—
78C2×D388C2311yescharacteristic
79C2×D388C2311yescharacteristic
80C2×D388C2311yescharacteristic
81C2×D388C2311yescharacteristic
82C2×D388C2311yescharacteristic
83C2×C22⋊D97—21no—
84C2×C22⋊D97—21no—
85C2×C22⋊D97—21no—
86C2×C22⋊D97—21no—
87D4×D97—44no—
88C2×C4×D97—21no—
89C2×C4×D97—21no—
90C2×C4×D97—21no—
91#13095500—22no—
92#13095500—22no—
93#13095500—41no—
94D776—22no—
95D776—22no—
96D776—22no—
97D776—22no—
98D776—41no—
99D776D414yescomplemented
100C4⋊Dic97C2311yescharacteristic
101C4⋊Dic97C2311yescharacteristic
102#13095490—21no—
103C22⋊Dic97—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488—21no—
107#13095488—21no—
108#13095488—21no—
109#13095488—21no—
110#13095488—21no—
111C2×D4×C97D411yescharacteristic
112C22×C4×C97C2311yescharacteristic
113C22×C4×C97C2311yescharacteristic
114M16×C97—22no—
115C2×C8×C97—22no—
116C2×C8×C97C2311yescharacteristic
117C2×C8×C97C2311yescharacteristic
118C2×C8×C97C2311yescharacteristic
119C2×C8×C97C2311yescharacteristic
120C4⋊C4×C97D411yescharacteristic
121C22⋊C4×C97D411yescharacteristic
122C22⋊C4×C97D411yescharacteristic
123C42×C97C2311yescharacteristic
124C23×C97C2×D411yescharacteristic
125C23×C97C2×D411yescharacteristic
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—21no—
130C22×D97—21no—
131C22×D97—42no—
132C22×D97—42no—
133C22×D97—42no—
134C22×D97—42no—
135C22×D97—42no—
136C22×D97—42no—
137C22×D97—81no—
138C22×D97—81no—
139C22×D97—81no—
140D4×C97—22no—
141C2×C4×C97—21no—
142C2×C4×C97—21no—
143C2×C4×C97—21no—
144C2×C4×C97—21no—
145C2×C4×C97C2×D411yescharacteristic
146C2×C4×C97C2×D411yescharacteristic
147C2×C4×C97C2×D411yescharacteristic
148C2×C4×C97C2×D411yescharacteristic
149C2×C4×C97C2411yescharacteristic
150C22⋊D97—42no—
151C22⋊D97—42no—
152C22⋊D97—42no—
153C22⋊D97—81no—
154C2×Dic97—21no—
155C2×Dic97—21no—
156C2×Dic97—21no—
157D388—21no—
158D388—21no—
159D388—21no—
160D388—21no—
161D388—21no—
162D388—21no—
163D388—21no—
164D388—21no—
165D388—42no—
166D388—42no—
167D388—42no—
168D388—42no—
169D388—81no—
170D388C2×D412yes—
171D388C2×D412yescomplemented
172C4×D97—42no—
173C4×D97—42no—
174C4×D97—81no—
175Dic194—21no—
176Dic194—21no—
177C8×C97—21no—
178C8×C97—21no—
179C8×C97—21no—
180C8×C97C2×D412yescomplemented
181C22×C97—21no—
182C22×C97—21no—
183C22×C97—21no—
184C22×C97—21no—
185C22×C97—21no—
186C22×C97AGL(1,ℤ/8)12yescomplemented
187C22×C97C22×D411yescharacteristic
188C2×D97—22no—
189C2×D97—22no—
190C2×D97—41no—
191C2×D97—41no—
192C2×D97—41no—
193C2×D97—41no—
194C2×D97—41no—
195C2×D97—41no—
196C2×D97—41no—
197C2×D97—41no—
198C2×D97—41no—
199C2×D97—41no—
200C2×D97—41no—
201C2×D97—41no—
202C2×D97—41no—
203C2×D97—84no—
204C2×D97—84no—
205C2×D97—84no—
206C4×C97—21no—
207C4×C97—21no—
208C4×C97—21no—
209C4×C97—21no—
210C4×C97C22×D411yescharacteristic
211C4×C97H(5,𝔽2)11yescharacteristic
212Dic97—41no—
213Dic97—41no—
214Dic97—41no—
215C2×C97—22no—
216C2×C97—41no—
217C2×C97D4211yescharacteristic
218C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C97#57611yescharacteristic
220D97—42no—
221D97—42no—
222D97—81no—
223D97—81no—
224D97—81no—
225#3264—971nomaximal, Sylow
226C97#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—971no—
228C22×D8—971no—
229#546—971no—
230D42—971no—
231#496—971no—
232#469—971no—
233#465—971no—
234#463—971no—
235#459—971no—
236#450—971no—
237#449—971no—
238#447—971no—
239#447—971no—
240#442—971no—
241#435D9711yescharacteristic, complemented
242C2×#41—971no—
243C22×D4—971no—
244C22×D4—971no—
245C22×D4—971no—
246C22×D4—1941no—
247AGL(1,ℤ/8)—1944no—
248C2×SD16—971no—
249C2×SD16—971no—
250C2×SD16—971no—
251C2×D8—971no—
252C2×D8—971no—
253C2×D8—971no—
254C2×D8—971no—
255C2×D8—971no—
256C2×D8—971no—
257C2×D8—1944no—
258C2×M16C2×D9711yescharacteristic, complemented
259C22×C8C2×D9711yescharacteristic, complemented
260#127—971no—
261#124—971no—
262#123—1941no—
263C22⋊Q8—971no—
264#121—971no—
265#121—971no—
266#121—971no—
267#121—1941no—
268#121—1941no—
269#120—1941no—
270#120—1941no—
271#120—1941no—
272C4×D4—971no—
273C4×D4—971no—
274C4×D4C2×D9711yescharacteristic
275C2×C22⋊C4—1941no—
276#106—971no—
277C4⋊C8C2×D9711yescharacteristic, complemented
278#103—971no—
279#102—971no—
280#102—971no—
281#102—971no—
282#102—971no—
283#102—971no—
284C22⋊C8C2×D9711yescharacteristic, complemented
285C22⋊C8C2×D9711yescharacteristic, complemented
286#97C2×D9711yescharacteristic, complemented
287C24—1941no—
288C24—1941no—
289C24—1941no—
290#41—3881no—
291#41—1942no—
292C2×Q8—971no—
293C2×D4—971no—
294C2×D4—971no—
295C2×D4—971no—
296C2×D4—971no—
297C2×D4—971no—
298C2×D4—3881no—
299C2×D4—3881no—
300C2×D4—3881no—
301C2×D4—3884no—
302C2×D4—1942no—
303C2×D4—1942no—
304C2×D4—1942no—
305C2×D4—1941no—
306C2×D4—1941no—
307C2×D4—1941no—
308C2×D4—1941no—
309C2×D4—1941no—
310C2×D4—1941no—
311C2×D4—1941no—
312C2×D4—1941no—
313C2×D4—1941no—
314C2×D4D38811yescharacteristic
315C22×C4—1941no—
316C22×C4—1941no—
317C22×C4—1941no—
318C22×C4C22×D9711yescharacteristic
319C22×C4C22×D9711yescharacteristic
320SD16—3881no—
321SD16—1942no—
322SD16—1942no—
323D8—974no—
324D8—3881no—
325D8—1942no—
326D8—1942no—
327D8—1942no—
328D8—1942no—
329M16—22no—
330C2×C8—22no—
331C2×C8C22×D9711yescharacteristic
332C2×C8C22×D9711yescharacteristic
333C2×C8C22×D9711yescharacteristic
334C2×C8C22×D9711yescharacteristic
335C4⋊C4—971no—
336C4⋊C4—971no—
337C4⋊C4—1941no—
338C4⋊C4D38811yescharacteristic
339C22⋊C4—1941no—
340C22⋊C4—1941no—
341C22⋊C4—1941no—
342C22⋊C4—1941no—
343C22⋊C4—1941no—
344C22⋊C4—1941no—
345C22⋊C4—1941no—
346C22⋊C4—1941no—
347C22⋊C4D38811yescharacteristic
348C22⋊C4D38811yescharacteristic
349C42C22×D9711yescharacteristic
350C23—7761no—
351C23—7761no—
352C23—7761no—
353C23—3882no—
354C23—3882no—
355C23—3882no—
356C23—3882no—
357C23—3882no—
358C23—3882no—
359C23—1941no—
360C23—1941no—
361C23—1941no—
362C23—1941no—
363C23—1941no—
364C23C2×D38811yescharacteristic
365C23C2×D38811yescharacteristic
366Q8—1941no—
367Q8—1941no—
368D4—22no—
369D4—7761no—
370D4—7761no—
371D4—972no—
372D4—972no—
373D4—3882no—
374D4—3882no—
375D4—3882no—
376D4—3882no—
377D4—3882no—
378D4—3882no—
379D4—3882no—
380D4—1941no—
381D4—1941no—
382D4—1941no—
383D4—1941no—
384D4—1941no—
385D4—1941no—
386D4—1941no—
387D4—1941no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—7761no—
393C2×C4—3882no—
394C2×C4—3882no—
395C2×C4—1941no—
396C2×C4—1941no—
397C2×C4—1941no—
398C2×C4C2×D38811yescharacteristic
399C2×C4C2×D38811yescharacteristic
400C2×C4C2×D38811yescharacteristic
401C2×C4C2×D38811yescharacteristic
402C2×C4C23×D9711yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8D4×D9712yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—7764no—
413C22—7764no—
414C22—7764no—
415C22—3881no—
416C22—3881no—
417C22—3881no—
418C22—3881no—
419C22—3881no—
420C22—3881no—
421C22—3881no—
422C22—3881no—
423C22—3881no—
424C22—3881no—
425C22—3881no—
426C22—3881no—
427C22—3881no—
428C22—1942no—
429C22—1942no—
430C22#1569083812yescomplemented
431C22C22×D38811yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—3881no—
437C4—3881no—
438C4—3881no—
439C4C2×D4×D9711yescharacteristic
440C4#1569085611yescharacteristic
441C2—22no—
442C2—41no—
443C2—7761no—
444C2—7761no—
445C2—7761no—
446C2—3882no—
447C2—3882no—
448C2D4×D38811yescharacteristic
449C2C2×#1569083811yescharacteristic
450C2#2026154011yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#435×C97C2
socleC22×C97C22×D4
hypercenter#435D97
hypocenterC97#3264
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. C22×D776C2semidirect
3. #20261381C2semidirect
4. D4×D388C2semidirect
5. #20261124C2semidirect
6. #20260988C2semidirect
7. #20260964C2semidirect
8. #20260951C2semidirect
9. #20260931C2semidirect
10. #20260888C2semidirect
11. #20260882C2semidirect
12. #20260871C2semidirect
13. #20260871C2semidirect
14. #20260848C2semidirect
15. #435×C97C2semidirect
16. C2×#13095511C22semidirect
18. C22×D388C22semidirect
19. C22×D388C22semidirect
22. C2×#13095500C22semidirect
23. C2×#13095500C22semidirect
24. C2×#13095500C22semidirect
28. C2×D776C22semidirect
29. C2×D776C22semidirect
30. C2×D776C22semidirect
31. C2×D776C22semidirect
32. #15690807C22semidirect
33. #15690798C22semidirect
35. #15690789C22semidirect
38. #15690784C22semidirect
39. #15690784C22semidirect
44. C4×D388C22semidirect
45. C4×D388C22semidirect
47. #15690742C22semidirect
48. #15690735C22semidirect
49. #15690732C22semidirect
50. #15690732C22semidirect
51. #15690732C22semidirect
52. #15690732C22semidirect
53. #15690732C22semidirect
54. C2×M16×C97C22semidirect
55. C22×C8×C97C22semidirect
57. C4⋊C8×C97C22semidirect
58. C22⋊C8×C97C22semidirect
59. C22⋊C8×C97C22semidirect
60. #97×C97C22semidirect
99. D776D4semidirect
171. D388C2×D4semidirect
180. C8×C97C2×D4semidirect
186. C22×C97AGL(1,ℤ/8)semidirect
226. C97#3264semidirect
241. #435D97semidirect
258. C2×M16C2×D97semidirect
259. C22×C8C2×D97semidirect
277. C4⋊C8C2×D97semidirect
284. C22⋊C8C2×D97semidirect
285. C22⋊C8C2×D97semidirect
286. #97C2×D97semidirect
406. C8D4×D97semidirect
430. C22#15690838semidirect

Maximal subgroups

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

subgroupindex
C2×#156908382
C22×D7762
#202613812
D4×D3882
#202611242
#202609882
#202609642
#202609512
#202609312
#202608882
#202608822
#202608712
#202608482
#435×C972
#326497