Finite Group Databasebeta

← #27153219

Subgroups — #27153219

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×#15690835C211yesmaximal, characteristic, complemented
2C2×D8×D97C211yesmaximal, characteristic, complemented
3#20261373C211yesmaximal, characteristic, complemented
4#20261369C211yesmaximal, characteristic, complemented
5#20261121C211yesmaximal, characteristic, complemented
6#20260986C211yesmaximal, characteristic, complemented
7#20260967C211yesmaximal, characteristic, complemented
8#20260953C211yesmaximal, characteristic, complemented
9#20260928C211yesmaximal, characteristic, complemented
10#20260884C211yesmaximal, characteristic, complemented
11#20260871C211yesmaximal, characteristic, complemented
12#20260870C211yesmaximal, characteristic, complemented
13#20260844C211yesmaximal, characteristic, complemented
14#20260809C211yesmaximal, characteristic, complemented
15#449×C97C211yesmaximal, characteristic, complemented
16C22×D388C2211yescharacteristic, complemented
17C2×D4×D97—21no—
18C2×D4×D97C2211yescharacteristic
19C2×D4×D97C2211yescharacteristic, complemented
20#15690835—24no—
21C2×#13095498C2211yescharacteristic, complemented
22C2×#13095498C2211yescharacteristic, complemented
23C2×D776C2211yescharacteristic
24C2×D776C2211yescharacteristic, complemented
25C2×#13095496C2211yescharacteristic
26C2×#13095496C2211yescharacteristic, complemented
27C2×#13095496C2211yescharacteristic, complemented
28D8×D97—24no—
29C2×#13095495C2211yescharacteristic, complemented
30C2×C8×D97C2211yescharacteristic, complemented
31#15690806C2211yescharacteristic, complemented
32#15690799C2211yescharacteristic, complemented
33#15690793—21no—
34#15690786C2211yescharacteristic, complemented
35#15690785C2211yescharacteristic
36#15690784C2211yescharacteristic, complemented
37#15690783—21no—
38#15690782—21no—
39#15690780—21no—
40#15690778—21no—
41#15690778—21no—
42D4×Dic97C2211yescharacteristic, complemented
43#15690772C2211yescharacteristic, complemented
44C4×C22⋊D97C2211yescharacteristic
45C2×#13095488—21no—
46#15690741C2211yescharacteristic, complemented
47#15690739C2211yescharacteristic, complemented
48#15690732C2211yescharacteristic, complemented
49#15690732C2211yescharacteristic, complemented
50#15690731C2211yescharacteristic, complemented
51#15690730C2211yescharacteristic, complemented
52#15690722C2211yescharacteristic, complemented
53#15690720C2211yescharacteristic, complemented
54C2×#41×C97C2211yescharacteristic, complemented
55C2×SD16×C97C2211yescharacteristic, complemented
56C2×D8×C97C2211yescharacteristic, complemented
57#121×C97C2211yescharacteristic, complemented
58#103×C97C2211yescharacteristic, complemented
59#102×C97C2211yescharacteristic, complemented
60C22⋊C8×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97—21no—
63C23×D97—21no—
64C2×D388—22no—
65C2×D388—21no—
66C2×D388—21no—
67C2×D388—41no—
68C2×D388C2311yescharacteristic
69C2×D388C2311yescharacteristic
70C2×D388C2311yescharacteristic
71C2×C22⋊D97—21no—
72C2×C22⋊D97—21no—
73C2×C22⋊D97—21no—
74C2×C22⋊D97—21no—
75C2×C22⋊D97—21no—
76C2×C22⋊D97D411yescharacteristic
77D4×D97—22no—
78D4×D97—22no—
79D4×D97—41no—
80D4×D97—41no—
81D4×D97—44no—
82C22×Dic97—21no—
83C2×C4×D97—21no—
84C2×C4×D97C2311yescharacteristic
85#13095498—22no—
86#13095498—22no—
87D776—22no—
88D776—41no—
89#13095496—22no—
90#13095496—22no—
91#13095496D414yescomplemented
92#13095495—22no—
93C2×#11715895C2311yescharacteristic
94C2×#11715895C2311yescharacteristic
95C8×D97—22no—
96C4⋊Dic97C2311yescharacteristic
97#13095490—21no—
98#13095490D411yescharacteristic
99C22⋊Dic97—21no—
100C22⋊Dic97D411yescharacteristic
101#13095488—21no—
102#13095488—21no—
103#13095488—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488—21no—
107#13095488D411yescharacteristic
108C4×Dic97C2311yescharacteristic
109#41×C97—22no—
110#41×C97—41no—
111C2×Q8×C97C2311yescharacteristic
112C2×D4×C97—21no—
113C2×D4×C97—21no—
114C2×D4×C97C2311yescharacteristic
115C2×D4×C97C2311yescharacteristic
116C22×C4×C97—21no—
117C22×C4×C97C2311yescharacteristic
118SD16×C97—41no—
119D8×C97—22no—
120C2×C8×C97C2311yescharacteristic
121C2×C8×C97C2311yescharacteristic
122C4⋊C4×C97C2311yescharacteristic
123C22⋊C4×C97—21no—
124C23×C97—21no—
125C23×C97—21no—
126C23×C97C2×D411yescharacteristic
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—21no—
130C22×D97—42no—
131C22×D97—42no—
132C22×D97—42no—
133C22×D97—42no—
134C22×D97—42no—
135C22×D97—42no—
136C22×D97—81no—
137C22×D97—81no—
138C22×D97—81no—
139C22×D97C2×D411yescharacteristic
140Q8×C97—21no—
141Q8×C97—21no—
142D4×C97—21no—
143D4×C97—21no—
144D4×C97—21no—
145D4×C97—42no—
146D4×C97—42no—
147D4×C97C2×D412yescomplemented
148C2×C4×C97—21no—
149C2×C4×C97—21no—
150C2×C4×C97—21no—
151C2×C4×C97—21no—
152C2×C4×C97—42no—
153C2×C4×C97C2×D411yescharacteristic
154C2×C4×C97C2411yescharacteristic
155C22⋊D97—22no—
156C22⋊D97—42no—
157C22⋊D97—42no—
158C22⋊D97—42no—
159C22⋊D97—81no—
160C22⋊D97—81no—
161C2×Dic97—21no—
162C2×Dic97—81no—
163C2×Dic97C2×D411yescharacteristic
164C2×Dic97C2×D411yescharacteristic
165C2×Dic97C2×D411yescharacteristic
166D388—21no—
167D388—21no—
168D388—21no—
169D388—21no—
170D388—21no—
171D388—42no—
172D388—42no—
173D388C2×D412yes—
174C4×D97—21no—
175C4×D97—21no—
176C4×D97—42no—
177C8×C97—21no—
178C8×C97—21no—
179#11715895—21no—
180#11715895C2×D412yescomplemented
181C22×C97—22no—
182C22×C97—21no—
183C22×C97—21no—
184C22×C97—21no—
185C22×C97—41no—
186C22×C97—41no—
187C22×C97—41no—
188C22×C97—41no—
189C22×C97—41no—
190C22×C97C22×D411yescharacteristic
191C2×D97—21no—
192C2×D97—21no—
193C2×D97—22no—
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—84no—
203C2×D97—84no—
204C2×D97—84no—
205C2×D97AGL(1,ℤ/8)12yescomplemented
206C4×C97—21no—
207C4×C97—41no—
208C4×C97—41no—
209C4×C97C22×D411yescharacteristic
210C4×C97H(5,𝔽2)11yescharacteristic
211Dic97—21no—
212Dic97—21no—
213Dic97—21no—
214Dic97—41no—
215C2×C97—41no—
216C2×C97—42no—
217C2×C97—81no—
218C2×C97D4211yescharacteristic
219C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C97#57611yescharacteristic
221D97—22no—
222D97—42no—
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#449D9711yescharacteristic, complemented
238#447—971no—
239#447—971no—
240#442—971no—
241#435—971no—
242C2×#41C2×D9711yescharacteristic, complemented
243C22×D4—971no—
244C22×D4—971no—
245C22×D4—971no—
246C22×D4—1941no—
247AGL(1,ℤ/8)—1944no—
248C2×SD16—971no—
249C2×SD16—971no—
250C2×SD16C2×D9711yescharacteristic, complemented
251C2×D8—971no—
252C2×D8—971no—
253C2×D8—971no—
254C2×D8—971no—
255C2×D8—971no—
256C2×D8—1944no—
257C2×D8C2×D9711yescharacteristic, complemented
258C2×M16—971no—
259C22×C8—971no—
260#127—971no—
261#124—971no—
262#123—1941no—
263C22⋊Q8—971no—
264#121—971no—
265#121—971no—
266#121—1941no—
267#121—1941no—
268#121C2×D9711yescharacteristic, complemented
269#120—1941no—
270#120—1941no—
271#120—1941no—
272C4×D4—971no—
273C4×D4—971no—
274C4×D4—971no—
275C2×C22⋊C4—1941no—
276#106—971no—
277C4⋊C8—971no—
278#103C2×D9711yescharacteristic, complemented
279#102—971no—
280#102—971no—
281#102—971no—
282#102—971no—
283#102C2×D9711yescharacteristic, complemented
284C22⋊C8—971no—
285C22⋊C8C2×D9711yescharacteristic, complemented
286#97—971no—
287C24—1941no—
288C24—1941no—
289C24—1941no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D9711yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—971no—
296C2×D4—971no—
297C2×D4—971no—
298C2×D4—971no—
299C2×D4—3881no—
300C2×D4—3884no—
301C2×D4—3881no—
302C2×D4—3881no—
303C2×D4—1942no—
304C2×D4—1941no—
305C2×D4—1941no—
306C2×D4—1941no—
307C2×D4—1941no—
308C2×D4—1941no—
309C2×D4—1942no—
310C2×D4—1941no—
311C2×D4—1942no—
312C2×D4—1941no—
313C2×D4C22×D9711yescharacteristic
314C2×D4C22×D9711yescharacteristic
315C22×C4—21no—
316C22×C4—971no—
317C22×C4—1941no—
318C22×C4—1941no—
319C22×C4C22×D9711yescharacteristic
320SD16—41no—
321SD16—1942no—
322SD16—1942no—
323D8—22no—
324D8—974no—
325D8—3881no—
326D8—1942no—
327D8—1942no—
328D8—1942no—
329M16—1942no—
330C2×C8—971no—
331C2×C8—971no—
332C2×C8—1942no—
333C2×C8C22×D9711yescharacteristic
334C2×C8C22×D9711yescharacteristic
335C4⋊C4—971no—
336C4⋊C4—971no—
337C4⋊C4—1941no—
338C4⋊C4C22×D9711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—971no—
341C22⋊C4—971no—
342C22⋊C4—1941no—
343C22⋊C4—1941no—
344C22⋊C4—1941no—
345C22⋊C4—1941no—
346C22⋊C4—1941no—
347C22⋊C4—1941no—
348C22⋊C4—1941no—
349C42—971no—
350C23—21no—
351C23—21no—
352C23—7761no—
353C23—7761no—
354C23—7761no—
355C23—971no—
356C23—3882no—
357C23—3882no—
358C23—3882no—
359C23—3882no—
360C23—3882no—
361C23—3882no—
362C23—1941no—
363C23—1941no—
364C23—1941no—
365C23D4×D9711yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—42no—
372D4—42no—
373D4—7761no—
374D4—7761no—
375D4—972no—
376D4—3882no—
377D4—3882no—
378D4—3882no—
379D4—3882no—
380D4—3882no—
381D4—1942no—
382D4—1941no—
383D4—1941no—
384D4—1941no—
385D4—1941no—
386D4—1941no—
387D4D4×D9712yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—7761no—
394C2×C4—971no—
395C2×C4—971no—
396C2×C4—971no—
397C2×C4—3882no—
398C2×C4—1941no—
399C2×C4—1941no—
400C2×C4—1941no—
401C2×C4D4×D9711yescharacteristic
402C2×C4C23×D9711yescharacteristic
403C8—21no—
404C8—21no—
405C8—972no—
406C8—1941no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—7764no—
417C22—7764no—
418C22—7764no—
419C22—972no—
420C22—3881no—
421C22—3881no—
422C22—3881no—
423C22—3881no—
424C22—3881no—
425C22—3881no—
426C22—3881no—
427C22—3881no—
428C22—1942no—
429C22—1941no—
430C22—1941no—
431C22C2×D4×D9711yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—3881no—
436C4—1941no—
437C4—1941no—
438C4—1941no—
439C4C2×D4×D9711yescharacteristic
440C4#1569085511yescharacteristic
441C2—41no—
442C2—42no—
443C2—81no—
444C2—7761no—
445C2—7761no—
446C2—3882no—
447C2—1942no—
448C2#2026136711yescharacteristic
449C2AGL(1,ℤ/8)×D9711yescharacteristic
450C2#2026153711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#449×C97C2
socleC22×C97C22×D4
hypercenter#449D97
hypocenterC97#3264
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. C2×D8×D97C2semidirect
3. #20261373C2semidirect
4. #20261369C2semidirect
5. #20261121C2semidirect
6. #20260986C2semidirect
7. #20260967C2semidirect
8. #20260953C2semidirect
9. #20260928C2semidirect
10. #20260884C2semidirect
11. #20260871C2semidirect
12. #20260870C2semidirect
13. #20260844C2semidirect
14. #20260809C2semidirect
15. #449×C97C2semidirect
16. C22×D388C22semidirect
19. C2×D4×D97C22semidirect
21. C2×#13095498C22semidirect
22. C2×#13095498C22semidirect
24. C2×D776C22semidirect
26. C2×#13095496C22semidirect
27. C2×#13095496C22semidirect
29. C2×#13095495C22semidirect
30. C2×C8×D97C22semidirect
31. #15690806C22semidirect
32. #15690799C22semidirect
34. #15690786C22semidirect
36. #15690784C22semidirect
42. D4×Dic97C22semidirect
43. #15690772C22semidirect
46. #15690741C22semidirect
47. #15690739C22semidirect
48. #15690732C22semidirect
49. #15690732C22semidirect
50. #15690731C22semidirect
51. #15690730C22semidirect
52. #15690722C22semidirect
53. #15690720C22semidirect
54. C2×#41×C97C22semidirect
55. C2×SD16×C97C22semidirect
56. C2×D8×C97C22semidirect
57. #121×C97C22semidirect
58. #103×C97C22semidirect
59. #102×C97C22semidirect
60. C22⋊C8×C97C22semidirect
91. #13095496D4semidirect
147. D4×C97C2×D4semidirect
180. #11715895C2×D4semidirect
205. C2×D97AGL(1,ℤ/8)semidirect
226. C97#3264semidirect
237. #449D97semidirect
242. C2×#41C2×D97semidirect
250. C2×SD16C2×D97semidirect
257. C2×D8C2×D97semidirect
268. #121C2×D97semidirect
278. #103C2×D97semidirect
283. #102C2×D97semidirect
285. C22⋊C8C2×D97semidirect
387. D4D4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
C2×D8×D972
#202613732
#202613692
#202611212
#202609862
#202609672
#202609532
#202609282
#202608842
#202608712
#202608702
#202608442
#202608092
#449×C972
#326497