Finite Group Databasebeta

← #27153211

Subgroups — #27153211

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×#15690834C211yesmaximal, characteristic, complemented
2C2×D8×D97C211yesmaximal, characteristic, complemented
3#20261380C211yesmaximal, characteristic, complemented
4#20261370C211yesmaximal, characteristic, complemented
5#20261122C211yesmaximal, characteristic, complemented
6#20260987C211yesmaximal, characteristic, complemented
7#20260963C211yesmaximal, characteristic, complemented
8#20260951C211yesmaximal, characteristic, complemented
9#20260887C211yesmaximal, characteristic, complemented
10#20260877C211yesmaximal, characteristic, complemented
11#20260870C211yesmaximal, characteristic, complemented
12#20260867C211yesmaximal, characteristic, complemented
13#20260845C211yesmaximal, characteristic, complemented
14#20260805C211yesmaximal, characteristic, complemented
15#459×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic, complemented
17C2×D4×D97—21no—
18C2×D4×D97C2211yescharacteristic
19C2×D4×D97C2211yescharacteristic, complemented
20C2×D4×D97C2211yescharacteristic, complemented
21#15690834—24no—
22C2×#13095500C2211yescharacteristic, complemented
23C2×#13095499C2211yescharacteristic, complemented
24C2×#13095499C2211yescharacteristic, complemented
25C2×D776C2211yescharacteristic
26C2×#13095496C2211yescharacteristic
27C2×#13095496C2211yescharacteristic, complemented
28C2×#13095496C2211yescharacteristic, complemented
29D8×D97—24no—
30C2×#13095494C2211yescharacteristic, complemented
31C2×C8×D97C2211yescharacteristic, complemented
32#15690798C2211yescharacteristic, complemented
33#15690794—21no—
34#15690790C2211yescharacteristic, complemented
35#15690785C2211yescharacteristic, complemented
36#15690784—21no—
37#15690781—21no—
38#15690781C2211yescharacteristic
39#15690781C2211yescharacteristic, complemented
40#15690780—21no—
41#15690778—21no—
42#15690778—21no—
43C4×D388C2211yescharacteristic
44C4×D388C2211yescharacteristic, complemented
45C2×#13095488—21no—
46#15690741C2211yescharacteristic, complemented
47#15690735C2211yescharacteristic, complemented
48#15690732C2211yescharacteristic, complemented
49#15690731C2211yescharacteristic, complemented
50#15690730C2211yescharacteristic, complemented
51#15690722C2211yescharacteristic, complemented
52#15690722C2211yescharacteristic, complemented
53#15690721C2211yescharacteristic, complemented
54C2×D8×C97C2211yescharacteristic, complemented
55C2×D8×C97C2211yescharacteristic, complemented
56#127×C97C2211yescharacteristic, complemented
57C4×D4×C97C2211yescharacteristic, complemented
58C4⋊C8×C97C2211yescharacteristic, complemented
59#102×C97C2211yescharacteristic, complemented
60#102×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97—21no—
63C23×D97—21no—
64#13095510—22no—
65#13095510—41no—
66C2×Dic194C2311yescharacteristic
67C2×D388—21no—
68C2×D388D411yescharacteristic
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⋊D97—21no—
77D4×D97—22no—
78D4×D97—22no—
79D4×D97—22no—
80D4×D97—41no—
81D4×D97—41no—
82D4×D97—41no—
83D4×D97—44no—
84C22×Dic97—21no—
85C2×C4×D97—21no—
86C2×C4×D97C2311yescharacteristic
87C2×C4×D97C2311yescharacteristic
88#13095500—22no—
89#13095499—22no—
90#13095499—41no—
91D776—22no—
92#13095496—22no—
93#13095496—41no—
94#13095496D414yescomplemented
95#13095494—22no—
96C2×#11715895C2311yescharacteristic
97C2×#11715895C2311yescharacteristic
98C8×D97—22no—
99C4⋊Dic97—21no—
100C4⋊Dic97D411yescharacteristic
101C4⋊Dic97C2311yescharacteristic
102C22⋊Dic97—21no—
103C22⋊Dic97—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488—21no—
107#13095488—21no—
108#13095488—21no—
109#13095488D411yescharacteristic
110#13095488D411yescharacteristic
111C2×D4×C97—21no—
112C2×D4×C97—21no—
113C2×D4×C97C2311yescharacteristic
114C2×D4×C97C2311yescharacteristic
115C2×D4×C97C2311yescharacteristic
116C22×C4×C97—21no—
117D8×C97—22no—
118D8×C97—22no—
119C2×C8×C97C2311yescharacteristic
120C2×C8×C97C2311yescharacteristic
121C4⋊C4×C97C2311yescharacteristic
122C22⋊C4×C97—21no—
123C42×C97C2311yescharacteristic
124C23×C97—21no—
125C23×C97—21no—
126C23×C97—21no—
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—42no—
130C22×D97—42no—
131C22×D97—42no—
132C22×D97—42no—
133C22×D97—42no—
134C22×D97—42no—
135C22×D97—81no—
136C22×D97—81no—
137C22×D97—81no—
138C22×D97C2×D411yescharacteristic
139C22×D97C2×D411yescharacteristic
140D4×C97—21no—
141D4×C97—21no—
142D4×C97—21no—
143D4×C97—21no—
144D4×C97—21no—
145D4×C97—42no—
146D4×C97—81no—
147D4×C97C2×D412yescomplemented
148C2×C4×C97—21no—
149C2×C4×C97—81no—
150C2×C4×C97C2×D411yescharacteristic
151C2×C4×C97C2×D411yescharacteristic
152C2×C4×C97C2411yescharacteristic
153C22⋊D97—42no—
154C22⋊D97—42no—
155C22⋊D97—42no—
156C22⋊D97—42no—
157C22⋊D97—42no—
158C22⋊D97—42no—
159C2×Dic97—21no—
160C2×Dic97—21no—
161C2×Dic97—42no—
162C2×Dic97C2×D411yescharacteristic
163C2×Dic97C2×D411yescharacteristic
164D388—22no—
165D388—21no—
166D388—21no—
167D388—21no—
168D388—81no—
169D388C2×D412yes—
170C4×D97—21no—
171C4×D97—21no—
172C4×D97—21no—
173C4×D97—21no—
174C4×D97—42no—
175Dic194—21no—
176Dic194—21no—
177C8×C97—21no—
178C8×C97—21no—
179#11715895—21no—
180#11715895C2×D412yescomplemented
181C22×C97—22no—
182C22×C97—41no—
183C22×C97—41no—
184C22×C97—41no—
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—21no—
194C2×D97—22no—
195C2×D97—21no—
196C2×D97—21no—
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—21no—
208C4×C97—41no—
209C4×C97C22×D411yescharacteristic
210C4×C97H(5,𝔽2)11yescharacteristic
211Dic97—21no—
212Dic97—21no—
213Dic97—41no—
214Dic97—41no—
215C2×C97—42no—
216C2×C97—81no—
217C2×C97—81no—
218C2×C97D4211yescharacteristic
219C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C97#57611yescharacteristic
221D97—22no—
222D97—41no—
223D97—42no—
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#459D9711yescharacteristic, complemented
236#450—971no—
237#449—971no—
238#447—971no—
239#447—971no—
240#442—971no—
241#435—971no—
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—1944no—
256C2×D8C2×D9711yescharacteristic, complemented
257C2×D8C2×D9711yescharacteristic, complemented
258C2×M16—971no—
259C22×C8—971no—
260#127C2×D9711yescharacteristic, complemented
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, complemented
275C2×C22⋊C4—1941no—
276#106—971no—
277C4⋊C8C2×D9711yescharacteristic, complemented
278#103—971no—
279#102—971no—
280#102—971no—
281#102—971no—
282#102C2×D9711yescharacteristic, complemented
283#102C2×D9711yescharacteristic, complemented
284C22⋊C8—971no—
285C22⋊C8—971no—
286#97—971no—
287C24—1941no—
288C24—1941no—
289C24—1941no—
290#41—3881no—
291#41—1942no—
292C2×Q8—971no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—971no—
296C2×D4—971no—
297C2×D4—971no—
298C2×D4—3884no—
299C2×D4—3881no—
300C2×D4—3881no—
301C2×D4—3881no—
302C2×D4—1941no—
303C2×D4—1941no—
304C2×D4—1941no—
305C2×D4—1941no—
306C2×D4—1942no—
307C2×D4—1942no—
308C2×D4—1941no—
309C2×D4—1941no—
310C2×D4—1941no—
311C2×D4—1942no—
312C2×D4C22×D9711yescharacteristic
313C2×D4C22×D9711yescharacteristic
314C2×D4C22×D9711yescharacteristic
315C22×C4—21no—
316C22×C4—971no—
317C22×C4—971no—
318C22×C4—1941no—
319C22×C4—1941no—
320SD16—3881no—
321SD16—1942no—
322SD16—1942no—
323D8—22no—
324D8—22no—
325D8—974no—
326D8—3881no—
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—
349C42C22×D9711yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—7761no—
354C23—7761no—
355C23—7761no—
356C23—971no—
357C23—971no—
358C23—3882no—
359C23—3882no—
360C23—3882no—
361C23—3882no—
362C23—3882no—
363C23—3882no—
364C23—1941no—
365C23—1941no—
366Q8—1941no—
367Q8—1941no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—81no—
375D4—7761no—
376D4—972no—
377D4—3882no—
378D4—3882no—
379D4—3882no—
380D4—3882no—
381D4—3882no—
382D4—3882no—
383D4—1942no—
384D4—1941no—
385D4—1941no—
386D4—1941no—
387D4D4×D9712yescomplemented
388C2×C4—21no—
389C2×C4—81no—
390C2×C4—971no—
391C2×C4—971no—
392C2×C4—3882no—
393C2×C4—3882no—
394C2×C4—1941no—
395C2×C4—1941no—
396C2×C4—1941no—
397C2×C4—1941no—
398C2×C4—1941no—
399C2×C4—1941no—
400C2×C4D4×D9711yescharacteristic
401C2×C4D4×D9711yescharacteristic
402C2×C4C23×D9711yescharacteristic
403C8—21no—
404C8—21no—
405C8—972no—
406C8—1941no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
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—1941no—
426C22—1941no—
427C22—1941no—
428C22—1942no—
429C22—1941no—
430C22—1941no—
431C22C2×D4×D9711yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—3881no—
436C4—3881no—
437C4—1941no—
438C4—1941no—
439C4C2×D4×D9711yescharacteristic
440C4#1569085611yescharacteristic
441C2—42no—
442C2—81no—
443C2—81no—
444C2—7761no—
445C2—3882no—
446C2—3881no—
447C2—1942no—
448C2#2026136911yescharacteristic
449C2AGL(1,ℤ/8)×D9711yescharacteristic
450C2#2026153611yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#459×C97C2
socleC22×C97C22×D4
hypercenter#459D97
hypocenterC97#3264
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#15690834C2semidirect
2. C2×D8×D97C2semidirect
3. #20261380C2semidirect
4. #20261370C2semidirect
5. #20261122C2semidirect
6. #20260987C2semidirect
7. #20260963C2semidirect
8. #20260951C2semidirect
9. #20260887C2semidirect
10. #20260877C2semidirect
11. #20260870C2semidirect
12. #20260867C2semidirect
13. #20260845C2semidirect
14. #20260805C2semidirect
15. #459×C97C2semidirect
16. C2×#13095510C22semidirect
19. C2×D4×D97C22semidirect
20. C2×D4×D97C22semidirect
22. C2×#13095500C22semidirect
23. C2×#13095499C22semidirect
24. C2×#13095499C22semidirect
27. C2×#13095496C22semidirect
28. C2×#13095496C22semidirect
30. C2×#13095494C22semidirect
31. C2×C8×D97C22semidirect
32. #15690798C22semidirect
34. #15690790C22semidirect
35. #15690785C22semidirect
39. #15690781C22semidirect
44. C4×D388C22semidirect
46. #15690741C22semidirect
47. #15690735C22semidirect
48. #15690732C22semidirect
49. #15690731C22semidirect
50. #15690730C22semidirect
51. #15690722C22semidirect
52. #15690722C22semidirect
53. #15690721C22semidirect
54. C2×D8×C97C22semidirect
55. C2×D8×C97C22semidirect
56. #127×C97C22semidirect
57. C4×D4×C97C22semidirect
58. C4⋊C8×C97C22semidirect
59. #102×C97C22semidirect
60. #102×C97C22semidirect
94. #13095496D4semidirect
147. D4×C97C2×D4semidirect
180. #11715895C2×D4semidirect
205. C2×D97AGL(1,ℤ/8)semidirect
226. C97#3264semidirect
235. #459D97semidirect
256. C2×D8C2×D97semidirect
257. C2×D8C2×D97semidirect
260. #127C2×D97semidirect
274. C4×D4C2×D97semidirect
277. C4⋊C8C2×D97semidirect
282. #102C2×D97semidirect
283. #102C2×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×#156908342
C2×D8×D972
#202613802
#202613702
#202611222
#202609872
#202609632
#202609512
#202608872
#202608772
#202608702
#202608672
#202608452
#202608052
#459×C972
#326497