Finite Group Databasebeta

← #27153215

Subgroups — #27153215

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×#15690837C211yesmaximal, characteristic, complemented
2C2×D8×D97C211yesmaximal, characteristic, complemented
3#20261378C211yesmaximal, characteristic, complemented
4#20261370C211yesmaximal, characteristic, complemented
5#20260985C211yesmaximal, characteristic, complemented
6#20260961C211yesmaximal, characteristic, complemented
7#20260947C211yesmaximal, characteristic, complemented
8#20260927C211yesmaximal, characteristic, complemented
9#20260884C211yesmaximal, characteristic, complemented
10#20260879C211yesmaximal, characteristic, complemented
11#20260867C211yesmaximal, characteristic, complemented
12#20260867C211yesmaximal, characteristic, complemented
13#20260848C211yesmaximal, characteristic, complemented
14#20260808C211yesmaximal, characteristic, complemented
15#496×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic, complemented
17C2×D4×D97—21no—
18C2×D4×D97C2211yescharacteristic
19C2×D4×D97C2211yescharacteristic, complemented
20C2×D4×D97C2211yescharacteristic, complemented
21#15690837—24no—
22C2×#13095498C2211yescharacteristic, complemented
23C2×D776C2211yescharacteristic
24C2×#13095496C2211yescharacteristic
25C2×#13095496C2211yescharacteristic, complemented
26C2×#13095496C2211yescharacteristic, complemented
27D8×D97—24no—
28C2×#13095494C2211yescharacteristic, complemented
29C2×C8×D97C2211yescharacteristic, complemented
30#15690792—21no—
31#15690786C2211yescharacteristic, complemented
32#15690783—21no—
33#15690781—21no—
34#15690781C2211yescharacteristic
35#15690781C2211yescharacteristic, complemented
36#15690781C2211yescharacteristic, complemented
37#15690780—21no—
38#15690778—21no—
39#15690778—21no—
40C4×D388C2211yescharacteristic
41C4×D388C2211yescharacteristic, complemented
42C4×D388C2211yescharacteristic, complemented
43C22⋊C4×D97—21no—
44#15690742C2211yescharacteristic, complemented
45C4⋊#11715895C2211yescharacteristic, complemented
46#15690734C2211yescharacteristic, complemented
47#15690732C2211yescharacteristic, complemented
48#15690732C2211yescharacteristic, complemented
49#15690731C2211yescharacteristic, complemented
50#15690731C2211yescharacteristic, complemented
51#15690731C2211yescharacteristic, complemented
52#15690722C2211yescharacteristic, complemented
53#15690722C2211yescharacteristic, complemented
54C2×SD16×C97C2211yescharacteristic, complemented
55C2×SD16×C97C2211yescharacteristic, complemented
56C2×D8×C97C2211yescharacteristic, complemented
57C2×D8×C97C2211yescharacteristic, complemented
58#127×C97C2211yescharacteristic, complemented
59#124×C97C2211yescharacteristic, complemented
60#97×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97—21no—
63C23×D97—21no—
64#13095512—22no—
65#13095512—41no—
66C2×D388—21no—
67C2×D388D411yescharacteristic
68C2×D388C2311yescharacteristic
69C2×D388C2311yescharacteristic
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C22⋊D97—21no—
73C2×C22⋊D97—21no—
74C2×C22⋊D97—21no—
75C2×C22⋊D97—21no—
76D4×D97—22no—
77D4×D97—22no—
78D4×D97—22no—
79D4×D97—41no—
80D4×D97—41no—
81D4×D97—41no—
82D4×D97—44no—
83C2×C4×D97—21no—
84C2×C4×D97—21no—
85C2×C4×D97—21no—
86C2×C4×D97C2311yescharacteristic
87C2×C4×D97C2311yescharacteristic
88#13095498—22no—
89D776D414yescomplemented
90#13095496—22no—
91#13095496—22no—
92#13095496—22no—
93#13095494—22no—
94C2×#11715895C2311yescharacteristic
95C2×#11715895C2311yescharacteristic
96C8×D97—22no—
97C4⋊Dic97D411yescharacteristic
98C4⋊Dic97C2311yescharacteristic
99C4⋊Dic97C2311yescharacteristic
100#13095490—21no—
101C22⋊Dic97—21no—
102C22⋊Dic97—21no—
103C22⋊Dic97—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488—21no—
107#13095488D411yescharacteristic
108#13095488D411yescharacteristic
109C2×Q8×C97C2311yescharacteristic
110C2×D4×C97—21no—
111C2×D4×C97—21no—
112C2×D4×C97C2311yescharacteristic
113C2×D4×C97C2311yescharacteristic
114C2×D4×C97C2311yescharacteristic
115SD16×C97—22no—
116SD16×C97—41no—
117D8×C97—22no—
118D8×C97—41no—
119C2×C8×C97C2311yescharacteristic
120C2×C8×C97C2311yescharacteristic
121C22⋊C4×C97—21no—
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
140Q8×C97—21no—
141Q8×C97—21no—
142D4×C97—21no—
143D4×C97—21no—
144D4×C97—21no—
145D4×C97—21no—
146D4×C97—21no—
147D4×C97—21no—
148D4×C97—42no—
149D4×C97—81no—
150C2×C4×C97—21no—
151C2×C4×C97C2×D411yescharacteristic
152C2×C4×C97C2×D411yescharacteristic
153C2×C4×C97C2411yescharacteristic
154C22⋊D97—42no—
155C22⋊D97—42no—
156C22⋊D97—42no—
157C22⋊D97—42no—
158C22⋊D97—42no—
159C22⋊D97—81no—
160C2×Dic97—21no—
161C2×Dic97—21no—
162C2×Dic97C2×D411yescharacteristic
163C2×Dic97C2×D411yescharacteristic
164D388—22no—
165D388—21no—
166D388—21no—
167D388—42no—
168D388C2×D412yes—
169D388C2×D412yescomplemented
170C4×D97—21no—
171C4×D97—21no—
172C4×D97—21no—
173C4×D97—21no—
174C4×D97—42no—
175C4×D97—42no—
176C4×D97—81no—
177C8×C97—21no—
178C8×C97C2×D412yescomplemented
179#11715895—21no—
180#11715895—21no—
181C22×C97—41no—
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—22no—
196C2×D97—21no—
197C2×D97—21no—
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—81no—
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—42no—
225#3264—971nomaximal, Sylow
226C97#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—971no—
228C22×D8—971no—
229#546—971no—
230D42—971no—
231#496D9711yescharacteristic, complemented
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#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×SD16C2×D9711yescharacteristic, complemented
250C2×SD16C2×D9711yescharacteristic, complemented
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#124C2×D9711yescharacteristic, complemented
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×D4—971no—
275C2×C22⋊C4—1941no—
276#106—971no—
277C4⋊C8—971no—
278#103—971no—
279#102—971no—
280#102—971no—
281#102—971no—
282#102—971no—
283#102—971no—
284C22⋊C8—971no—
285C22⋊C8—971no—
286#97C2×D9711yescharacteristic, complemented
287C24—1941no—
288C24—1941no—
289C24—1941no—
290#41—3881no—
291#41—1942no—
292C2×Q8C22×D9711yescharacteristic
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—1942no—
309C2×D4—1941no—
310C2×D4—1941no—
311C2×D4—1941no—
312C2×D4C22×D9711yescharacteristic
313C2×D4C22×D9711yescharacteristic
314C2×D4C22×D9711yescharacteristic
315C22×C4—971no—
316C22×C4—971no—
317C22×C4—1941no—
318C22×C4—1941no—
319C22×C4—1941no—
320SD16—22no—
321SD16—41no—
322SD16—1942no—
323D8—22no—
324D8—41no—
325D8—974no—
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—971no—
338C4⋊C4—1941no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—971no—
342C22⋊C4—971no—
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—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—7761no—
377D4—972no—
378D4—972no—
379D4—3882no—
380D4—3882no—
381D4—3882no—
382D4—3882no—
383D4—3882no—
384D4—3882no—
385D4—1942no—
386D4—1941no—
387D4—1941no—
388C2×C4—21no—
389C2×C4—7761no—
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—1941no—
405C8—1941no—
406C8D4×D9712yescomplemented
407C22—41no—
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—1941no—
425C22—1941no—
426C22—1942no—
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#1569085511yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—3882no—
445C2—3881no—
446C2—3882no—
447C2—1942no—
448C2#2026137011yescharacteristic
449C2AGL(1,ℤ/8)×D9711yescharacteristic
450C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#496×C97C2
socleC22×C97C22×D4
hypercenter#496D97
hypocenterC97#3264
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. C2×D8×D97C2semidirect
3. #20261378C2semidirect
4. #20261370C2semidirect
5. #20260985C2semidirect
6. #20260961C2semidirect
7. #20260947C2semidirect
8. #20260927C2semidirect
9. #20260884C2semidirect
10. #20260879C2semidirect
11. #20260867C2semidirect
12. #20260867C2semidirect
13. #20260848C2semidirect
14. #20260808C2semidirect
15. #496×C97C2semidirect
16. C2×#13095512C22semidirect
19. C2×D4×D97C22semidirect
20. C2×D4×D97C22semidirect
22. C2×#13095498C22semidirect
25. C2×#13095496C22semidirect
26. C2×#13095496C22semidirect
28. C2×#13095494C22semidirect
29. C2×C8×D97C22semidirect
31. #15690786C22semidirect
35. #15690781C22semidirect
36. #15690781C22semidirect
41. C4×D388C22semidirect
42. C4×D388C22semidirect
44. #15690742C22semidirect
45. C4⋊#11715895C22semidirect
46. #15690734C22semidirect
47. #15690732C22semidirect
48. #15690732C22semidirect
49. #15690731C22semidirect
50. #15690731C22semidirect
51. #15690731C22semidirect
52. #15690722C22semidirect
53. #15690722C22semidirect
54. C2×SD16×C97C22semidirect
55. C2×SD16×C97C22semidirect
56. C2×D8×C97C22semidirect
57. C2×D8×C97C22semidirect
58. #127×C97C22semidirect
59. #124×C97C22semidirect
60. #97×C97C22semidirect
89. D776D4semidirect
169. D388C2×D4semidirect
178. C8×C97C2×D4semidirect
205. C2×D97AGL(1,ℤ/8)semidirect
226. C97#3264semidirect
231. #496D97semidirect
249. C2×SD16C2×D97semidirect
250. C2×SD16C2×D97semidirect
256. C2×D8C2×D97semidirect
257. C2×D8C2×D97semidirect
260. #127C2×D97semidirect
261. #124C2×D97semidirect
286. #97C2×D97semidirect
406. C8D4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
C2×D8×D972
#202613782
#202613702
#202609852
#202609612
#202609472
#202609272
#202608842
#202608792
#202608672
#202608482
#202608082
#496×C972
#326497