Finite Group Databasebeta

← #31792268

Subgroups — #31792268

Order 17536.

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×#17984115C211yesmaximal, characteristic, complemented
2C2×D8×D137C211yesmaximal, characteristic, complemented
3#22578046C211yesmaximal, characteristic, complemented
4#22578038C211yesmaximal, characteristic, complemented
5#22577653C211yesmaximal, characteristic, complemented
6#22577629C211yesmaximal, characteristic, complemented
7#22577615C211yesmaximal, characteristic, complemented
8#22577595C211yesmaximal, characteristic, complemented
9#22577552C211yesmaximal, characteristic, complemented
10#22577547C211yesmaximal, characteristic, complemented
11#22577535C211yesmaximal, characteristic, complemented
12#22577535C211yesmaximal, characteristic, complemented
13#22577516C211yesmaximal, characteristic, complemented
14#22577476C211yesmaximal, characteristic, complemented
15#496×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic, complemented
17C2×D4×D137—21no—
18C2×D4×D137C2211yescharacteristic
19C2×D4×D137C2211yescharacteristic, complemented
20C2×D4×D137C2211yescharacteristic, complemented
21#17984115—24no—
22C2×#14555497C2211yescharacteristic, complemented
23C2×D1096C2211yescharacteristic
24C2×#14555495C2211yescharacteristic
25C2×#14555495C2211yescharacteristic, complemented
26C2×#14555495C2211yescharacteristic, complemented
27D8×D137—24no—
28C2×#14555493C2211yescharacteristic, complemented
29C2×C8×D137C2211yescharacteristic, complemented
30#17984070—21no—
31#17984064C2211yescharacteristic, complemented
32#17984061—21no—
33#17984059—21no—
34#17984059C2211yescharacteristic
35#17984059C2211yescharacteristic, complemented
36#17984059C2211yescharacteristic, complemented
37#17984058—21no—
38#17984056—21no—
39#17984056—21no—
40C4×D548C2211yescharacteristic
41C4×D548C2211yescharacteristic, complemented
42C4×D548C2211yescharacteristic, complemented
43C22⋊C4×D137—21no—
44#17984020C2211yescharacteristic, complemented
45C4⋊#11764040C2211yescharacteristic, complemented
46#17984012C2211yescharacteristic, complemented
47#17984010C2211yescharacteristic, complemented
48#17984010C2211yescharacteristic, complemented
49#17984009C2211yescharacteristic, complemented
50#17984009C2211yescharacteristic, complemented
51#17984009C2211yescharacteristic, complemented
52#17984000C2211yescharacteristic, complemented
53#17984000C2211yescharacteristic, complemented
54C2×SD16×C137C2211yescharacteristic, complemented
55C2×SD16×C137C2211yescharacteristic, complemented
56C2×D8×C137C2211yescharacteristic, complemented
57C2×D8×C137C2211yescharacteristic, complemented
58#127×C137C2211yescharacteristic, complemented
59#124×C137C2211yescharacteristic, complemented
60#97×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C23×D137—21no—
63C23×D137—21no—
64#14555511—22no—
65#14555511—41no—
66C2×D548—21no—
67C2×D548D411yescharacteristic
68C2×D548C2311yescharacteristic
69C2×D548C2311yescharacteristic
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73C2×C22⋊D137—21no—
74C2×C22⋊D137—21no—
75C2×C22⋊D137—21no—
76D4×D137—22no—
77D4×D137—22no—
78D4×D137—22no—
79D4×D137—41no—
80D4×D137—41no—
81D4×D137—41no—
82D4×D137—44no—
83C2×C4×D137—21no—
84C2×C4×D137—21no—
85C2×C4×D137—21no—
86C2×C4×D137C2311yescharacteristic
87C2×C4×D137C2311yescharacteristic
88#14555497—22no—
89D1096D414yescomplemented
90#14555495—22no—
91#14555495—22no—
92#14555495—22no—
93#14555493—22no—
94C2×#11764040C2311yescharacteristic
95C2×#11764040C2311yescharacteristic
96C8×D137—22no—
97C4⋊Dic137D411yescharacteristic
98C4⋊Dic137C2311yescharacteristic
99C4⋊Dic137C2311yescharacteristic
100#14555489—21no—
101C22⋊Dic137—21no—
102C22⋊Dic137—21no—
103C22⋊Dic137—21no—
104#14555487—21no—
105#14555487—21no—
106#14555487—21no—
107#14555487D411yescharacteristic
108#14555487D411yescharacteristic
109C2×Q8×C137C2311yescharacteristic
110C2×D4×C137—21no—
111C2×D4×C137—21no—
112C2×D4×C137C2311yescharacteristic
113C2×D4×C137C2311yescharacteristic
114C2×D4×C137C2311yescharacteristic
115SD16×C137—22no—
116SD16×C137—41no—
117D8×C137—22no—
118D8×C137—41no—
119C2×C8×C137C2311yescharacteristic
120C2×C8×C137C2311yescharacteristic
121C22⋊C4×C137—21no—
122C22⋊C4×C137—21no—
123C42×C137C2311yescharacteristic
124C22×D137—21no—
125C22×D137—21no—
126C22×D137—42no—
127C22×D137—42no—
128C22×D137—42no—
129C22×D137—42no—
130C22×D137—42no—
131C22×D137—42no—
132C22×D137—81no—
133C22×D137—81no—
134C22×D137—81no—
135C22×D137C2×D411yescharacteristic
136C22×D137C2×D411yescharacteristic
137D548—22no—
138D548—21no—
139D548—21no—
140D548—42no—
141D548C2×D412yes—
142D548C2×D412yescomplemented
143C22⋊D137—42no—
144C22⋊D137—42no—
145C22⋊D137—42no—
146C22⋊D137—42no—
147C22⋊D137—42no—
148C22⋊D137—81no—
149C2×Dic137—21no—
150C2×Dic137—21no—
151C2×Dic137C2×D411yescharacteristic
152C2×Dic137C2×D411yescharacteristic
153C4×D137—21no—
154C4×D137—21no—
155C4×D137—21no—
156C4×D137—21no—
157C4×D137—42no—
158C4×D137—42no—
159C4×D137—81no—
160#11764040—21no—
161#11764040—21no—
162C23×C137—21no—
163C23×C137—21no—
164C23×C137—21no—
165Q8×C137—21no—
166Q8×C137—21no—
167D4×C137—21no—
168D4×C137—21no—
169D4×C137—21no—
170D4×C137—21no—
171D4×C137—21no—
172D4×C137—21no—
173D4×C137—42no—
174D4×C137—81no—
175C2×C4×C137—21no—
176C2×C4×C137C2×D411yescharacteristic
177C2×C4×C137C2×D411yescharacteristic
178C2×C4×C137C2411yescharacteristic
179C8×C137—21no—
180C8×C137C2×D412yescomplemented
181C22×C137—41no—
182C22×C137—41no—
183C22×C137—41no—
184C22×C137—41no—
185C22×C137—41no—
186C22×C137—41no—
187C22×C137—41no—
188C22×C137—41no—
189C22×C137—41no—
190C22×C137C22×D411yescharacteristic
191C2×D137—21no—
192C2×D137—21no—
193C2×D137—21no—
194C2×D137—22no—
195C2×D137—22no—
196C2×D137—21no—
197C2×D137—21no—
198C2×D137—41no—
199C2×D137—41no—
200C2×D137—41no—
201C2×D137—41no—
202C2×D137—84no—
203C2×D137—84no—
204C2×D137—84no—
205C2×D137AGL(1,ℤ/8)12yescomplemented
206C4×C137—21no—
207C4×C137—21no—
208C4×C137—41no—
209C4×C137C22×D411yescharacteristic
210C4×C137H(5,𝔽2)11yescharacteristic
211Dic137—21no—
212Dic137—21no—
213Dic137—41no—
214Dic137—41no—
215C2×C137—81no—
216C2×C137—81no—
217C2×C137—81no—
218C2×C137D4211yescharacteristic
219C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C137#57611yescharacteristic
221D137—22no—
222D137—41no—
223D137—42no—
224D137—42no—
225C137#326411yescharacteristic, complemented, Sylow
226#3264—1371nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1371no—
228C22×D8—1371no—
229#546—1371no—
230D42—1371no—
231#496D13711yescharacteristic, complemented
232#469—1371no—
233#465—1371no—
234#463—1371no—
235#459—1371no—
236#450—1371no—
237#449—1371no—
238#447—1371no—
239#447—1371no—
240#442—1371no—
241#435—1371no—
242C2×#41—1371no—
243C22×D4—2741no—
244C22×D4—1371no—
245C22×D4—1371no—
246C22×D4—1371no—
247AGL(1,ℤ/8)—2744no—
248C2×SD16—1371no—
249C2×SD16C2×D13711yescharacteristic, complemented
250C2×SD16C2×D13711yescharacteristic, complemented
251C2×D8—2744no—
252C2×D8—1371no—
253C2×D8—1371no—
254C2×D8—1371no—
255C2×D8—1371no—
256C2×D8C2×D13711yescharacteristic, complemented
257C2×D8C2×D13711yescharacteristic, complemented
258C2×M16—1371no—
259C22×C8—1371no—
260#127C2×D13711yescharacteristic, complemented
261#124C2×D13711yescharacteristic, complemented
262#123—2741no—
263C22⋊Q8—1371no—
264#121—2741no—
265#121—2741no—
266#121—1371no—
267#121—1371no—
268#121—1371no—
269#120—2741no—
270#120—2741no—
271#120—2741no—
272C4×D4—1371no—
273C4×D4—1371no—
274C4×D4—1371no—
275C2×C22⋊C4—2741no—
276#106—1371no—
277C4⋊C8—1371no—
278#103—1371no—
279#102—1371no—
280#102—1371no—
281#102—1371no—
282#102—1371no—
283#102—1371no—
284C22⋊C8—1371no—
285C22⋊C8—1371no—
286#97C2×D13711yescharacteristic, complemented
287C24—2741no—
288C24—2741no—
289C24—2741no—
290#41—2742no—
291#41—5481no—
292C2×Q8C22×D13711yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—2741no—
296C2×D4—2742no—
297C2×D4—2742no—
298C2×D4—2742no—
299C2×D4—2741no—
300C2×D4—2741no—
301C2×D4—2741no—
302C2×D4—2741no—
303C2×D4—2741no—
304C2×D4—2741no—
305C2×D4—5481no—
306C2×D4—5481no—
307C2×D4—5481no—
308C2×D4—5484no—
309C2×D4—1371no—
310C2×D4—1371no—
311C2×D4—1371no—
312C2×D4C22×D13711yescharacteristic
313C2×D4C22×D13711yescharacteristic
314C2×D4C22×D13711yescharacteristic
315C22×C4—2741no—
316C22×C4—2741no—
317C22×C4—2741no—
318C22×C4—1371no—
319C22×C4—1371no—
320SD16—22no—
321SD16—41no—
322SD16—2742no—
323D8—22no—
324D8—41no—
325D8—2742no—
326D8—2742no—
327D8—2742no—
328D8—1374no—
329M16—2742no—
330C2×C8—2742no—
331C2×C8—1371no—
332C2×C8—1371no—
333C2×C8C22×D13711yescharacteristic
334C2×C8C22×D13711yescharacteristic
335C4⋊C4—2741no—
336C4⋊C4—1371no—
337C4⋊C4—1371no—
338C4⋊C4—1371no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—2741no—
342C22⋊C4—2741no—
343C22⋊C4—2741no—
344C22⋊C4—2741no—
345C22⋊C4—2741no—
346C22⋊C4—2741no—
347C22⋊C4—1371no—
348C22⋊C4—1371no—
349C42C22×D13711yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—2741no—
354C23—2741no—
355C23—5482no—
356C23—5482no—
357C23—5482no—
358C23—5482no—
359C23—5482no—
360C23—5482no—
361C23—10961no—
362C23—10961no—
363C23—10961no—
364C23—1371no—
365C23—1371no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—2742no—
377D4—2741no—
378D4—2741no—
379D4—5482no—
380D4—5482no—
381D4—5482no—
382D4—5482no—
383D4—5482no—
384D4—5482no—
385D4—10961no—
386D4—1372no—
387D4—1372no—
388C2×C4—21no—
389C2×C4—2741no—
390C2×C4—2741no—
391C2×C4—2741no—
392C2×C4—2741no—
393C2×C4—2741no—
394C2×C4—2741no—
395C2×C4—5482no—
396C2×C4—5482no—
397C2×C4—10961no—
398C2×C4—1371no—
399C2×C4—1371no—
400C2×C4D4×D13711yescharacteristic
401C2×C4D4×D13711yescharacteristic
402C2×C4C23×D13711yescharacteristic
403C8—21no—
404C8—2741no—
405C8—2741no—
406C8D4×D13712yescomplemented
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—2741no—
417C22—2741no—
418C22—2741no—
419C22—2741no—
420C22—2741no—
421C22—2742no—
422C22—2742no—
423C22—5481no—
424C22—5481no—
425C22—5481no—
426C22—5481no—
427C22—10964no—
428C22—10964no—
429C22—10964no—
430C22—1372no—
431C22C2×D4×D13711yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—2741no—
436C4—2741no—
437C4—5481no—
438C4—5481no—
439C4C2×D4×D13711yescharacteristic
440C4#1798413311yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—2742no—
445C2—5481no—
446C2—5482no—
447C2—5482no—
448C2#2257803811yescharacteristic
449C2AGL(1,ℤ/8)×D13711yescharacteristic
450C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#496×C137C2
socleC22×C137C22×D4
hypercenter#496D137
hypocenterC137#3264
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#17984115C2semidirect
2. C2×D8×D137C2semidirect
3. #22578046C2semidirect
4. #22578038C2semidirect
5. #22577653C2semidirect
6. #22577629C2semidirect
7. #22577615C2semidirect
8. #22577595C2semidirect
9. #22577552C2semidirect
10. #22577547C2semidirect
11. #22577535C2semidirect
12. #22577535C2semidirect
13. #22577516C2semidirect
14. #22577476C2semidirect
15. #496×C137C2semidirect
16. C2×#14555511C22semidirect
19. C2×D4×D137C22semidirect
20. C2×D4×D137C22semidirect
22. C2×#14555497C22semidirect
25. C2×#14555495C22semidirect
26. C2×#14555495C22semidirect
28. C2×#14555493C22semidirect
29. C2×C8×D137C22semidirect
31. #17984064C22semidirect
35. #17984059C22semidirect
36. #17984059C22semidirect
41. C4×D548C22semidirect
42. C4×D548C22semidirect
44. #17984020C22semidirect
45. C4⋊#11764040C22semidirect
46. #17984012C22semidirect
47. #17984010C22semidirect
48. #17984010C22semidirect
49. #17984009C22semidirect
50. #17984009C22semidirect
51. #17984009C22semidirect
52. #17984000C22semidirect
53. #17984000C22semidirect
54. C2×SD16×C137C22semidirect
55. C2×SD16×C137C22semidirect
56. C2×D8×C137C22semidirect
57. C2×D8×C137C22semidirect
58. #127×C137C22semidirect
59. #124×C137C22semidirect
60. #97×C137C22semidirect
89. D1096D4semidirect
142. D548C2×D4semidirect
180. C8×C137C2×D4semidirect
205. C2×D137AGL(1,ℤ/8)semidirect
225. C137#3264semidirect
231. #496D137semidirect
249. C2×SD16C2×D137semidirect
250. C2×SD16C2×D137semidirect
256. C2×D8C2×D137semidirect
257. C2×D8C2×D137semidirect
260. #127C2×D137semidirect
261. #124C2×D137semidirect
286. #97C2×D137semidirect
406. C8D4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841152
C2×D8×D1372
#225780462
#225780382
#225776532
#225776292
#225776152
#225775952
#225775522
#225775472
#225775352
#225775162
#225774762
#496×C1372
#3264137