Finite Group Databasebeta

← #31792275

Subgroups — #31792275

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
4D4×D548C211yesmaximal, characteristic, complemented
5#22577793C211yesmaximal, characteristic, complemented
6#22577655C211yesmaximal, characteristic, complemented
7#22577635C211yesmaximal, characteristic, complemented
8#22577599C211yesmaximal, characteristic, complemented
9#22577552C211yesmaximal, characteristic, complemented
10#22577551C211yesmaximal, characteristic, complemented
11#22577538C211yesmaximal, characteristic, complemented
12#22577535C211yesmaximal, characteristic, complemented
13#22577513C211yesmaximal, characteristic, complemented
14#22577473C211yesmaximal, characteristic, complemented
15#463×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic, complemented
17C22×D548—21no—
18C2×D4×D137C2211yescharacteristic
19C2×D4×D137C2211yescharacteristic, complemented
20C2×D4×D137C2211yescharacteristic, complemented
21#17984115—24no—
22C2×#14555497C2211yescharacteristic, complemented
23C2×#14555497C2211yescharacteristic, complemented
24C2×D1096C2211yescharacteristic, complemented
25C2×D1096C2211yescharacteristic, complemented
26C2×#14555495C2211yescharacteristic
27C2×#14555495C2211yescharacteristic, complemented
28C2×#14555495C2211yescharacteristic, complemented
29D8×D137—24no—
30C2×#14555493C2211yescharacteristic, complemented
31C2×C8×D137C2211yescharacteristic, complemented
32#17984085C2211yescharacteristic, complemented
33#17984070—21no—
34#17984064C2211yescharacteristic, complemented
35#17984063C2211yescharacteristic
36#17984063C2211yescharacteristic, complemented
37#17984062—21no—
38#17984061—21no—
39#17984059C2211yescharacteristic, complemented
40#17984058—21no—
41#17984058—21no—
42#17984058—21no—
43C4×D548C2211yescharacteristic
44C4×D548C2211yescharacteristic, complemented
45C22⋊C4×D137—21no—
46#17984019C2211yescharacteristic, complemented
47#17984010C2211yescharacteristic, complemented
48#17984010C2211yescharacteristic, complemented
49#17984009C2211yescharacteristic, complemented
50#17984008C2211yescharacteristic, complemented
51#17984000C2211yescharacteristic, complemented
52#17984000C2211yescharacteristic, complemented
53#17983999C2211yescharacteristic, complemented
54C2×SD16×C137C2211yescharacteristic, complemented
55C2×D8×C137C2211yescharacteristic
56#124×C137C2211yescharacteristic, complemented
57C4×D4×C137C2211yescharacteristic, complemented
58C4⋊C8×C137C2211yescharacteristic, complemented
59#103×C137C2211yescharacteristic, complemented
60#102×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C23×D137—21no—
63C23×D137—21no—
64#14555511—22no—
65#14555511—41no—
66C2×D548—21no—
67C2×D548—21no—
68C2×D548—21no—
69C2×D548—21no—
70C2×D548—21no—
71C2×D548—44no—
72C2×D548D411yescharacteristic
73C2×D548C2311yescharacteristic
74C2×D548C2311yescharacteristic
75C2×D548C2311yescharacteristic
76C2×C22⋊D137—21no—
77C2×C22⋊D137—21no—
78C2×C22⋊D137—21no—
79C2×C22⋊D137—21no—
80D4×D137—22no—
81D4×D137—22no—
82D4×D137—22no—
83D4×D137—41no—
84D4×D137—41no—
85D4×D137—41no—
86C2×C4×D137—21no—
87C2×C4×D137—21no—
88C2×C4×D137C2311yescharacteristic
89C2×C4×D137C2311yescharacteristic
90#14555497—22no—
91#14555497—41no—
92D1096—22no—
93D1096—22no—
94#14555495—22no—
95#14555495—41no—
96#14555495D414yescomplemented
97#14555493—22no—
98C2×#11764040C2311yescharacteristic
99C2×#11764040C2311yescharacteristic
100C8×D137—22no—
101C4⋊Dic137D411yescharacteristic
102C4⋊Dic137C2311yescharacteristic
103#14555489—21no—
104C22⋊Dic137—21no—
105#14555487—21no—
106#14555487—21no—
107#14555487—21no—
108#14555487—21no—
109#14555487D411yescharacteristic
110#14555487D411yescharacteristic
111C2×Q8×C137C2311yescharacteristic
112C2×D4×C137C2311yescharacteristic
113C2×D4×C137C2311yescharacteristic
114C22×C4×C137—21no—
115SD16×C137—22no—
116D8×C137—22no—
117C2×C8×C137C2311yescharacteristic
118C2×C8×C137C2311yescharacteristic
119C4⋊C4×C137C2311yescharacteristic
120C22⋊C4×C137—21no—
121C22⋊C4×C137—21no—
122C22⋊C4×C137—21no—
123C42×C137C2311yescharacteristic
124C22×D137—21no—
125C22×D137—21no—
126C22×D137—21no—
127C22×D137—42no—
128C22×D137—42no—
129C22×D137—42no—
130C22×D137—42no—
131C22×D137—42no—
132C22×D137—42no—
133C22×D137—81no—
134C22×D137—81no—
135C22×D137—81no—
136C22×D137C2×D411yescharacteristic
137C22×D137C2×D411yescharacteristic
138D548—22no—
139D548—21no—
140D548—21no—
141D548—21no—
142D548—21no—
143D548—21no—
144D548—42no—
145D548—42no—
146D548—42no—
147D548—42no—
148D548—81no—
149D548C2×D412yescomplemented
150C22⋊D137—42no—
151C22⋊D137—42no—
152C22⋊D137—42no—
153C22⋊D137—81no—
154C2×Dic137—21no—
155C2×Dic137C2×D411yescharacteristic
156C2×Dic137C2×D411yescharacteristic
157C4×D137—21no—
158C4×D137—21no—
159C4×D137—21no—
160C4×D137—21no—
161C4×D137—42no—
162C4×D137—81no—
163#11764040—21no—
164#11764040C2×D412yescomplemented
165C23×C137—21no—
166C23×C137—21no—
167Q8×C137—21no—
168Q8×C137—21no—
169D4×C137—21no—
170D4×C137—21no—
171D4×C137—21no—
172D4×C137C2×D412yes—
173C2×C4×C137—21no—
174C2×C4×C137—21no—
175C2×C4×C137—42no—
176C2×C4×C137C2×D411yescharacteristic
177C2×C4×C137C2×D411yescharacteristic
178C2×C4×C137C2411yescharacteristic
179C8×C137—21no—
180C8×C137—21no—
181C22×C137—22no—
182C22×C137—41no—
183C22×C137—41no—
184C22×C137—41no—
185C22×C137—41no—
186C22×C137—41no—
187C22×C137C22×D411yescharacteristic
188C2×D137—21no—
189C2×D137—21no—
190C2×D137—21no—
191C2×D137—22no—
192C2×D137—21no—
193C2×D137—21no—
194C2×D137—41no—
195C2×D137—41no—
196C2×D137—41no—
197C2×D137—41no—
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×C137—41no—
210C4×C137C22×D411yescharacteristic
211C4×C137H(5,𝔽2)11yescharacteristic
212Dic137—21no—
213Dic137—21no—
214Dic137—41no—
215C2×C137—42no—
216C2×C137—81no—
217C2×C137D4211yescharacteristic
218C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C137#57611yescharacteristic
220D137—22no—
221D137—41no—
222D137—42no—
223D137—81no—
224D137—81no—
225C137#326411yescharacteristic, complemented, Sylow
226#3264—1371nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1371no—
228C22×D8—1371no—
229#546—1371no—
230D42—1371no—
231#496—1371no—
232#469—1371no—
233#465—1371no—
234#463D13711yescharacteristic, complemented
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×SD16—1371no—
250C2×SD16C2×D13711yescharacteristic, complemented
251C2×D8—2744no—
252C2×D8—1371no—
253C2×D8—1371no—
254C2×D8—1371no—
255C2×D8—1371no—
256C2×D8—1371no—
257C2×D8C2×D13711yescharacteristic
258C2×M16—1371no—
259C22×C8—1371no—
260#127—1371no—
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×D4C2×D13711yescharacteristic, complemented
275C2×C22⋊C4—2741no—
276#106—1371no—
277C4⋊C8C2×D13711yescharacteristic, complemented
278#103C2×D13711yescharacteristic, complemented
279#102—1371no—
280#102—1371no—
281#102—1371no—
282#102—1371no—
283#102C2×D13711yescharacteristic, complemented
284C22⋊C8—1371no—
285C22⋊C8—1371no—
286#97—1371no—
287C24—2741no—
288C24—2741no—
289C24—2741no—
290#41—2742no—
291#41—5481no—
292C2×Q8C22×D13711yescharacteristic
293C2×D4—2741no—
294C2×D4—2741no—
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—5484no—
306C2×D4—5481no—
307C2×D4—5481no—
308C2×D4—5481no—
309C2×D4—1371no—
310C2×D4—1371no—
311C2×D4—1371no—
312C2×D4—1371no—
313C2×D4C22×D13711yescharacteristic
314C2×D4C22×D13711yescharacteristic
315C22×C4—21no—
316C22×C4—2741no—
317C22×C4—2741no—
318C22×C4—1371no—
319C22×C4—1371no—
320SD16—22no—
321SD16—2742no—
322SD16—5481no—
323D8—22no—
324D8—2742no—
325D8—2742no—
326D8—2742no—
327D8—5481no—
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⋊C4C22×D13711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
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—2741no—
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—2742no—
372D4—2741no—
373D4—2741no—
374D4—2741no—
375D4—2741no—
376D4—2741no—
377D4—5482no—
378D4—5482no—
379D4—5482no—
380D4—5482no—
381D4—5482no—
382D4—5482no—
383D4—5482no—
384D4—10961no—
385D4—10961no—
386D4—1372no—
387D4D4×D13712yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—2741no—
392C2×C4—2741no—
393C2×C4—2741no—
394C2×C4—2741no—
395C2×C4—2741no—
396C2×C4—5482no—
397C2×C4—10961no—
398C2×C4—1371no—
399C2×C4—1371no—
400C2×C4D4×D13711yescharacteristic
401C2×C4D4×D13711yescharacteristic
402C2×C4C23×D13711yescharacteristic
403C8—21no—
404C8—21no—
405C8—2741no—
406C8—1372no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—2741no—
414C22—2741no—
415C22—2741no—
416C22—2741no—
417C22—2741no—
418C22—2742no—
419C22—5481no—
420C22—5481no—
421C22—5481no—
422C22—5481no—
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—41no—
436C4—2741no—
437C4—2741no—
438C4—5481no—
439C4C2×D4×D13711yescharacteristic
440C4#1798413411yescharacteristic
441C2—42no—
442C2—81no—
443C2—2742no—
444C2—5481no—
445C2—5482no—
446C2—10961no—
447C2—10961no—
448C2#2257803711yescharacteristic
449C2AGL(1,ℤ/8)×D13711yescharacteristic
450C2#2257820511yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#463×C137C2
socleC22×C137C22×D4
hypercenter#463D137
hypocenterC137#3264
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#17984115C2semidirect
2. C2×D8×D137C2semidirect
3. #22578046C2semidirect
4. D4×D548C2semidirect
5. #22577793C2semidirect
6. #22577655C2semidirect
7. #22577635C2semidirect
8. #22577599C2semidirect
9. #22577552C2semidirect
10. #22577551C2semidirect
11. #22577538C2semidirect
12. #22577535C2semidirect
13. #22577513C2semidirect
14. #22577473C2semidirect
15. #463×C137C2semidirect
16. C2×#14555511C22semidirect
19. C2×D4×D137C22semidirect
20. C2×D4×D137C22semidirect
22. C2×#14555497C22semidirect
23. C2×#14555497C22semidirect
24. C2×D1096C22semidirect
25. C2×D1096C22semidirect
27. C2×#14555495C22semidirect
28. C2×#14555495C22semidirect
30. C2×#14555493C22semidirect
31. C2×C8×D137C22semidirect
32. #17984085C22semidirect
34. #17984064C22semidirect
36. #17984063C22semidirect
39. #17984059C22semidirect
44. C4×D548C22semidirect
46. #17984019C22semidirect
47. #17984010C22semidirect
48. #17984010C22semidirect
49. #17984009C22semidirect
50. #17984008C22semidirect
51. #17984000C22semidirect
52. #17984000C22semidirect
53. #17983999C22semidirect
54. C2×SD16×C137C22semidirect
56. #124×C137C22semidirect
57. C4×D4×C137C22semidirect
58. C4⋊C8×C137C22semidirect
59. #103×C137C22semidirect
60. #102×C137C22semidirect
96. #14555495D4semidirect
149. D548C2×D4semidirect
164. #11764040C2×D4semidirect
205. C2×D137AGL(1,ℤ/8)semidirect
225. C137#3264semidirect
234. #463D137semidirect
250. C2×SD16C2×D137semidirect
261. #124C2×D137semidirect
274. C4×D4C2×D137semidirect
277. C4⋊C8C2×D137semidirect
278. #103C2×D137semidirect
283. #102C2×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
D4×D5482
#225777932
#225776552
#225776352
#225775992
#225775522
#225775512
#225775382
#225775352
#225775132
#225774732
#463×C1372
#3264137