Finite Group Databasebeta

← #31792264

Subgroups — #31792264

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×#17984112C211yesmaximal, characteristic, complemented
2C2×D8×D137C211yesmaximal, characteristic, complemented
3#22578048C211yesmaximal, characteristic, complemented
4#22578038C211yesmaximal, characteristic, complemented
5#22577790C211yesmaximal, characteristic, complemented
6#22577655C211yesmaximal, characteristic, complemented
7#22577631C211yesmaximal, characteristic, complemented
8#22577619C211yesmaximal, characteristic, complemented
9#22577555C211yesmaximal, characteristic, complemented
10#22577545C211yesmaximal, characteristic, complemented
11#22577538C211yesmaximal, characteristic, complemented
12#22577535C211yesmaximal, characteristic, complemented
13#22577513C211yesmaximal, characteristic, complemented
14#22577473C211yesmaximal, characteristic, complemented
15#459×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic, complemented
17C2×D4×D137—21no—
18C2×D4×D137C2211yescharacteristic
19C2×D4×D137C2211yescharacteristic, complemented
20C2×D4×D137C2211yescharacteristic, complemented
21#17984112—24no—
22C2×#14555499C2211yescharacteristic, complemented
23C2×#14555498C2211yescharacteristic, complemented
24C2×#14555498C2211yescharacteristic, complemented
25C2×D1096C2211yescharacteristic
26C2×#14555495C2211yescharacteristic
27C2×#14555495C2211yescharacteristic, complemented
28C2×#14555495C2211yescharacteristic, complemented
29D8×D137—24no—
30C2×#14555493C2211yescharacteristic, complemented
31C2×C8×D137C2211yescharacteristic, complemented
32#17984076C2211yescharacteristic, complemented
33#17984072—21no—
34#17984068C2211yescharacteristic, complemented
35#17984063C2211yescharacteristic, complemented
36#17984062—21no—
37#17984059—21no—
38#17984059C2211yescharacteristic
39#17984059C2211yescharacteristic, complemented
40#17984058—21no—
41#17984056—21no—
42#17984056—21no—
43C4×D548C2211yescharacteristic
44C4×D548C2211yescharacteristic, complemented
45C2×#14555487—21no—
46#17984019C2211yescharacteristic, complemented
47#17984013C2211yescharacteristic, complemented
48#17984010C2211yescharacteristic, complemented
49#17984009C2211yescharacteristic, complemented
50#17984008C2211yescharacteristic, complemented
51#17984000C2211yescharacteristic, complemented
52#17984000C2211yescharacteristic, complemented
53#17983999C2211yescharacteristic, complemented
54C2×D8×C137C2211yescharacteristic, complemented
55C2×D8×C137C2211yescharacteristic, complemented
56#127×C137C2211yescharacteristic, complemented
57C4×D4×C137C2211yescharacteristic, complemented
58C4⋊C8×C137C2211yescharacteristic, complemented
59#102×C137C2211yescharacteristic, complemented
60#102×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C23×D137—21no—
63C23×D137—21no—
64#14555509—22no—
65#14555509—41no—
66C2×Dic274C2311yescharacteristic
67C2×D548—21no—
68C2×D548D411yescharacteristic
69C2×D548C2311yescharacteristic
70C2×D548C2311yescharacteristic
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73C2×C22⋊D137—21no—
74C2×C22⋊D137—21no—
75C2×C22⋊D137—21no—
76C2×C22⋊D137—21no—
77D4×D137—22no—
78D4×D137—22no—
79D4×D137—22no—
80D4×D137—41no—
81D4×D137—41no—
82D4×D137—41no—
83D4×D137—44no—
84C22×Dic137—21no—
85C2×C4×D137—21no—
86C2×C4×D137C2311yescharacteristic
87C2×C4×D137C2311yescharacteristic
88#14555499—22no—
89#14555498—22no—
90#14555498—41no—
91D1096—22no—
92#14555495—22no—
93#14555495—41no—
94#14555495D414yescomplemented
95#14555493—22no—
96C2×#11764040C2311yescharacteristic
97C2×#11764040C2311yescharacteristic
98C8×D137—22no—
99C4⋊Dic137—21no—
100C4⋊Dic137D411yescharacteristic
101C4⋊Dic137C2311yescharacteristic
102C22⋊Dic137—21no—
103C22⋊Dic137—21no—
104#14555487—21no—
105#14555487—21no—
106#14555487—21no—
107#14555487—21no—
108#14555487—21no—
109#14555487D411yescharacteristic
110#14555487D411yescharacteristic
111C2×D4×C137—21no—
112C2×D4×C137—21no—
113C2×D4×C137C2311yescharacteristic
114C2×D4×C137C2311yescharacteristic
115C2×D4×C137C2311yescharacteristic
116C22×C4×C137—21no—
117D8×C137—22no—
118D8×C137—22no—
119C2×C8×C137C2311yescharacteristic
120C2×C8×C137C2311yescharacteristic
121C4⋊C4×C137C2311yescharacteristic
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
137Dic274—21no—
138Dic274—21no—
139D548—22no—
140D548—21no—
141D548—21no—
142D548—21no—
143D548—81no—
144D548C2×D412yes—
145C22⋊D137—42no—
146C22⋊D137—42no—
147C22⋊D137—42no—
148C22⋊D137—42no—
149C22⋊D137—42no—
150C22⋊D137—42no—
151C2×Dic137—21no—
152C2×Dic137—21no—
153C2×Dic137—42no—
154C2×Dic137C2×D411yescharacteristic
155C2×Dic137C2×D411yescharacteristic
156C4×D137—21no—
157C4×D137—21no—
158C4×D137—21no—
159C4×D137—21no—
160C4×D137—42no—
161#11764040—21no—
162#11764040C2×D412yescomplemented
163C23×C137—21no—
164C23×C137—21no—
165C23×C137—21no—
166D4×C137—21no—
167D4×C137—21no—
168D4×C137—21no—
169D4×C137—21no—
170D4×C137—21no—
171D4×C137—42no—
172D4×C137—81no—
173D4×C137C2×D412yescomplemented
174C2×C4×C137—21no—
175C2×C4×C137—81no—
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×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—21no—
196C2×D137—21no—
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×C137C22×D411yescharacteristic
210C4×C137H(5,𝔽2)11yescharacteristic
211Dic137—21no—
212Dic137—21no—
213Dic137—41no—
214Dic137—41no—
215C2×C137—42no—
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—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#463—1371no—
235#459D13711yescharacteristic, complemented
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×SD16—1371no—
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#124—1371no—
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#103—1371no—
279#102—1371no—
280#102—1371no—
281#102—1371no—
282#102C2×D13711yescharacteristic, complemented
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×Q8—1371no—
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—21no—
316C22×C4—2741no—
317C22×C4—2741no—
318C22×C4—1371no—
319C22×C4—1371no—
320SD16—2742no—
321SD16—2742no—
322SD16—5481no—
323D8—22no—
324D8—22no—
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—2741no—
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—2741no—
367Q8—2741no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—81no—
375D4—2742no—
376D4—2741no—
377D4—2741no—
378D4—2741no—
379D4—5482no—
380D4—5482no—
381D4—5482no—
382D4—5482no—
383D4—5482no—
384D4—5482no—
385D4—10961no—
386D4—1372no—
387D4D4×D13712yescomplemented
388C2×C4—21no—
389C2×C4—81no—
390C2×C4—2741no—
391C2×C4—2741no—
392C2×C4—2741no—
393C2×C4—2741no—
394C2×C4—2741no—
395C2×C4—2741no—
396C2×C4—5482no—
397C2×C4—5482no—
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—41no—
414C22—41no—
415C22—41no—
416C22—2741no—
417C22—2741no—
418C22—2741no—
419C22—2741no—
420C22—2741no—
421C22—2742no—
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—2741no—
436C4—2741no—
437C4—5481no—
438C4—5481no—
439C4C2×D4×D13711yescharacteristic
440C4#1798413411yescharacteristic
441C2—42no—
442C2—81no—
443C2—81no—
444C2—2742no—
445C2—5481no—
446C2—5482no—
447C2—10961no—
448C2#2257803711yescharacteristic
449C2AGL(1,ℤ/8)×D13711yescharacteristic
450C2#2257820411yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#459×C137C2
socleC22×C137C22×D4
hypercenter#459D137
hypocenterC137#3264
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#17984112C2semidirect
2. C2×D8×D137C2semidirect
3. #22578048C2semidirect
4. #22578038C2semidirect
5. #22577790C2semidirect
6. #22577655C2semidirect
7. #22577631C2semidirect
8. #22577619C2semidirect
9. #22577555C2semidirect
10. #22577545C2semidirect
11. #22577538C2semidirect
12. #22577535C2semidirect
13. #22577513C2semidirect
14. #22577473C2semidirect
15. #459×C137C2semidirect
16. C2×#14555509C22semidirect
19. C2×D4×D137C22semidirect
20. C2×D4×D137C22semidirect
22. C2×#14555499C22semidirect
23. C2×#14555498C22semidirect
24. C2×#14555498C22semidirect
27. C2×#14555495C22semidirect
28. C2×#14555495C22semidirect
30. C2×#14555493C22semidirect
31. C2×C8×D137C22semidirect
32. #17984076C22semidirect
34. #17984068C22semidirect
35. #17984063C22semidirect
39. #17984059C22semidirect
44. C4×D548C22semidirect
46. #17984019C22semidirect
47. #17984013C22semidirect
48. #17984010C22semidirect
49. #17984009C22semidirect
50. #17984008C22semidirect
51. #17984000C22semidirect
52. #17984000C22semidirect
53. #17983999C22semidirect
54. C2×D8×C137C22semidirect
55. C2×D8×C137C22semidirect
56. #127×C137C22semidirect
57. C4×D4×C137C22semidirect
58. C4⋊C8×C137C22semidirect
59. #102×C137C22semidirect
60. #102×C137C22semidirect
94. #14555495D4semidirect
162. #11764040C2×D4semidirect
173. D4×C137C2×D4semidirect
205. C2×D137AGL(1,ℤ/8)semidirect
225. C137#3264semidirect
235. #459D137semidirect
256. C2×D8C2×D137semidirect
257. C2×D8C2×D137semidirect
260. #127C2×D137semidirect
274. C4×D4C2×D137semidirect
277. C4⋊C8C2×D137semidirect
282. #102C2×D137semidirect
283. #102C2×D137semidirect
387. D4D4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841122
C2×D8×D1372
#225780482
#225780382
#225777902
#225776552
#225776312
#225776192
#225775552
#225775452
#225775382
#225775352
#225775132
#225774732
#459×C1372
#3264137