Finite Group Databasebeta

← #31792274

Subgroups — #31792274

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×#17984116C211yesmaximal, characteristic, complemented
2C22×D1096C211yesmaximal, characteristic, complemented
3#22578049C211yesmaximal, characteristic, complemented
4D4×D548C211yesmaximal, characteristic, complemented
5#22577792C211yesmaximal, characteristic, complemented
6#22577656C211yesmaximal, characteristic, complemented
7#22577632C211yesmaximal, characteristic, complemented
8#22577619C211yesmaximal, characteristic, complemented
9#22577599C211yesmaximal, characteristic, complemented
10#22577556C211yesmaximal, characteristic, complemented
11#22577550C211yesmaximal, characteristic, complemented
12#22577539C211yesmaximal, characteristic, complemented
13#22577539C211yesmaximal, characteristic, complemented
14#22577516C211yesmaximal, characteristic, complemented
15#435×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic, complemented
17C22×D548C2211yescharacteristic
18C22×D548C2211yescharacteristic, complemented
19C22×D548C2211yescharacteristic, complemented
20C2×D4×D137—21no—
21#17984116—24no—
22C2×#14555499C2211yescharacteristic, complemented
23C2×#14555499C2211yescharacteristic, complemented
24C2×#14555499C2211yescharacteristic, complemented
25C2×D1096—24no—
26C2×D1096C2211yescharacteristic
27C2×D1096C2211yescharacteristic
28C2×D1096C2211yescharacteristic, complemented
29C2×D1096C2211yescharacteristic, complemented
30C2×D1096C2211yescharacteristic, complemented
31C2×D1096C2211yescharacteristic, complemented
32#17984085C2211yescharacteristic, complemented
33#17984076C2211yescharacteristic, complemented
34#17984073—21no—
35#17984067C2211yescharacteristic, complemented
36#17984063—21no—
37#17984062C2211yescharacteristic
38#17984062C2211yescharacteristic, complemented
39#17984062C2211yescharacteristic, complemented
40#17984061—21no—
41#17984058—21no—
42#17984058—21no—
43#17984056—21no—
44C4×D548C2211yescharacteristic, complemented
45C4×D548C2211yescharacteristic, complemented
46C22⋊C4×D137—21no—
47#17984020C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984010C2211yescharacteristic, complemented
50#17984010C2211yescharacteristic, complemented
51#17984010C2211yescharacteristic, complemented
52#17984010C2211yescharacteristic, complemented
53#17984010C2211yescharacteristic, complemented
54C2×M16×C137C2211yescharacteristic, complemented
55C22×C8×C137C2211yescharacteristic, complemented
56C4×D4×C137C2211yescharacteristic
57C4⋊C8×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic, complemented
59C22⋊C8×C137C2211yescharacteristic, complemented
60#97×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C23×D137—21no—
63C23×D137—21no—
64#14555510—22no—
65#14555510—41no—
66C2×Dic274C2311yescharacteristic
67C2×D548—22no—
68C2×D548—22no—
69C2×D548—22no—
70C2×D548—21no—
71C2×D548—21no—
72C2×D548—21no—
73C2×D548—21no—
74C2×D548—21no—
75C2×D548—41no—
76C2×D548—41no—
77C2×D548—41no—
78C2×D548C2311yescharacteristic
79C2×D548C2311yescharacteristic
80C2×D548C2311yescharacteristic
81C2×D548C2311yescharacteristic
82C2×D548C2311yescharacteristic
83C2×C22⋊D137—21no—
84C2×C22⋊D137—21no—
85C2×C22⋊D137—21no—
86C2×C22⋊D137—21no—
87D4×D137—44no—
88C2×C4×D137—21no—
89C2×C4×D137—21no—
90C2×C4×D137—21no—
91#14555499—22no—
92#14555499—22no—
93#14555499—41no—
94D1096—22no—
95D1096—22no—
96D1096—22no—
97D1096—22no—
98D1096—41no—
99D1096D414yescomplemented
100C4⋊Dic137C2311yescharacteristic
101C4⋊Dic137C2311yescharacteristic
102#14555489—21no—
103C22⋊Dic137—21no—
104#14555487—21no—
105#14555487—21no—
106#14555487—21no—
107#14555487—21no—
108#14555487—21no—
109#14555487—21no—
110#14555487—21no—
111C2×D4×C137D411yescharacteristic
112C22×C4×C137C2311yescharacteristic
113C22×C4×C137C2311yescharacteristic
114M16×C137—22no—
115C2×C8×C137—22no—
116C2×C8×C137C2311yescharacteristic
117C2×C8×C137C2311yescharacteristic
118C2×C8×C137C2311yescharacteristic
119C2×C8×C137C2311yescharacteristic
120C4⋊C4×C137D411yescharacteristic
121C22⋊C4×C137D411yescharacteristic
122C22⋊C4×C137D411yescharacteristic
123C42×C137C2311yescharacteristic
124C22×D137—21no—
125C22×D137—21no—
126C22×D137—21no—
127C22×D137—21no—
128C22×D137—21no—
129C22×D137—42no—
130C22×D137—42no—
131C22×D137—42no—
132C22×D137—42no—
133C22×D137—42no—
134C22×D137—42no—
135C22×D137—81no—
136C22×D137—81no—
137C22×D137—81no—
138Dic274—21no—
139Dic274—21no—
140D548—21no—
141D548—21no—
142D548—21no—
143D548—21no—
144D548—21no—
145D548—21no—
146D548—21no—
147D548—21no—
148D548—42no—
149D548—42no—
150D548—42no—
151D548—42no—
152D548—81no—
153D548C2×D412yes—
154D548C2×D412yescomplemented
155C22⋊D137—42no—
156C22⋊D137—42no—
157C22⋊D137—42no—
158C22⋊D137—81no—
159C2×Dic137—21no—
160C2×Dic137—21no—
161C2×Dic137—21no—
162C4×D137—42no—
163C4×D137—42no—
164C4×D137—81no—
165C23×C137C2×D411yescharacteristic
166C23×C137C2×D411yescharacteristic
167D4×C137—22no—
168C2×C4×C137—21no—
169C2×C4×C137—21no—
170C2×C4×C137—21no—
171C2×C4×C137—21no—
172C2×C4×C137C2×D411yescharacteristic
173C2×C4×C137C2×D411yescharacteristic
174C2×C4×C137C2×D411yescharacteristic
175C2×C4×C137C2×D411yescharacteristic
176C2×C4×C137C2411yescharacteristic
177C8×C137—21no—
178C8×C137—21no—
179C8×C137—21no—
180C8×C137C2×D412yescomplemented
181C22×C137—21no—
182C22×C137—21no—
183C22×C137—21no—
184C22×C137—21no—
185C22×C137—21no—
186C22×C137AGL(1,ℤ/8)12yescomplemented
187C22×C137C22×D411yescharacteristic
188C2×D137—22no—
189C2×D137—22no—
190C2×D137—41no—
191C2×D137—41no—
192C2×D137—41no—
193C2×D137—41no—
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—41no—
203C2×D137—84no—
204C2×D137—84no—
205C2×D137—84no—
206C4×C137—21no—
207C4×C137—21no—
208C4×C137—21no—
209C4×C137—21no—
210C4×C137C22×D411yescharacteristic
211C4×C137H(5,𝔽2)11yescharacteristic
212Dic137—41no—
213Dic137—41no—
214Dic137—41no—
215C2×C137—22no—
216C2×C137—41no—
217C2×C137D4211yescharacteristic
218C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C137#57611yescharacteristic
220D137—42no—
221D137—42no—
222D137—81no—
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#463—1371no—
235#459—1371no—
236#450—1371no—
237#449—1371no—
238#447—1371no—
239#447—1371no—
240#442—1371no—
241#435D13711yescharacteristic, complemented
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×D8—1371no—
257C2×D8—1371no—
258C2×M16C2×D13711yescharacteristic, complemented
259C22×C8C2×D13711yescharacteristic, complemented
260#127—1371no—
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
275C2×C22⋊C4—2741no—
276#106—1371no—
277C4⋊C8C2×D13711yescharacteristic, complemented
278#103—1371no—
279#102—1371no—
280#102—1371no—
281#102—1371no—
282#102—1371no—
283#102—1371no—
284C22⋊C8C2×D13711yescharacteristic, complemented
285C22⋊C8C2×D13711yescharacteristic, complemented
286#97C2×D13711yescharacteristic, complemented
287C24—2741no—
288C24—2741no—
289C24—2741no—
290#41—2742no—
291#41—5481no—
292C2×Q8—1371no—
293C2×D4—2742no—
294C2×D4—2742no—
295C2×D4—2742no—
296C2×D4—2741no—
297C2×D4—2741no—
298C2×D4—2741no—
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×D4—1371no—
313C2×D4—1371no—
314C2×D4D54811yescharacteristic
315C22×C4—2741no—
316C22×C4—2741no—
317C22×C4—2741no—
318C22×C4C22×D13711yescharacteristic
319C22×C4C22×D13711yescharacteristic
320SD16—2742no—
321SD16—2742no—
322SD16—5481no—
323D8—2742no—
324D8—2742no—
325D8—2742no—
326D8—2742no—
327D8—5481no—
328D8—1374no—
329M16—22no—
330C2×C8—22no—
331C2×C8C22×D13711yescharacteristic
332C2×C8C22×D13711yescharacteristic
333C2×C8C22×D13711yescharacteristic
334C2×C8C22×D13711yescharacteristic
335C4⋊C4—2741no—
336C4⋊C4—1371no—
337C4⋊C4—1371no—
338C4⋊C4D54811yescharacteristic
339C22⋊C4—2741no—
340C22⋊C4—2741no—
341C22⋊C4—2741no—
342C22⋊C4—2741no—
343C22⋊C4—2741no—
344C22⋊C4—2741no—
345C22⋊C4—2741no—
346C22⋊C4—2741no—
347C22⋊C4D54811yescharacteristic
348C22⋊C4D54811yescharacteristic
349C42C22×D13711yescharacteristic
350C23—2741no—
351C23—2741no—
352C23—2741no—
353C23—2741no—
354C23—2741no—
355C23—5482no—
356C23—5482no—
357C23—5482no—
358C23—5482no—
359C23—5482no—
360C23—5482no—
361C23—10961no—
362C23—10961no—
363C23—10961no—
364C23C2×D54811yescharacteristic
365C23C2×D54811yescharacteristic
366Q8—2741no—
367Q8—2741no—
368D4—22no—
369D4—2741no—
370D4—2741no—
371D4—2741no—
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—
387D4—1372no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—2741no—
393C2×C4—2741no—
394C2×C4—2741no—
395C2×C4—5482no—
396C2×C4—5482no—
397C2×C4—10961no—
398C2×C4C2×D54811yescharacteristic
399C2×C4C2×D54811yescharacteristic
400C2×C4C2×D54811yescharacteristic
401C2×C4C2×D54811yescharacteristic
402C2×C4C23×D13711yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8D4×D13712yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—2742no—
413C22—2742no—
414C22—5481no—
415C22—5481no—
416C22—5481no—
417C22—5481no—
418C22—5481no—
419C22—5481no—
420C22—5481no—
421C22—5481no—
422C22—5481no—
423C22—5481no—
424C22—5481no—
425C22—5481no—
426C22—5481no—
427C22—10964no—
428C22—10964no—
429C22—10964no—
430C22#1798411612yescomplemented
431C22C22×D54811yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—5481no—
437C4—5481no—
438C4—5481no—
439C4C2×D4×D13711yescharacteristic
440C4#1798413411yescharacteristic
441C2—22no—
442C2—41no—
443C2—5482no—
444C2—5482no—
445C2—10961no—
446C2—10961no—
447C2—10961no—
448C2D4×D54811yescharacteristic
449C2C2×#1798411611yescharacteristic
450C2#2257820811yescharacteristic
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#435×C137C2
socleC22×C137C22×D4
hypercenter#435D137
hypocenterC137#3264
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#17984116C2semidirect
2. C22×D1096C2semidirect
3. #22578049C2semidirect
4. D4×D548C2semidirect
5. #22577792C2semidirect
6. #22577656C2semidirect
7. #22577632C2semidirect
8. #22577619C2semidirect
9. #22577599C2semidirect
10. #22577556C2semidirect
11. #22577550C2semidirect
12. #22577539C2semidirect
13. #22577539C2semidirect
14. #22577516C2semidirect
15. #435×C137C2semidirect
16. C2×#14555510C22semidirect
18. C22×D548C22semidirect
19. C22×D548C22semidirect
22. C2×#14555499C22semidirect
23. C2×#14555499C22semidirect
24. C2×#14555499C22semidirect
28. C2×D1096C22semidirect
29. C2×D1096C22semidirect
30. C2×D1096C22semidirect
31. C2×D1096C22semidirect
32. #17984085C22semidirect
33. #17984076C22semidirect
35. #17984067C22semidirect
38. #17984062C22semidirect
39. #17984062C22semidirect
44. C4×D548C22semidirect
45. C4×D548C22semidirect
47. #17984020C22semidirect
48. #17984013C22semidirect
49. #17984010C22semidirect
50. #17984010C22semidirect
51. #17984010C22semidirect
52. #17984010C22semidirect
53. #17984010C22semidirect
54. C2×M16×C137C22semidirect
55. C22×C8×C137C22semidirect
57. C4⋊C8×C137C22semidirect
58. C22⋊C8×C137C22semidirect
59. C22⋊C8×C137C22semidirect
60. #97×C137C22semidirect
99. D1096D4semidirect
154. D548C2×D4semidirect
180. C8×C137C2×D4semidirect
186. C22×C137AGL(1,ℤ/8)semidirect
225. C137#3264semidirect
241. #435D137semidirect
258. C2×M16C2×D137semidirect
259. C22×C8C2×D137semidirect
277. C4⋊C8C2×D137semidirect
284. C22⋊C8C2×D137semidirect
285. C22⋊C8C2×D137semidirect
286. #97C2×D137semidirect
406. C8D4×D137semidirect
430. C22#17984116semidirect

Maximal subgroups

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

subgroupindex
C2×#179841162
C22×D10962
#225780492
D4×D5482
#225777922
#225776562
#225776322
#225776192
#225775992
#225775562
#225775502
#225775392
#225775162
#435×C1372
#3264137