Finite Group Databasebeta

← #31792270

Subgroups — #31792270

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×#17984113C211yesmaximal, characteristic, complemented
2C22×#14555495C211yesmaximal, characteristic, complemented
3D4×D548C211yesmaximal, characteristic, complemented
4#22577793C211yesmaximal, characteristic, complemented
5#22577654C211yesmaximal, characteristic, complemented
6#22577634C211yesmaximal, characteristic, complemented
7#22577615C211yesmaximal, characteristic, complemented
8#22577598C211yesmaximal, characteristic, complemented
9#22577553C211yesmaximal, characteristic, complemented
10#22577548C211yesmaximal, characteristic, complemented
11#22577537C211yesmaximal, characteristic, complemented
12#22577536C211yesmaximal, characteristic, complemented
13#22577513C211yesmaximal, characteristic, complemented
14#22577475C211yesmaximal, characteristic, complemented
15#546×C137C211yesmaximal, characteristic, complemented
16C22×D548C2211yescharacteristic
17C22×D548C2211yescharacteristic, complemented
18C2×D4×D137—21no—
19#17984113—24no—
20C2×#14555497C2211yescharacteristic, complemented
21C2×#14555497C2211yescharacteristic, complemented
22C2×#14555497C2211yescharacteristic, complemented
23C2×#14555495—24no—
24C2×#14555495C2211yescharacteristic
25C2×#14555495C2211yescharacteristic
26C2×#14555495C2211yescharacteristic, complemented
27C2×#14555495C2211yescharacteristic, complemented
28C2×#14555495C2211yescharacteristic, complemented
29C2×#14555495C2211yescharacteristic, complemented
30C2×#14555494C2211yescharacteristic, complemented
31C22×#11764040C2211yescharacteristic, complemented
32#17984085C2211yescharacteristic, complemented
33#17984063—21no—
34#17984062C2211yescharacteristic
35#17984062C2211yescharacteristic, complemented
36#17984058—21no—
37#17984058—21no—
38C4×D548C2211yescharacteristic, complemented
39#17984019C2211yescharacteristic, complemented
40C4⋊#11764040C2211yescharacteristic, complemented
41#17984012C2211yescharacteristic, complemented
42#17984009C2211yescharacteristic, complemented
43#17984009C2211yescharacteristic, complemented
44#17984008C2211yescharacteristic, complemented
45#17984008C2211yescharacteristic, complemented
46#17984008C2211yescharacteristic, complemented
47C22⋊#11764040C2211yescharacteristic, complemented
48C22⋊#11764040C2211yescharacteristic, complemented
49#17983999C2211yescharacteristic, complemented
50C2×#41×C137C2211yescharacteristic, complemented
51C22×D4×C137C2211yescharacteristic, complemented
52#124×C137C2211yescharacteristic, complemented
53#123×C137—21no—
54C22⋊Q8×C137C2211yescharacteristic, complemented
55#121×C137—21no—
56#121×C137C2211yescharacteristic, complemented
57#120×C137—21no—
58C4×D4×C137C2211yescharacteristic
59C4×D4×C137C2211yescharacteristic, complemented
60C2×C22⋊C4×C137—21no—
61C23×D137—21no—
62C23×D137—21no—
63C2×D548—22no—
64C2×D548—22no—
65C2×D548—21no—
66C2×D548—21no—
67C2×D548—21no—
68C2×D548—21no—
69C2×D548—41no—
70C2×D548—41no—
71C2×D548C2311yescharacteristic
72C2×D548C2311yescharacteristic
73C2×D548C2311yescharacteristic
74C2×C22⋊D137—21no—
75C2×C22⋊D137—21no—
76D4×D137—44no—
77C2×C4×D137—21no—
78#14555497—22no—
79#14555497—22no—
80#14555497—41no—
81#14555495—22no—
82#14555495—22no—
83#14555495—22no—
84#14555495—22no—
85#14555495—41no—
86#14555495D414yescomplemented
87#14555494—22no—
88C2×#11764040—22no—
89C2×#11764040C2311yescharacteristic
90C2×#11764040C2311yescharacteristic
91C2×#11764040C2311yescharacteristic
92C2×#11764040C2311yescharacteristic
93C4⋊Dic137C2311yescharacteristic
94#14555487—21no—
95#14555487—21no—
96#14555487—21no—
97C24×C137—21no—
98#41×C137—22no—
99#41×C137—41no—
100C2×Q8×C137C2311yescharacteristic
101C2×D4×C137—22no—
102C2×D4×C137—21no—
103C2×D4×C137—21no—
104C2×D4×C137—21no—
105C2×D4×C137—41no—
106C2×D4×C137D411yescharacteristic
107C2×D4×C137C2311yescharacteristic
108C2×D4×C137C2311yescharacteristic
109C22×C4×C137—21no—
110C22×C4×C137—21no—
111C22×C4×C137C2311yescharacteristic
112C22×C4×C137C2311yescharacteristic
113C4⋊C4×C137—21no—
114C4⋊C4×C137D411yescharacteristic
115C4⋊C4×C137C2311yescharacteristic
116C22⋊C4×C137—21no—
117C22⋊C4×C137—21no—
118C22⋊C4×C137—21no—
119C22⋊C4×C137—21no—
120C22⋊C4×C137—21no—
121C22⋊C4×C137D411yescharacteristic
122C22⋊C4×C137D411yescharacteristic
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—81no—
132C22×D137—81no—
133D548—21no—
134D548—21no—
135D548—21no—
136D548—21no—
137D548—21no—
138D548—42no—
139D548—42no—
140D548—42no—
141D548—81no—
142D548C2×D412yes—
143C22⋊D137—42no—
144C22⋊D137—42no—
145C2×Dic137—21no—
146C4×D137—42no—
147#11764040—21no—
148#11764040—21no—
149#11764040—21no—
150#11764040C2×D412yescomplemented
151C23×C137—21no—
152C23×C137—21no—
153C23×C137—42no—
154C23×C137—42no—
155C23×C137—81no—
156C23×C137C2×D411yescharacteristic
157C23×C137C2×D411yescharacteristic
158Q8×C137—21no—
159Q8×C137—21no—
160D4×C137—22no—
161D4×C137—21no—
162D4×C137—21no—
163D4×C137—21no—
164D4×C137—42no—
165D4×C137—42no—
166D4×C137—81no—
167D4×C137C2×D412yescomplemented
168C2×C4×C137—21no—
169C2×C4×C137—21no—
170C2×C4×C137—21no—
171C2×C4×C137—21no—
172C2×C4×C137—21no—
173C2×C4×C137—21no—
174C2×C4×C137—42no—
175C2×C4×C137—81no—
176C2×C4×C137C2×D411yescharacteristic
177C2×C4×C137C2×D411yescharacteristic
178C2×C4×C137C2×D411yescharacteristic
179C2×C4×C137C2×D411yescharacteristic
180C2×C4×C137C2411yescharacteristic
181C22×C137—21no—
182C22×C137—21no—
183C22×C137—21no—
184C22×C137—22no—
185C22×C137—21no—
186C22×C137—21no—
187C22×C137—41no—
188C22×C137—41no—
189C22×C137—41no—
190C22×C137—41no—
191C22×C137—41no—
192C22×C137—84no—
193C22×C137AGL(1,ℤ/8)12yescomplemented
194C22×C137C22×D411yescharacteristic
195C2×D137—22no—
196C2×D137—41no—
197C2×D137—41no—
198C2×D137—41no—
199C2×D137—41no—
200C2×D137—41no—
201C2×D137—41no—
202C2×D137—41no—
203C2×D137—41no—
204C2×D137—84no—
205C2×D137—84no—
206C4×C137—21no—
207C4×C137—21no—
208C4×C137—21no—
209C4×C137—21no—
210C4×C137—41no—
211C4×C137—41no—
212C4×C137C22×D411yescharacteristic
213C4×C137H(5,𝔽2)11yescharacteristic
214Dic137—41no—
215C2×C137—22no—
216C2×C137—41no—
217C2×C137—42no—
218C2×C137—81no—
219C2×C137D4211yescharacteristic
220C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C137#57611yescharacteristic
222D137—42no—
223D137—81no—
224D137—81no—
225C137#326411yescharacteristic, complemented, Sylow
226#3264—1371nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1371no—
228C22×D8—1371no—
229#546D13711yescharacteristic, complemented
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#435—1371no—
242C2×#41C2×D13711yescharacteristic, complemented
243C22×D4—2741no—
244C22×D4—1371no—
245C22×D4—1371no—
246C22×D4C2×D13711yescharacteristic, complemented
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×M16—1371no—
259C22×C8—1371no—
260#127—1371no—
261#124C2×D13711yescharacteristic, complemented
262#123—21no—
263C22⋊Q8C2×D13711yescharacteristic, complemented
264#121—21no—
265#121—2741no—
266#121—1371no—
267#121—1371no—
268#121C2×D13711yescharacteristic, complemented
269#120—21no—
270#120—2741no—
271#120—2741no—
272C4×D4—1371no—
273C4×D4C2×D13711yescharacteristic
274C4×D4C2×D13711yescharacteristic, complemented
275C2×C22⋊C4—21no—
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#97—1371no—
287C24—21no—
288C24—2741no—
289C24—2741no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D13711yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—2742no—
299C2×D4—2742no—
300C2×D4—2741no—
301C2×D4—2741no—
302C2×D4—2741no—
303C2×D4—2741no—
304C2×D4—2741no—
305C2×D4—2741no—
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—21no—
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—2742no—
330C2×C8—2742no—
331C2×C8—1371no—
332C2×C8—1371no—
333C2×C8—1371no—
334C2×C8—1371no—
335C4⋊C4—21no—
336C4⋊C4—1371no—
337C4⋊C4C22⋊D13711yescharacteristic
338C4⋊C4C22×D13711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—2741no—
345C22⋊C4—2741no—
346C22⋊C4—2741no—
347C22⋊C4C22⋊D13711yescharacteristic
348C22⋊C4C22⋊D13711yescharacteristic
349C42C22×D13711yescharacteristic
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—2741no—
356C23—2741no—
357C23—2741no—
358C23—5482no—
359C23—5482no—
360C23—5482no—
361C23—5482no—
362C23—10961no—
363C23—10961no—
364C23C2×C22⋊D13711yescharacteristic
365C23C2×C22⋊D13711yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—2741no—
376D4—2741no—
377D4—2741no—
378D4—2741no—
379D4—2741no—
380D4—5482no—
381D4—5482no—
382D4—5482no—
383D4—5482no—
384D4—5482no—
385D4—10961no—
386D4—1372no—
387D4D4×D13712yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—2741no—
397C2×C4—5482no—
398C2×C4C2×C22⋊D13711yescharacteristic
399C2×C4C2×C22⋊D13711yescharacteristic
400C2×C4C2×C22⋊D13711yescharacteristic
401C2×C4C2×C22⋊D13711yescharacteristic
402C2×C4C23×D13711yescharacteristic
403C8—2741no—
404C8—2741no—
405C8—2741no—
406C8—1372no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—2742no—
420C22—5481no—
421C22—5481no—
422C22—5481no—
423C22—5481no—
424C22—5481no—
425C22—5481no—
426C22—5481no—
427C22—5481no—
428C22—10964no—
429C22—10964no—
430C22#1798411312yescomplemented
431C22C22×C22⋊D13711yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—5481no—
439C4C2×D4×D13711yescharacteristic
440C4#1798413311yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—5482no—
446C2—10961no—
447C2—10961no—
448C2D4×C22⋊D13711yescharacteristic
449C2C2×#1798411311yescharacteristic
450C2#2257820611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#546×C137C2
socleC22×C137C22×D4
hypercenter#546D137
hypocenterC137#3264
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#17984113C2semidirect
2. C22×#14555495C2semidirect
3. D4×D548C2semidirect
4. #22577793C2semidirect
5. #22577654C2semidirect
6. #22577634C2semidirect
7. #22577615C2semidirect
8. #22577598C2semidirect
9. #22577553C2semidirect
10. #22577548C2semidirect
11. #22577537C2semidirect
12. #22577536C2semidirect
13. #22577513C2semidirect
14. #22577475C2semidirect
15. #546×C137C2semidirect
17. C22×D548C22semidirect
20. C2×#14555497C22semidirect
21. C2×#14555497C22semidirect
22. C2×#14555497C22semidirect
26. C2×#14555495C22semidirect
27. C2×#14555495C22semidirect
28. C2×#14555495C22semidirect
29. C2×#14555495C22semidirect
30. C2×#14555494C22semidirect
31. C22×#11764040C22semidirect
32. #17984085C22semidirect
35. #17984062C22semidirect
38. C4×D548C22semidirect
39. #17984019C22semidirect
40. C4⋊#11764040C22semidirect
41. #17984012C22semidirect
42. #17984009C22semidirect
43. #17984009C22semidirect
44. #17984008C22semidirect
45. #17984008C22semidirect
46. #17984008C22semidirect
47. C22⋊#11764040C22semidirect
48. C22⋊#11764040C22semidirect
49. #17983999C22semidirect
50. C2×#41×C137C22semidirect
51. C22×D4×C137C22semidirect
52. #124×C137C22semidirect
54. C22⋊Q8×C137C22semidirect
56. #121×C137C22semidirect
59. C4×D4×C137C22semidirect
86. #14555495D4semidirect
150. #11764040C2×D4semidirect
167. D4×C137C2×D4semidirect
193. C22×C137AGL(1,ℤ/8)semidirect
225. C137#3264semidirect
229. #546D137semidirect
242. C2×#41C2×D137semidirect
246. C22×D4C2×D137semidirect
261. #124C2×D137semidirect
263. C22⋊Q8C2×D137semidirect
268. #121C2×D137semidirect
274. C4×D4C2×D137semidirect
387. D4D4×D137semidirect
430. C22#17984113semidirect

Maximal subgroups

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

subgroupindex
C2×#179841132
C22×#145554952
D4×D5482
#225777932
#225776542
#225776342
#225776152
#225775982
#225775532
#225775482
#225775372
#225775362
#225775132
#225774752
#546×C1372
#3264137