Finite Group Databasebeta

← #31792263

Subgroups — #31792263

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×D8×D137C211yesmaximal, characteristic, complemented
2#22578041C211yesmaximal, characteristic, complemented
3D4×C22⋊D137C211yesmaximal, characteristic, complemented
4#22577789C211yesmaximal, characteristic, complemented
5#22577656C211yesmaximal, characteristic, complemented
6#22577629C211yesmaximal, characteristic, complemented
7#22577620C211yesmaximal, characteristic, complemented
8#22577596C211yesmaximal, characteristic, complemented
9#22577552C211yesmaximal, characteristic, complemented
10#22577548C211yesmaximal, characteristic, complemented
11#22577537C211yesmaximal, characteristic, complemented
12#22577535C211yesmaximal, characteristic, complemented
13#22577515C211yesmaximal, characteristic, complemented
14#22577472C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C137C211yesmaximal, characteristic, complemented
16C22×C22⋊D137—21no—
17C2×D4×D137C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic, complemented
19C2×#14555497C2211yescharacteristic, complemented
20C2×D1096C2211yescharacteristic, complemented
21C2×#14555495C2211yescharacteristic
22C2×#14555495C2211yescharacteristic, complemented
23C2×#14555495C2211yescharacteristic, complemented
24D8×D137—24no—
25C2×C8×D137C2211yescharacteristic, complemented
26#17984084C2211yescharacteristic, complemented
27#17984077C2211yescharacteristic, complemented
28#17984071—21no—
29#17984064C2211yescharacteristic, complemented
30#17984062C2211yescharacteristic, complemented
31#17984062C2211yescharacteristic, complemented
32#17984060—21no—
33#17984060—21no—
34#17984059C2211yescharacteristic
35#17984057—21no—
36#17984056—21no—
37#17984056—21no—
38D4×Dic137C2211yescharacteristic, complemented
39D4×Dic137C2211yescharacteristic, complemented
40C4×C22⋊D137C2211yescharacteristic
41C2×#14555487—21no—
42#17984020C2211yescharacteristic, complemented
43#17984017C2211yescharacteristic, complemented
44#17984012C2211yescharacteristic, complemented
45#17984010C2211yescharacteristic, complemented
46#17984010C2211yescharacteristic, complemented
47#17984009C2211yescharacteristic, complemented
48#17984009C2211yescharacteristic, complemented
49#17984009C2211yescharacteristic, complemented
50C22⋊#11764040C2211yescharacteristic, complemented
51#17984000C2211yescharacteristic, complemented
52#17983998C2211yescharacteristic, complemented
53C2×#41×C137C2211yescharacteristic, complemented
54C22×D4×C137C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C137—24no—
56C2×SD16×C137C2211yescharacteristic, complemented
57C2×SD16×C137C2211yescharacteristic, complemented
58C2×D8×C137C2211yescharacteristic
59C2×D8×C137C2211yescharacteristic, complemented
60C2×M16×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C23×D137—21no—
63C2×D548C2311yescharacteristic
64C2×D548C2311yescharacteristic
65C2×C22⋊D137—21no—
66C2×C22⋊D137—21no—
67C2×C22⋊D137—21no—
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C22⋊D137—44no—
73C2×C22⋊D137D411yescharacteristic
74D4×D137—22no—
75D4×D137—22no—
76D4×D137—41no—
77D4×D137—41no—
78C22×Dic137—21no—
79C22×Dic137—21no—
80C2×C4×D137C2311yescharacteristic
81#14555497—41no—
82D1096—22no—
83#14555495—22no—
84#14555495—22no—
85#14555495—41no—
86C2×#11764040C2311yescharacteristic
87C2×#11764040C2311yescharacteristic
88C8×D137—22no—
89C4⋊Dic137C2311yescharacteristic
90C4⋊Dic137C2311yescharacteristic
91#14555489—21no—
92#14555489D411yescharacteristic
93C22⋊Dic137—21no—
94C22⋊Dic137—21no—
95C22⋊Dic137—21no—
96C22⋊Dic137D411yescharacteristic
97#14555487—21no—
98#14555487—21no—
99#14555487—21no—
100#14555487—21no—
101#14555487—21no—
102#14555487D411yescharacteristic
103C4×Dic137C2311yescharacteristic
104C24×C137—21no—
105#41×C137—22no—
106#41×C137—41no—
107C2×Q8×C137C2311yescharacteristic
108C2×D4×C137—22no—
109C2×D4×C137—21no—
110C2×D4×C137—21no—
111C2×D4×C137—41no—
112C2×D4×C137C2311yescharacteristic
113C2×D4×C137C2311yescharacteristic
114C2×D4×C137C2311yescharacteristic
115C22×C4×C137—21no—
116C22×C4×C137C2311yescharacteristic
117SD16×C137—22no—
118SD16×C137—22no—
119D8×C137—22no—
120D8×C137D414yescomplemented
121M16×C137—22no—
122C2×C8×C137C2311yescharacteristic
123C2×C8×C137C2311yescharacteristic
124C22×D137—21no—
125C22×D137—21no—
126C22×D137—42no—
127C22×D137—42no—
128C22×D137—42no—
129C22×D137—42no—
130C22×D137—81no—
131C22×D137—81no—
132C22×D137C2×D411yescharacteristic
133D548—21no—
134D548—21no—
135D548—21no—
136D548—21no—
137C22⋊D137—22no—
138C22⋊D137—42no—
139C22⋊D137—42no—
140C22⋊D137—42no—
141C22⋊D137—42no—
142C22⋊D137—42no—
143C22⋊D137—81no—
144C22⋊D137—81no—
145C2×Dic137—21no—
146C2×Dic137—21no—
147C2×Dic137—42no—
148C2×Dic137—81no—
149C2×Dic137C2×D411yescharacteristic
150C2×Dic137C2×D411yescharacteristic
151C2×Dic137C2×D411yescharacteristic
152C4×D137—21no—
153C4×D137—21no—
154#11764040—21no—
155#11764040—21no—
156C23×C137—21no—
157C23×C137—21no—
158C23×C137—21no—
159C23×C137—42no—
160C23×C137—42no—
161C23×C137—81no—
162C23×C137C2×D411yescharacteristic
163Q8×C137—21no—
164Q8×C137—21no—
165D4×C137—21no—
166D4×C137—21no—
167D4×C137—21no—
168D4×C137—21no—
169D4×C137—42no—
170D4×C137—42no—
171D4×C137C2×D412yes—
172D4×C137C2×D412yescomplemented
173C2×C4×C137—21no—
174C2×C4×C137—21no—
175C2×C4×C137—21no—
176C2×C4×C137—42no—
177C2×C4×C137C2×D411yescharacteristic
178C2×C4×C137C2411yescharacteristic
179C8×C137—21no—
180C8×C137C2×D412yescomplemented
181C22×C137—21no—
182C22×C137—22no—
183C22×C137—22no—
184C22×C137—21no—
185C22×C137—21no—
186C22×C137—41no—
187C22×C137—41no—
188C22×C137—41no—
189C22×C137—41no—
190C22×C137—41no—
191C22×C137—41no—
192C22×C137—41no—
193C22×C137—84no—
194C22×C137C22×D411yescharacteristic
195C2×D137—21no—
196C2×D137—21no—
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×D137AGL(1,ℤ/8)12yescomplemented
206C4×C137—21no—
207C4×C137—41no—
208C4×C137C22×D411yescharacteristic
209C4×C137H(5,𝔽2)11yescharacteristic
210Dic137—21no—
211Dic137—21no—
212Dic137—21no—
213Dic137—41no—
214Dic137—41no—
215C2×C137—42no—
216C2×C137—42no—
217C2×C137—41no—
218C2×C137—81no—
219C2×C137D4211yescharacteristic
220C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C137#57611yescharacteristic
222D137—22no—
223D137—81no—
224D137—81no—
225C137#326411yescharacteristic, complemented, Sylow
226#3264—1371nomaximal, Sylow
227C2×AGL(1,ℤ/8)D13711yescharacteristic, complemented
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#435—1371no—
242C2×#41C2×D13711yescharacteristic, complemented
243C22×D4—2741no—
244C22×D4—1371no—
245C22×D4—1371no—
246C22×D4C2×D13711yescharacteristic, complemented
247AGL(1,ℤ/8)—24no—
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
257C2×D8C2×D13711yescharacteristic, complemented
258C2×M16C2×D13711yescharacteristic, complemented
259C22×C8—1371no—
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×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#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—41no—
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—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—2741no—
317C22×C4—2741no—
318C22×C4—1371no—
319C22×C4C22×D13711yescharacteristic
320SD16—22no—
321SD16—22no—
322SD16—5481no—
323D8—22no—
324D8—2742no—
325D8—2742no—
326D8—2742no—
327D8—5481no—
328D8C22⋊D13714yescomplemented
329M16—22no—
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—2741no—
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—
349C42—1371no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—81no—
356C23—2741no—
357C23—2741no—
358C23—5482no—
359C23—5482no—
360C23—5482no—
361C23—5482no—
362C23—10961no—
363C23—10961no—
364C23—1371no—
365C23D4×D13711yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—2742no—
375D4—2741no—
376D4—2741no—
377D4—2741no—
378D4—2741no—
379D4—5482no—
380D4—5482no—
381D4—5482no—
382D4—5482no—
383D4—5482no—
384D4—10961no—
385D4—10961no—
386D4C2×C22⋊D13712yes—
387D4C2×C22⋊D13712yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
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×C4—1371no—
401C2×C4D4×D13711yescharacteristic
402C2×C4C23×D13711yescharacteristic
403C8—21no—
404C8—2741no—
405C8—2741no—
406C8C2×C22⋊D13712yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—84no—
420C22—2741no—
421C22—2741no—
422C22—5481no—
423C22—5481no—
424C22—5481no—
425C22—5481no—
426C22—5481no—
427C22—5481no—
428C22—10964no—
429C22—10964no—
430C22—1372no—
431C22C2×D4×D13711yescharacteristic
432C4—21no—
433C4—41no—
434C4—2741no—
435C4—2741no—
436C4—2741no—
437C4—5481no—
438C4—5481no—
439C4C22×C22⋊D13711yescharacteristic
440C4#1798413311yescharacteristic
441C2—41no—
442C2—42no—
443C2—42no—
444C2—81no—
445C2—2742no—
446C2—10961no—
447C2—10961no—
448C2D4×C22⋊D13711yescharacteristic
449C2AGL(1,ℤ/8)×D13711yescharacteristic
450C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×AGL(1,ℤ/8)×C137C2
socleC22×C137C22×D4
hypercenterC2×AGL(1,ℤ/8)D137
hypocenterC137#3264
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×D8×D137C2semidirect
2. #22578041C2semidirect
3. D4×C22⋊D137C2semidirect
4. #22577789C2semidirect
5. #22577656C2semidirect
6. #22577629C2semidirect
7. #22577620C2semidirect
8. #22577596C2semidirect
9. #22577552C2semidirect
10. #22577548C2semidirect
11. #22577537C2semidirect
12. #22577535C2semidirect
13. #22577515C2semidirect
14. #22577472C2semidirect
15. C2×AGL(1,ℤ/8)×C137C2semidirect
18. C2×D4×D137C22semidirect
19. C2×#14555497C22semidirect
20. C2×D1096C22semidirect
22. C2×#14555495C22semidirect
23. C2×#14555495C22semidirect
25. C2×C8×D137C22semidirect
26. #17984084C22semidirect
27. #17984077C22semidirect
29. #17984064C22semidirect
30. #17984062C22semidirect
31. #17984062C22semidirect
38. D4×Dic137C22semidirect
39. D4×Dic137C22semidirect
42. #17984020C22semidirect
43. #17984017C22semidirect
44. #17984012C22semidirect
45. #17984010C22semidirect
46. #17984010C22semidirect
47. #17984009C22semidirect
48. #17984009C22semidirect
49. #17984009C22semidirect
50. C22⋊#11764040C22semidirect
51. #17984000C22semidirect
52. #17983998C22semidirect
53. C2×#41×C137C22semidirect
54. C22×D4×C137C22semidirect
56. C2×SD16×C137C22semidirect
57. C2×SD16×C137C22semidirect
59. C2×D8×C137C22semidirect
60. C2×M16×C137C22semidirect
120. D8×C137D4semidirect
172. D4×C137C2×D4semidirect
180. C8×C137C2×D4semidirect
205. C2×D137AGL(1,ℤ/8)semidirect
225. C137#3264semidirect
227. C2×AGL(1,ℤ/8)D137semidirect
242. C2×#41C2×D137semidirect
246. C22×D4C2×D137semidirect
249. C2×SD16C2×D137semidirect
250. C2×SD16C2×D137semidirect
257. C2×D8C2×D137semidirect
258. C2×M16C2×D137semidirect
328. D8C22⋊D137semidirect
387. D4C2×C22⋊D137semidirect
406. C8C2×C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1372
#225780412
D4×C22⋊D1372
#225777892
#225776562
#225776292
#225776202
#225775962
#225775522
#225775482
#225775372
#225775352
#225775152
#225774722
C2×AGL(1,ℤ/8)×C1372
#3264137