Finite Group Databasebeta

← #31792283

Subgroups — #31792283

Order 17536.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984112C211yesmaximal, characteristic, complemented
2C2×SD16×D137C211yesmaximal, characteristic, complemented
3#22578061C211yesmaximal, characteristic, complemented
4#22578038C211yesmaximal, characteristic, complemented
5#22577646C211yesmaximal, characteristic, complemented
6#22577636C211yesmaximal, characteristic, complemented
7#22577615C211yesmaximal, characteristic, complemented
8#22577600C211yesmaximal, characteristic, complemented
9#22577552C211yesmaximal, characteristic, complemented
10#22577545C211yesmaximal, characteristic, complemented
11#22577540C211yesmaximal, characteristic, complemented
12#22577535C211yesmaximal, characteristic, complemented
13#22577505C211yesmaximal, characteristic, complemented
14#22577476C211yesmaximal, characteristic, complemented
15#496×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic, complemented
17C2×Q8×D137C2211yescharacteristic, complemented
18C2×D4×D137—21no—
19C2×D4×D137C2211yescharacteristic
20C2×D4×D137C2211yescharacteristic
21#17984112—24no—
22C2×#14555499C2211yescharacteristic
23C2×#14555498C2211yescharacteristic, complemented
24C2×#14555498C2211yescharacteristic, complemented
25C2×#14555497C2211yescharacteristic
26SD16×D137—24no—
27C2×#14555495C2211yescharacteristic
28C2×#14555493C2211yescharacteristic, complemented
29C2×C8×D137C2211yescharacteristic, complemented
30#17984074—21no—
31#17984065—21no—
32#17984064C2211yescharacteristic, complemented
33#17984061—21no—
34#17984059—21no—
35#17984059C2211yescharacteristic
36#17984059C2211yescharacteristic
37#17984059C2211yescharacteristic, complemented
38#17984056—21no—
39#17984056—21no—
40C4×Dic274C2211yescharacteristic, complemented
41C4×D548C2211yescharacteristic
42C4×D548C2211yescharacteristic
43#17984048—21no—
44#17984022C2211yescharacteristic, complemented
45C4⋊#11764040C2211yescharacteristic, complemented
46#17984013C2211yescharacteristic, complemented
47#17984012C2211yescharacteristic, complemented
48#17984010C2211yescharacteristic
49#17984009C2211yescharacteristic, complemented
50#17984009C2211yescharacteristic, complemented
51#17984009C2211yescharacteristic, complemented
52#17984000C2211yescharacteristic, complemented
53#17984000C2211yescharacteristic, complemented
54C2×SD16×C137C2211yescharacteristic, complemented
55C2×SD16×C137C2211yescharacteristic, complemented
56C2×D8×C137C2211yescharacteristic, complemented
57C2×D8×C137C2211yescharacteristic, complemented
58#127×C137C2211yescharacteristic
59#124×C137C2211yescharacteristic, complemented
60#97×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C23×D137—21no—
63#14555509—22no—
64#14555509—41no—
65C2×Dic274—21no—
66C2×Dic274C2311yescharacteristic
67Q8×D137—22no—
68Q8×D137—41no—
69C2×D548D411yescharacteristic
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—41no—
80D4×D137—41no—
81D4×D137—44no—
82C22×Dic137—21no—
83C2×C4×D137—21no—
84C2×C4×D137—21no—
85C2×C4×D137C2311yescharacteristic
86C2×C4×D137C2311yescharacteristic
87#14555499D414yescomplemented
88#14555498—22no—
89#14555498—22no—
90#14555497—22no—
91#14555495—22no—
92#14555493—22no—
93C2×#11764040C2311yescharacteristic
94C2×#11764040C2311yescharacteristic
95C8×D137—22no—
96C4⋊Dic137D411yescharacteristic
97C4⋊Dic137C2311yescharacteristic
98C4⋊Dic137C2311yescharacteristic
99#14555489—21no—
100#14555489—21no—
101C22⋊Dic137—21no—
102C22⋊Dic137—21no—
103C22⋊Dic137—21no—
104#14555487—21no—
105#14555487—21no—
106#14555487D411yescharacteristic
107#14555487D411yescharacteristic
108C4×Dic137—21no—
109C2×Q8×C137C2311yescharacteristic
110C2×D4×C137—21no—
111C2×D4×C137—21no—
112C2×D4×C137C2311yescharacteristic
113C2×D4×C137C2311yescharacteristic
114C2×D4×C137C2311yescharacteristic
115SD16×C137—22no—
116SD16×C137—41no—
117D8×C137—22no—
118D8×C137—41no—
119C2×C8×C137C2311yescharacteristic
120C2×C8×C137C2311yescharacteristic
121C22⋊C4×C137—21no—
122C22⋊C4×C137—21no—
123C42×C137C2311yescharacteristic
124C22×D137—21no—
125C22×D137—42no—
126C22×D137—42no—
127C22×D137—42no—
128C22×D137—42no—
129C22×D137—81no—
130C22×D137—81no—
131C22×D137C2×D411yescharacteristic
132C22×D137C2×D411yescharacteristic
133Dic274—21no—
134Dic274—42no—
135Dic274C2×D412yescomplemented
136D548—22no—
137D548—21no—
138D548C2×D412yes—
139C22⋊D137—42no—
140C22⋊D137—42no—
141C22⋊D137—42no—
142C22⋊D137—42no—
143C22⋊D137—42no—
144C22⋊D137—42no—
145C2×Dic137—21no—
146C2×Dic137—21no—
147C2×Dic137—21no—
148C2×Dic137—42no—
149C2×Dic137C2×D411yescharacteristic
150C2×Dic137C2×D411yescharacteristic
151C4×D137—21no—
152C4×D137—21no—
153C4×D137—21no—
154C4×D137—21no—
155C4×D137—42no—
156C4×D137—42no—
157#11764040—21no—
158#11764040—21no—
159C23×C137—21no—
160C23×C137—21no—
161C23×C137—21no—
162Q8×C137—21no—
163Q8×C137—21no—
164D4×C137—21no—
165D4×C137—21no—
166D4×C137—21no—
167D4×C137—21no—
168D4×C137—21no—
169D4×C137—21no—
170D4×C137—42no—
171D4×C137—81no—
172C2×C4×C137—21no—
173C2×C4×C137C2×D411yescharacteristic
174C2×C4×C137C2×D411yescharacteristic
175C2×C4×C137C2411yescharacteristic
176C8×C137—21no—
177C8×C137C2×D412yescomplemented
178C22×C137—41no—
179C22×C137—41no—
180C22×C137—41no—
181C22×C137—41no—
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—84no—
197C2×D137—84no—
198C2×D137AGL(1,ℤ/8)12yescomplemented
199C4×C137—21no—
200C4×C137—21no—
201C4×C137—41no—
202C4×C137C22×D411yescharacteristic
203C4×C137H(5,𝔽2)11yescharacteristic
204Dic137—21no—
205Dic137—21no—
206Dic137—22no—
207Dic137—41no—
208Dic137—41no—
209C2×C137—81no—
210C2×C137—81no—
211C2×C137—81no—
212C2×C137D4211yescharacteristic
213C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C137#57711yescharacteristic
215D137—22no—
216D137—41no—
217D137—42no—
218C137#326511yescharacteristic, complemented, Sylow
219#3265—1371nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1371no—
221C22×SD16—1371no—
222#548—1371no—
223D42—1371no—
224#496D13711yescharacteristic, complemented
225#468—1371no—
226#466—1371no—
227#463—1371no—
228#460—1371no—
229#450—1371no—
230#449—1371no—
231#448—1371no—
232#447—1371no—
233#440—1371no—
234#435—1371no—
235C2×#41—1371no—
236C22×Q8—1371no—
237C22×D4—2741no—
238C22×D4—1371no—
239C22×D4—1371no—
240AGL(1,ℤ/8)—2744no—
241C2×SD16—2744no—
242C2×SD16—1371no—
243C2×SD16—1371no—
244C2×SD16—1371no—
245C2×SD16—1371no—
246C2×SD16C2×D13711yescharacteristic, complemented
247C2×SD16C2×D13711yescharacteristic, complemented
248C2×D8—1371no—
249C2×D8C2×D13711yescharacteristic, complemented
250C2×D8C2×D13711yescharacteristic, complemented
251C2×M16—1371no—
252C22×C8—1371no—
253#127C2×D13711yescharacteristic
254#124—2741no—
255#124C2×D13711yescharacteristic, complemented
256C22⋊Q8—2741no—
257C22⋊Q8—1371no—
258#121—2741no—
259#121—2741no—
260#121—1371no—
261#121—1371no—
262#121—1371no—
263#120—2741no—
264#120—2741no—
265C4×Q8—1371no—
266C4×D4—2741no—
267C4×D4—1371no—
268C4×D4—1371no—
269#107—1371no—
270C4⋊C8—1371no—
271#103—1371no—
272#103—1371no—
273#102—1371no—
274#102—1371no—
275#102—1371no—
276#102—1371no—
277C22⋊C8—1371no—
278C22⋊C8—1371no—
279#97C2×D13711yescharacteristic, complemented
280C24—2741no—
281C24—2741no—
282#41—2742no—
283#41—5481no—
284C2×Q8—2741no—
285C2×Q8—2742no—
286C2×Q8—5481no—
287C2×Q8—1371no—
288C2×Q8C22×D13711yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—2742no—
292C2×D4—2742no—
293C2×D4—2741no—
294C2×D4—2741no—
295C2×D4—2741no—
296C2×D4—2741no—
297C2×D4—2741no—
298C2×D4—2741no—
299C2×D4—5481no—
300C2×D4—5481no—
301C2×D4—5484no—
302C2×D4—1371no—
303C2×D4—1371no—
304C2×D4C22×D13711yescharacteristic
305C2×D4C22×D13711yescharacteristic
306C2×D4C22×D13711yescharacteristic
307C22×C4—2741no—
308C22×C4—2741no—
309C22×C4—2741no—
310C22×C4—1371no—
311C22×C4—1371no—
312SD16—22no—
313SD16—41no—
314SD16—2742no—
315SD16—2742no—
316SD16—2742no—
317SD16—1374no—
318D8—22no—
319D8—41no—
320D8—2742no—
321M16—2742no—
322C2×C8—2742no—
323C2×C8—1371no—
324C2×C8—1371no—
325C2×C8C22×D13711yescharacteristic
326C2×C8C22×D13711yescharacteristic
327C4⋊C4—2741no—
328C4⋊C4—2741no—
329C4⋊C4—1371no—
330C4⋊C4—1371no—
331C4⋊C4—1371no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—2741no—
335C22⋊C4—2741no—
336C22⋊C4—2741no—
337C22⋊C4—2741no—
338C22⋊C4—2741no—
339C22⋊C4—1371no—
340C22⋊C4—1371no—
341C42—2741no—
342C42C22×D13711yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—2741no—
347C23—5482no—
348C23—5482no—
349C23—5482no—
350C23—5482no—
351C23—10961no—
352C23—10961no—
353C23—1371no—
354C23—1371no—
355Q8—21no—
356Q8—21no—
357Q8—2741no—
358Q8—5482no—
359Q8—1372no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—2742no—
369D4—2741no—
370D4—5482no—
371D4—5482no—
372D4—5482no—
373D4—5482no—
374D4—5482no—
375D4—5482no—
376D4—1372no—
377C2×C4—21no—
378C2×C4—2741no—
379C2×C4—2741no—
380C2×C4—2741no—
381C2×C4—2741no—
382C2×C4—2741no—
383C2×C4—2741no—
384C2×C4—2741no—
385C2×C4—5482no—
386C2×C4—5482no—
387C2×C4—5482no—
388C2×C4—1371no—
389C2×C4—1371no—
390C2×C4D4×D13711yescharacteristic
391C2×C4D4×D13711yescharacteristic
392C2×C4C23×D13711yescharacteristic
393C8—21no—
394C8—2741no—
395C8—2741no—
396C8D4×D13712yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—2741no—
407C22—2741no—
408C22—2741no—
409C22—2741no—
410C22—2741no—
411C22—2742no—
412C22—5481no—
413C22—5481no—
414C22—10964no—
415C22—10964no—
416C22—1372no—
417C22C2×D4×D13711yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—2741no—
422C4—2741no—
423C4—2742no—
424C4—5481no—
425C4—5481no—
426C4C2×D4×D13711yescharacteristic
427C4#1798413311yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—2742no—
432C2—5481no—
433C2—5482no—
434C2#2257803811yescharacteristic
435C2AGL(1,ℤ/8)×D13711yescharacteristic
436C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#496×C137C2
socleC22×C137C22×D4
hypercenter#496D137
hypocenterC137#3265
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#17984112C2semidirect
2. C2×SD16×D137C2semidirect
3. #22578061C2semidirect
4. #22578038C2semidirect
5. #22577646C2semidirect
6. #22577636C2semidirect
7. #22577615C2semidirect
8. #22577600C2semidirect
9. #22577552C2semidirect
10. #22577545C2semidirect
11. #22577540C2semidirect
12. #22577535C2semidirect
13. #22577505C2semidirect
14. #22577476C2semidirect
15. #496×C137C2semidirect
16. C2×#14555509C22semidirect
17. C2×Q8×D137C22semidirect
23. C2×#14555498C22semidirect
24. C2×#14555498C22semidirect
28. C2×#14555493C22semidirect
29. C2×C8×D137C22semidirect
32. #17984064C22semidirect
37. #17984059C22semidirect
40. C4×Dic274C22semidirect
44. #17984022C22semidirect
45. C4⋊#11764040C22semidirect
46. #17984013C22semidirect
47. #17984012C22semidirect
49. #17984009C22semidirect
50. #17984009C22semidirect
51. #17984009C22semidirect
52. #17984000C22semidirect
53. #17984000C22semidirect
54. C2×SD16×C137C22semidirect
55. C2×SD16×C137C22semidirect
56. C2×D8×C137C22semidirect
57. C2×D8×C137C22semidirect
59. #124×C137C22semidirect
60. #97×C137C22semidirect
87. #14555499D4semidirect
135. Dic274C2×D4semidirect
177. C8×C137C2×D4semidirect
198. C2×D137AGL(1,ℤ/8)semidirect
218. C137#3265semidirect
224. #496D137semidirect
246. C2×SD16C2×D137semidirect
247. C2×SD16C2×D137semidirect
249. C2×D8C2×D137semidirect
250. C2×D8C2×D137semidirect
255. #124C2×D137semidirect
279. #97C2×D137semidirect
396. C8D4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841122
C2×SD16×D1372
#225780612
#225780382
#225776462
#225776362
#225776152
#225776002
#225775522
#225775452
#225775402
#225775352
#225775052
#225774762
#496×C1372
#3265137