Finite Group Databasebeta

← #31792282

Subgroups — #31792282

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×#17984114C211yesmaximal, characteristic, complemented
2C2×SD16×D137C211yesmaximal, characteristic, complemented
3#22578063C211yesmaximal, characteristic, complemented
4D4×C22⋊D137C211yesmaximal, characteristic, complemented
5#22577789C211yesmaximal, characteristic, complemented
6#22577651C211yesmaximal, characteristic, complemented
7#22577638C211yesmaximal, characteristic, complemented
8#22577620C211yesmaximal, characteristic, complemented
9#22577602C211yesmaximal, characteristic, complemented
10#22577550C211yesmaximal, characteristic, complemented
11#22577540C211yesmaximal, characteristic, complemented
12#22577538C211yesmaximal, characteristic, complemented
13#22577500C211yesmaximal, characteristic, complemented
14#22577477C211yesmaximal, characteristic, complemented
15#450×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic, complemented
17C2×Q8×D137C2211yescharacteristic, complemented
18C22×C22⋊D137—21no—
19C2×D4×D137C2211yescharacteristic
20#17984114—24no—
21C2×#14555499C2211yescharacteristic, complemented
22C2×#14555498C2211yescharacteristic
23C2×#14555498C2211yescharacteristic, complemented
24C2×#14555497C2211yescharacteristic, complemented
25SD16×D137—24no—
26C2×D1096C2211yescharacteristic, complemented
27C2×#14555495C2211yescharacteristic
28C2×#14555495C2211yescharacteristic, complemented
29C2×#14555494C2211yescharacteristic, complemented
30C2×C8×D137C2211yescharacteristic, complemented
31#17984084C2211yescharacteristic
32#17984077C2211yescharacteristic, complemented
33#17984074—21no—
34#17984066—21no—
35#17984063C2211yescharacteristic, complemented
36#17984062C2211yescharacteristic
37#17984061—21no—
38#17984060—21no—
39#17984059C2211yescharacteristic
40#17984057—21no—
41#17984056—21no—
42#17984053C2211yescharacteristic, complemented
43D4×Dic137C2211yescharacteristic
44#17984050—21no—
45C4×C22⋊D137C2211yescharacteristic
46#17984021C2211yescharacteristic, complemented
47#17984017C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984012C2211yescharacteristic, complemented
50#17984010C2211yescharacteristic, complemented
51#17984008C2211yescharacteristic, complemented
52#17984000C2211yescharacteristic, complemented
53#17983998C2211yescharacteristic, complemented
54C22×D4×C137C2211yescharacteristic
55C2×SD16×C137C2211yescharacteristic
56C2×SD16×C137C2211yescharacteristic, complemented
57C22⋊Q8×C137C2211yescharacteristic, complemented
58#102×C137C2211yescharacteristic
59#102×C137C2211yescharacteristic, complemented
60C22⋊C8×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62#14555510—22no—
63#14555510—41no—
64C2×Dic274—21no—
65C2×Dic274C2311yescharacteristic
66Q8×D137—22no—
67Q8×D137—41no—
68C2×D548—21no—
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—
77C2×C22⋊D137—44no—
78C2×C22⋊D137D411yescharacteristic
79D4×D137—22no—
80D4×D137—41no—
81C22×Dic137—21no—
82C2×C4×D137—21no—
83C2×C4×D137—21no—
84C2×C4×D137C2311yescharacteristic
85#14555499—22no—
86#14555498—22no—
87#14555498D414yescomplemented
88#14555497—22no—
89D1096—41no—
90#14555495—22no—
91#14555495—22no—
92#14555494—22no—
93C2×#11764040C2311yescharacteristic
94C2×#11764040C2311yescharacteristic
95C8×D137—22no—
96C4⋊Dic137C2311yescharacteristic
97#14555489—21no—
98#14555489D411yescharacteristic
99C22⋊Dic137—21no—
100C22⋊Dic137—21no—
101C22⋊Dic137D411yescharacteristic
102#14555487—21no—
103#14555487—21no—
104#14555487—21no—
105#14555487—21no—
106#14555487D411yescharacteristic
107C4×Dic137—21no—
108C4×Dic137C2311yescharacteristic
109C24×C137—21no—
110C2×Q8×C137C2311yescharacteristic
111C2×D4×C137—22no—
112C2×D4×C137—21no—
113C2×D4×C137—41no—
114C2×D4×C137C2311yescharacteristic
115C2×D4×C137C2311yescharacteristic
116C22×C4×C137C2311yescharacteristic
117SD16×C137—22no—
118SD16×C137—41no—
119C2×C8×C137C2311yescharacteristic
120C2×C8×C137C2311yescharacteristic
121C4⋊C4×C137—21no—
122C4⋊C4×C137C2311yescharacteristic
123C22⋊C4×C137—21no—
124C22×D137—21no—
125C22×D137—21no—
126C22×D137—42no—
127C22×D137—42no—
128C22×D137—81no—
129C22×D137C2×D411yescharacteristic
130Dic274—21no—
131Dic274—42no—
132Dic274C2×D412yescomplemented
133D548—21no—
134D548—21no—
135D548—21no—
136D548—21no—
137D548—42no—
138C22⋊D137—22no—
139C22⋊D137—42no—
140C22⋊D137—42no—
141C22⋊D137—42no—
142C22⋊D137—42no—
143C22⋊D137—42no—
144C22⋊D137—81no—
145C2×Dic137—21no—
146C2×Dic137—21no—
147C2×Dic137—42no—
148C2×Dic137C2×D411yescharacteristic
149C2×Dic137C2×D411yescharacteristic
150C2×Dic137C2×D411yescharacteristic
151C4×D137—21no—
152C4×D137—21no—
153C4×D137—42no—
154C4×D137—42no—
155#11764040—21no—
156#11764040C2×D412yescomplemented
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—42no—
169D4×C137C2×D412yes—
170C2×C4×C137—21no—
171C2×C4×C137—21no—
172C2×C4×C137—21no—
173C2×C4×C137—21no—
174C2×C4×C137C2×D411yescharacteristic
175C2×C4×C137C2411yescharacteristic
176C8×C137—21no—
177C8×C137—21no—
178C22×C137—21no—
179C22×C137—22no—
180C22×C137—21no—
181C22×C137—21no—
182C22×C137—41no—
183C22×C137—41no—
184C22×C137—41no—
185C22×C137—41no—
186C22×C137—41no—
187C22×C137—84no—
188C22×C137C22×D411yescharacteristic
189C2×D137—21no—
190C2×D137—21no—
191C2×D137—41no—
192C2×D137—41no—
193C2×D137—41no—
194C2×D137—41no—
195C2×D137—41no—
196C2×D137—41no—
197C2×D137—84no—
198C2×D137AGL(1,ℤ/8)12yescomplemented
199C4×C137—21no—
200C4×C137—41no—
201C4×C137—41no—
202C4×C137C22×D411yescharacteristic
203C4×C137H(5,𝔽2)11yescharacteristic
204Dic137—21no—
205Dic137—22no—
206Dic137—21no—
207Dic137—21no—
208Dic137—41no—
209C2×C137—42no—
210C2×C137—41no—
211C2×C137—81no—
212C2×C137D4211yescharacteristic
213C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C137#57711yescharacteristic
215D137—22no—
216D137—81no—
217D137—81no—
218C137#326511yescharacteristic, complemented, Sylow
219#3265—1371nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1371no—
221C22×SD16—1371no—
222#548—1371no—
223D42—1371no—
224#496—1371no—
225#468—1371no—
226#466—1371no—
227#463—1371no—
228#460—1371no—
229#450D13711yescharacteristic, complemented
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×D4C2×D13711yescharacteristic
240AGL(1,ℤ/8)—2744no—
241C2×SD16—2744no—
242C2×SD16—1371no—
243C2×SD16—1371no—
244C2×SD16—1371no—
245C2×SD16—1371no—
246C2×SD16C2×D13711yescharacteristic
247C2×SD16C2×D13711yescharacteristic, complemented
248C2×D8—1371no—
249C2×D8—1371no—
250C2×D8—1371no—
251C2×M16—1371no—
252C22×C8—1371no—
253#127—1371no—
254#124—2741no—
255#124—1371no—
256C22⋊Q8—2741no—
257C22⋊Q8C2×D13711yescharacteristic, complemented
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#102C2×D13711yescharacteristic
276#102C2×D13711yescharacteristic, complemented
277C22⋊C8—1371no—
278C22⋊C8C2×D13711yescharacteristic, complemented
279#97—1371no—
280C24—21no—
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—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—2741no—
293C2×D4—2742no—
294C2×D4—2741no—
295C2×D4—2741no—
296C2×D4—2741no—
297C2×D4—2741no—
298C2×D4—2741no—
299C2×D4—2741no—
300C2×D4—5481no—
301C2×D4—5484no—
302C2×D4—1371no—
303C2×D4—1371no—
304C2×D4—1371no—
305C2×D4C22×D13711yescharacteristic
306C2×D4C22×D13711yescharacteristic
307C22×C4—2741no—
308C22×C4—2741no—
309C22×C4—2741no—
310C22×C4—1371no—
311C22×C4C22×D13711yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—2742no—
315SD16—2742no—
316SD16—2742no—
317SD16—1374no—
318D8—2742no—
319D8—2742no—
320D8—5481no—
321M16—2742no—
322C2×C8—2742no—
323C2×C8—1371no—
324C2×C8—1371no—
325C2×C8C22×D13711yescharacteristic
326C2×C8C22×D13711yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—2741no—
329C4⋊C4—1371no—
330C4⋊C4—1371no—
331C4⋊C4C22×D13711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—2741no—
334C22⋊C4—2741no—
335C22⋊C4—2741no—
336C22⋊C4—2741no—
337C22⋊C4—2741no—
338C22⋊C4—2741no—
339C22⋊C4—1371no—
340C22⋊C4—1371no—
341C42—2741no—
342C42—1371no—
343C23—21no—
344C23—21no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—2741no—
349C23—2741no—
350C23—5482no—
351C23—5482no—
352C23—10961no—
353C23—1371no—
354C23D4×D13711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—2741no—
358Q8—5482no—
359Q8—1372no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—2742no—
365D4—2741no—
366D4—2741no—
367D4—2741no—
368D4—2741no—
369D4—5482no—
370D4—5482no—
371D4—5482no—
372D4—5482no—
373D4—5482no—
374D4—5482no—
375D4—10961no—
376D4D4×D13712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
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×C4—1371no—
391C2×C4D4×D13711yescharacteristic
392C2×C4C23×D13711yescharacteristic
393C8—21no—
394C8—21no—
395C8—2741no—
396C8—1372no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—84no—
407C22—2741no—
408C22—2741no—
409C22—5481no—
410C22—5481no—
411C22—5481no—
412C22—5481no—
413C22—5481no—
414C22—5481no—
415C22—10964no—
416C22—1372no—
417C22C2×D4×D13711yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—2741no—
422C4—2741no—
423C4—2741no—
424C4—2742no—
425C4—5481no—
426C4C2×D4×D13711yescharacteristic
427C4#1798413311yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—2742no—
432C2—10961no—
433C2—10961no—
434C2#2257803511yescharacteristic
435C2AGL(1,ℤ/8)×D13711yescharacteristic
436C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#450×C137C2
socleC22×C137C22×D4
hypercenter#450D137
hypocenterC137#3265
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#17984114C2semidirect
2. C2×SD16×D137C2semidirect
3. #22578063C2semidirect
4. D4×C22⋊D137C2semidirect
5. #22577789C2semidirect
6. #22577651C2semidirect
7. #22577638C2semidirect
8. #22577620C2semidirect
9. #22577602C2semidirect
10. #22577550C2semidirect
11. #22577540C2semidirect
12. #22577538C2semidirect
13. #22577500C2semidirect
14. #22577477C2semidirect
15. #450×C137C2semidirect
16. C2×#14555510C22semidirect
17. C2×Q8×D137C22semidirect
21. C2×#14555499C22semidirect
23. C2×#14555498C22semidirect
24. C2×#14555497C22semidirect
26. C2×D1096C22semidirect
28. C2×#14555495C22semidirect
29. C2×#14555494C22semidirect
30. C2×C8×D137C22semidirect
32. #17984077C22semidirect
35. #17984063C22semidirect
42. #17984053C22semidirect
46. #17984021C22semidirect
47. #17984017C22semidirect
48. #17984013C22semidirect
49. #17984012C22semidirect
50. #17984010C22semidirect
51. #17984008C22semidirect
52. #17984000C22semidirect
53. #17983998C22semidirect
56. C2×SD16×C137C22semidirect
57. C22⋊Q8×C137C22semidirect
59. #102×C137C22semidirect
60. C22⋊C8×C137C22semidirect
87. #14555498D4semidirect
132. Dic274C2×D4semidirect
156. #11764040C2×D4semidirect
198. C2×D137AGL(1,ℤ/8)semidirect
218. C137#3265semidirect
229. #450D137semidirect
247. C2×SD16C2×D137semidirect
257. C22⋊Q8C2×D137semidirect
276. #102C2×D137semidirect
278. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841142
C2×SD16×D1372
#225780632
D4×C22⋊D1372
#225777892
#225776512
#225776382
#225776202
#225776022
#225775502
#225775402
#225775382
#225775002
#225774772
#450×C1372
#3265137