Finite Group Databasebeta

← #18006193

Subgroups — #18006193

Order 4736.

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×#11924564C211yesmaximal, characteristic, complemented
2C2×#11924563C211yesmaximal, characteristic, complemented
3#14558185C211yesmaximal, characteristic, complemented
4#14558167C211yesmaximal, characteristic, complemented
5#14557904C211yesmaximal, characteristic, complemented
6#14557780C211yesmaximal, characteristic, complemented
7#14557737C211yesmaximal, characteristic, complemented
8#14557727C211yesmaximal, characteristic, complemented
9#14557685C211yesmaximal, characteristic, complemented
10#14557683C211yesmaximal, characteristic, complemented
11#14557681C211yesmaximal, characteristic, complemented
12#14557676C211yesmaximal, characteristic, complemented
13#14557615C211yesmaximal, characteristic, complemented
14#14557609C211yesmaximal, characteristic, complemented
15C37×#468C211yesmaximal, characteristic, complemented
16C2×#10596966C2211yescharacteristic, complemented
17C2×#10596962C2211yescharacteristic, complemented
18C22×D148C2211yescharacteristic
19C2×D4×D37—21no—
20C2×D4×D37C2211yescharacteristic
21#11924564—24no—
22#11924563—24no—
23C2×#10596930C2211yescharacteristic
24C2×#10596930C2211yescharacteristic, complemented
25C2×#10596940C2211yescharacteristic, complemented
26C2×#10596941C2211yescharacteristic
27C2×#10596941C2211yescharacteristic, complemented
28C2×D296C2211yescharacteristic
29C2×D296C2211yescharacteristic, complemented
30C2×#10596939C2211yescharacteristic, complemented
31C2×#10596929C2211yescharacteristic, complemented
32#11924534C2211yescharacteristic, complemented
33#11924532C2211yescharacteristic
34#11924521—21no—
35#11924518—21no—
36#11924516C2211yescharacteristic, complemented
37#11924511C2211yescharacteristic
38#11924509—21no—
39#11924509—21no—
40#11924506—21no—
41#11924504—21no—
42#11924498C2211yescharacteristic
43#11924498C2211yescharacteristic, complemented
44C4×C22⋊D37C2211yescharacteristic
45C4⋊C4×D37—21no—
46#11924465C2211yescharacteristic, complemented
47#11924459C2211yescharacteristic, complemented
48#11924456C2211yescharacteristic
49#11924456C2211yescharacteristic, complemented
50#11924456C2211yescharacteristic, complemented
51C22⋊#64436C2211yescharacteristic, complemented
52#11924448C2211yescharacteristic, complemented
53C8×Dic37C2211yescharacteristic, complemented
54C2×SD16×C37C2211yescharacteristic, complemented
55C2×M16×C37C2211yescharacteristic, complemented
56C22⋊Q8×C37C2211yescharacteristic, complemented
57#121×C37C2211yescharacteristic
58#107×C37C2211yescharacteristic, complemented
59#103×C37C2211yescharacteristic, complemented
60#102×C37C2211yescharacteristic, complemented
61C23×D37—21no—
62C23×D37—21no—
63C2×Q8×C37C2311yescharacteristic
64C2×D4×C37—21no—
65C2×D4×C37C2311yescharacteristic
66C22×C4×C37C2311yescharacteristic
67C2×C22⋊D37—21no—
68C2×C22⋊D37—21no—
69C2×C22⋊D37—21no—
70C2×C22⋊D37—21no—
71C2×C22⋊D37D411yescharacteristic
72#10596966—22no—
73#10596966—41no—
74D4×D37—22no—
75D4×D37—41no—
76D4×D37—44no—
77#10596962—22no—
78#10596962—41no—
79C2×D148—22no—
80C2×D148—21no—
81C2×D148—21no—
82C2×D148—21no—
83C2×D148—41no—
84C2×D148C2311yescharacteristic
85C2×D148C2311yescharacteristic
86C2×D148C2311yescharacteristic
87C2×C4×D37—21no—
88C2×C4×D37—21no—
89C2×C4×D37—21no—
90C2×C4×D37—21no—
91C2×C4×D37C2311yescharacteristic
92C2×Dic74C2311yescharacteristic
93SD16×C37—22no—
94M16×C37—22no—
95C2×C8×C37C2311yescharacteristic
96C2×C8×C37C2311yescharacteristic
97C4⋊C4×C37—21no—
98C4⋊C4×C37C2311yescharacteristic
99C4⋊C4×C37C2311yescharacteristic
100C22⋊C4×C37—21no—
101C22⋊C4×C37—21no—
102C22⋊Dic37D411yescharacteristic
103#10596941—22no—
104#10596941—41no—
105#10596940—41no—
106#10596939—22no—
107#10596938—21no—
108#10596938—21no—
109#10596938—21no—
110#10596938—21no—
111#10596938D411yescharacteristic
112C4⋊Dic37—21no—
113#10596936—21no—
114#10596936—21no—
115#10596936D411yescharacteristic
116C4×Dic37C2311yescharacteristic
117C2×#64436C2311yescharacteristic
118C2×#64436C2311yescharacteristic
119D296—22no—
120D296D414yescomplemented
121#10596930—22no—
122#10596930—22no—
123#10596929—22no—
124C23×C37—21no—
125C23×C37C2×D411yescharacteristic
126C22×D37—21no—
127C22×D37—21no—
128C22×D37—21no—
129C22×D37—42no—
130C22×D37—42no—
131C22×D37—42no—
132C22×D37—42no—
133C22×D37—81no—
134C22×D37—81no—
135C22×D37C2×D411yescharacteristic
136Q8×C37—21no—
137Q8×C37—21no—
138D4×C37—21no—
139D4×C37—21no—
140D4×C37—42no—
141C2×C4×C37—21no—
142C2×C4×C37—21no—
143C2×C4×C37—21no—
144C2×C4×C37—21no—
145C2×C4×C37—21no—
146C2×C4×C37C2×D411yescharacteristic
147C2×C4×C37C2411yescharacteristic
148C22⋊D37—22no—
149C22⋊D37—42no—
150C22⋊D37—42no—
151C22⋊D37—42no—
152C22⋊D37—81no—
153C2×Dic37—21no—
154C2×Dic37C2×D411yescharacteristic
155C2×Dic37C2×D411yescharacteristic
156C2×Dic37C2×D411yescharacteristic
157D148—21no—
158D148—21no—
159D148—21no—
160D148—21no—
161D148—42no—
162D148—42no—
163D148—42no—
164D148C2×D412yes—
165D148C2×D412yescomplemented
166C4×D37—21no—
167C4×D37—21no—
168C4×D37—42no—
169C4×D37—42no—
170C4×D37—42no—
171C4×D37—42no—
172Dic74—21no—
173Dic74—21no—
174C8×C37—21no—
175C8×C37C2×D412yescomplemented
176#64436—21no—
177#64436—21no—
178C22×C37—21no—
179C22×C37—21no—
180C22×C37—21no—
181C22×C37—41no—
182C22×C37—41no—
183C22×C37—41no—
184C22×C37C22×D411yescharacteristic
185C2×D37—21no—
186C2×D37—22no—
187C2×D37—22no—
188C2×D37—21no—
189C2×D37—21no—
190C2×D37—41no—
191C2×D37—41no—
192C2×D37—41no—
193C2×D37—41no—
194C2×D37—41no—
195C2×D37—41no—
196C2×D37—41no—
197C2×D37—84no—
198C2×D37—84no—
199C4×C37—21no—
200C4×C37—41no—
201C4×C37—41no—
202C4×C37—41no—
203C4×C37C22×D411yescharacteristic
204C4×C37H(5,𝔽2)11yescharacteristic
205Dic37—21no—
206Dic37—21no—
207Dic37—41no—
208Dic37AGL(1,ℤ/8)12yescomplemented
209#3272—371nomaximal, Sylow
210C2×C37—41no—
211C2×C37—81no—
212C2×C37D4211yescharacteristic
213C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C37#57711yescharacteristic
215D37—41no—
216D37—42no—
217D37—42no—
218D37—81no—
219C2×AGL(1,ℤ/8)—371no—
220C2×AGL(1,ℤ/8)—371no—
221#547—371no—
222D42—371no—
223#492—371no—
224#468—371no—
225#468D3711yescharacteristic, complemented
226#461—371no—
227#459—371no—
228#450—371no—
229#450—371no—
230#449—371no—
231#449—371no—
232C4×D8—371no—
233#436—371no—
234C37#327211yescharacteristic, complemented, Sylow
235C2×#41—371no—
236C2×#41—371no—
237C22×D4—371no—
238C22×D4—371no—
239C22×D4—741no—
240AGL(1,ℤ/8)—744no—
241AGL(1,ℤ/8)—744no—
242C2×SD16—371no—
243C2×SD16—371no—
244C2×SD16—371no—
245C2×SD16—371no—
246C2×SD16—371no—
247C2×SD16C2×D3711yescharacteristic, complemented
248C2×D8—371no—
249C2×D8—371no—
250C2×D8—371no—
251C2×M16—371no—
252C2×M16C2×D3711yescharacteristic, complemented
253C4⋊Q8—371no—
254#127—371no—
255#123—741no—
256#123—741no—
257C22⋊Q8—371no—
258C22⋊Q8C2×D3711yescharacteristic, complemented
259#121—371no—
260#121—741no—
261#121—741no—
262#121C2×D3711yescharacteristic
263#120—741no—
264#120—741no—
265C4×D4—371no—
266C4×D4—371no—
267C4×D4—371no—
268C2×C4⋊C4—741no—
269#107C2×D3711yescharacteristic, complemented
270C4⋊C8—371no—
271#103—371no—
272#103C2×D3711yescharacteristic, complemented
273#102—371no—
274#102—371no—
275#102—371no—
276#102C2×D3711yescharacteristic, complemented
277C22⋊C8—371no—
278C22⋊C8—371no—
279C4×C8—371no—
280C24—741no—
281C24—741no—
282#41—742no—
283#41—742no—
284#41—1481no—
285#41—1481no—
286C2×Q8—371no—
287C2×Q8C22×D3711yescharacteristic
288C2×D4—21no—
289C2×D4—371no—
290C2×D4—371no—
291C2×D4—371no—
292C2×D4—371no—
293C2×D4—742no—
294C2×D4—741no—
295C2×D4—741no—
296C2×D4—741no—
297C2×D4—741no—
298C2×D4—741no—
299C2×D4—742no—
300C2×D4—741no—
301C2×D4—741no—
302C2×D4—1481no—
303C2×D4—1481no—
304C2×D4—1484no—
305C2×D4C22×D3711yescharacteristic
306C22×C4—371no—
307C22×C4—741no—
308C22×C4—741no—
309C22×C4—741no—
310C22×C4—741no—
311C22×C4C22×D3711yescharacteristic
312SD16—22no—
313SD16—742no—
314SD16—742no—
315SD16—742no—
316SD16—1481no—
317SD16—1481no—
318D8—374no—
319D8—742no—
320D8—742no—
321M16—22no—
322M16—742no—
323C2×C8—371no—
324C2×C8—371no—
325C2×C8C22×D3711yescharacteristic
326C2×C8C22×D3711yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—371no—
329C4⋊C4—741no—
330C4⋊C4—741no—
331C4⋊C4—741no—
332C4⋊C4C22×D3711yescharacteristic
333C4⋊C4C22×D3711yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—371no—
337C22⋊C4—371no—
338C22⋊C4—741no—
339C22⋊C4—741no—
340C22⋊C4—741no—
341C22⋊C4—741no—
342C42—371no—
343C23—21no—
344C23—371no—
345C23—2961no—
346C23—2961no—
347C23—741no—
348C23—741no—
349C23—741no—
350C23—1482no—
351C23—1482no—
352C23—1482no—
353C23—1482no—
354C23D4×D3711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—741no—
358Q8—741no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—372no—
363D4—372no—
364D4—2961no—
365D4—742no—
366D4—741no—
367D4—741no—
368D4—741no—
369D4—741no—
370D4—1482no—
371D4—1482no—
372D4—1482no—
373D4—1482no—
374D4—1482no—
375D4—1482no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—371no—
382C2×C4—371no—
383C2×C4—371no—
384C2×C4—741no—
385C2×C4—741no—
386C2×C4—741no—
387C2×C4—1482no—
388C2×C4—1482no—
389C2×C4—1482no—
390C2×C4—1482no—
391C2×C4D4×D3711yescharacteristic
392C2×C4C23×D3711yescharacteristic
393C8—21no—
394C8—741no—
395C8—741no—
396C8D4×D3712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—2964no—
404C22—2964no—
405C22—741no—
406C22—741no—
407C22—741no—
408C22—742no—
409C22—742no—
410C22—1481no—
411C22—1481no—
412C22—1481no—
413C22—1481no—
414C22—1481no—
415C22—1481no—
416C22—1481no—
417C22C2×D4×D3711yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—372no—
423C4—741no—
424C4—741no—
425C4—1481no—
426C4C2×D4×D3711yescharacteristic
427C4#1192458211yescharacteristic
428C2—41no—
429C2—81no—
430C2—2961no—
431C2—1481no—
432C2—1482no—
433C2—1482no—
434C2#1455816711yescharacteristic
435C2AGL(1,ℤ/8)×D3711yescharacteristic
436C2#1455834211yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#468C2
socleC22×C37C22×D4
hypercenter#468D37
hypocenterC37#3272
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#11924564C2semidirect
2. C2×#11924563C2semidirect
3. #14558185C2semidirect
4. #14558167C2semidirect
5. #14557904C2semidirect
6. #14557780C2semidirect
7. #14557737C2semidirect
8. #14557727C2semidirect
9. #14557685C2semidirect
10. #14557683C2semidirect
11. #14557681C2semidirect
12. #14557676C2semidirect
13. #14557615C2semidirect
14. #14557609C2semidirect
15. C37×#468C2semidirect
16. C2×#10596966C22semidirect
17. C2×#10596962C22semidirect
24. C2×#10596930C22semidirect
25. C2×#10596940C22semidirect
27. C2×#10596941C22semidirect
29. C2×D296C22semidirect
30. C2×#10596939C22semidirect
31. C2×#10596929C22semidirect
32. #11924534C22semidirect
36. #11924516C22semidirect
43. #11924498C22semidirect
46. #11924465C22semidirect
47. #11924459C22semidirect
49. #11924456C22semidirect
50. #11924456C22semidirect
51. C22⋊#64436C22semidirect
52. #11924448C22semidirect
53. C8×Dic37C22semidirect
54. C2×SD16×C37C22semidirect
55. C2×M16×C37C22semidirect
56. C22⋊Q8×C37C22semidirect
58. #107×C37C22semidirect
59. #103×C37C22semidirect
60. #102×C37C22semidirect
120. D296D4semidirect
165. D148C2×D4semidirect
175. C8×C37C2×D4semidirect
208. Dic37AGL(1,ℤ/8)semidirect
225. #468D37semidirect
234. C37#3272semidirect
247. C2×SD16C2×D37semidirect
252. C2×M16C2×D37semidirect
258. C22⋊Q8C2×D37semidirect
269. #107C2×D37semidirect
272. #103C2×D37semidirect
276. #102C2×D37semidirect
396. C8D4×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245642
C2×#119245632
#145581852
#145581672
#145579042
#145577802
#145577372
#145577272
#145576852
#145576832
#145576812
#145576762
#145576152
#145576092
C37×#4682
#327237