Finite Group Databasebeta

← #18006075

Subgroups — #18006075

Order 4736.

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×#11924563C211yesmaximal, characteristic, complemented
2C2×D8×D37C211yesmaximal, characteristic, complemented
3#14558176C211yesmaximal, characteristic, complemented
4#14558168C211yesmaximal, characteristic, complemented
5#14557783C211yesmaximal, characteristic, complemented
6#14557759C211yesmaximal, characteristic, complemented
7#14557745C211yesmaximal, characteristic, complemented
8#14557725C211yesmaximal, characteristic, complemented
9#14557682C211yesmaximal, characteristic, complemented
10#14557677C211yesmaximal, characteristic, complemented
11#14557665C211yesmaximal, characteristic, complemented
12#14557665C211yesmaximal, characteristic, complemented
13#14557646C211yesmaximal, characteristic, complemented
14#14557606C211yesmaximal, characteristic, complemented
15C37×#496C211yesmaximal, characteristic, complemented
16C2×#10596966C2211yescharacteristic, complemented
17C2×D4×D37—21no—
18C2×D4×D37C2211yescharacteristic
19C2×D4×D37C2211yescharacteristic, complemented
20C2×D4×D37C2211yescharacteristic, complemented
21#11924563—24no—
22C2×#10596941C2211yescharacteristic, complemented
23C2×D296C2211yescharacteristic
24C2×#10596939C2211yescharacteristic
25C2×#10596939C2211yescharacteristic, complemented
26C2×#10596939C2211yescharacteristic, complemented
27D8×D37—24no—
28C2×#10596929C2211yescharacteristic, complemented
29C2×C8×D37C2211yescharacteristic, complemented
30#11924518—21no—
31#11924512C2211yescharacteristic, complemented
32#11924509—21no—
33#11924507—21no—
34#11924507C2211yescharacteristic
35#11924507C2211yescharacteristic, complemented
36#11924507C2211yescharacteristic, complemented
37#11924506—21no—
38#11924504—21no—
39#11924504—21no—
40C4×D148C2211yescharacteristic
41C4×D148C2211yescharacteristic, complemented
42C4×D148C2211yescharacteristic, complemented
43C22⋊C4×D37—21no—
44#11924468C2211yescharacteristic, complemented
45C4⋊#64436C2211yescharacteristic, complemented
46#11924460C2211yescharacteristic, complemented
47#11924458C2211yescharacteristic, complemented
48#11924458C2211yescharacteristic, complemented
49#11924457C2211yescharacteristic, complemented
50#11924457C2211yescharacteristic, complemented
51#11924457C2211yescharacteristic, complemented
52#11924448C2211yescharacteristic, complemented
53#11924448C2211yescharacteristic, complemented
54C2×SD16×C37C2211yescharacteristic, complemented
55C2×SD16×C37C2211yescharacteristic, complemented
56C2×D8×C37C2211yescharacteristic, complemented
57C2×D8×C37C2211yescharacteristic, complemented
58#127×C37C2211yescharacteristic, complemented
59#124×C37C2211yescharacteristic, complemented
60#97×C37C2211yescharacteristic, complemented
61C23×D37—21no—
62C23×D37—21no—
63C23×D37—21no—
64C2×Q8×C37C2311yescharacteristic
65C2×D4×C37—21no—
66C2×D4×C37—21no—
67C2×D4×C37C2311yescharacteristic
68C2×D4×C37C2311yescharacteristic
69C2×D4×C37C2311yescharacteristic
70C2×C22⋊D37—21no—
71C2×C22⋊D37—21no—
72C2×C22⋊D37—21no—
73C2×C22⋊D37—21no—
74C2×C22⋊D37—21no—
75C2×C22⋊D37—21no—
76#10596966—22no—
77#10596966—41no—
78D4×D37—22no—
79D4×D37—22no—
80D4×D37—22no—
81D4×D37—41no—
82D4×D37—41no—
83D4×D37—41no—
84D4×D37—44no—
85C2×D148—21no—
86C2×D148D411yescharacteristic
87C2×D148C2311yescharacteristic
88C2×D148C2311yescharacteristic
89C2×C4×D37—21no—
90C2×C4×D37—21no—
91C2×C4×D37—21no—
92C2×C4×D37C2311yescharacteristic
93C2×C4×D37C2311yescharacteristic
94SD16×C37—22no—
95SD16×C37—41no—
96D8×C37—22no—
97D8×C37—41no—
98C2×C8×C37C2311yescharacteristic
99C2×C8×C37C2311yescharacteristic
100C22⋊C4×C37—21no—
101C22⋊C4×C37—21no—
102C42×C37C2311yescharacteristic
103C22⋊Dic37—21no—
104C22⋊Dic37—21no—
105C22⋊Dic37—21no—
106#10596941—22no—
107#10596939—22no—
108#10596939—22no—
109#10596939—22no—
110#10596938—21no—
111#10596938—21no—
112#10596938—21no—
113#10596938D411yescharacteristic
114#10596938D411yescharacteristic
115C4⋊Dic37D411yescharacteristic
116C4⋊Dic37C2311yescharacteristic
117C4⋊Dic37C2311yescharacteristic
118#10596936—21no—
119C2×#64436C2311yescharacteristic
120C2×#64436C2311yescharacteristic
121D296D414yescomplemented
122#10596929—22no—
123C8×D37—22no—
124C23×C37—21no—
125C23×C37—21no—
126C23×C37—21no—
127C22×D37—21no—
128C22×D37—21no—
129C22×D37—42no—
130C22×D37—42no—
131C22×D37—42no—
132C22×D37—42no—
133C22×D37—42no—
134C22×D37—42no—
135C22×D37—81no—
136C22×D37—81no—
137C22×D37—81no—
138C22×D37C2×D411yescharacteristic
139C22×D37C2×D411yescharacteristic
140Q8×C37—21no—
141Q8×C37—21no—
142D4×C37—21no—
143D4×C37—21no—
144D4×C37—21no—
145D4×C37—21no—
146D4×C37—21no—
147D4×C37—21no—
148D4×C37—42no—
149D4×C37—81no—
150C2×C4×C37—21no—
151C2×C4×C37C2×D411yescharacteristic
152C2×C4×C37C2×D411yescharacteristic
153C2×C4×C37C2411yescharacteristic
154C22⋊D37—42no—
155C22⋊D37—42no—
156C22⋊D37—42no—
157C22⋊D37—42no—
158C22⋊D37—42no—
159C22⋊D37—81no—
160C2×Dic37—21no—
161C2×Dic37—21no—
162C2×Dic37C2×D411yescharacteristic
163C2×Dic37C2×D411yescharacteristic
164D148—22no—
165D148—21no—
166D148—21no—
167D148—42no—
168D148C2×D412yes—
169D148C2×D412yescomplemented
170C4×D37—21no—
171C4×D37—21no—
172C4×D37—21no—
173C4×D37—21no—
174C4×D37—42no—
175C4×D37—42no—
176C4×D37—81no—
177C8×C37—21no—
178C8×C37C2×D412yescomplemented
179#64436—21no—
180#64436—21no—
181C22×C37—41no—
182C22×C37—41no—
183C22×C37—41no—
184C22×C37—41no—
185C22×C37—41no—
186C22×C37—41no—
187C22×C37—41no—
188C22×C37—41no—
189C22×C37—41no—
190C22×C37C22×D411yescharacteristic
191C2×D37—21no—
192C2×D37—21no—
193C2×D37—21no—
194C2×D37—22no—
195C2×D37—22no—
196C2×D37—21no—
197C2×D37—21no—
198C2×D37—41no—
199C2×D37—41no—
200C2×D37—41no—
201C2×D37—41no—
202C2×D37—84no—
203C2×D37—84no—
204C2×D37—84no—
205C2×D37AGL(1,ℤ/8)12yescomplemented
206C4×C37—21no—
207C4×C37—21no—
208C4×C37—41no—
209C4×C37C22×D411yescharacteristic
210C4×C37H(5,𝔽2)11yescharacteristic
211Dic37—21no—
212Dic37—21no—
213Dic37—41no—
214Dic37—41no—
215#3264—371nomaximal, Sylow
216C2×C37—81no—
217C2×C37—81no—
218C2×C37—81no—
219C2×C37D4211yescharacteristic
220C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C37#57611yescharacteristic
222D37—22no—
223D37—41no—
224D37—42no—
225D37—42no—
226C2×AGL(1,ℤ/8)—371no—
227C22×D8—371no—
228#546—371no—
229D42—371no—
230#496D3711yescharacteristic, complemented
231#469—371no—
232#465—371no—
233#463—371no—
234#459—371no—
235#450—371no—
236#449—371no—
237#447—371no—
238#447—371no—
239#442—371no—
240#435—371no—
241C37#326411yescharacteristic, complemented, Sylow
242C2×#41—371no—
243C22×D4—371no—
244C22×D4—371no—
245C22×D4—371no—
246C22×D4—741no—
247AGL(1,ℤ/8)—744no—
248C2×SD16—371no—
249C2×SD16C2×D3711yescharacteristic, complemented
250C2×SD16C2×D3711yescharacteristic, complemented
251C2×D8—371no—
252C2×D8—371no—
253C2×D8—371no—
254C2×D8—371no—
255C2×D8—744no—
256C2×D8C2×D3711yescharacteristic, complemented
257C2×D8C2×D3711yescharacteristic, complemented
258C2×M16—371no—
259C22×C8—371no—
260#127C2×D3711yescharacteristic, complemented
261#124C2×D3711yescharacteristic, complemented
262#123—741no—
263C22⋊Q8—371no—
264#121—371no—
265#121—371no—
266#121—371no—
267#121—741no—
268#121—741no—
269#120—741no—
270#120—741no—
271#120—741no—
272C4×D4—371no—
273C4×D4—371no—
274C4×D4—371no—
275C2×C22⋊C4—741no—
276#106—371no—
277C4⋊C8—371no—
278#103—371no—
279#102—371no—
280#102—371no—
281#102—371no—
282#102—371no—
283#102—371no—
284C22⋊C8—371no—
285C22⋊C8—371no—
286#97C2×D3711yescharacteristic, complemented
287C24—741no—
288C24—741no—
289C24—741no—
290#41—742no—
291#41—1481no—
292C2×Q8C22×D3711yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—371no—
296C2×D4—371no—
297C2×D4—371no—
298C2×D4—741no—
299C2×D4—741no—
300C2×D4—741no—
301C2×D4—741no—
302C2×D4—742no—
303C2×D4—742no—
304C2×D4—742no—
305C2×D4—741no—
306C2×D4—741no—
307C2×D4—741no—
308C2×D4—1481no—
309C2×D4—1481no—
310C2×D4—1481no—
311C2×D4—1484no—
312C2×D4C22×D3711yescharacteristic
313C2×D4C22×D3711yescharacteristic
314C2×D4C22×D3711yescharacteristic
315C22×C4—371no—
316C22×C4—371no—
317C22×C4—741no—
318C22×C4—741no—
319C22×C4—741no—
320SD16—22no—
321SD16—41no—
322SD16—742no—
323D8—22no—
324D8—41no—
325D8—374no—
326D8—742no—
327D8—742no—
328D8—742no—
329M16—742no—
330C2×C8—371no—
331C2×C8—371no—
332C2×C8—742no—
333C2×C8C22×D3711yescharacteristic
334C2×C8C22×D3711yescharacteristic
335C4⋊C4—371no—
336C4⋊C4—371no—
337C4⋊C4—371no—
338C4⋊C4—741no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—371no—
342C22⋊C4—371no—
343C22⋊C4—741no—
344C22⋊C4—741no—
345C22⋊C4—741no—
346C22⋊C4—741no—
347C22⋊C4—741no—
348C22⋊C4—741no—
349C42C22×D3711yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—371no—
354C23—371no—
355C23—2961no—
356C23—2961no—
357C23—2961no—
358C23—741no—
359C23—741no—
360C23—1482no—
361C23—1482no—
362C23—1482no—
363C23—1482no—
364C23—1482no—
365C23—1482no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—372no—
377D4—372no—
378D4—2961no—
379D4—742no—
380D4—741no—
381D4—741no—
382D4—1482no—
383D4—1482no—
384D4—1482no—
385D4—1482no—
386D4—1482no—
387D4—1482no—
388C2×C4—21no—
389C2×C4—371no—
390C2×C4—371no—
391C2×C4—2961no—
392C2×C4—741no—
393C2×C4—741no—
394C2×C4—741no—
395C2×C4—741no—
396C2×C4—741no—
397C2×C4—741no—
398C2×C4—1482no—
399C2×C4—1482no—
400C2×C4D4×D3711yescharacteristic
401C2×C4D4×D3711yescharacteristic
402C2×C4C23×D3711yescharacteristic
403C8—21no—
404C8—741no—
405C8—741no—
406C8D4×D3712yescomplemented
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—372no—
417C22—2964no—
418C22—2964no—
419C22—2964no—
420C22—741no—
421C22—741no—
422C22—741no—
423C22—741no—
424C22—741no—
425C22—742no—
426C22—742no—
427C22—1481no—
428C22—1481no—
429C22—1481no—
430C22—1481no—
431C22C2×D4×D3711yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—741no—
436C4—741no—
437C4—1481no—
438C4—1481no—
439C4C2×D4×D3711yescharacteristic
440C4#1192458111yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—742no—
445C2—1481no—
446C2—1482no—
447C2—1482no—
448C2#1455816811yescharacteristic
449C2AGL(1,ℤ/8)×D3711yescharacteristic
450C2#1455833711yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#496C2
socleC22×C37C22×D4
hypercenter#496D37
hypocenterC37#3264
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#11924563C2semidirect
2. C2×D8×D37C2semidirect
3. #14558176C2semidirect
4. #14558168C2semidirect
5. #14557783C2semidirect
6. #14557759C2semidirect
7. #14557745C2semidirect
8. #14557725C2semidirect
9. #14557682C2semidirect
10. #14557677C2semidirect
11. #14557665C2semidirect
12. #14557665C2semidirect
13. #14557646C2semidirect
14. #14557606C2semidirect
15. C37×#496C2semidirect
16. C2×#10596966C22semidirect
19. C2×D4×D37C22semidirect
20. C2×D4×D37C22semidirect
22. C2×#10596941C22semidirect
25. C2×#10596939C22semidirect
26. C2×#10596939C22semidirect
28. C2×#10596929C22semidirect
29. C2×C8×D37C22semidirect
31. #11924512C22semidirect
35. #11924507C22semidirect
36. #11924507C22semidirect
41. C4×D148C22semidirect
42. C4×D148C22semidirect
44. #11924468C22semidirect
45. C4⋊#64436C22semidirect
46. #11924460C22semidirect
47. #11924458C22semidirect
48. #11924458C22semidirect
49. #11924457C22semidirect
50. #11924457C22semidirect
51. #11924457C22semidirect
52. #11924448C22semidirect
53. #11924448C22semidirect
54. C2×SD16×C37C22semidirect
55. C2×SD16×C37C22semidirect
56. C2×D8×C37C22semidirect
57. C2×D8×C37C22semidirect
58. #127×C37C22semidirect
59. #124×C37C22semidirect
60. #97×C37C22semidirect
121. D296D4semidirect
169. D148C2×D4semidirect
178. C8×C37C2×D4semidirect
205. C2×D37AGL(1,ℤ/8)semidirect
230. #496D37semidirect
241. C37#3264semidirect
249. C2×SD16C2×D37semidirect
250. C2×SD16C2×D37semidirect
256. C2×D8C2×D37semidirect
257. C2×D8C2×D37semidirect
260. #127C2×D37semidirect
261. #124C2×D37semidirect
286. #97C2×D37semidirect
406. C8D4×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245632
C2×D8×D372
#145581762
#145581682
#145577832
#145577592
#145577452
#145577252
#145576822
#145576772
#145576652
#145576462
#145576062
C37×#4962
#326437