Finite Group Databasebeta

← #18006078

Subgroups — #18006078

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×#11924560C211yesmaximal, characteristic, complemented
2C2×D8×D37C211yesmaximal, characteristic, complemented
3#14558178C211yesmaximal, characteristic, complemented
4D4×D148C211yesmaximal, characteristic, complemented
5#14557922C211yesmaximal, characteristic, complemented
6#14557787C211yesmaximal, characteristic, complemented
7#14557763C211yesmaximal, characteristic, complemented
8#14557748C211yesmaximal, characteristic, complemented
9#14557728C211yesmaximal, characteristic, complemented
10#14557685C211yesmaximal, characteristic, complemented
11#14557679C211yesmaximal, characteristic, complemented
12#14557668C211yesmaximal, characteristic, complemented
13#14557668C211yesmaximal, characteristic, complemented
14#14557606C211yesmaximal, characteristic, complemented
15C37×#442C211yesmaximal, characteristic, complemented
16C2×#10596964C2211yescharacteristic, complemented
17C22×D148—21no—
18C2×D4×D37C2211yescharacteristic
19C2×D4×D37C2211yescharacteristic, complemented
20C2×D4×D37C2211yescharacteristic, complemented
21#11924560—24no—
22C2×#10596930C2211yescharacteristic, complemented
23C2×#10596930C2211yescharacteristic, complemented
24C2×#10596940C2211yescharacteristic, complemented
25C2×D296C2211yescharacteristic, complemented
26C2×D296C2211yescharacteristic, complemented
27C2×#10596939C2211yescharacteristic
28C2×#10596939C2211yescharacteristic, complemented
29C2×#10596939C2211yescharacteristic, complemented
30D8×D37—24no—
31C2×#10596929C2211yescharacteristic, complemented
32C2×C8×D37C2211yescharacteristic, complemented
33#11924533C2211yescharacteristic, complemented
34#11924524C2211yescharacteristic, complemented
35#11924520—21no—
36#11924516C2211yescharacteristic, complemented
37#11924511C2211yescharacteristic
38#11924511C2211yescharacteristic, complemented
39#11924511C2211yescharacteristic, complemented
40#11924510—21no—
41#11924510—21no—
42#11924506—21no—
43#11924506—21no—
44#11924506—21no—
45C4×D148C2211yescharacteristic
46C2×#10596938—21no—
47C4⋊#64436C2211yescharacteristic, complemented
48#11924459C2211yescharacteristic, complemented
49#11924456C2211yescharacteristic, complemented
50#11924456C2211yescharacteristic, complemented
51#11924456C2211yescharacteristic, complemented
52#11924448C2211yescharacteristic, complemented
53#11924448C2211yescharacteristic, complemented
54C2×D8×C37C2211yescharacteristic
55C4×D4×C37C2211yescharacteristic, complemented
56C4×D4×C37C2211yescharacteristic, complemented
57#106×C37C2211yescharacteristic, complemented
58#102×C37C2211yescharacteristic, complemented
59#102×C37C2211yescharacteristic, complemented
60#97×C37C2211yescharacteristic, complemented
61C23×D37—21no—
62C23×D37—21no—
63C23×D37—21no—
64C2×D4×C37C2311yescharacteristic
65C2×D4×C37C2311yescharacteristic
66C22×C4×C37—21no—
67C22×C4×C37—21no—
68C2×C22⋊D37—21no—
69C2×C22⋊D37—21no—
70C2×C22⋊D37—21no—
71C2×C22⋊D37—21no—
72C22×Dic37—21no—
73#10596964—22no—
74#10596964—41no—
75D4×D37—22no—
76D4×D37—22no—
77D4×D37—22no—
78D4×D37—41no—
79D4×D37—41no—
80D4×D37—41no—
81C2×D148—21no—
82C2×D148—21no—
83C2×D148—21no—
84C2×D148—21no—
85C2×D148—21no—
86C2×D148—44no—
87C2×D148D411yescharacteristic
88C2×D148C2311yescharacteristic
89C2×D148C2311yescharacteristic
90C2×D148C2311yescharacteristic
91C2×C4×D37C2311yescharacteristic
92C2×C4×D37C2311yescharacteristic
93C2×Dic74C2311yescharacteristic
94D8×C37D414yescomplemented
95C2×C8×C37C2311yescharacteristic
96C2×C8×C37C2311yescharacteristic
97C4⋊C4×C37C2311yescharacteristic
98C4⋊C4×C37C2311yescharacteristic
99C22⋊C4×C37—21no—
100C22⋊C4×C37—21no—
101C42×C37C2311yescharacteristic
102#10596940—22no—
103#10596939—22no—
104#10596939—22no—
105#10596939—22no—
106#10596938—21no—
107#10596938—21no—
108#10596938—21no—
109#10596938—21no—
110#10596938—21no—
111#10596938—21no—
112#10596938D411yescharacteristic
113#10596938D411yescharacteristic
114C4⋊Dic37—21no—
115C4⋊Dic37D411yescharacteristic
116C2×#64436C2311yescharacteristic
117C2×#64436C2311yescharacteristic
118D296—22no—
119D296—41no—
120#10596930—22no—
121#10596930—41no—
122#10596929—22no—
123C8×D37—22no—
124C23×C37—21no—
125C23×C37—21no—
126C22×D37—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
140D4×C37—21no—
141D4×C37—21no—
142D4×C37C2×D412yes—
143D4×C37C2×D412yescomplemented
144C2×C4×C37—21no—
145C2×C4×C37—21no—
146C2×C4×C37—42no—
147C2×C4×C37—81no—
148C2×C4×C37C2×D411yescharacteristic
149C2×C4×C37C2×D411yescharacteristic
150C2×C4×C37C2411yescharacteristic
151C22⋊D37—42no—
152C22⋊D37—42no—
153C22⋊D37—42no—
154C22⋊D37—42no—
155C2×Dic37—21no—
156C2×Dic37—42no—
157C2×Dic37C2×D411yescharacteristic
158C2×Dic37C2×D411yescharacteristic
159D148—22no—
160D148—21no—
161D148—21no—
162D148—21no—
163D148—21no—
164D148—21no—
165D148—21no—
166D148—42no—
167D148—42no—
168D148—42no—
169D148—81no—
170D148—81no—
171C4×D37—21no—
172C4×D37—21no—
173C4×D37—21no—
174C4×D37—21no—
175Dic74—21no—
176Dic74—21no—
177C8×C37—21no—
178C8×C37C2×D412yescomplemented
179#64436—21no—
180#64436—21no—
181C22×C37—22no—
182C22×C37—22no—
183C22×C37—41no—
184C22×C37—41no—
185C22×C37—41no—
186C22×C37—41no—
187C22×C37C22×D411yescharacteristic
188C2×D37—21no—
189C2×D37—21no—
190C2×D37—21no—
191C2×D37—21no—
192C2×D37—21no—
193C2×D37—41no—
194C2×D37—41no—
195C2×D37—41no—
196C2×D37—41no—
197C2×D37—41no—
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×C37—41no—
210C4×C37C22×D411yescharacteristic
211C4×C37H(5,𝔽2)11yescharacteristic
212Dic37—21no—
213Dic37—21no—
214Dic37—41no—
215#3264—371nomaximal, Sylow
216C2×C37—42no—
217C2×C37—42no—
218C2×C37D4211yescharacteristic
219C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C37#57611yescharacteristic
221D37—22no—
222D37—41no—
223D37—81no—
224D37—81no—
225D37—81no—
226C2×AGL(1,ℤ/8)—371no—
227C22×D8—371no—
228#546—371no—
229D42—371no—
230#496—371no—
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#442D3711yescharacteristic, complemented
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×SD16—371no—
250C2×SD16—371no—
251C2×D8—371no—
252C2×D8—371no—
253C2×D8—371no—
254C2×D8—371no—
255C2×D8—371no—
256C2×D8—744no—
257C2×D8C2×D3711yescharacteristic
258C2×M16—371no—
259C22×C8—371no—
260#127—371no—
261#124—371no—
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×D4C2×D3711yescharacteristic, complemented
274C4×D4C2×D3711yescharacteristic, complemented
275C2×C22⋊C4—741no—
276#106C2×D3711yescharacteristic, complemented
277C4⋊C8—371no—
278#103—371no—
279#102—371no—
280#102—371no—
281#102—371no—
282#102C2×D3711yescharacteristic, complemented
283#102C2×D3711yescharacteristic, complemented
284C22⋊C8—371no—
285C22⋊C8—371no—
286#97C2×D3711yescharacteristic, complemented
287C24—741no—
288C24—741no—
289C24—741no—
290#41—742no—
291#41—1481no—
292C2×Q8—371no—
293C2×D4—371no—
294C2×D4—371no—
295C2×D4—371no—
296C2×D4—371no—
297C2×D4—741no—
298C2×D4—741no—
299C2×D4—741no—
300C2×D4—741no—
301C2×D4—741no—
302C2×D4—741no—
303C2×D4—742no—
304C2×D4—742no—
305C2×D4—742no—
306C2×D4—741no—
307C2×D4—741no—
308C2×D4—741no—
309C2×D4—1484no—
310C2×D4—1481no—
311C2×D4—1481no—
312C2×D4—1481no—
313C2×D4C22×D3711yescharacteristic
314C2×D4C22×D3711yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—371no—
318C22×C4—371no—
319C22×C4—741no—
320SD16—742no—
321SD16—742no—
322SD16—1481no—
323D8—742no—
324D8—742no—
325D8—742no—
326D8—742no—
327D8—1481no—
328D8D14814yescomplemented
329M16—742no—
330C2×C8—371no—
331C2×C8—371no—
332C2×C8—742no—
333C2×C8C22×D3711yescharacteristic
334C2×C8C22×D3711yescharacteristic
335C4⋊C4—371no—
336C4⋊C4—741no—
337C4⋊C4C22×D3711yescharacteristic
338C4⋊C4C22×D3711yescharacteristic
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—371no—
353C23—371no—
354C23—2961no—
355C23—2961no—
356C23—2961no—
357C23—741no—
358C23—741no—
359C23—741no—
360C23—1482no—
361C23—1482no—
362C23—1482no—
363C23—1482no—
364C23—1482no—
365C23—1482no—
366Q8—741no—
367Q8—741no—
368D4—21no—
369D4—21no—
370D4—2961no—
371D4—2961no—
372D4—742no—
373D4—741no—
374D4—741no—
375D4—741no—
376D4—741no—
377D4—741no—
378D4—741no—
379D4—1482no—
380D4—1482no—
381D4—1482no—
382D4—1482no—
383D4—1482no—
384D4—1482no—
385D4—1482no—
386D4C2×D14812yes—
387D4C2×D14812yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—371no—
393C2×C4—371no—
394C2×C4—741no—
395C2×C4—741no—
396C2×C4—741no—
397C2×C4—741no—
398C2×C4—741no—
399C2×C4—1482no—
400C2×C4D4×D3711yescharacteristic
401C2×C4D4×D3711yescharacteristic
402C2×C4C23×D3711yescharacteristic
403C8—21no—
404C8—741no—
405C8—741no—
406C8C2×D14812yescomplemented
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—372no—
414C22—2964no—
415C22—2964no—
416C22—2964no—
417C22—741no—
418C22—741no—
419C22—741no—
420C22—741no—
421C22—741no—
422C22—1481no—
423C22—1481no—
424C22—1481no—
425C22—1481no—
426C22—1481no—
427C22—1481no—
428C22—1481no—
429C22—1481no—
430C22—1481no—
431C22C2×D4×D3711yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—741no—
437C4—741no—
438C4—1481no—
439C4C22×D14811yescharacteristic
440C4#1192458211yescharacteristic
441C2—42no—
442C2—42no—
443C2—2961no—
444C2—2961no—
445C2—2961no—
446C2—742no—
447C2—1481no—
448C2D4×D14811yescharacteristic
449C2AGL(1,ℤ/8)×D3711yescharacteristic
450C2#1455833711yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#442C2
socleC22×C37C22×D4
hypercenter#442D37
hypocenterC37#3264
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#11924560C2semidirect
2. C2×D8×D37C2semidirect
3. #14558178C2semidirect
4. D4×D148C2semidirect
5. #14557922C2semidirect
6. #14557787C2semidirect
7. #14557763C2semidirect
8. #14557748C2semidirect
9. #14557728C2semidirect
10. #14557685C2semidirect
11. #14557679C2semidirect
12. #14557668C2semidirect
13. #14557668C2semidirect
14. #14557606C2semidirect
15. C37×#442C2semidirect
16. C2×#10596964C22semidirect
19. C2×D4×D37C22semidirect
20. C2×D4×D37C22semidirect
22. C2×#10596930C22semidirect
23. C2×#10596930C22semidirect
24. C2×#10596940C22semidirect
25. C2×D296C22semidirect
26. C2×D296C22semidirect
28. C2×#10596939C22semidirect
29. C2×#10596939C22semidirect
31. C2×#10596929C22semidirect
32. C2×C8×D37C22semidirect
33. #11924533C22semidirect
34. #11924524C22semidirect
36. #11924516C22semidirect
38. #11924511C22semidirect
39. #11924511C22semidirect
47. C4⋊#64436C22semidirect
48. #11924459C22semidirect
49. #11924456C22semidirect
50. #11924456C22semidirect
51. #11924456C22semidirect
52. #11924448C22semidirect
53. #11924448C22semidirect
55. C4×D4×C37C22semidirect
56. C4×D4×C37C22semidirect
57. #106×C37C22semidirect
58. #102×C37C22semidirect
59. #102×C37C22semidirect
60. #97×C37C22semidirect
94. D8×C37D4semidirect
143. D4×C37C2×D4semidirect
178. C8×C37C2×D4semidirect
205. C2×D37AGL(1,ℤ/8)semidirect
239. #442D37semidirect
241. C37#3264semidirect
273. C4×D4C2×D37semidirect
274. C4×D4C2×D37semidirect
276. #106C2×D37semidirect
282. #102C2×D37semidirect
283. #102C2×D37semidirect
286. #97C2×D37semidirect
328. D8D148semidirect
387. D4C2×D148semidirect
406. C8C2×D148semidirect

Maximal subgroups

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

subgroupindex
C2×#119245602
C2×D8×D372
#145581782
D4×D1482
#145579222
#145577872
#145577632
#145577482
#145577282
#145576852
#145576792
#145576682
#145576062
C37×#4422
#326437