Finite Group Databasebeta

← #18006077

Subgroups — #18006077

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×#11924561C211yesmaximal, characteristic, complemented
2C22×#10596939C211yesmaximal, characteristic, complemented
3D4×D148C211yesmaximal, characteristic, complemented
4#14557923C211yesmaximal, characteristic, complemented
5#14557784C211yesmaximal, characteristic, complemented
6#14557764C211yesmaximal, characteristic, complemented
7#14557745C211yesmaximal, characteristic, complemented
8#14557728C211yesmaximal, characteristic, complemented
9#14557683C211yesmaximal, characteristic, complemented
10#14557678C211yesmaximal, characteristic, complemented
11#14557667C211yesmaximal, characteristic, complemented
12#14557666C211yesmaximal, characteristic, complemented
13#14557643C211yesmaximal, characteristic, complemented
14#14557605C211yesmaximal, characteristic, complemented
15C37×#546C211yesmaximal, characteristic, complemented
16C22×D148C2211yescharacteristic
17C22×D148C2211yescharacteristic, complemented
18C2×D4×D37—21no—
19#11924561—24no—
20C2×#10596941C2211yescharacteristic, complemented
21C2×#10596941C2211yescharacteristic, complemented
22C2×#10596941C2211yescharacteristic, complemented
23C2×#10596939—24no—
24C2×#10596939C2211yescharacteristic
25C2×#10596939C2211yescharacteristic
26C2×#10596939C2211yescharacteristic, complemented
27C2×#10596939C2211yescharacteristic, complemented
28C2×#10596939C2211yescharacteristic, complemented
29C2×#10596939C2211yescharacteristic, complemented
30C2×#10596934C2211yescharacteristic, complemented
31C22×#64436C2211yescharacteristic, complemented
32#11924533C2211yescharacteristic, complemented
33#11924511—21no—
34#11924510C2211yescharacteristic
35#11924510C2211yescharacteristic, complemented
36#11924506—21no—
37#11924506—21no—
38C4×D148C2211yescharacteristic, complemented
39#11924467C2211yescharacteristic, complemented
40C4⋊#64436C2211yescharacteristic, complemented
41#11924460C2211yescharacteristic, complemented
42#11924457C2211yescharacteristic, complemented
43#11924457C2211yescharacteristic, complemented
44#11924456C2211yescharacteristic, complemented
45#11924456C2211yescharacteristic, complemented
46#11924456C2211yescharacteristic, complemented
47C22⋊#64436C2211yescharacteristic, complemented
48C22⋊#64436C2211yescharacteristic, complemented
49#11924447C2211yescharacteristic, complemented
50C2×#41×C37C2211yescharacteristic, complemented
51C22×D4×C37C2211yescharacteristic, complemented
52#124×C37C2211yescharacteristic, complemented
53#123×C37—21no—
54C22⋊Q8×C37C2211yescharacteristic, complemented
55#121×C37—21no—
56#121×C37C2211yescharacteristic, complemented
57#120×C37—21no—
58C4×D4×C37C2211yescharacteristic
59C4×D4×C37C2211yescharacteristic, complemented
60C2×C22⋊C4×C37—21no—
61C24×C37—21no—
62C23×D37—21no—
63C23×D37—21no—
64#41×C37—22no—
65#41×C37—41no—
66C2×Q8×C37C2311yescharacteristic
67C2×D4×C37—22no—
68C2×D4×C37—21no—
69C2×D4×C37—21no—
70C2×D4×C37—21no—
71C2×D4×C37—41no—
72C2×D4×C37D411yescharacteristic
73C2×D4×C37C2311yescharacteristic
74C2×D4×C37C2311yescharacteristic
75C22×C4×C37—21no—
76C22×C4×C37—21no—
77C22×C4×C37C2311yescharacteristic
78C22×C4×C37C2311yescharacteristic
79C2×C22⋊D37—21no—
80C2×C22⋊D37—21no—
81D4×D37—44no—
82C2×D148—22no—
83C2×D148—22no—
84C2×D148—21no—
85C2×D148—21no—
86C2×D148—21no—
87C2×D148—21no—
88C2×D148—41no—
89C2×D148—41no—
90C2×D148C2311yescharacteristic
91C2×D148C2311yescharacteristic
92C2×D148C2311yescharacteristic
93C2×C4×D37—21no—
94C4⋊C4×C37—21no—
95C4⋊C4×C37D411yescharacteristic
96C4⋊C4×C37C2311yescharacteristic
97C22⋊C4×C37—21no—
98C22⋊C4×C37—21no—
99C22⋊C4×C37—21no—
100C22⋊C4×C37—21no—
101C22⋊C4×C37—21no—
102C22⋊C4×C37D411yescharacteristic
103C22⋊C4×C37D411yescharacteristic
104C42×C37C2311yescharacteristic
105#10596941—22no—
106#10596941—22no—
107#10596941—41no—
108#10596939—22no—
109#10596939—22no—
110#10596939—22no—
111#10596939—22no—
112#10596939—41no—
113#10596939D414yescomplemented
114#10596938—21no—
115#10596938—21no—
116#10596938—21no—
117C4⋊Dic37C2311yescharacteristic
118#10596934—22no—
119C2×#64436—22no—
120C2×#64436C2311yescharacteristic
121C2×#64436C2311yescharacteristic
122C2×#64436C2311yescharacteristic
123C2×#64436C2311yescharacteristic
124C23×C37—21no—
125C23×C37—21no—
126C23×C37—42no—
127C23×C37—42no—
128C23×C37—81no—
129C23×C37C2×D411yescharacteristic
130C23×C37C2×D411yescharacteristic
131C22×D37—21no—
132C22×D37—21no—
133C22×D37—21no—
134C22×D37—42no—
135C22×D37—42no—
136C22×D37—42no—
137C22×D37—42no—
138C22×D37—81no—
139C22×D37—81no—
140Q8×C37—21no—
141Q8×C37—21no—
142D4×C37—22no—
143D4×C37—21no—
144D4×C37—21no—
145D4×C37—21no—
146D4×C37—42no—
147D4×C37—42no—
148D4×C37—81no—
149D4×C37C2×D412yescomplemented
150C2×C4×C37—21no—
151C2×C4×C37—21no—
152C2×C4×C37—21no—
153C2×C4×C37—21no—
154C2×C4×C37—21no—
155C2×C4×C37—21no—
156C2×C4×C37—42no—
157C2×C4×C37—81no—
158C2×C4×C37C2×D411yescharacteristic
159C2×C4×C37C2×D411yescharacteristic
160C2×C4×C37C2×D411yescharacteristic
161C2×C4×C37C2×D411yescharacteristic
162C2×C4×C37C2411yescharacteristic
163C22⋊D37—42no—
164C22⋊D37—42no—
165C2×Dic37—21no—
166D148—21no—
167D148—21no—
168D148—21no—
169D148—21no—
170D148—21no—
171D148—42no—
172D148—42no—
173D148—42no—
174D148—81no—
175D148C2×D412yes—
176C4×D37—42no—
177#64436—21no—
178#64436—21no—
179#64436—21no—
180#64436C2×D412yescomplemented
181C22×C37—21no—
182C22×C37—21no—
183C22×C37—21no—
184C22×C37—22no—
185C22×C37—21no—
186C22×C37—21no—
187C22×C37—41no—
188C22×C37—41no—
189C22×C37—41no—
190C22×C37—41no—
191C22×C37—41no—
192C22×C37—84no—
193C22×C37AGL(1,ℤ/8)12yescomplemented
194C22×C37C22×D411yescharacteristic
195C2×D37—22no—
196C2×D37—41no—
197C2×D37—41no—
198C2×D37—41no—
199C2×D37—41no—
200C2×D37—41no—
201C2×D37—41no—
202C2×D37—41no—
203C2×D37—41no—
204C2×D37—84no—
205C2×D37—84no—
206C4×C37—21no—
207C4×C37—21no—
208C4×C37—21no—
209C4×C37—21no—
210C4×C37—41no—
211C4×C37—41no—
212C4×C37C22×D411yescharacteristic
213C4×C37H(5,𝔽2)11yescharacteristic
214Dic37—41no—
215#3264—371nomaximal, Sylow
216C2×C37—22no—
217C2×C37—41no—
218C2×C37—42no—
219C2×C37—81no—
220C2×C37D4211yescharacteristic
221C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C37#57611yescharacteristic
223D37—42no—
224D37—81no—
225D37—81no—
226C2×AGL(1,ℤ/8)—371no—
227C22×D8—371no—
228#546D3711yescharacteristic, complemented
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#442—371no—
240#435—371no—
241C37#326411yescharacteristic, complemented, Sylow
242C2×#41C2×D3711yescharacteristic, complemented
243C22×D4—371no—
244C22×D4—371no—
245C22×D4—741no—
246C22×D4C2×D3711yescharacteristic, complemented
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—371no—
257C2×D8—744no—
258C2×M16—371no—
259C22×C8—371no—
260#127—371no—
261#124C2×D3711yescharacteristic, complemented
262#123—21no—
263C22⋊Q8C2×D3711yescharacteristic, complemented
264#121—21no—
265#121—371no—
266#121—371no—
267#121—741no—
268#121C2×D3711yescharacteristic, complemented
269#120—21no—
270#120—741no—
271#120—741no—
272C4×D4—371no—
273C4×D4C2×D3711yescharacteristic
274C4×D4C2×D3711yescharacteristic, complemented
275C2×C22⋊C4—21no—
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#97—371no—
287C24—21no—
288C24—741no—
289C24—741no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D3711yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—371no—
299C2×D4—371no—
300C2×D4—371no—
301C2×D4—742no—
302C2×D4—742no—
303C2×D4—741no—
304C2×D4—741no—
305C2×D4—741no—
306C2×D4—741no—
307C2×D4—741no—
308C2×D4—741no—
309C2×D4—1481no—
310C2×D4—1481no—
311C2×D4—1484no—
312C2×D4C22⋊D3711yescharacteristic
313C2×D4C22×D3711yescharacteristic
314C2×D4C22×D3711yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—741no—
318C22×C4C22×D3711yescharacteristic
319C22×C4C22×D3711yescharacteristic
320SD16—742no—
321SD16—742no—
322SD16—1481no—
323D8—374no—
324D8—742no—
325D8—742no—
326D8—742no—
327D8—742no—
328D8—1481no—
329M16—742no—
330C2×C8—371no—
331C2×C8—371no—
332C2×C8—371no—
333C2×C8—371no—
334C2×C8—742no—
335C4⋊C4—21no—
336C4⋊C4—371no—
337C4⋊C4C22⋊D3711yescharacteristic
338C4⋊C4C22×D3711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—741no—
345C22⋊C4—741no—
346C22⋊C4—741no—
347C22⋊C4C22⋊D3711yescharacteristic
348C22⋊C4C22⋊D3711yescharacteristic
349C42C22×D3711yescharacteristic
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—2961no—
356C23—2961no—
357C23—741no—
358C23—741no—
359C23—741no—
360C23—1482no—
361C23—1482no—
362C23—1482no—
363C23—1482no—
364C23C2×C22⋊D3711yescharacteristic
365C23C2×C22⋊D3711yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—372no—
376D4—2961no—
377D4—741no—
378D4—741no—
379D4—741no—
380D4—741no—
381D4—741no—
382D4—1482no—
383D4—1482no—
384D4—1482no—
385D4—1482no—
386D4—1482no—
387D4D4×D3712yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—741no—
397C2×C4—1482no—
398C2×C4C2×C22⋊D3711yescharacteristic
399C2×C4C2×C22⋊D3711yescharacteristic
400C2×C4C2×C22⋊D3711yescharacteristic
401C2×C4C2×C22⋊D3711yescharacteristic
402C2×C4C23×D3711yescharacteristic
403C8—372no—
404C8—741no—
405C8—741no—
406C8—741no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—2964no—
420C22—2964no—
421C22—742no—
422C22—1481no—
423C22—1481no—
424C22—1481no—
425C22—1481no—
426C22—1481no—
427C22—1481no—
428C22—1481no—
429C22—1481no—
430C22#1192456112yescomplemented
431C22C22×C22⋊D3711yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—1481no—
439C4C2×D4×D3711yescharacteristic
440C4#1192458111yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—2961no—
446C2—2961no—
447C2—1482no—
448C2D4×C22⋊D3711yescharacteristic
449C2C2×#1192456111yescharacteristic
450C2#1455833611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#546C2
socleC22×C37C22×D4
hypercenter#546D37
hypocenterC37#3264
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#11924561C2semidirect
2. C22×#10596939C2semidirect
3. D4×D148C2semidirect
4. #14557923C2semidirect
5. #14557784C2semidirect
6. #14557764C2semidirect
7. #14557745C2semidirect
8. #14557728C2semidirect
9. #14557683C2semidirect
10. #14557678C2semidirect
11. #14557667C2semidirect
12. #14557666C2semidirect
13. #14557643C2semidirect
14. #14557605C2semidirect
15. C37×#546C2semidirect
17. C22×D148C22semidirect
20. C2×#10596941C22semidirect
21. C2×#10596941C22semidirect
22. C2×#10596941C22semidirect
26. C2×#10596939C22semidirect
27. C2×#10596939C22semidirect
28. C2×#10596939C22semidirect
29. C2×#10596939C22semidirect
30. C2×#10596934C22semidirect
31. C22×#64436C22semidirect
32. #11924533C22semidirect
35. #11924510C22semidirect
38. C4×D148C22semidirect
39. #11924467C22semidirect
40. C4⋊#64436C22semidirect
41. #11924460C22semidirect
42. #11924457C22semidirect
43. #11924457C22semidirect
44. #11924456C22semidirect
45. #11924456C22semidirect
46. #11924456C22semidirect
47. C22⋊#64436C22semidirect
48. C22⋊#64436C22semidirect
49. #11924447C22semidirect
50. C2×#41×C37C22semidirect
51. C22×D4×C37C22semidirect
52. #124×C37C22semidirect
54. C22⋊Q8×C37C22semidirect
56. #121×C37C22semidirect
59. C4×D4×C37C22semidirect
113. #10596939D4semidirect
149. D4×C37C2×D4semidirect
180. #64436C2×D4semidirect
193. C22×C37AGL(1,ℤ/8)semidirect
228. #546D37semidirect
241. C37#3264semidirect
242. C2×#41C2×D37semidirect
246. C22×D4C2×D37semidirect
261. #124C2×D37semidirect
263. C22⋊Q8C2×D37semidirect
268. #121C2×D37semidirect
274. C4×D4C2×D37semidirect
387. D4D4×D37semidirect
430. C22#11924561semidirect

Maximal subgroups

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

subgroupindex
C2×#119245612
C22×#105969392
D4×D1482
#145579232
#145577842
#145577642
#145577452
#145577282
#145576832
#145576782
#145576672
#145576662
#145576432
#145576052
C37×#5462
#326437