Finite Group Databasebeta

← #18003901

Subgroups — #18003901

Order 4736.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924581C211yesmaximal, characteristic, complemented
2C2×#11924563C211yesmaximal, characteristic, complemented
3C2×#11924557C211yesmaximal, characteristic, complemented
4#14558113C211yesmaximal, characteristic, complemented
5#14557693C211yesmaximal, characteristic, complemented
6#14557692C211yesmaximal, characteristic, complemented
7#14557682C211yesmaximal, characteristic, complemented
8#14557680C211yesmaximal, characteristic, complemented
9#14557679C211yesmaximal, characteristic, complemented
10#14557668C211yesmaximal, characteristic, complemented
11#14557667C211yesmaximal, characteristic, complemented
12#14557531C211yesmaximal, characteristic, complemented
13#102×D37C211yesmaximal, characteristic, complemented
14#14557498C211yesmaximal, characteristic, complemented
15C37×#450C211yesmaximal, characteristic, complemented
16#11924581—24no—
17#11924581—24no—
18C2×#10596966C2211yescharacteristic, complemented
19C2×#10596962C2211yescharacteristic
20C2×#10596962C2211yescharacteristic
21C2×#10596964—21no—
22C2×#10596964C2211yescharacteristic
23C2×#10596964C2211yescharacteristic
24C22×C22⋊D37—21no—
25C22×C22⋊D37—21no—
26C2×D4×D37—21no—
27C2×D4×D37C2211yescharacteristic
28C2×D4×D37C2211yescharacteristic
29#11924563—24no—
30#11924557—24no—
31C2×#10596942C2211yescharacteristic, complemented
32C2×#10596930C2211yescharacteristic, complemented
33C2×#10596941C2211yescharacteristic, complemented
34C2×D296C2211yescharacteristic, complemented
35C2×#10596939C2211yescharacteristic, complemented
36C2×#10596939C2211yescharacteristic, complemented
37C2×#10596929C2211yescharacteristic, complemented
38C2×C8×D37C2211yescharacteristic, complemented
39#11924514—21no—
40#11924512C2211yescharacteristic, complemented
41#11924511—21no—
42#11924511C2211yescharacteristic, complemented
43#11924510C2211yescharacteristic, complemented
44#11924509—21no—
45#11924509—21no—
46C4⋊C4×D37C2211yescharacteristic, complemented
47#11924461C2211yescharacteristic, complemented
48#11924459C2211yescharacteristic, complemented
49#11924458C2211yescharacteristic, complemented
50#11924457C2211yescharacteristic, complemented
51#11924457C2211yescharacteristic, complemented
52#11924456C2211yescharacteristic, complemented
53C22⋊#64436C2211yescharacteristic, complemented
54#11924448C2211yescharacteristic, complemented
55#11924448C2211yescharacteristic, complemented
56C22×D4×C37C2211yescharacteristic
57C2×SD16×C37C2211yescharacteristic, complemented
58C2×SD16×C37C2211yescharacteristic, complemented
59C22⋊Q8×C37C2211yescharacteristic, complemented
60#102×C37C2211yescharacteristic, complemented
61#102×C37C2211yescharacteristic, complemented
62C22⋊C8×C37C2211yescharacteristic, complemented
63C24×C37—21no—
64C23×D37—21no—
65C23×D37—21no—
66C2×Q8×C37C2311yescharacteristic
67C2×D4×C37—22no—
68C2×D4×C37—22no—
69C2×D4×C37—21no—
70C2×D4×C37C2311yescharacteristic
71C2×D4×C37C2311yescharacteristic
72C22×C4×C37C2311yescharacteristic
73C2×C22⋊D37—28no—
74C2×C22⋊D37—28no—
75C2×C22⋊D37—21no—
76C2×C22⋊D37—21no—
77C2×C22⋊D37—21no—
78C2×C22⋊D37—21no—
79C2×C22⋊D37—21no—
80C2×C22⋊D37—21no—
81C2×C22⋊D37D411yescharacteristic
82C2×C22⋊D37D411yescharacteristic
83C22×Dic37—21no—
84C22×Dic37—21no—
85#10596966—22no—
86#10596966—22no—
87#10596964—28no—
88#10596964—22no—
89#10596964—22no—
90#10596964—22no—
91#10596964D414yescomplemented
92D4×D37—28no—
93D4×D37—22no—
94D4×D37—22no—
95D4×D37—22no—
96D4×D37D414yescomplemented
97#10596962—22no—
98#10596962—22no—
99#10596962—22no—
100#10596962—22no—
101C2×D148—21no—
102C2×D148—21no—
103C2×D148D411yescharacteristic
104C2×D148C2311yescharacteristic
105C2×D148C2311yescharacteristic
106C2×C4×D37—21no—
107C2×C4×D37—21no—
108C2×C4×D37—21no—
109C2×C4×D37C2311yescharacteristic
110C2×C4×D37C2311yescharacteristic
111C2×Dic74D411yescharacteristic
112C2×Dic74C2311yescharacteristic
113SD16×C37—22no—
114SD16×C37—22no—
115C2×C8×C37C2311yescharacteristic
116C2×C8×C37C2311yescharacteristic
117C4⋊C4×C37—21no—
118C4⋊C4×C37C2311yescharacteristic
119C22⋊C4×C37—21no—
120#10596942—22no—
121#10596941—22no—
122#10596939—22no—
123#10596939—22no—
124#10596938—21no—
125#10596938—21no—
126#10596938—21no—
127C4⋊Dic37C2311yescharacteristic
128#10596936—21no—
129#10596936—21no—
130C2×#64436C2311yescharacteristic
131C2×#64436C2311yescharacteristic
132D296—22no—
133#10596930—22no—
134#10596929—22no—
135C8×D37—22no—
136C23×C37—24no—
137C23×C37—21no—
138C23×C37—21no—
139C23×C37—81no—
140C23×C37—81no—
141C23×C37C2×D411yescharacteristic
142C22×D37—21no—
143C22×D37—24no—
144C22×D37—24no—
145C22×D37—21no—
146C22×D37—81no—
147C22×D37—81no—
148C22×D37—81no—
149C22×D37—81no—
150C22×D37C2×D411yescharacteristic
151C22×D37C2×D411yescharacteristic
152Q8×C37—21no—
153Q8×C37—21no—
154D4×C37—24no—
155D4×C37—21no—
156D4×C37—21no—
157D4×C37C2×D412yes—
158D4×C37C2×D412yes—
159C2×C4×C37—21no—
160C2×C4×C37—21no—
161C2×C4×C37—21no—
162C2×C4×C37—21no—
163C2×C4×C37C2×D411yescharacteristic
164C2×C4×C37C2411yescharacteristic
165C22⋊D37—24no—
166C22⋊D37—24no—
167C22⋊D37—24no—
168C22⋊D37—22no—
169C22⋊D37—24no—
170C22⋊D37—22no—
171C22⋊D37—24no—
172C22⋊D37—81no—
173C2×Dic37—24no—
174C2×Dic37—21no—
175C2×Dic37—24no—
176C2×Dic37—21no—
177C2×Dic37C2×D411yescharacteristic
178C2×Dic37C2×D411yescharacteristic
179D148—22no—
180D148—21no—
181D148—21no—
182D148—21no—
183D148—42no—
184D148—81no—
185D148C2×D412yes—
186C4×D37—24no—
187C4×D37—21no—
188C4×D37—21no—
189C4×D37—42no—
190C4×D37—81no—
191C4×D37C2×D412yes—
192C4×D37C2×D412yes—
193Dic74—21no—
194Dic74—22no—
195Dic74C2×D412yes—
196C8×C37—21no—
197C8×C37—21no—
198#64436—21no—
199#64436—21no—
200C22×C37—22no—
201C22×C37—22no—
202C22×C37—21no—
203C22×C37—21no—
204C22×C37—21no—
205C22×C37—41no—
206C22×C37—41no—
207C22×C37—41no—
208C22×C37—41no—
209C22×C37—84no—
210C22×C37C22×D411yescharacteristic
211C2×D37—21no—
212C2×D37—21no—
213C2×D37—21no—
214C2×D37—22no—
215C2×D37—21no—
216C2×D37—21no—
217C2×D37—41no—
218C2×D37—41no—
219C2×D37—41no—
220C2×D37—41no—
221C2×D37—41no—
222C2×D37—84no—
223C2×D37—84no—
224C2×D37#12012yes—
225C4×C37—21no—
226C4×C37—41no—
227C4×C37—41no—
228C4×C37C22×D411yescharacteristic
229C4×C37C22×D411yescharacteristic
230Dic37—21no—
231Dic37—22no—
232Dic37—41no—
233Dic37#12012yes—
234#3012—371nomaximal, Sylow
235C2×C37—42no—
236C2×C37—42no—
237C2×C37—41no—
238C2×C37C2×#12011yescharacteristic
239C2×C37#57611yescharacteristic
240C2×C37#57711yescharacteristic
241D37—22no—
242D37—41no—
243D37—42no—
244D37—81no—
245C2×H(5,𝔽2)—371no—
246C2×AGL(1,ℤ/8)—371no—
247C2×#135—371no—
248#537—371no—
249#452—371no—
250#452—371no—
251#450—371no—
252#450D3711yescharacteristic, complemented
253#449—371no—
254#449—371no—
255#447—371no—
256#447—371no—
257#417—371no—
258C2×#102—371no—
259#408—371no—
260C37#301211yescharacteristic, complemented, Sylow
261H(5,𝔽2)—744no—
262H(5,𝔽2)—744no—
263C2×#41—371no—
264C2×#41—371no—
265C2×#41—371no—
266C2×#41—371no—
267C2×#41—371no—
268C2×#41—741no—
269C22×D4—371no—
270C22×D4—371no—
271C22×D4—741no—
272C22×D4—741no—
273C22×D4—741no—
274C22×D4C2×D3711yescharacteristic
275AGL(1,ℤ/8)—744no—
276#135—744no—
277C2×Q16—371no—
278C2×SD16—371no—
279C2×SD16—371no—
280C2×SD16C2×D3711yescharacteristic, complemented
281C2×SD16C2×D3711yescharacteristic, complemented
282C2×D8—371no—
283C2×D8—371no—
284C2×D8—371no—
285C2×M16—371no—
286C22×C8—371no—
287C22⋊Q8—371no—
288C22⋊Q8—741no—
289C22⋊Q8C2×D3711yescharacteristic, complemented
290#121—371no—
291#121—371no—
292#121—741no—
293#121—741no—
294#121—741no—
295C2×C4⋊C4—371no—
296#103—371no—
297#103—371no—
298#102—371no—
299#102—371no—
300#102—371no—
301#102—371no—
302#102C2×D3711yescharacteristic, complemented
303#102C2×D3711yescharacteristic, complemented
304C22⋊C8—371no—
305C22⋊C8—371no—
306C22⋊C8—371no—
307C22⋊C8C2×D3711yescharacteristic, complemented
308C24—21no—
309C24—741no—
310C24—741no—
311#41—374no—
312#41—742no—
313#41—742no—
314#41—742no—
315#41—742no—
316#41—742no—
317#41—742no—
318#41—742no—
319#41—742no—
320#41—748no—
321#41—742no—
322C2×Q8—371no—
323C2×Q8—371no—
324C2×Q8C22×D3711yescharacteristic
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—374no—
329C2×D4—371no—
330C2×D4—371no—
331C2×D4—371no—
332C2×D4—371no—
333C2×D4—371no—
334C2×D4—741no—
335C2×D4—741no—
336C2×D4—742no—
337C2×D4—741no—
338C2×D4—741no—
339C2×D4—742no—
340C2×D4—748no—
341C2×D4—741no—
342C2×D4—741no—
343C2×D4—741no—
344C2×D4—748no—
345C2×D4—748no—
346C2×D4—741no—
347C2×D4—742no—
348C2×D4C22×D3711yescharacteristic
349C2×D4C22×D3711yescharacteristic
350C22×C4—371no—
351C22×C4—371no—
352C22×C4—741no—
353C22×C4—741no—
354C22×C4—741no—
355C22×C4—741no—
356C22×C4—741no—
357C22×C4C22×D3711yescharacteristic
358Q16—742no—
359SD16—22no—
360SD16—22no—
361SD16—742no—
362SD16—742no—
363D8—742no—
364D8—742no—
365D8—742no—
366M16—742no—
367C2×C8—371no—
368C2×C8—371no—
369C2×C8—742no—
370C2×C8C22×D3711yescharacteristic
371C2×C8C22×D3711yescharacteristic
372C4⋊C4—21no—
373C4⋊C4—371no—
374C4⋊C4—741no—
375C4⋊C4—741no—
376C4⋊C4C22×D3711yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—741no—
379C22⋊C4—741no—
380C22⋊C4—741no—
381C23—24no—
382C23—21no—
383C23—21no—
384C23—81no—
385C23—81no—
386C23—371no—
387C23—371no—
388C23—2961no—
389C23—2961no—
390C23—2961no—
391C23—2961no—
392C23—744no—
393C23—744no—
394C23—741no—
395C23—741no—
396C23D4×D3711yescharacteristic
397Q8—21no—
398Q8—21no—
399Q8—372no—
400Q8—742no—
401Q8—741no—
402D4—24no—
403D4—21no—
404D4—21no—
405D4—372no—
406D4—2961no—
407D4—2961no—
408D4—742no—
409D4—742no—
410D4—744no—
411D4—742no—
412D4—744no—
413D4—744no—
414D4—744no—
415D4—744no—
416D4—741no—
417D4—741no—
418D4—741no—
419D4—1482no—
420D4D4×D3712yes—
421D4D4×D3712yes—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—371no—
427C2×C4—372no—
428C2×C4—372no—
429C2×C4—371no—
430C2×C4—2961no—
431C2×C4—744no—
432C2×C4—744no—
433C2×C4—744no—
434C2×C4—741no—
435C2×C4—741no—
436C2×C4—741no—
437C2×C4—741no—
438C2×C4—1482no—
439C2×C4D4×D3711yescharacteristic
440C2×C4C23×D3711yescharacteristic
441C8—21no—
442C8—21no—
443C8—741no—
444C8—741no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—21no—
449C22—21no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—84no—
455C22—372no—
456C22—2964no—
457C22—2964no—
458C22—741no—
459C22—741no—
460C22—742no—
461C22—741no—
462C22—741no—
463C22—741no—
464C22—1481no—
465C22—1481no—
466C22—1481no—
467C22—1481no—
468C22—1481no—
469C22C2×D4×D3711yescharacteristic
470C4—21no—
471C4—41no—
472C4—41no—
473C4—372no—
474C4—741no—
475C4—742no—
476C4—1481no—
477C4C2×D4×D3711yescharacteristic
478C4C2×D4×D3711yescharacteristic
479C2—41no—
480C2—42no—
481C2—42no—
482C2—2961no—
483C2—742no—
484C2—1481no—
485C2—1482no—
486C2#120×D3711yescharacteristic
487C2#1455833711yescharacteristic
488C2#1455833911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#450C2
socleC22×C37C22×D4
hypercenter#450D37
hypocenterC37#3012
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#11924581C2semidirect
2. C2×#11924563C2semidirect
3. C2×#11924557C2semidirect
4. #14558113C2semidirect
5. #14557693C2semidirect
6. #14557692C2semidirect
7. #14557682C2semidirect
8. #14557680C2semidirect
9. #14557679C2semidirect
10. #14557668C2semidirect
11. #14557667C2semidirect
12. #14557531C2semidirect
13. #102×D37C2semidirect
14. #14557498C2semidirect
15. C37×#450C2semidirect
18. C2×#10596966C22semidirect
31. C2×#10596942C22semidirect
32. C2×#10596930C22semidirect
33. C2×#10596941C22semidirect
34. C2×D296C22semidirect
35. C2×#10596939C22semidirect
36. C2×#10596939C22semidirect
37. C2×#10596929C22semidirect
38. C2×C8×D37C22semidirect
40. #11924512C22semidirect
42. #11924511C22semidirect
43. #11924510C22semidirect
46. C4⋊C4×D37C22semidirect
47. #11924461C22semidirect
48. #11924459C22semidirect
49. #11924458C22semidirect
50. #11924457C22semidirect
51. #11924457C22semidirect
52. #11924456C22semidirect
53. C22⋊#64436C22semidirect
54. #11924448C22semidirect
55. #11924448C22semidirect
57. C2×SD16×C37C22semidirect
58. C2×SD16×C37C22semidirect
59. C22⋊Q8×C37C22semidirect
60. #102×C37C22semidirect
61. #102×C37C22semidirect
62. C22⋊C8×C37C22semidirect
91. #10596964D4semidirect
96. D4×D37D4semidirect
252. #450D37semidirect
260. C37#3012semidirect
280. C2×SD16C2×D37semidirect
281. C2×SD16C2×D37semidirect
289. C22⋊Q8C2×D37semidirect
302. #102C2×D37semidirect
303. #102C2×D37semidirect
307. C22⋊C8C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245812
C2×#119245632
C2×#119245572
#145581132
#145576932
#145576922
#145576822
#145576802
#145576792
#145576682
#145576672
#145575312
#102×D372
#145574982
C37×#4502
#301237