Finite Group Databasebeta

← #18006096

Subgroups — #18006096

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
2C22×#10596930C211yesmaximal, characteristic, complemented
3#14558192C211yesmaximal, characteristic, complemented
4D4×D148C211yesmaximal, characteristic, complemented
5#14557922C211yesmaximal, characteristic, complemented
6#14557779C211yesmaximal, characteristic, complemented
7#14557769C211yesmaximal, characteristic, complemented
8#14557749C211yesmaximal, characteristic, complemented
9#14557734C211yesmaximal, characteristic, complemented
10#14557686C211yesmaximal, characteristic, complemented
11#14557680C211yesmaximal, characteristic, complemented
12#14557674C211yesmaximal, characteristic, complemented
13#14557669C211yesmaximal, characteristic, complemented
14#14557635C211yesmaximal, characteristic, complemented
15C37×#435C211yesmaximal, characteristic, complemented
16C2×#10596962C2211yescharacteristic, complemented
17C22×Dic74C2211yescharacteristic, complemented
18C22×D148C2211yescharacteristic
19C22×D148C2211yescharacteristic
20C2×D4×D37—21no—
21#11924564—24no—
22C2×#10596930—24no—
23C2×#10596930C2211yescharacteristic
24C2×#10596930C2211yescharacteristic
25C2×#10596930C2211yescharacteristic, complemented
26C2×#10596930C2211yescharacteristic, complemented
27C2×#10596930C2211yescharacteristic, complemented
28C2×#10596930C2211yescharacteristic, complemented
29C2×D296C2211yescharacteristic
30C2×D296C2211yescharacteristic, complemented
31C2×D296C2211yescharacteristic, complemented
32#11924533C2211yescharacteristic
33#11924524C2211yescharacteristic, complemented
34#11924522—21no—
35#11924515C2211yescharacteristic, complemented
36#11924514—21no—
37#11924511—21no—
38#11924510C2211yescharacteristic
39#11924510C2211yescharacteristic
40#11924510C2211yescharacteristic, complemented
41#11924508—21no—
42#11924506—21no—
43#11924506—21no—
44C4×Dic74C2211yescharacteristic, complemented
45C4×D148C2211yescharacteristic
46#11924496—21no—
47#11924470C2211yescharacteristic, complemented
48#11924461C2211yescharacteristic, complemented
49#11924461C2211yescharacteristic, complemented
50#11924458C2211yescharacteristic
51#11924458C2211yescharacteristic, complemented
52#11924458C2211yescharacteristic, complemented
53#11924458C2211yescharacteristic, complemented
54C2×M16×C37C2211yescharacteristic, complemented
55C22×C8×C37C2211yescharacteristic, complemented
56C4×D4×C37C2211yescharacteristic
57C4⋊C8×C37C2211yescharacteristic, complemented
58C22⋊C8×C37C2211yescharacteristic, complemented
59C22⋊C8×C37C2211yescharacteristic, complemented
60#97×C37C2211yescharacteristic, complemented
61C23×D37—21no—
62C23×D37—21no—
63C2×D4×C37D411yescharacteristic
64C22×C4×C37C2311yescharacteristic
65C22×C4×C37C2311yescharacteristic
66C2×C22⋊D37—21no—
67C2×C22⋊D37—21no—
68C2×C22⋊D37—21no—
69C2×C22⋊D37—21no—
70C22×Dic37—21no—
71D4×D37—44no—
72#10596962—22no—
73#10596962—41no—
74C2×D148—22no—
75C2×D148—22no—
76C2×D148—21no—
77C2×D148—21no—
78C2×D148—21no—
79C2×D148—21no—
80C2×D148—41no—
81C2×D148—41no—
82C2×D148C2311yescharacteristic
83C2×D148C2311yescharacteristic
84C2×D148C2311yescharacteristic
85C2×D148C2311yescharacteristic
86C2×C4×D37—21no—
87C2×C4×D37—21no—
88C2×Dic74—22no—
89C2×Dic74—21no—
90C2×Dic74—41no—
91C2×Dic74C2311yescharacteristic
92C2×Dic74C2311yescharacteristic
93M16×C37—22no—
94C2×C8×C37—22no—
95C2×C8×C37C2311yescharacteristic
96C2×C8×C37C2311yescharacteristic
97C2×C8×C37C2311yescharacteristic
98C2×C8×C37C2311yescharacteristic
99C4⋊C4×C37D411yescharacteristic
100C22⋊C4×C37D411yescharacteristic
101C22⋊C4×C37D411yescharacteristic
102C42×C37C2311yescharacteristic
103C22⋊Dic37—21no—
104#10596938—21no—
105#10596938—21no—
106#10596938—21no—
107#10596938—21no—
108#10596938—21no—
109#10596938—21no—
110C4⋊Dic37C2311yescharacteristic
111C4⋊Dic37C2311yescharacteristic
112#10596936—21no—
113#10596936—21no—
114C4×Dic37—21no—
115D296—22no—
116D296—22no—
117D296—41no—
118#10596930—22no—
119#10596930—22no—
120#10596930—22no—
121#10596930—22no—
122#10596930—41no—
123#10596930D414yescomplemented
124C23×C37C2×D411yescharacteristic
125C23×C37C2×D411yescharacteristic
126C22×D37—21no—
127C22×D37—21no—
128C22×D37—21no—
129C22×D37—21no—
130C22×D37—42no—
131C22×D37—42no—
132C22×D37—42no—
133C22×D37—42no—
134C22×D37—81no—
135C22×D37—81no—
136D4×C37—22no—
137C2×C4×C37—21no—
138C2×C4×C37—21no—
139C2×C4×C37—21no—
140C2×C4×C37—21no—
141C2×C4×C37C2×D411yescharacteristic
142C2×C4×C37C2×D411yescharacteristic
143C2×C4×C37C2×D411yescharacteristic
144C2×C4×C37C2×D411yescharacteristic
145C2×C4×C37C2411yescharacteristic
146C22⋊D37—42no—
147C22⋊D37—42no—
148C22⋊D37—42no—
149C22⋊D37—42no—
150C2×Dic37—21no—
151C2×Dic37—21no—
152C2×Dic37—21no—
153C2×Dic37—21no—
154C2×Dic37—42no—
155D148—21no—
156D148—21no—
157D148—21no—
158D148—21no—
159D148—21no—
160D148—21no—
161D148—21no—
162D148—42no—
163D148—42no—
164D148—42no—
165D148—81no—
166D148C2×D412yes—
167C4×D37—42no—
168C4×D37—42no—
169Dic74—21no—
170Dic74—21no—
171Dic74—21no—
172Dic74—42no—
173Dic74C2×D412yescomplemented
174C8×C37—21no—
175C8×C37—21no—
176C8×C37—21no—
177C8×C37C2×D412yescomplemented
178C22×C37—21no—
179C22×C37—21no—
180C22×C37—21no—
181C22×C37—21no—
182C22×C37—21no—
183C22×C37AGL(1,ℤ/8)12yescomplemented
184C22×C37C22×D411yescharacteristic
185C2×D37—22no—
186C2×D37—41no—
187C2×D37—41no—
188C2×D37—41no—
189C2×D37—41no—
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—21no—
201C4×C37—21no—
202C4×C37—21no—
203C4×C37C22×D411yescharacteristic
204C4×C37H(5,𝔽2)11yescharacteristic
205Dic37—22no—
206Dic37—41no—
207Dic37—41no—
208Dic37—41no—
209#3265—371nomaximal, Sylow
210C2×C37—22no—
211C2×C37—41no—
212C2×C37D4211yescharacteristic
213C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C37#57711yescharacteristic
215D37—42no—
216D37—81no—
217D37—81no—
218D37—81no—
219C2×AGL(1,ℤ/8)—371no—
220C22×SD16—371no—
221#548—371no—
222D42—371no—
223#496—371no—
224#468—371no—
225#466—371no—
226#463—371no—
227#460—371no—
228#450—371no—
229#449—371no—
230#448—371no—
231#447—371no—
232#440—371no—
233#435D3711yescharacteristic, complemented
234C37#326511yescharacteristic, complemented, Sylow
235C2×#41—371no—
236C22×Q8—371no—
237C22×D4—371no—
238C22×D4—371no—
239C22×D4—741no—
240AGL(1,ℤ/8)—744no—
241C2×SD16—371no—
242C2×SD16—371no—
243C2×SD16—371no—
244C2×SD16—371no—
245C2×SD16—371no—
246C2×SD16—371no—
247C2×SD16—744no—
248C2×D8—371no—
249C2×D8—371no—
250C2×D8—371no—
251C2×M16C2×D3711yescharacteristic, complemented
252C22×C8C2×D3711yescharacteristic, complemented
253#127—371no—
254#124—371no—
255#124—741no—
256C22⋊Q8—371no—
257C22⋊Q8—741no—
258#121—371no—
259#121—371no—
260#121—371no—
261#121—741no—
262#121—741no—
263#120—741no—
264#120—741no—
265C4×Q8—371no—
266C4×D4—371no—
267C4×D4—741no—
268C4×D4C2×D3711yescharacteristic
269#107—371no—
270C4⋊C8C2×D3711yescharacteristic, complemented
271#103—371no—
272#103—371no—
273#102—371no—
274#102—371no—
275#102—371no—
276#102—371no—
277C22⋊C8C2×D3711yescharacteristic, complemented
278C22⋊C8C2×D3711yescharacteristic, complemented
279#97C2×D3711yescharacteristic, complemented
280C24—741no—
281C24—741no—
282#41—742no—
283#41—1481no—
284C2×Q8—371no—
285C2×Q8—371no—
286C2×Q8—742no—
287C2×Q8—741no—
288C2×Q8—1481no—
289C2×D4—371no—
290C2×D4—371no—
291C2×D4—371no—
292C2×D4—371no—
293C2×D4—742no—
294C2×D4—742no—
295C2×D4—741no—
296C2×D4—741no—
297C2×D4—741no—
298C2×D4—741no—
299C2×D4—741no—
300C2×D4—741no—
301C2×D4—741no—
302C2×D4—741no—
303C2×D4—1481no—
304C2×D4—1481no—
305C2×D4—1484no—
306C2×D4D14811yescharacteristic
307C22×C4—741no—
308C22×C4—741no—
309C22×C4—741no—
310C22×C4C22×D3711yescharacteristic
311C22×C4C22×D3711yescharacteristic
312SD16—374no—
313SD16—742no—
314SD16—742no—
315SD16—742no—
316SD16—742no—
317SD16—1481no—
318D8—742no—
319D8—742no—
320D8—1481no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D3711yescharacteristic
324C2×C8C22×D3711yescharacteristic
325C2×C8C22×D3711yescharacteristic
326C2×C8C22×D3711yescharacteristic
327C4⋊C4—371no—
328C4⋊C4—371no—
329C4⋊C4—741no—
330C4⋊C4—741no—
331C4⋊C4D14811yescharacteristic
332C22⋊C4—741no—
333C22⋊C4—741no—
334C22⋊C4—741no—
335C22⋊C4—741no—
336C22⋊C4—741no—
337C22⋊C4—741no—
338C22⋊C4—741no—
339C22⋊C4D14811yescharacteristic
340C22⋊C4D14811yescharacteristic
341C42—741no—
342C42C22×D3711yescharacteristic
343C23—2961no—
344C23—2961no—
345C23—741no—
346C23—741no—
347C23—741no—
348C23—741no—
349C23—1482no—
350C23—1482no—
351C23—1482no—
352C23—1482no—
353C23C2×D14811yescharacteristic
354C23C2×D14811yescharacteristic
355Q8—372no—
356Q8—741no—
357Q8—741no—
358Q8—741no—
359Q8—1482no—
360D4—22no—
361D4—372no—
362D4—2961no—
363D4—741no—
364D4—741no—
365D4—741no—
366D4—741no—
367D4—741no—
368D4—741no—
369D4—741no—
370D4—1482no—
371D4—1482no—
372D4—1482no—
373D4—1482no—
374D4—1482no—
375D4—1482no—
376D4—1482no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—741no—
382C2×C4—741no—
383C2×C4—741no—
384C2×C4—741no—
385C2×C4—1482no—
386C2×C4—1482no—
387C2×C4—1482no—
388C2×C4C2×D14811yescharacteristic
389C2×C4C2×D14811yescharacteristic
390C2×C4C2×D14811yescharacteristic
391C2×C4C2×D14811yescharacteristic
392C2×C4C23×D3711yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D3712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—2964no—
403C22—2964no—
404C22—742no—
405C22—1481no—
406C22—1481no—
407C22—1481no—
408C22—1481no—
409C22—1481no—
410C22—1481no—
411C22—1481no—
412C22—1481no—
413C22—1481no—
414C22—1481no—
415C22—1481no—
416C22#1192456412yescomplemented
417C22C22×D14811yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—742no—
423C4—1481no—
424C4—1481no—
425C4—1481no—
426C4C2×D4×D3711yescharacteristic
427C4#1192458211yescharacteristic
428C2—22no—
429C2—41no—
430C2—2961no—
431C2—2961no—
432C2—2961no—
433C2—1482no—
434C2D4×D14811yescharacteristic
435C2C2×#1192456411yescharacteristic
436C2#1455834511yescharacteristic
subgroupG/H
centerC22C22×D148
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#435C2
socleC22×C37C22×D4
hypercenter#435D37
hypocenterC37#3265
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#11924564C2semidirect
2. C22×#10596930C2semidirect
3. #14558192C2semidirect
4. D4×D148C2semidirect
5. #14557922C2semidirect
6. #14557779C2semidirect
7. #14557769C2semidirect
8. #14557749C2semidirect
9. #14557734C2semidirect
10. #14557686C2semidirect
11. #14557680C2semidirect
12. #14557674C2semidirect
13. #14557669C2semidirect
14. #14557635C2semidirect
15. C37×#435C2semidirect
16. C2×#10596962C22semidirect
17. C22×Dic74C22semidirect
25. C2×#10596930C22semidirect
26. C2×#10596930C22semidirect
27. C2×#10596930C22semidirect
28. C2×#10596930C22semidirect
30. C2×D296C22semidirect
31. C2×D296C22semidirect
33. #11924524C22semidirect
35. #11924515C22semidirect
40. #11924510C22semidirect
44. C4×Dic74C22semidirect
47. #11924470C22semidirect
48. #11924461C22semidirect
49. #11924461C22semidirect
51. #11924458C22semidirect
52. #11924458C22semidirect
53. #11924458C22semidirect
54. C2×M16×C37C22semidirect
55. C22×C8×C37C22semidirect
57. C4⋊C8×C37C22semidirect
58. C22⋊C8×C37C22semidirect
59. C22⋊C8×C37C22semidirect
60. #97×C37C22semidirect
123. #10596930D4semidirect
173. Dic74C2×D4semidirect
177. C8×C37C2×D4semidirect
183. C22×C37AGL(1,ℤ/8)semidirect
233. #435D37semidirect
234. C37#3265semidirect
251. C2×M16C2×D37semidirect
252. C22×C8C2×D37semidirect
270. C4⋊C8C2×D37semidirect
277. C22⋊C8C2×D37semidirect
278. C22⋊C8C2×D37semidirect
279. #97C2×D37semidirect
396. C8D4×D37semidirect
416. C22#11924564semidirect

Maximal subgroups

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

subgroupindex
C2×#119245642
C22×#105969302
#145581922
D4×D1482
#145579222
#145577792
#145577692
#145577492
#145577342
#145576862
#145576802
#145576742
#145576692
#145576352
C37×#4352
#326537