Finite Group Databasebeta

← #16860483

Subgroups — #16860483

Order 3968.

1424 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521886C211yesmaximal, characteristic, complemented
2#14521884C211yesmaximal, characteristic, complemented
3#14521865C213yesmaximal, complemented
4#14521864C211yesmaximal, characteristic, complemented
5#14521861C211yesmaximal, characteristic, complemented
6#14521856C213yesmaximal, complemented
7#14521855C213yesmaximal, complemented
8#14521853C213yesmaximal, complemented
9#14521852C211yesmaximal, characteristic, complemented
10#14521848C213yesmaximal, complemented
11#14521845C213yesmaximal, complemented
12C2×#11758493C213yesmaximal, complemented
13#135×D31C213yesmaximal, complemented
14#131×D31C211yesmaximal, characteristic, complemented
15C31×#577C211yesmaximal, characteristic, complemented
16C31×2−1+4C2211yescharacteristic, complemented
17C31×H(5,𝔽2)C2211yescharacteristic, complemented
18#11758567—21no—
19#11758567C2213yescomplemented
20#11758567C2211yescharacteristic, complemented
21#11758566—23no—
22#11758566C2213yescomplemented
23#11758566C2211yescharacteristic, complemented
24#41×D31—21no—
25#41×D31—23no—
26#41×D31C2211yescharacteristic
27#41×D31C2213yescomplemented
28#41×D31C2213yescomplemented
29#11758564—21no—
30#11758564C2213yescomplemented
31C2×#92499—21no—
32C2×#92499C2213yescomplemented
33#11758561—23no—
34#11758561C2213yescomplemented
35C2×#92497—23no—
36C2×#92497C2213yescomplemented
37C31×#137C2213yescomplemented
38C31×AGL(1,ℤ/8)C2213yescomplemented
39C31×#135C2213yescomplemented
40C2×SD16×C31C2213yescomplemented
41C31×#131C2211yescharacteristic, complemented
42#11758528C2213yescomplemented
43#11758527C2213yescomplemented
44#11758527C2213yescomplemented
45#11758526C2213yescomplemented
46#11758525C2211yescharacteristic, complemented
47C2×#92482C2213yescomplemented
48#11758519C2213yescomplemented
49#11758509C2213yescomplemented
50C2×#92479C2213yescomplemented
51#11758496C2213yescomplemented
52#11758495C2216yescomplemented
53Q16×D31C2213yescomplemented
54#11758493C2216yescomplemented
55#11758493C2216yescomplemented
56#11758493C2211yescharacteristic, complemented
57#11758493C2213yescomplemented
58#11758493C2216yescomplemented
59#11758492C2216yescomplemented
60#11758491C2216yescomplemented
61SD16×D31C2213yescomplemented
62SD16×D31C2213yescomplemented
63#11758489C2213yescomplemented
64#11758488C2216yescomplemented
65D8×D31C2213yescomplemented
66#11758486C2213yescomplemented
67#11758485C2213yescomplemented
68#11758484C2213yescomplemented
69M16×D31C2213yescomplemented
70#11758481C2213yescomplemented
71C2×#92470C2213yescomplemented
72#11758478C2213yescomplemented
73C2×C8×D31C2213yescomplemented
74#41×C31—21no—
75#41×C31—23no—
76#41×C31C2313yes—
77#41×C31C2313yes—
78#41×C31C2311yescharacteristic
79C2×Q8×C31—21no—
80C2×Q8×C31C2313yes—
81C2×D4×C31—23no—
82C2×D4×C31C2313yes—
83C2×C22⋊D31—23no—
84C2×C22⋊D31—26no—
85C22×Dic31—23no—
86#92499—21no—
87#92499—23no—
88#92499—23no—
89#92499—23no—
90#92499D411yescharacteristic
91#92499C2316yes—
92#92499C2313yes—
93#92499C2311yescharacteristic
94Q8×D31—21no—
95Q8×D31—23no—
96Q8×D31D411yescharacteristic
97Q8×D31C2313yes—
98Q8×D31C2313yes—
99#92497—23no—
100#92497—26no—
101#92497—23no—
102#92497—23no—
103#92497—23no—
104#92497—23no—
105#92497—23no—
106#92497D413yes—
107#92497C2316yes—
108#92497C2311yescharacteristic
109#92497C2313yes—
110D4×D31—23no—
111D4×D31—23no—
112D4×D31—23no—
113D4×D31—26no—
114D4×D31—23no—
115D4×D31D413yes—
116D4×D31C2313yes—
117D4×D31C2313yes—
118#92495—26no—
119#92495—23no—
120#92495—23no—
121#92495—23no—
122#92495—23no—
123#92495C2313yes—
124#92495C2313yes—
125#92495C2313yes—
126C2×D124—23no—
127C2×D124—23no—
128C2×D124C2313yes—
129C2×C4×D31—23no—
130C2×C4×D31—23no—
131C2×C4×D31—23no—
132C2×C4×D31C2313yes—
133C2×Dic62—23no—
134C2×Dic62C2313yes—
135Q16×C31C2313yes—
136SD16×C31C2311yescharacteristic
137SD16×C31C2313yes—
138SD16×C31C2316yes—
139D8×C31C2313yes—
140M16×C31C2313yes—
141C2×C8×C31C2313yes—
142#92482C2316yes—
143#92482C2313yes—
144#92482C2311yescharacteristic
145#92481C2313yes—
146#92481C2313yes—
147#92480C2313yes—
148#92480C2313yes—
149#92479C2311yescharacteristic
150#92479C2313yes—
151#92479C2316yes—
152#92474C2313yes—
153C2×C31⋊C8C2313yes—
154Dic124C2313yes—
155D248C2313yes—
156#92470C2313yes—
157#92470C2316yes—
158#92470C2311yescharacteristic
159#92469C2316yes—
160C8×D31C2311yescharacteristic
161C8×D31C2313yes—
162C8×D31C2313yes—
163C8×D31C2313yes—
164C23×C31—23no—
165C22×D31—23no—
166C22×D31—23no—
167C22×D31—23no—
168C22×D31C2×D413yes—
169Q8×C31—23no—
170Q8×C31C2×D411yescharacteristic
171Q8×C31C2411yescharacteristic
172Q8×C31C2413yes—
173D4×C31—23no—
174D4×C31—23no—
175D4×C31—23no—
176D4×C31C2×D413yes—
177D4×C31C2413yes—
178D4×C31C2411yescharacteristic
179C2×C4×C31—23no—
180C2×C4×C31—23no—
181C2×C4×C31—23no—
182C2×C4×C31C2413yes—
183C22⋊D31—23no—
184C22⋊D31—23no—
185C22⋊D31—21no—
186C22⋊D31—26no—
187C22⋊D31—23no—
188C22⋊D31—23no—
189C22⋊D31—23no—
190C22⋊D31C2×D416yes—
191C2×Dic31—23no—
192C2×Dic31—26no—
193C2×Dic31—21no—
194C2×Dic31—23no—
195C2×Dic31C2×D413yes—
196D124—26no—
197D124—23no—
198D124—21no—
199D124—23no—
200D124C2×D413yes—
201D124C2413yes—
202D124C2411yescharacteristic
203C4×D31—23no—
204C4×D31—23no—
205C4×D31—23no—
206C4×D31—23no—
207C4×D31—21no—
208C4×D31—23no—
209C4×D31—26no—
210C4×D31C2×D413yes—
211C4×D31C2×D413yes—
212C4×D31C2411yescharacteristic
213C4×D31C2413yes—
214Dic62—23no—
215Dic62—23no—
216Dic62—23no—
217Dic62C2×D413yes—
218Dic62C2413yes—
219Dic62C2411yescharacteristic
220C8×C31C2411yescharacteristic
221C8×C31C2413yes—
222C31⋊C8C2411yescharacteristic
223C31⋊C8C2413yes—
224#3584—311nomaximal, Sylow
225C22×C31—23no—
226C22×C31—21no—
227C22×C31—83no—
228C22×C31C22×D413yes—
229C2×D31—23no—
230C2×D31—21no—
231C2×D31—43no—
232C2×D31—83no—
233C2×D31—83no—
234C2×D31—83no—
235C2×D31C22×D413yes—
236C2×D31C22×D411yescharacteristic
237C4×C31—21no—
238C4×C31—23no—
239C4×C31C22×D413yes—
240C4×C31C2511yescharacteristic
241Dic31—21no—
242Dic31—23no—
243Dic31C22×D413yes—
244Dic31C22×D411yescharacteristic
245#585—311no—
246#585—311no—
247#578—311no—
248#578—313no—
249#577—313no—
250#577—313no—
251#577—311no—
252#577D3111yescharacteristic, complemented
253#576—311no—
254#576—313no—
255#575—313no—
256#575—313no—
257C2×#135—313no—
258C2×#135—313no—
259C2×#131—311no—
260C2×C31—23no—
261C2×C31—41no—
262C2×C31—43no—
263C2×C31C23×D411yescharacteristic
264D31—21no—
265D31—23no—
266D31—43no—
267D31—41no—
2682−1+4—313no—
2692−1+4—311no—
2702−1+4—313no—
2712−1+4—621no—
2722−1+4—621no—
2732−1+4C2×D3111yescharacteristic, complemented
274H(5,𝔽2)—313no—
275H(5,𝔽2)—311no—
276H(5,𝔽2)—313no—
277H(5,𝔽2)—623no—
278H(5,𝔽2)—623no—
279H(5,𝔽2)C2×D3111yescharacteristic, complemented
280C2×#41—311no—
281C2×#41—313no—
282C2×#41—313no—
283C2×#41—313no—
284C2×#41—313no—
285C2×#41—623no—
286C2×#41—621no—
287C2×#41—621no—
288C2×#41—623no—
289#137—313no—
290#137—313no—
291#137—316no—
292#137—316no—
293#137—313no—
294#137C2×D3113yescomplemented
295AGL(1,ℤ/8)—313no—
296AGL(1,ℤ/8)—313no—
297AGL(1,ℤ/8)—316no—
298AGL(1,ℤ/8)—316no—
299AGL(1,ℤ/8)—313no—
300AGL(1,ℤ/8)C2×D3113yescomplemented
301#135—313no—
302#135—313no—
303#135—313no—
304#135—311no—
305#135—313no—
306#135—316no—
307#135—316no—
308#135—316no—
309#135—313no—
310#135—313no—
311#135C2×D3113yescomplemented
312C2×Q16—313no—
313C2×Q16—313no—
314C2×SD16—313no—
315C2×SD16—313no—
316C2×SD16—313no—
317C2×SD16C2×D3113yescomplemented
318C2×D8—313no—
319C2×D8—313no—
320#131—311no—
321#131—313no—
322#131—313no—
323#131C2×D3111yescharacteristic, complemented
324C2×M16—313no—
325C22×C8—313no—
326C31#358411yescharacteristic, complemented, Sylow
327#41—21no—
328#41—23no—
329#41—311no—
330#41—313no—
331#41—313no—
332#41—311no—
333#41—313no—
334#41—316no—
335#41—313no—
336#41—311no—
337#41—316no—
338#41—313no—
339#41—313no—
340#41—623no—
341#41—623no—
342#41—626no—
343#41—623no—
344#41—623no—
345#41—623no—
346#41—623no—
347#41—623no—
348#41—623no—
349#41—623no—
350#41—623no—
351#41—626no—
352#41—623no—
353#41—623no—
354#41—621no—
355#41—623no—
356#41C22×D3113yes—
357#41C22×D3113yes—
358#41C22×D3111yescharacteristic
359C2×Q8—21no—
360C2×Q8—311no—
361C2×Q8—313no—
362C2×Q8—313no—
363C2×Q8—313no—
364C2×Q8—623no—
365C2×Q8—623no—
366C2×Q8—621no—
367C2×Q8C22×D3113yes—
368C2×D4—23no—
369C2×D4—313no—
370C2×D4—313no—
371C2×D4—313no—
372C2×D4—313no—
373C2×D4—623no—
374C2×D4—623no—
375C2×D4—626no—
376C2×D4—623no—
377C2×D4—623no—
378C2×D4—623no—
379C2×D4—626no—
380C2×D4—623no—
381C2×D4—623no—
382C2×D4C22×D3113yes—
383C22×C4—313no—
384C22×C4—623no—
385C22×C4—623no—
386C22×C4—623no—
387C22×C4—623no—
388Q16—311no—
389Q16—313no—
390Q16—316no—
391Q16—313no—
392Q16C22×D3113yes—
393SD16—313no—
394SD16—313no—
395SD16—313no—
396SD16—313no—
397SD16—313no—
398SD16—311no—
399SD16—316no—
400SD16C22×D3111yescharacteristic
401SD16C22×D3113yes—
402SD16C22×D3116yes—
403D8—313no—
404D8—311no—
405D8—316no—
406D8—313no—
407D8C22×D3113yes—
408M16—313no—
409M16—316no—
410M16C22×D3113yes—
411C2×C8—313no—
412C2×C8—311no—
413C2×C8—313no—
414C2×C8—313no—
415C2×C8—313no—
416C2×C8C22×D3113yes—
417C23—23no—
418C23—313no—
419C23—623no—
420C23—623no—
421C23—623no—
422Q8—23no—
423Q8—313no—
424Q8—313no—
425Q8—311no—
426Q8—623no—
427Q8—623no—
428Q8—623no—
429Q8D4×D3111yescharacteristic
430Q8C23×D3111yescharacteristic
431Q8C23×D3113yes—
432D4—23no—
433D4—23no—
434D4—23no—
435D4—316no—
436D4—313no—
437D4—313no—
438D4—311no—
439D4—621no—
440D4—623no—
441D4—626no—
442D4—623no—
443D4—623no—
444D4—621no—
445D4—623no—
446D4—626no—
447D4—623no—
448D4—623no—
449D4—623no—
450D4D4×D3113yes—
451D4C23×D3113yes—
452D4C23×D3111yescharacteristic
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—23no—
456C2×C4—313no—
457C2×C4—313no—
458C2×C4—313no—
459C2×C4—311no—
460C2×C4—313no—
461C2×C4—621no—
462C2×C4—623no—
463C2×C4—626no—
464C2×C4—623no—
465C2×C4—623no—
466C2×C4—621no—
467C2×C4—623no—
468C2×C4—626no—
469C2×C4—623no—
470C2×C4—623no—
471C2×C4—623no—
472C2×C4C23×D3113yes—
473C8—311no—
474C8—313no—
475C8C23×D3111yescharacteristic
476C8C23×D3113yes—
477C22—21no—
478C22—23no—
479C22—83no—
480C22—311no—
481C22—313no—
482C22—623no—
483C22—621no—
484C22—1243no—
485C22—2483no—
486C22—2483no—
487C22—2483no—
488C22C2×D4×D3113yes—
489C4—21no—
490C4—23no—
491C4—311no—
492C4—313no—
493C4—621no—
494C4—623no—
495C4C2×D4×D3113yes—
496C4C24×D3111yescharacteristic
497C2—23no—
498C2—43no—
499C2—41no—
500C2—621no—
501C2—623no—
502C2—1243no—
503C2—1241no—
504C2C22×D4×D3111yescharacteristic
subgroupG/H
centerC2C22×D4×D31
commutator, cosocleC4×C31C25
FrattiniC4C24×D31
FittingC31×#577C2
socleC2×C31C23×D4
hypercenter#577D31
hypocenterC31#3584
LCSG ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ #41 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #14521886C2semidirect
2. #14521884C2semidirect
3. #14521865C2semidirect
4. #14521864C2semidirect
5. #14521861C2semidirect
6. #14521856C2semidirect
7. #14521855C2semidirect
8. #14521853C2semidirect
9. #14521852C2semidirect
10. #14521848C2semidirect
11. #14521845C2semidirect
12. C2×#11758493C2semidirect
13. #135×D31C2semidirect
14. #131×D31C2semidirect
15. C31×#577C2semidirect
16. C31×2−1+4C22semidirect
17. C31×H(5,𝔽2)C22semidirect
19. #11758567C22semidirect
20. #11758567C22semidirect
22. #11758566C22semidirect
23. #11758566C22semidirect
27. #41×D31C22semidirect
28. #41×D31C22semidirect
30. #11758564C22semidirect
32. C2×#92499C22semidirect
34. #11758561C22semidirect
36. C2×#92497C22semidirect
37. C31×#137C22semidirect
38. C31×AGL(1,ℤ/8)C22semidirect
39. C31×#135C22semidirect
40. C2×SD16×C31C22semidirect
41. C31×#131C22semidirect
42. #11758528C22semidirect
43. #11758527C22semidirect
44. #11758527C22semidirect
45. #11758526C22semidirect
46. #11758525C22semidirect
47. C2×#92482C22semidirect
48. #11758519C22semidirect
49. #11758509C22semidirect
50. C2×#92479C22semidirect
51. #11758496C22semidirect
52. #11758495C22semidirect
53. Q16×D31C22semidirect
54. #11758493C22semidirect
55. #11758493C22semidirect
56. #11758493C22semidirect
57. #11758493C22semidirect
58. #11758493C22semidirect
59. #11758492C22semidirect
60. #11758491C22semidirect
61. SD16×D31C22semidirect
62. SD16×D31C22semidirect
63. #11758489C22semidirect
64. #11758488C22semidirect
65. D8×D31C22semidirect
66. #11758486C22semidirect
67. #11758485C22semidirect
68. #11758484C22semidirect
69. M16×D31C22semidirect
70. #11758481C22semidirect
71. C2×#92470C22semidirect
72. #11758478C22semidirect
73. C2×C8×D31C22semidirect
252. #577D31semidirect
273. 2−1+4C2×D31semidirect
279. H(5,𝔽2)C2×D31semidirect
294. #137C2×D31semidirect
300. AGL(1,ℤ/8)C2×D31semidirect
311. #135C2×D31semidirect
317. C2×SD16C2×D31semidirect
323. #131C2×D31semidirect
326. C31#3584semidirect

Maximal subgroups

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

subgroupindex
#145218862
#145218842
#145218652
#145218642
#145218612
#145218562
#145218552
#145218532
#145218522
#145218482
#145218452
C2×#117584932
#135×D312
#131×D312
C31×#5772
#358431