Finite Group Databasebeta

← #16860496

Subgroups — #16860496

Order 3968.

1412 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521886C211yesmaximal, characteristic, complemented
22−1+4×D31C211yesmaximal, characteristic, complemented
3#14521865C213yesmaximal, complemented
4#14521864C211yesmaximal, characteristic, complemented
5#14521862C213yesmaximal, complemented
6#14521861C211yesmaximal, characteristic, complemented
7#14521858C213yesmaximal, complemented
8#14521857C211yesmaximal, characteristic, complemented
9#14521856C213yesmaximal, complemented
10#14521848C213yesmaximal, complemented
11#14521846C213yesmaximal, complemented
12C2×#11758495C213yesmaximal, complemented
13#137×D31C213yesmaximal, complemented
14#14521814C211yesmaximal, characteristic, complemented
15C31×#578C211yesmaximal, characteristic, complemented
16C31×2−1+4C2211yescharacteristic, complemented
17C31×2−1+4C2211yescharacteristic, complemented
18#11758567—23no—
19#11758567C2213yescomplemented
20#11758567C2211yescharacteristic, complemented
21#11758566—23no—
22#11758566C2213yescomplemented
23#11758566C2211yescharacteristic, complemented
24#41×D31—23no—
25#41×D31—23no—
26#41×D31C2213yes—
27#41×D31C2211yescharacteristic
28#41×D31C2213yescomplemented
29#11758564—21no—
30#11758564—21no—
31#11758564C2213yescomplemented
32#11758564C2213yescomplemented
33C2×#92499—21no—
34C2×#92499C2213yes—
35C2×Q8×D31—21no—
36C2×Q8×D31C2213yescomplemented
37C31×#137C2213yescomplemented
38C31×#137C2213yescomplemented
39C31×#135C2213yescomplemented
40C2×Q16×C31C2213yescomplemented
41C31×#131C2211yescharacteristic, complemented
42#11758528C2213yescomplemented
43#11758527C2213yescomplemented
44#11758527C2213yescomplemented
45#11758526C2213yescomplemented
46#11758525C2211yescharacteristic, complemented
47C2×#92482C2213yescomplemented
48#11758519C2213yescomplemented
49#11758519C2213yescomplemented
50C2×#92481C2213yescomplemented
51#11758496C2216yescomplemented
52#11758495C2216yescomplemented
53#11758495C2216yescomplemented
54#11758495C2213yescomplemented
55#11758495C2216yescomplemented
56#11758495C2213yescomplemented
57Q16×D31C2216yescomplemented
58#11758493C2216yescomplemented
59#11758492C2213yescomplemented
60#11758492C2213yescomplemented
61#11758491C2213yescomplemented
62#11758491C2213yescomplemented
63SD16×D31C2216yescomplemented
64#11758488C2213yescomplemented
65#11758488C2211yescharacteristic, complemented
66#11758486C2213yescomplemented
67#11758485C2213yescomplemented
68#11758484C2213yescomplemented
69M16×D31C2213yescomplemented
70#11758481C2213yescomplemented
71C2×#92470C2213yescomplemented
72#11758478C2213yescomplemented
73C2×#92469C2213yescomplemented
74#41×C31—23no—
75#41×C31—23no—
76#41×C31C2313yes—
77#41×C31C2313yes—
78#41×C31C2311yescharacteristic
79C2×Q8×C31—21no—
80C2×Q8×C31—21no—
81C2×Q8×C31C2313yes—
82C2×Q8×C31C2313yes—
83#92499—21no—
84#92499—23no—
85#92499—23no—
86#92499—23no—
87#92499—23no—
88#92499D411yescharacteristic
89#92499C2313yes—
90#92499C2313yes—
91#92499C2316yes—
92Q8×D31—21no—
93Q8×D31—23no—
94Q8×D31—23no—
95Q8×D31—23no—
96Q8×D31—23no—
97Q8×D31D411yescharacteristic
98Q8×D31C2316yes—
99Q8×D31C2313yes—
100Q8×D31C2313yes—
101#92497—23no—
102#92497—26no—
103#92497—23no—
104#92497—23no—
105#92497D413yes—
106#92497C2311yescharacteristic
107#92497C2313yes—
108D4×D31—23no—
109D4×D31—23no—
110D4×D31—23no—
111D4×D31—26no—
112D4×D31D413yes—
113D4×D31C2313yes—
114D4×D31C2311yescharacteristic
115#92495—26no—
116#92495—26no—
117#92495—23no—
118#92495—23no—
119#92495—23no—
120#92495—23no—
121#92495C2313yes—
122#92495C2313yes—
123#92495C2313yes—
124C2×D124—23no—
125C2×D124—23no—
126C2×D124C2313yes—
127C2×C4×D31—23no—
128C2×C4×D31—23no—
129C2×C4×D31—23no—
130C2×C4×D31—23no—
131C2×C4×D31C2313yes—
132C2×Dic62—23no—
133C2×Dic62—23no—
134C2×Dic62C2313yes—
135Q16×C31C2316yes—
136Q16×C31C2313yes—
137SD16×C31C2313yes—
138SD16×C31C2313yes—
139D8×C31C2311yescharacteristic
140M16×C31C2313yes—
141C2×C8×C31C2313yes—
142#92482C2313yes—
143#92482C2313yes—
144#92482C2316yes—
145#92481C2316yes—
146#92481C2313yes—
147#92481C2313yes—
148#92480C2313yes—
149#92480C2311yescharacteristic
150#92479C2311yescharacteristic
151#92479C2313yes—
152#92474C2313yes—
153C2×C31⋊C8C2313yes—
154Dic124C2313yes—
155D248C2313yes—
156#92470C2313yes—
157#92470C2316yes—
158#92470C2311yescharacteristic
159#92469C2311yescharacteristic
160#92469C2313yes—
161#92469C2313yes—
162#92469C2313yes—
163C8×D31C2316yes—
164C22×D31—23no—
165C22×D31—21no—
166C22×D31—23no—
167C22×D31C2×D413yes—
168Q8×C31—23no—
169Q8×C31—23no—
170Q8×C31C2×D411yescharacteristic
171Q8×C31C2413yes—
172Q8×C31C2413yes—
173D4×C31—23no—
174D4×C31—23no—
175D4×C31C2×D413yes—
176D4×C31C2411yescharacteristic
177D4×C31C2411yescharacteristic
178C2×C4×C31—23no—
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—23no—
187C22⋊D31—23no—
188C22⋊D31—21no—
189C22⋊D31C2×D416yes—
190C2×Dic31—23no—
191C2×Dic31—23no—
192C2×Dic31—21no—
193C2×Dic31C2×D413yes—
194D124—23no—
195D124—26no—
196D124—23no—
197D124—23no—
198D124C2×D413yes—
199D124C2413yes—
200D124C2411yescharacteristic
201C4×D31—23no—
202C4×D31—23no—
203C4×D31—26no—
204C4×D31—23no—
205C4×D31—23no—
206C4×D31—23no—
207C4×D31—26no—
208C4×D31—23no—
209C4×D31C2×D413yes—
210C4×D31C2×D413yes—
211C4×D31C2413yes—
212C4×D31C2411yescharacteristic
213Dic62—23no—
214Dic62—26no—
215Dic62—23no—
216Dic62—23no—
217Dic62C2×D413yes—
218Dic62C2413yes—
219Dic62C2411yescharacteristic
220C8×C31C2413yes—
221C8×C31C2411yescharacteristic
222C31⋊C8C2413yes—
223C31⋊C8C2411yescharacteristic
224#3586—311nomaximal, Sylow
225C22×C31—21no—
226C22×C31—21no—
227C22×C31C22×D413yes—
228C2×D31—21no—
229C2×D31—23no—
230C2×D31—43no—
231C2×D31—42no—
232C2×D31—83no—
233C2×D31—83no—
234C2×D31C22×D411yescharacteristic
235C2×D31C22×D413yes—
236C4×C31—23no—
237C4×C31—23no—
238C4×C31C22×D413yes—
239C4×C31C2511yescharacteristic
240Dic31—23no—
241Dic31—21no—
242Dic31C22×D413yes—
243Dic31C22×D411yescharacteristic
244#585—311no—
245C2×2−1+4—311no—
246#578—311no—
247#578—313no—
248#578—313no—
249#578D3111yescharacteristic, complemented
250#577—311no—
251#577—313no—
252#577—313no—
253#577—311no—
254#575—313no—
255#575—313no—
256C2×#137—313no—
257C2×#137—313no—
258#568—311no—
259C2×C31—23no—
260C2×C31—41no—
261C2×C31—41no—
262C2×C31C23×D411yescharacteristic
263D31—21no—
264D31—23no—
265D31—41no—
266D31—43no—
2672−1+4—313no—
2682−1+4—311no—
2692−1+4—313no—
2702−1+4—313no—
2712−1+4—621no—
2722−1+4—621no—
2732−1+4—623no—
2742−1+4C2×D3111yescharacteristic, complemented
2752−1+4C2×D3111yescharacteristic, complemented
276H(5,𝔽2)—313no—
277H(5,𝔽2)—311no—
278H(5,𝔽2)—623no—
279C2×#41—311no—
280C2×#41—313no—
281C2×#41—313no—
282C2×#41—313no—
283C2×#41—623no—
284C2×#41—621no—
285C2×#41—623no—
286C22×Q8—313no—
287C22×Q8—621no—
288#137—313no—
289#137—313no—
290#137—313no—
291#137—316no—
292#137—313no—
293#137—316no—
294#137—316no—
295#137—313no—
296#137—313no—
297#137—313no—
298#137—313no—
299#137C2×D3113yescomplemented
300#137C2×D3113yescomplemented
301AGL(1,ℤ/8)—313no—
302AGL(1,ℤ/8)—313no—
303AGL(1,ℤ/8)—313no—
304AGL(1,ℤ/8)—311no—
305AGL(1,ℤ/8)—313no—
306AGL(1,ℤ/8)—313no—
307#135—313no—
308#135—313no—
309#135—316no—
310#135—316no—
311#135—313no—
312#135C2×D3113yescomplemented
313C2×Q16—313no—
314C2×Q16—316no—
315C2×Q16C2×D3113yescomplemented
316C2×SD16—313no—
317C2×SD16—316no—
318C2×SD16—313no—
319#131—311no—
320#131—313no—
321#131—313no—
322#131C2×D3111yescharacteristic, complemented
323C2×M16—313no—
324C2×M16—313no—
325C31#358611yescharacteristic, complemented, Sylow
326#41—23no—
327#41—23no—
328#41—311no—
329#41—313no—
330#41—313no—
331#41—313no—
332#41—316no—
333#41—313no—
334#41—313no—
335#41—311no—
336#41—313no—
337#41—313no—
338#41—623no—
339#41—623no—
340#41—626no—
341#41—626no—
342#41—623no—
343#41—623no—
344#41—623no—
345#41—623no—
346#41—623no—
347#41—623no—
348#41—626no—
349#41—623no—
350#41—623no—
351#41—621no—
352#41—623no—
353#41C22×D3113yes—
354#41C22×D3113yes—
355#41C22×D3111yescharacteristic
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—311no—
359C2×Q8—316no—
360C2×Q8—313no—
361C2×Q8—313no—
362C2×Q8—313no—
363C2×Q8—623no—
364C2×Q8—623no—
365C2×Q8—623no—
366C2×Q8—623no—
367C2×Q8—623no—
368C2×Q8—623no—
369C2×Q8—621no—
370C2×Q8C22×D3113yes—
371C2×Q8C22×D3113yes—
372C2×D4—313no—
373C2×D4—313no—
374C2×D4—311no—
375C2×D4—313no—
376C2×D4—623no—
377C2×D4—623no—
378C2×D4—626no—
379C2×D4—623no—
380C2×D4—623no—
381C2×D4—623no—
382C22×C4—313no—
383C22×C4—623no—
384C22×C4—623no—
385C22×C4—623no—
386C22×C4—623no—
387Q16—313no—
388Q16—316no—
389Q16—313no—
390Q16—313no—
391Q16C22×D3116yes—
392Q16C22×D3113yes—
393SD16—316no—
394SD16—313no—
395SD16—313no—
396SD16—313no—
397SD16—311no—
398SD16—316no—
399SD16—313no—
400SD16—311no—
401SD16C22×D3113yes—
402SD16C22×D3113yes—
403D8—311no—
404D8—313no—
405D8—313no—
406D8C22×D3111yescharacteristic
407M16—313no—
408M16—313no—
409M16—311no—
410M16—313no—
411M16—313no—
412M16C22×D3113yes—
413C2×C8—313no—
414C2×C8—316no—
415C2×C8C22×D3113yes—
416C23—313no—
417C23—621no—
418C23—623no—
419C23—623no—
420Q8—23no—
421Q8—23no—
422Q8—313no—
423Q8—313no—
424Q8—311no—
425Q8—623no—
426Q8—626no—
427Q8—623no—
428Q8—623no—
429Q8D4×D3111yescharacteristic
430Q8C23×D3113yes—
431Q8C23×D3113yes—
432D4—23no—
433D4—23no—
434D4—316no—
435D4—313no—
436D4—311no—
437D4—313no—
438D4—623no—
439D4—626no—
440D4—623no—
441D4—623no—
442D4—621no—
443D4—621no—
444D4—623no—
445D4—623no—
446D4—623no—
447D4—623no—
448D4D4×D3113yes—
449D4C23×D3111yescharacteristic
450D4C23×D3111yescharacteristic
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—313no—
456C2×C4—313no—
457C2×C4—313no—
458C2×C4—313no—
459C2×C4—311no—
460C2×C4—623no—
461C2×C4—626no—
462C2×C4—623no—
463C2×C4—623no—
464C2×C4—623no—
465C2×C4—626no—
466C2×C4—621no—
467C2×C4—623no—
468C2×C4—623no—
469C2×C4—623no—
470C2×C4—623no—
471C2×C4C23×D3113yes—
472C8—313no—
473C8—311no—
474C8C23×D3113yes—
475C8C23×D3111yescharacteristic
476C22—21no—
477C22—21no—
478C22—311no—
479C22—313no—
480C22—621no—
481C22—623no—
482C22—1243no—
483C22—1242no—
484C22—2483no—
485C22—2483no—
486C22C2×D4×D3113yes—
487C4—23no—
488C4—23no—
489C4—311no—
490C4—313no—
491C4—621no—
492C4—623no—
493C4C2×D4×D3113yes—
494C4C24×D3111yescharacteristic
495C2—23no—
496C2—41no—
497C2—41no—
498C2—621no—
499C2—623no—
500C2—1241no—
501C2—1243no—
502C2C22×D4×D3111yescharacteristic
subgroupG/H
centerC2C22×D4×D31
commutator, cosocleC4×C31C25
FrattiniC4C24×D31
FittingC31×#578C2
socleC2×C31C23×D4
hypercenter#578D31
hypocenterC31#3586
LCSG ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ #41 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #14521886C2semidirect
2. 2−1+4×D31C2semidirect
3. #14521865C2semidirect
4. #14521864C2semidirect
5. #14521862C2semidirect
6. #14521861C2semidirect
7. #14521858C2semidirect
8. #14521857C2semidirect
9. #14521856C2semidirect
10. #14521848C2semidirect
11. #14521846C2semidirect
12. C2×#11758495C2semidirect
13. #137×D31C2semidirect
14. #14521814C2semidirect
15. C31×#578C2semidirect
16. C31×2−1+4C22semidirect
17. C31×2−1+4C22semidirect
19. #11758567C22semidirect
20. #11758567C22semidirect
22. #11758566C22semidirect
23. #11758566C22semidirect
28. #41×D31C22semidirect
31. #11758564C22semidirect
32. #11758564C22semidirect
36. C2×Q8×D31C22semidirect
37. C31×#137C22semidirect
38. C31×#137C22semidirect
39. C31×#135C22semidirect
40. C2×Q16×C31C22semidirect
41. C31×#131C22semidirect
42. #11758528C22semidirect
43. #11758527C22semidirect
44. #11758527C22semidirect
45. #11758526C22semidirect
46. #11758525C22semidirect
47. C2×#92482C22semidirect
48. #11758519C22semidirect
49. #11758519C22semidirect
50. C2×#92481C22semidirect
51. #11758496C22semidirect
52. #11758495C22semidirect
53. #11758495C22semidirect
54. #11758495C22semidirect
55. #11758495C22semidirect
56. #11758495C22semidirect
57. Q16×D31C22semidirect
58. #11758493C22semidirect
59. #11758492C22semidirect
60. #11758492C22semidirect
61. #11758491C22semidirect
62. #11758491C22semidirect
63. SD16×D31C22semidirect
64. #11758488C22semidirect
65. #11758488C22semidirect
66. #11758486C22semidirect
67. #11758485C22semidirect
68. #11758484C22semidirect
69. M16×D31C22semidirect
70. #11758481C22semidirect
71. C2×#92470C22semidirect
72. #11758478C22semidirect
73. C2×#92469C22semidirect
249. #578D31semidirect
274. 2−1+4C2×D31semidirect
275. 2−1+4C2×D31semidirect
299. #137C2×D31semidirect
300. #137C2×D31semidirect
312. #135C2×D31semidirect
315. C2×Q16C2×D31semidirect
322. #131C2×D31semidirect
325. C31#3586semidirect

Maximal subgroups

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

subgroupindex
#145218862
2−1+4×D312
#145218652
#145218642
#145218622
#145218612
#145218582
#145218572
#145218562
#145218482
#145218462
C2×#117584952
#137×D312
#145218142
C31×#5782
#358631