Finite Group Databasebeta

← #16860493

Subgroups — #16860493

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#14521884C211yesmaximal, characteristic, complemented
22−1+4×D31C211yesmaximal, characteristic, complemented
3#14521866C211yesmaximal, characteristic, complemented
4#14521865C213yesmaximal, complemented
5#14521864C211yesmaximal, characteristic, complemented
6#14521863C213yesmaximal, complemented
7#14521860C211yesmaximal, characteristic, complemented
8#14521859C213yesmaximal, complemented
9#14521855C213yesmaximal, complemented
10#14521846C213yesmaximal, complemented
11#14521845C213yesmaximal, complemented
12C2×#11758492C213yesmaximal, complemented
13#137×D31C213yesmaximal, complemented
14#14521814C211yesmaximal, characteristic, complemented
15C31×#577C211yesmaximal, characteristic, complemented
16C31×2−1+4C2211yescharacteristic, complemented
17C31×H(5,𝔽2)C2211yescharacteristic, complemented
18#11758567—21no—
19#11758567—23no—
20#11758567C2213yescomplemented
21#11758567C2213yescomplemented
22#11758567C2211yescharacteristic, complemented
23#11758567C2211yescharacteristic, complemented
24#41×D31—21no—
25#41×D31—23no—
26#41×D31C2213yes—
27#41×D31C2211yescharacteristic
28#41×D31C2213yescomplemented
29#11758564—21no—
30#11758564C2213yescomplemented
31C2×Q8×D31—21no—
32C2×Q8×D31C2213yescomplemented
33#11758561—23no—
34#11758561C2213yescomplemented
35C2×#92497—23no—
36C2×#92497C2213yes—
37C31×#137C2213yescomplemented
38C31×AGL(1,ℤ/8)C2213yescomplemented
39C31×#135C2213yescomplemented
40C2×SD16×C31C2213yescomplemented
41C31×#131C2211yescharacteristic, complemented
42#11758528C2213yescomplemented
43#11758528C2213yescomplemented
44#11758527C2213yescomplemented
45#11758527C2213yescomplemented
46#11758525C2211yescharacteristic, complemented
47C2×#92482C2213yescomplemented
48#11758519C2213yescomplemented
49C2×#92480C2213yescomplemented
50#11758509C2213yescomplemented
51#11758495C2213yescomplemented
52#11758495C2213yescomplemented
53Q16×D31C2216yescomplemented
54#11758493C2216yescomplemented
55#11758492C2216yescomplemented
56#11758492C2213yescomplemented
57#11758492C2216yescomplemented
58#11758492C2216yescomplemented
59#11758492C2213yescomplemented
60#11758491C2211yescharacteristic, complemented
61#11758491C2213yescomplemented
62SD16×D31C2216yescomplemented
63#11758489C2216yescomplemented
64#11758488C2213yescomplemented
65#11758488C2213yescomplemented
66#11758486C2213yescomplemented
67#11758486C2213yescomplemented
68#11758484C2213yescomplemented
69M16×D31C2213yescomplemented
70C2×Dic124C2213yescomplemented
71#11758481C2213yescomplemented
72#11758478C2213yescomplemented
73C2×#92469C2213yescomplemented
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—23no—
87#92499D411yescharacteristic
88#92499C2313yes—
89#92499C2311yescharacteristic
90Q8×D31—21no—
91Q8×D31—21no—
92Q8×D31—23no—
93Q8×D31—23no—
94Q8×D31—23no—
95Q8×D31D411yescharacteristic
96Q8×D31C2316yes—
97Q8×D31C2313yes—
98Q8×D31C2313yes—
99#92497—23no—
100#92497—23no—
101#92497—26no—
102#92497—26no—
103#92497—23no—
104#92497—23no—
105#92497—23no—
106#92497—23no—
107#92497—23no—
108#92497D413yes—
109#92497C2313yes—
110#92497C2313yes—
111#92497C2316yes—
112D4×D31—23no—
113D4×D31—23no—
114D4×D31—23no—
115D4×D31D413yes—
116D4×D31C2311yescharacteristic
117D4×D31C2313yes—
118#92495—26no—
119#92495—23no—
120#92495—23no—
121#92495—23no—
122#92495—23no—
123#92495C2313yes—
124#92495C2313yes—
125#92495C2313yes—
126C2×C4×D31—23no—
127C2×C4×D31—23no—
128C2×C4×D31—23no—
129C2×C4×D31C2313yes—
130C2×Dic62—23no—
131C2×Dic62—23no—
132C2×Dic62—23no—
133C2×Dic62C2313yes—
134C2×Dic62C2313yes—
135Q16×C31C2313yes—
136SD16×C31C2316yes—
137SD16×C31C2313yes—
138SD16×C31C2311yescharacteristic
139D8×C31C2313yes—
140M16×C31C2313yes—
141C2×C8×C31C2313yes—
142#92482C2313yes—
143#92482C2316yes—
144#92482C2313yes—
145#92481C2313yes—
146#92481C2311yescharacteristic
147#92480C2313yes—
148#92480C2313yes—
149#92480C2316yes—
150#92479C2311yescharacteristic
151#92479C2313yes—
152#92474C2313yes—
153C2×C31⋊C8C2313yes—
154Dic124C2313yes—
155Dic124C2316yes—
156D248C2311yescharacteristic
157#92470C2313yes—
158#92470C2313yes—
159#92469C2311yescharacteristic
160#92469C2313yes—
161#92469C2313yes—
162#92469C2313yes—
163C8×D31C2316yes—
164C23×C31—23no—
165C22×D31—21no—
166C22×D31—23no—
167C22×D31C2×D413yes—
168Q8×C31—23no—
169Q8×C31C2×D411yescharacteristic
170Q8×C31C2413yes—
171Q8×C31C2411yescharacteristic
172D4×C31—23no—
173D4×C31—23no—
174D4×C31—23no—
175D4×C31C2×D413yes—
176D4×C31C2411yescharacteristic
177D4×C31C2413yes—
178C2×C4×C31—23no—
179C2×C4×C31—23no—
180C2×C4×C31—23no—
181C2×C4×C31C2413yes—
182C22⋊D31—23no—
183C22⋊D31—23no—
184C22⋊D31—23no—
185C22⋊D31—23no—
186C22⋊D31—26no—
187C22⋊D31—21no—
188C22⋊D31—23no—
189C22⋊D31C2×D416yes—
190C2×Dic31—23no—
191C2×Dic31—23no—
192C2×Dic31—23no—
193C2×Dic31—26no—
194C2×Dic31—23no—
195C2×Dic31C2×D413yes—
196D124—21no—
197D124—23no—
198D124C2×D413yes—
199D124C2411yescharacteristic
200D124C2411yescharacteristic
201C4×D31—23no—
202C4×D31—23no—
203C4×D31—26no—
204C4×D31—23no—
205C4×D31—23no—
206C4×D31—21no—
207C4×D31—23no—
208C4×D31C2×D413yes—
209C4×D31C2×D413yes—
210C4×D31C2413yes—
211C4×D31C2411yescharacteristic
212Dic62—23no—
213Dic62—23no—
214Dic62—23no—
215Dic62—26no—
216Dic62—23no—
217Dic62C2×D413yes—
218Dic62C2413yes—
219Dic62C2413yes—
220C8×C31C2413yes—
221C8×C31C2411yescharacteristic
222C31⋊C8C2413yes—
223C31⋊C8C2411yescharacteristic
224#3586—311nomaximal, Sylow
225C22×C31—21no—
226C22×C31—23no—
227C22×C31—83no—
228C22×C31C22×D413yes—
229C2×D31—21no—
230C2×D31—21no—
231C2×D31—43no—
232C2×D31—42no—
233C2×D31—83no—
234C2×D31C22×D411yescharacteristic
235C2×D31C22×D413yes—
236C4×C31—21no—
237C4×C31—23no—
238C4×C31C22×D413yes—
239C4×C31C2511yescharacteristic
240Dic31—23no—
241Dic31—23no—
242Dic31C22×D413yes—
243Dic31C22×D411yescharacteristic
244#585—311no—
245C2×2−1+4—311no—
246#578—311no—
247#578—313no—
248#578—313no—
249#578—311no—
250#577—311no—
251#577—313no—
252#577—313no—
253#577D3111yescharacteristic, complemented
254#575—313no—
255#575—313no—
256C2×#137—313no—
257C2×#137—313no—
258#568—311no—
259C2×C31—23no—
260C2×C31—41no—
261C2×C31—43no—
262C2×C31C23×D411yescharacteristic
263D31—21no—
264D31—23no—
265D31—41no—
266D31—41no—
2672−1+4—313no—
2682−1+4—313no—
2692−1+4—311no—
2702−1+4—311no—
2712−1+4—313no—
2722−1+4—621no—
2732−1+4—621no—
2742−1+4—623no—
2752−1+4C2×D3111yescharacteristic, complemented
276H(5,𝔽2)—313no—
277H(5,𝔽2)—623no—
278H(5,𝔽2)C2×D3111yescharacteristic, complemented
279C2×#41—311no—
280C2×#41—313no—
281C2×#41—313no—
282C2×#41—313no—
283C2×#41—623no—
284C2×#41—623no—
285C2×#41—621no—
286C22×Q8—313no—
287C22×Q8—621no—
288#137—313no—
289#137—313no—
290#137—313no—
291#137—313no—
292#137—313no—
293#137—316no—
294#137—313no—
295#137—316no—
296#137—316no—
297#137—313no—
298#137—313no—
299#137—313no—
300#137C2×D3113yescomplemented
301AGL(1,ℤ/8)—313no—
302AGL(1,ℤ/8)—311no—
303AGL(1,ℤ/8)—313no—
304AGL(1,ℤ/8)—313no—
305AGL(1,ℤ/8)—313no—
306AGL(1,ℤ/8)C2×D3113yescomplemented
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×Q16—313no—
316C2×SD16—313no—
317C2×SD16—316no—
318C2×SD16C2×D3113yescomplemented
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—21no—
327#41—23no—
328#41—311no—
329#41—313no—
330#41—313no—
331#41—313no—
332#41—311no—
333#41—313no—
334#41—316no—
335#41—313no—
336#41—313no—
337#41—313no—
338#41—623no—
339#41—623no—
340#41—623no—
341#41—626no—
342#41—623no—
343#41—623no—
344#41—623no—
345#41—623no—
346#41—623no—
347#41—626no—
348#41—626no—
349#41—623no—
350#41—623no—
351#41—623no—
352#41—623no—
353#41C22×D3113yes—
354#41C22×D3113yes—
355#41C22×D3111yescharacteristic
356C2×Q8—21no—
357C2×Q8—311no—
358C2×Q8—316no—
359C2×Q8—313no—
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×Q8—621no—
371C2×Q8C22×D3113yes—
372C2×D4—23no—
373C2×D4—313no—
374C2×D4—313no—
375C2×D4—311no—
376C2×D4—623no—
377C2×D4—623no—
378C2×D4—623no—
379C2×D4—626no—
380C2×D4—623no—
381C2×D4C22×D3113yes—
382C22×C4—313no—
383C22×C4—623no—
384C22×C4—623no—
385C22×C4—623no—
386C22×C4—623no—
387Q16—313no—
388Q16—313no—
389Q16—316no—
390Q16—316no—
391Q16—313no—
392Q16C22×D3113yes—
393SD16—313no—
394SD16—311no—
395SD16—313no—
396SD16—316no—
397SD16—313no—
398SD16—313no—
399SD16—313no—
400SD16C22×D3116yes—
401SD16C22×D3113yes—
402SD16C22×D3111yescharacteristic
403D8—313no—
404D8—311no—
405D8—311no—
406D8C22×D3113yes—
407M16—313no—
408M16—313no—
409M16—311no—
410M16—313no—
411M16—313no—
412M16C22×D3113yes—
413C2×C8—313no—
414C2×C8—316no—
415C2×C8C22×D3113yes—
416C23—23no—
417C23—313no—
418C23—621no—
419C23—623no—
420Q8—23no—
421Q8—313no—
422Q8—313no—
423Q8—313no—
424Q8—623no—
425Q8—626no—
426Q8—623no—
427Q8—623no—
428Q8—623no—
429Q8D4×D3111yescharacteristic
430Q8C23×D3113yes—
431Q8C23×D3111yescharacteristic
432D4—23no—
433D4—23no—
434D4—23no—
435D4—316no—
436D4—313no—
437D4—311no—
438D4—311no—
439D4—621no—
440D4—623no—
441D4—623no—
442D4—621no—
443D4—623no—
444D4—626no—
445D4—623no—
446D4—623no—
447D4—623no—
448D4D4×D3113yes—
449D4C23×D3111yescharacteristic
450D4C23×D3113yes—
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—313no—
455C2×C4—313no—
456C2×C4—313no—
457C2×C4—313no—
458C2×C4—311no—
459C2×C4—621no—
460C2×C4—623no—
461C2×C4—623no—
462C2×C4—623no—
463C2×C4—626no—
464C2×C4—623no—
465C2×C4—626no—
466C2×C4—623no—
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—23no—
478C22—83no—
479C22—311no—
480C22—313no—
481C22—621no—
482C22—621no—
483C22—1243no—
484C22—1242no—
485C22—2483no—
486C22C2×D4×D3113yes—
487C4—21no—
488C4—23no—
489C4—311no—
490C4—313no—
491C4—623no—
492C4—623no—
493C4C2×D4×D3113yes—
494C4C24×D3111yescharacteristic
495C2—23no—
496C2—43no—
497C2—41no—
498C2—621no—
499C2—623no—
500C2—1241no—
501C2—1241no—
502C2C22×D4×D3111yescharacteristic
subgroupG/H
centerC2C22×D4×D31
commutator, cosocleC4×C31C25
FrattiniC4C24×D31
FittingC31×#577C2
socleC2×C31C23×D4
hypercenter#577D31
hypocenterC31#3586
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. #14521884C2semidirect
2. 2−1+4×D31C2semidirect
3. #14521866C2semidirect
4. #14521865C2semidirect
5. #14521864C2semidirect
6. #14521863C2semidirect
7. #14521860C2semidirect
8. #14521859C2semidirect
9. #14521855C2semidirect
10. #14521846C2semidirect
11. #14521845C2semidirect
12. C2×#11758492C2semidirect
13. #137×D31C2semidirect
14. #14521814C2semidirect
15. C31×#577C2semidirect
16. C31×2−1+4C22semidirect
17. C31×H(5,𝔽2)C22semidirect
20. #11758567C22semidirect
21. #11758567C22semidirect
22. #11758567C22semidirect
23. #11758567C22semidirect
28. #41×D31C22semidirect
30. #11758564C22semidirect
32. C2×Q8×D31C22semidirect
34. #11758561C22semidirect
37. C31×#137C22semidirect
38. C31×AGL(1,ℤ/8)C22semidirect
39. C31×#135C22semidirect
40. C2×SD16×C31C22semidirect
41. C31×#131C22semidirect
42. #11758528C22semidirect
43. #11758528C22semidirect
44. #11758527C22semidirect
45. #11758527C22semidirect
46. #11758525C22semidirect
47. C2×#92482C22semidirect
48. #11758519C22semidirect
49. C2×#92480C22semidirect
50. #11758509C22semidirect
51. #11758495C22semidirect
52. #11758495C22semidirect
53. Q16×D31C22semidirect
54. #11758493C22semidirect
55. #11758492C22semidirect
56. #11758492C22semidirect
57. #11758492C22semidirect
58. #11758492C22semidirect
59. #11758492C22semidirect
60. #11758491C22semidirect
61. #11758491C22semidirect
62. SD16×D31C22semidirect
63. #11758489C22semidirect
64. #11758488C22semidirect
65. #11758488C22semidirect
66. #11758486C22semidirect
67. #11758486C22semidirect
68. #11758484C22semidirect
69. M16×D31C22semidirect
70. C2×Dic124C22semidirect
71. #11758481C22semidirect
72. #11758478C22semidirect
73. C2×#92469C22semidirect
253. #577D31semidirect
275. 2−1+4C2×D31semidirect
278. H(5,𝔽2)C2×D31semidirect
300. #137C2×D31semidirect
306. AGL(1,ℤ/8)C2×D31semidirect
312. #135C2×D31semidirect
318. C2×SD16C2×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
#145218842
2−1+4×D312
#145218662
#145218652
#145218642
#145218632
#145218602
#145218592
#145218552
#145218462
#145218452
C2×#117584922
#137×D312
#145218142
C31×#5772
#358631