Finite Group Databasebeta

← #27156521

Subgroups — #27156521

Order 12416.

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#20261572C211yesmaximal, characteristic, complemented
22−1+4×D97C211yesmaximal, characteristic, complemented
3#20261551C213yesmaximal, complemented
4#20261550C211yesmaximal, characteristic, complemented
5#20261548C213yesmaximal, complemented
6#20261547C211yesmaximal, characteristic, complemented
7#20261544C213yesmaximal, complemented
8#20261543C211yesmaximal, characteristic, complemented
9#20261542C213yesmaximal, complemented
10#20261534C213yesmaximal, complemented
11#20261532C213yesmaximal, complemented
12C2×#15690842C213yesmaximal, complemented
13#137×D97C213yesmaximal, complemented
14#20261500C211yesmaximal, characteristic, complemented
15#578×C97C211yesmaximal, characteristic, complemented
16#15690858—21no—
17#15690858—21no—
18#15690858C2213yescomplemented
19#15690858C2213yescomplemented
20#15690857—23no—
21#15690857C2211yescharacteristic, complemented
22#15690857C2213yescomplemented
23#15690856—23no—
24#15690856C2211yescharacteristic, complemented
25#15690856C2213yescomplemented
26C2×#13095512—21no—
27C2×#13095512C2213yes—
28#41×D97—23no—
29#41×D97—23no—
30#41×D97C2213yes—
31#41×D97C2211yescharacteristic
32#41×D97C2213yescomplemented
33C2×Q8×D97—21no—
34C2×Q8×D97C2213yescomplemented
35#15690843C2213yescomplemented
36#15690842C2216yescomplemented
37#15690842C2213yescomplemented
38#15690842C2216yescomplemented
39#15690842C2216yescomplemented
40#15690842C2213yescomplemented
41#15690841C2213yescomplemented
42#15690841C2213yescomplemented
43#15690840C2213yescomplemented
44#15690839C2213yescomplemented
45#15690839C2213yescomplemented
46#15690838C2213yescomplemented
47#15690837C2213yescomplemented
48#15690837C2213yescomplemented
49#15690835C2213yescomplemented
50#15690834C2211yescharacteristic, complemented
51#15690834C2213yescomplemented
52#15690833C2216yescomplemented
53#15690832C2213yescomplemented
54#15690831C2216yescomplemented
55#15690830C2213yescomplemented
56#15690830C2213yescomplemented
57C2×#13095501C2213yescomplemented
58Q16×D97C2216yescomplemented
59C2×#13095500C2213yescomplemented
60C2×#13095498C2213yescomplemented
61SD16×D97C2216yescomplemented
62#15690818C2213yescomplemented
63#15690817C2211yescharacteristic, complemented
64#15690816C2213yescomplemented
65C2×#13095494C2213yescomplemented
66M16×D97C2213yescomplemented
672−1+4×C97C2211yescharacteristic, complemented
682−1+4×C97C2211yescharacteristic, complemented
69#137×C97C2213yescomplemented
70#137×C97C2213yescomplemented
71#135×C97C2213yescomplemented
72C2×Q16×C97C2213yescomplemented
73#131×C97C2211yescharacteristic, complemented
74#13095512—21no—
75#13095512—23no—
76#13095512—23no—
77#13095512—23no—
78#13095512—23no—
79#13095512D411yescharacteristic
80#13095512C2316yes—
81#13095512C2313yes—
82#13095512C2313yes—
83#13095511—26no—
84#13095511—26no—
85#13095511—23no—
86#13095511—23no—
87#13095511—23no—
88#13095511—23no—
89#13095511C2313yes—
90#13095511C2313yes—
91#13095511C2313yes—
92#13095510—23no—
93#13095510—26no—
94#13095510—23no—
95#13095510—23no—
96#13095510D413yes—
97#13095510C2313yes—
98#13095510C2311yescharacteristic
99C2×Dic194—23no—
100C2×Dic194—23no—
101C2×Dic194C2313yes—
102Q8×D97—21no—
103Q8×D97—23no—
104Q8×D97—23no—
105Q8×D97—23no—
106Q8×D97—23no—
107Q8×D97D411yescharacteristic
108Q8×D97C2316yes—
109Q8×D97C2313yes—
110Q8×D97C2313yes—
111C2×D388—23no—
112C2×D388—23no—
113C2×D388C2313yes—
114D4×D97—23no—
115D4×D97—23no—
116D4×D97—23no—
117D4×D97—26no—
118D4×D97D413yes—
119D4×D97C2313yes—
120D4×D97C2311yescharacteristic
121C2×C4×D97—23no—
122C2×C4×D97—23no—
123C2×C4×D97—23no—
124C2×C4×D97—23no—
125C2×C4×D97C2313yes—
126Dic388C2313yes—
127#13095501C2313yes—
128#13095501C2316yes—
129#13095501C2313yes—
130#13095500C2316yes—
131#13095500C2313yes—
132#13095500C2311yescharacteristic
133#13095499C2313yes—
134#13095499C2311yescharacteristic
135#13095498C2316yes—
136#13095498C2313yes—
137#13095498C2313yes—
138D776C2313yes—
139#13095496C2311yescharacteristic
140#13095496C2313yes—
141#13095495C2313yes—
142#13095494C2313yes—
143#13095494C2311yescharacteristic
144#13095494C2313yes—
145#13095494C2313yes—
146C2×#11715895C2313yes—
147C8×D97C2316yes—
148#41×C97—23no—
149#41×C97—23no—
150#41×C97C2313yes—
151#41×C97C2313yes—
152#41×C97C2311yescharacteristic
153C2×Q8×C97—21no—
154C2×Q8×C97—21no—
155C2×Q8×C97C2313yes—
156C2×Q8×C97C2313yes—
157Q16×C97C2316yes—
158Q16×C97C2313yes—
159SD16×C97C2313yes—
160SD16×C97C2313yes—
161D8×C97C2311yescharacteristic
162M16×C97C2313yes—
163C2×C8×C97C2313yes—
164C22×D97—23no—
165C22×D97—21no—
166C22×D97—23no—
167C22×D97C2×D413yes—
168Q8×C97—23no—
169Q8×C97—23no—
170Q8×C97C2×D411yescharacteristic
171Q8×C97C2413yes—
172Q8×C97C2413yes—
173D4×C97—23no—
174D4×C97—23no—
175D4×C97C2×D413yes—
176D4×C97C2411yescharacteristic
177D4×C97C2411yescharacteristic
178C2×C4×C97—23no—
179C2×C4×C97—23no—
180C2×C4×C97—23no—
181C2×C4×C97—23no—
182C2×C4×C97C2413yes—
183C22⋊D97—23no—
184C22⋊D97—23no—
185C22⋊D97—21no—
186C22⋊D97—23no—
187C22⋊D97—23no—
188C22⋊D97—21no—
189C22⋊D97C2×D416yes—
190C2×Dic97—23no—
191C2×Dic97—21no—
192C2×Dic97—23no—
193C2×Dic97C2×D413yes—
194D388—23no—
195D388—26no—
196D388—23no—
197D388—23no—
198D388C2×D413yes—
199D388C2411yescharacteristic
200D388C2413yes—
201C4×D97—23no—
202C4×D97—23no—
203C4×D97—26no—
204C4×D97—23no—
205C4×D97—23no—
206C4×D97—23no—
207C4×D97—26no—
208C4×D97—23no—
209C4×D97C2×D413yes—
210C4×D97C2×D413yes—
211C4×D97C2413yes—
212C4×D97C2411yescharacteristic
213Dic194—23no—
214Dic194—26no—
215Dic194—23no—
216Dic194—23no—
217Dic194C2×D413yes—
218Dic194C2413yes—
219Dic194C2411yescharacteristic
220C8×C97C2411yescharacteristic
221C8×C97C2413yes—
222#11715895C2413yes—
223#11715895C2411yescharacteristic
224C22×C97—21no—
225C22×C97—21no—
226C22×C97C22×D413yes—
227C2×D97—21no—
228C2×D97—23no—
229C2×D97—43no—
230C2×D97—42no—
231C2×D97—83no—
232C2×D97—83no—
233C2×D97C22×D411yescharacteristic
234C2×D97C22×D413yes—
235C4×C97—23no—
236C4×C97—23no—
237C4×C97C22×D413yes—
238C4×C97C2511yescharacteristic
239Dic97—23no—
240Dic97—21no—
241Dic97C22×D413yes—
242Dic97C22×D411yescharacteristic
243C2×C97—23no—
244C2×C97—41no—
245C2×C97—41no—
246C2×C97C23×D411yescharacteristic
247D97—21no—
248D97—23no—
249D97—41no—
250D97—43no—
251#3586—971nomaximal, Sylow
252C97#358611yescharacteristic, complemented, Sylow
253#585—971no—
254C2×2−1+4—971no—
255#578—971no—
256#578—973no—
257#578—973no—
258#578D9711yescharacteristic, complemented
259#577—971no—
260#577—973no—
261#577—973no—
262#577—971no—
263#575—973no—
264#575—973no—
265C2×#137—973no—
266C2×#137—973no—
267#568—971no—
2682−1+4—973no—
2692−1+4—971no—
2702−1+4—973no—
2712−1+4—973no—
2722−1+4—1941no—
2732−1+4—1941no—
2742−1+4—1943no—
2752−1+4C2×D9711yescharacteristic, complemented
2762−1+4C2×D9711yescharacteristic, complemented
277H(5,𝔽2)—973no—
278H(5,𝔽2)—971no—
279H(5,𝔽2)—1943no—
280C2×#41—971no—
281C2×#41—973no—
282C2×#41—973no—
283C2×#41—973no—
284C2×#41—1943no—
285C2×#41—1941no—
286C2×#41—1943no—
287C22×Q8—973no—
288C22×Q8—1941no—
289#137—973no—
290#137—973no—
291#137—973no—
292#137—976no—
293#137—973no—
294#137—976no—
295#137—976no—
296#137—973no—
297#137—973no—
298#137—973no—
299#137—973no—
300#137C2×D9713yescomplemented
301#137C2×D9713yescomplemented
302AGL(1,ℤ/8)—973no—
303AGL(1,ℤ/8)—973no—
304AGL(1,ℤ/8)—973no—
305AGL(1,ℤ/8)—971no—
306AGL(1,ℤ/8)—973no—
307AGL(1,ℤ/8)—973no—
308#135—973no—
309#135—973no—
310#135—976no—
311#135—976no—
312#135—973no—
313#135C2×D9713yescomplemented
314C2×Q16—973no—
315C2×Q16—976no—
316C2×Q16C2×D9713yescomplemented
317C2×SD16—973no—
318C2×SD16—976no—
319C2×SD16—973no—
320#131—971no—
321#131—973no—
322#131—973no—
323#131C2×D9711yescharacteristic, complemented
324C2×M16—973no—
325C2×M16—973no—
326#41—23no—
327#41—23no—
328#41—971no—
329#41—973no—
330#41—973no—
331#41—973no—
332#41—976no—
333#41—973no—
334#41—973no—
335#41—971no—
336#41—973no—
337#41—973no—
338#41—1943no—
339#41—1943no—
340#41—1946no—
341#41—1946no—
342#41—1943no—
343#41—1943no—
344#41—1943no—
345#41—1943no—
346#41—1943no—
347#41—1943no—
348#41—1946no—
349#41—1943no—
350#41—1943no—
351#41—1941no—
352#41—1943no—
353#41C22×D9713yes—
354#41C22×D9713yes—
355#41C22×D9711yescharacteristic
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—971no—
359C2×Q8—976no—
360C2×Q8—973no—
361C2×Q8—973no—
362C2×Q8—973no—
363C2×Q8—1943no—
364C2×Q8—1943no—
365C2×Q8—1943no—
366C2×Q8—1943no—
367C2×Q8—1943no—
368C2×Q8—1943no—
369C2×Q8—1941no—
370C2×Q8C22×D9713yes—
371C2×Q8C22×D9713yes—
372C2×D4—973no—
373C2×D4—973no—
374C2×D4—971no—
375C2×D4—973no—
376C2×D4—1943no—
377C2×D4—1943no—
378C2×D4—1946no—
379C2×D4—1943no—
380C2×D4—1943no—
381C2×D4—1943no—
382C22×C4—973no—
383C22×C4—1943no—
384C22×C4—1943no—
385C22×C4—1943no—
386C22×C4—1943no—
387Q16—976no—
388Q16—973no—
389Q16—973no—
390Q16—973no—
391Q16C22×D9716yes—
392Q16C22×D9713yes—
393SD16—976no—
394SD16—973no—
395SD16—973no—
396SD16—973no—
397SD16—971no—
398SD16—976no—
399SD16—973no—
400SD16—971no—
401SD16C22×D9713yes—
402SD16C22×D9713yes—
403D8—971no—
404D8—973no—
405D8—973no—
406D8C22×D9711yescharacteristic
407M16—973no—
408M16—973no—
409M16—971no—
410M16—973no—
411M16—973no—
412M16C22×D9713yes—
413C2×C8—973no—
414C2×C8—976no—
415C2×C8C22×D9713yes—
416C23—973no—
417C23—1943no—
418C23—1941no—
419C23—1943no—
420Q8—23no—
421Q8—23no—
422Q8—973no—
423Q8—973no—
424Q8—971no—
425Q8—1943no—
426Q8—1946no—
427Q8—1943no—
428Q8—1943no—
429Q8D4×D9711yescharacteristic
430Q8C23×D9713yes—
431Q8C23×D9713yes—
432D4—23no—
433D4—23no—
434D4—976no—
435D4—973no—
436D4—971no—
437D4—973no—
438D4—1943no—
439D4—1946no—
440D4—1943no—
441D4—1943no—
442D4—1941no—
443D4—1941no—
444D4—1943no—
445D4—1943no—
446D4—1943no—
447D4—1943no—
448D4D4×D9713yes—
449D4C23×D9711yescharacteristic
450D4C23×D9711yescharacteristic
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—973no—
456C2×C4—973no—
457C2×C4—973no—
458C2×C4—973no—
459C2×C4—971no—
460C2×C4—1943no—
461C2×C4—1946no—
462C2×C4—1943no—
463C2×C4—1943no—
464C2×C4—1943no—
465C2×C4—1946no—
466C2×C4—1941no—
467C2×C4—1943no—
468C2×C4—1943no—
469C2×C4—1943no—
470C2×C4—1943no—
471C2×C4C23×D9713yes—
472C8—971no—
473C8—973no—
474C8C23×D9713yes—
475C8C23×D9711yescharacteristic
476C22—21no—
477C22—21no—
478C22—7763no—
479C22—7763no—
480C22—971no—
481C22—973no—
482C22—3883no—
483C22—3882no—
484C22—1941no—
485C22—1943no—
486C22C2×D4×D9713yes—
487C4—23no—
488C4—23no—
489C4—971no—
490C4—973no—
491C4—1941no—
492C4—1943no—
493C4C2×D4×D9713yes—
494C4C24×D9711yescharacteristic
495C2—23no—
496C2—41no—
497C2—41no—
498C2—3883no—
499C2—3881no—
500C2—1943no—
501C2—1941no—
502C2C22×D4×D9711yescharacteristic
subgroupG/H
centerC2C22×D4×D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
Fitting#578×C97C2
socleC2×C97C23×D4
hypercenter#578D97
hypocenterC97#3586
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ #41 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #20261572C2semidirect
2. 2−1+4×D97C2semidirect
3. #20261551C2semidirect
4. #20261550C2semidirect
5. #20261548C2semidirect
6. #20261547C2semidirect
7. #20261544C2semidirect
8. #20261543C2semidirect
9. #20261542C2semidirect
10. #20261534C2semidirect
11. #20261532C2semidirect
12. C2×#15690842C2semidirect
13. #137×D97C2semidirect
14. #20261500C2semidirect
15. #578×C97C2semidirect
18. #15690858C22semidirect
19. #15690858C22semidirect
21. #15690857C22semidirect
22. #15690857C22semidirect
24. #15690856C22semidirect
25. #15690856C22semidirect
32. #41×D97C22semidirect
34. C2×Q8×D97C22semidirect
35. #15690843C22semidirect
36. #15690842C22semidirect
37. #15690842C22semidirect
38. #15690842C22semidirect
39. #15690842C22semidirect
40. #15690842C22semidirect
41. #15690841C22semidirect
42. #15690841C22semidirect
43. #15690840C22semidirect
44. #15690839C22semidirect
45. #15690839C22semidirect
46. #15690838C22semidirect
47. #15690837C22semidirect
48. #15690837C22semidirect
49. #15690835C22semidirect
50. #15690834C22semidirect
51. #15690834C22semidirect
52. #15690833C22semidirect
53. #15690832C22semidirect
54. #15690831C22semidirect
55. #15690830C22semidirect
56. #15690830C22semidirect
57. C2×#13095501C22semidirect
58. Q16×D97C22semidirect
59. C2×#13095500C22semidirect
60. C2×#13095498C22semidirect
61. SD16×D97C22semidirect
62. #15690818C22semidirect
63. #15690817C22semidirect
64. #15690816C22semidirect
65. C2×#13095494C22semidirect
66. M16×D97C22semidirect
67. 2−1+4×C97C22semidirect
68. 2−1+4×C97C22semidirect
69. #137×C97C22semidirect
70. #137×C97C22semidirect
71. #135×C97C22semidirect
72. C2×Q16×C97C22semidirect
73. #131×C97C22semidirect
252. C97#3586semidirect
258. #578D97semidirect
275. 2−1+4C2×D97semidirect
276. 2−1+4C2×D97semidirect
300. #137C2×D97semidirect
301. #137C2×D97semidirect
313. #135C2×D97semidirect
316. C2×Q16C2×D97semidirect
323. #131C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615722
2−1+4×D972
#202615512
#202615502
#202615482
#202615472
#202615442
#202615432
#202615422
#202615342
#202615322
C2×#156908422
#137×D972
#202615002
#578×C972
#358697