Finite Group Databasebeta

← #27156511

Subgroups — #27156511

Order 12416.

1464 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261570C211yesmaximal, characteristic, complemented
2H(5,𝔽2)×D97C211yesmaximal, characteristic, complemented
3#20261547C211yesmaximal, characteristic, complemented
4#20261546C211yesmaximal, characteristic, complemented
5#20261545C213yesmaximal, complemented
6#20261542C213yesmaximal, complemented
7#20261539C213yesmaximal, complemented
8#20261538C211yesmaximal, characteristic, complemented
9#20261536C213yesmaximal, complemented
10#20261532C213yesmaximal, complemented
11#20261531C213yesmaximal, complemented
12C2×#15690834C213yesmaximal, complemented
13AGL(1,ℤ/8)×D97C213yesmaximal, complemented
14#20261500C211yesmaximal, characteristic, complemented
15#576×C97C211yesmaximal, characteristic, complemented
16#15690857—21no—
17#15690857C2211yescharacteristic, complemented
18#15690857C2213yescomplemented
19#15690856—21no—
20#15690856C2211yescharacteristic, complemented
21#15690856C2213yescomplemented
22#15690855—23no—
23#15690855—23no—
24#15690855C2213yescomplemented
25#15690855C2213yescomplemented
26C2×#13095510—23no—
27C2×#13095510C2213yescomplemented
28#41×D97—21no—
29#41×D97—21no—
30#41×D97C2211yescharacteristic
31#41×D97C2213yescomplemented
32#41×D97C2213yescomplemented
33C2×D4×D97—23no—
34C2×D4×D97C2213yescomplemented
35#15690843C2213yescomplemented
36#15690842C2211yescharacteristic, complemented
37#15690842C2213yescomplemented
38#15690840C2213yescomplemented
39#15690839C2213yescomplemented
40#15690839C2213yescomplemented
41#15690838C2213yescomplemented
42#15690837C2213yescomplemented
43#15690837C2213yescomplemented
44#15690836C2213yescomplemented
45#15690836C2213yescomplemented
46#15690835C2213yescomplemented
47#15690834C2216yescomplemented
48#15690834C2213yescomplemented
49#15690834C2216yescomplemented
50#15690834C2216yescomplemented
51#15690834C2213yescomplemented
52#15690832C2213yescomplemented
53#15690831C2216yescomplemented
54#15690830C2213yescomplemented
55#15690830C2213yescomplemented
56#15690829C2216yescomplemented
57C2×#13095500C2213yescomplemented
58C2×#13095499C2213yescomplemented
59SD16×D97C2216yescomplemented
60C2×#13095496C2213yescomplemented
61D8×D97C2216yescomplemented
62#15690818C2213yescomplemented
63#15690817C2211yescharacteristic, complemented
64#15690816C2213yescomplemented
65C2×#13095494C2213yescomplemented
66M16×D97C2213yescomplemented
67H(5,𝔽2)×C97C2211yescharacteristic, complemented
68H(5,𝔽2)×C97C2211yescharacteristic, complemented
69AGL(1,ℤ/8)×C97C2213yescomplemented
70AGL(1,ℤ/8)×C97C2213yescomplemented
71#135×C97C2213yescomplemented
72C2×D8×C97C2213yescomplemented
73#131×C97C2211yescharacteristic, complemented
74C23×D97—23no—
75#13095512—21no—
76#13095512D411yescharacteristic
77#13095512C2313yes—
78#13095512C2311yescharacteristic
79#13095511—23no—
80#13095511—23no—
81#13095511—23no—
82#13095511—23no—
83#13095511C2313yes—
84#13095511C2313yes—
85#13095511C2313yescomplemented
86#13095510—23no—
87#13095510—26no—
88#13095510—23no—
89#13095510—23no—
90#13095510—23no—
91#13095510—23no—
92#13095510—23no—
93#13095510—23no—
94#13095510D413yes—
95#13095510C2313yes—
96#13095510C2313yes—
97#13095510C2316yescomplemented
98C2×Dic194—23no—
99C2×Dic194C2313yescomplemented
100Q8×D97—21no—
101Q8×D97D411yescharacteristic
102Q8×D97C2311yescharacteristic
103Q8×D97C2313yescomplemented
104C2×D388—23no—
105C2×D388C2313yes—
106C2×C22⋊D97—23no—
107C2×C22⋊D97—26no—
108C2×C22⋊D97—26no—
109C2×C22⋊D97—23no—
110D4×D97—23no—
111D4×D97—23no—
112D4×D97—23no—
113D4×D97—26no—
114D4×D97—23no—
115D4×D97—23no—
116D4×D97—23no—
117D4×D97—23no—
118D4×D97D413yes—
119D4×D97C2316yes—
120D4×D97C2313yes—
121D4×D97C2313yescomplemented
122C22×Dic97—23no—
123C2×C4×D97—23no—
124C2×C4×D97—23no—
125C2×C4×D97C2313yes—
126Dic388C2313yescomplemented
127#13095501C2311yescharacteristic, complemented
128#13095501C2313yescomplemented
129#13095500C2316yescomplemented
130#13095500C2313yescomplemented
131#13095500C2311yescharacteristic, complemented
132#13095499C2316yescomplemented
133#13095499C2313yescomplemented
134#13095499C2313yescomplemented
135#13095498C2313yescomplemented
136#13095498C2311yescharacteristic, complemented
137D776C2313yescomplemented
138#13095496C2313yescomplemented
139#13095496C2316yescomplemented
140#13095496C2313yescomplemented
141#13095495C2313yescomplemented
142#13095494C2313yes—
143#13095494C2311yescharacteristic
144#13095494C2313yescomplemented
145#13095494C2313yescomplemented
146C2×#11715895C2313yescomplemented
147C8×D97C2316yescomplemented
148#41×C97—21no—
149#41×C97—21no—
150#41×C97C2313yes—
151#41×C97C2311yescharacteristic
152#41×C97C2313yescomplemented
153C2×D4×C97—23no—
154C2×D4×C97—23no—
155C2×D4×C97C2313yes—
156C2×D4×C97C2313yescomplemented
157Q16×C97C2311yescharacteristic, complemented
158SD16×C97C2313yescomplemented
159SD16×C97C2313yescomplemented
160D8×C97C2316yescomplemented
161D8×C97C2313yescomplemented
162M16×C97C2313yescomplemented
163C2×C8×C97C2313yescomplemented
164C23×C97—23no—
165C23×C97—23no—
166C22×D97—23no—
167C22×D97—23no—
168C22×D97—23no—
169C22×D97—26no—
170C22×D97—86no—
171C22×D97C2×D413yes—
172Q8×C97C2×D411yescharacteristic
173Q8×C97C2411yescharacteristic
174Q8×C97C2411yescharacteristic
175D4×C97—23no—
176D4×C97—23no—
177D4×C97—23no—
178D4×C97—23no—
179D4×C97C2×D413yes—
180D4×C97C2413yes—
181D4×C97C2413yes—
182C2×C4×C97—23no—
183C2×C4×C97—23no—
184C2×C4×C97C2413yes—
185C22⋊D97—23no—
186C22⋊D97—23no—
187C22⋊D97—23no—
188C22⋊D97—23no—
189C22⋊D97—26no—
190C22⋊D97—26no—
191C22⋊D97—23no—
192C22⋊D97—23no—
193C22⋊D97C2×D416yes—
194C2×Dic97—23no—
195C2×Dic97—23no—
196C2×Dic97—23no—
197C2×Dic97—26no—
198C2×Dic97C2×D413yes—
199D388—23no—
200D388—21no—
201D388—23no—
202D388C2×D413yes—
203D388C2411yescharacteristic
204D388C2413yes—
205C4×D97—21no—
206C4×D97—23no—
207C4×D97—23no—
208C4×D97—23no—
209C4×D97—21no—
210C4×D97—23no—
211C4×D97C2×D413yes—
212C4×D97C2×D413yes—
213C4×D97C2413yes—
214C4×D97C2411yescharacteristic
215Dic194—23no—
216Dic194—21no—
217Dic194—23no—
218Dic194C2×D413yes—
219Dic194C2413yes—
220Dic194C2411yescharacteristic
221C8×C97C2411yescharacteristic
222C8×C97C2413yes—
223#11715895C2413yes—
224#11715895C2411yescharacteristic
225C22×C97—23no—
226C22×C97—23no—
227C22×C97—83no—
228C22×C97—83no—
229C22×C97C22×D413yes—
230C2×D97—21no—
231C2×D97—23no—
232C2×D97—43no—
233C2×D97—46no—
234C2×D97—83no—
235C2×D97—86no—
236C2×D97—83no—
237C2×D97C22×D411yescharacteristic
238C2×D97C22×D413yes—
239C4×C97—21no—
240C4×C97—21no—
241C4×C97C22×D413yes—
242C4×C97C2511yescharacteristic
243Dic97—23no—
244Dic97—21no—
245Dic97C22×D413yes—
246Dic97C22×D411yescharacteristic
247C2×C97—23no—
248C2×C97—43no—
249C2×C97—43no—
250C2×C97C23×D411yescharacteristic
251D97—21no—
252D97—23no—
253D97—43no—
254D97—41no—
255#3585—971nomaximal, Sylow
256C97#358511yescharacteristic, complemented, Sylow
257#585—971no—
258C2×H(5,𝔽2)—971no—
259#577—971no—
260#577—973no—
261#577—973no—
262#577—971no—
263#576—971no—
264#576—973no—
265#576—973no—
266#576D9711yescharacteristic, complemented
267#575—973no—
268#575—973no—
269C2×AGL(1,ℤ/8)—973no—
270C2×AGL(1,ℤ/8)—973no—
271#568—971no—
2722−1+4—971no—
2732−1+4—973no—
2742−1+4—1941no—
275H(5,𝔽2)—973no—
276H(5,𝔽2)—971no—
277H(5,𝔽2)—973no—
278H(5,𝔽2)—973no—
279H(5,𝔽2)—1943no—
280H(5,𝔽2)—1943no—
281H(5,𝔽2)—1941no—
282H(5,𝔽2)C2×D9711yescharacteristic, complemented
283H(5,𝔽2)C2×D9711yescharacteristic, complemented
284C2×#41—971no—
285C2×#41—973no—
286C2×#41—973no—
287C2×#41—973no—
288C2×#41—1943no—
289C2×#41—1941no—
290C2×#41—1941no—
291C22×D4—973no—
292C22×D4—1943no—
293#137—973no—
294#137—971no—
295#137—973no—
296#137—973no—
297#137—973no—
298#137—973no—
299AGL(1,ℤ/8)—973no—
300AGL(1,ℤ/8)—973no—
301AGL(1,ℤ/8)—973no—
302AGL(1,ℤ/8)—973no—
303AGL(1,ℤ/8)—973no—
304AGL(1,ℤ/8)—976no—
305AGL(1,ℤ/8)—973no—
306AGL(1,ℤ/8)—976no—
307AGL(1,ℤ/8)—976no—
308AGL(1,ℤ/8)—973no—
309AGL(1,ℤ/8)—973no—
310AGL(1,ℤ/8)C2×D9713yescomplemented
311AGL(1,ℤ/8)C2×D9713yescomplemented
312#135—973no—
313#135—973no—
314#135—976no—
315#135—976no—
316#135—973no—
317#135C2×D9713yescomplemented
318C2×SD16—973no—
319C2×SD16—976no—
320C2×SD16—973no—
321C2×D8—973no—
322C2×D8—976no—
323C2×D8C2×D9713yescomplemented
324#131—971no—
325#131—973no—
326#131—973no—
327#131C2×D9711yescharacteristic, complemented
328C2×M16—973no—
329C2×M16—973no—
330C24—1943no—
331#41—21no—
332#41—21no—
333#41—971no—
334#41—973no—
335#41—973no—
336#41—973no—
337#41—971no—
338#41—973no—
339#41—973no—
340#41—976no—
341#41—973no—
342#41—973no—
343#41—1943no—
344#41—1943no—
345#41—1943no—
346#41—1943no—
347#41—1943no—
348#41—1943no—
349#41—1943no—
350#41—1943no—
351#41—1946no—
352#41—1943no—
353#41—1943no—
354#41—1941no—
355#41—1943no—
356#41C22×D9713yes—
357#41C22×D9711yescharacteristic
358#41C22×D9713yescomplemented
359C2×Q8—971no—
360C2×Q8—971no—
361C2×Q8—973no—
362C2×Q8—973no—
363C2×Q8—1943no—
364C2×Q8—1941no—
365C2×D4—23no—
366C2×D4—23no—
367C2×D4—973no—
368C2×D4—976no—
369C2×D4—973no—
370C2×D4—973no—
371C2×D4—973no—
372C2×D4—1943no—
373C2×D4—1943no—
374C2×D4—1943no—
375C2×D4—1943no—
376C2×D4—1946no—
377C2×D4—1943no—
378C2×D4—1943no—
379C2×D4—1943no—
380C2×D4—1943no—
381C2×D4—1946no—
382C2×D4—1946no—
383C2×D4—1943no—
384C2×D4—1943no—
385C2×D4C22×D9713yes—
386C2×D4C22×D9713yescomplemented
387C22×C4—973no—
388C22×C4—1943no—
389C22×C4—1943no—
390C22×C4—1943no—
391Q16—973no—
392Q16—971no—
393Q16—973no—
394Q16C22×D9711yescharacteristic, complemented
395SD16—973no—
396SD16—971no—
397SD16—976no—
398SD16—973no—
399SD16—973no—
400SD16—976no—
401SD16—973no—
402SD16—971no—
403SD16C22×D9713yescomplemented
404SD16C22×D9713yescomplemented
405D8—973no—
406D8—976no—
407D8—973no—
408D8—973no—
409D8C22×D9716yescomplemented
410D8C22×D9713yescomplemented
411M16—973no—
412M16—973no—
413M16—971no—
414M16—973no—
415M16—973no—
416M16C22×D9713yescomplemented
417C2×C8—973no—
418C2×C8—976no—
419C2×C8C22×D9713yescomplemented
420C23—23no—
421C23—23no—
422C23—7766no—
423C23—973no—
424C23—1943no—
425C23—1946no—
426C23—1943no—
427C23—1943no—
428Q8—973no—
429Q8—973no—
430Q8—971no—
431Q8—1943no—
432Q8—1941no—
433Q8—1943no—
434Q8D4×D9711yescharacteristic
435Q8C23×D9711yescharacteristic
436Q8C23×D9711yescharacteristic
437D4—23no—
438D4—23no—
439D4—23no—
440D4—23no—
441D4—976no—
442D4—973no—
443D4—971no—
444D4—973no—
445D4—1943no—
446D4—1941no—
447D4—1943no—
448D4—1946no—
449D4—1943no—
450D4—1943no—
451D4—1943no—
452D4—1946no—
453D4—1943no—
454D4—1943no—
455D4—1943no—
456D4D4×D9713yes—
457D4C23×D9713yes—
458D4C23×D9713yes—
459C2×C4—23no—
460C2×C4—23no—
461C2×C4—973no—
462C2×C4—973no—
463C2×C4—973no—
464C2×C4—973no—
465C2×C4—971no—
466C2×C4—1941no—
467C2×C4—1943no—
468C2×C4—1941no—
469C2×C4—1943no—
470C2×C4—1943no—
471C2×C4—1943no—
472C2×C4—1946no—
473C2×C4—1943no—
474C2×C4—1943no—
475C2×C4—1943no—
476C2×C4C23×D9713yes—
477C8—971no—
478C8—973no—
479C8C23×D9713yes—
480C8C23×D9711yescharacteristic
481C22—23no—
482C22—23no—
483C22—83no—
484C22—83no—
485C22—7763no—
486C22—7766no—
487C22—7763no—
488C22—971no—
489C22—973no—
490C22—3883no—
491C22—3886no—
492C22—1941no—
493C22—1943no—
494C22C2×D4×D9713yes—
495C4—21no—
496C4—21no—
497C4—971no—
498C4—973no—
499C4—1941no—
500C4—1943no—
501C4C2×D4×D9713yes—
502C4C24×D9711yescharacteristic
503C2—23no—
504C2—43no—
505C2—43no—
506C2—3883no—
507C2—3881no—
508C2—1943no—
509C2—1941no—
510C2C22×D4×D9711yescharacteristic
subgroupG/H
centerC2C22×D4×D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
Fitting#576×C97C2
socleC2×C97C23×D4
hypercenter#576D97
hypocenterC97#3585
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ #41 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #20261570C2semidirect
2. H(5,𝔽2)×D97C2semidirect
3. #20261547C2semidirect
4. #20261546C2semidirect
5. #20261545C2semidirect
6. #20261542C2semidirect
7. #20261539C2semidirect
8. #20261538C2semidirect
9. #20261536C2semidirect
10. #20261532C2semidirect
11. #20261531C2semidirect
12. C2×#15690834C2semidirect
13. AGL(1,ℤ/8)×D97C2semidirect
14. #20261500C2semidirect
15. #576×C97C2semidirect
17. #15690857C22semidirect
18. #15690857C22semidirect
20. #15690856C22semidirect
21. #15690856C22semidirect
24. #15690855C22semidirect
25. #15690855C22semidirect
27. C2×#13095510C22semidirect
31. #41×D97C22semidirect
32. #41×D97C22semidirect
34. C2×D4×D97C22semidirect
35. #15690843C22semidirect
36. #15690842C22semidirect
37. #15690842C22semidirect
38. #15690840C22semidirect
39. #15690839C22semidirect
40. #15690839C22semidirect
41. #15690838C22semidirect
42. #15690837C22semidirect
43. #15690837C22semidirect
44. #15690836C22semidirect
45. #15690836C22semidirect
46. #15690835C22semidirect
47. #15690834C22semidirect
48. #15690834C22semidirect
49. #15690834C22semidirect
50. #15690834C22semidirect
51. #15690834C22semidirect
52. #15690832C22semidirect
53. #15690831C22semidirect
54. #15690830C22semidirect
55. #15690830C22semidirect
56. #15690829C22semidirect
57. C2×#13095500C22semidirect
58. C2×#13095499C22semidirect
59. SD16×D97C22semidirect
60. C2×#13095496C22semidirect
61. D8×D97C22semidirect
62. #15690818C22semidirect
63. #15690817C22semidirect
64. #15690816C22semidirect
65. C2×#13095494C22semidirect
66. M16×D97C22semidirect
67. H(5,𝔽2)×C97C22semidirect
68. H(5,𝔽2)×C97C22semidirect
69. AGL(1,ℤ/8)×C97C22semidirect
70. AGL(1,ℤ/8)×C97C22semidirect
71. #135×C97C22semidirect
72. C2×D8×C97C22semidirect
73. #131×C97C22semidirect
85. #13095511C23semidirect
97. #13095510C23semidirect
99. C2×Dic194C23semidirect
103. Q8×D97C23semidirect
121. D4×D97C23semidirect
126. Dic388C23semidirect
127. #13095501C23semidirect
128. #13095501C23semidirect
129. #13095500C23semidirect
130. #13095500C23semidirect
131. #13095500C23semidirect
132. #13095499C23semidirect
133. #13095499C23semidirect
134. #13095499C23semidirect
135. #13095498C23semidirect
136. #13095498C23semidirect
137. D776C23semidirect
138. #13095496C23semidirect
139. #13095496C23semidirect
140. #13095496C23semidirect
141. #13095495C23semidirect
144. #13095494C23semidirect
145. #13095494C23semidirect
146. C2×#11715895C23semidirect
147. C8×D97C23semidirect
152. #41×C97C23semidirect
156. C2×D4×C97C23semidirect
157. Q16×C97C23semidirect
158. SD16×C97C23semidirect
159. SD16×C97C23semidirect
160. D8×C97C23semidirect
161. D8×C97C23semidirect
162. M16×C97C23semidirect
163. C2×C8×C97C23semidirect
256. C97#3585semidirect
266. #576D97semidirect
282. H(5,𝔽2)C2×D97semidirect
283. H(5,𝔽2)C2×D97semidirect
310. AGL(1,ℤ/8)C2×D97semidirect
311. AGL(1,ℤ/8)C2×D97semidirect
317. #135C2×D97semidirect
323. C2×D8C2×D97semidirect
327. #131C2×D97semidirect
358. #41C22×D97semidirect
386. C2×D4C22×D97semidirect
394. Q16C22×D97semidirect
403. SD16C22×D97semidirect
404. SD16C22×D97semidirect
409. D8C22×D97semidirect
410. D8C22×D97semidirect
416. M16C22×D97semidirect
419. C2×C8C22×D97semidirect

Maximal subgroups

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

subgroupindex
#202615702
H(5,𝔽2)×D972
#202615472
#202615462
#202615452
#202615422
#202615392
#202615382
#202615362
#202615322
#202615312
C2×#156908342
AGL(1,ℤ/8)×D972
#202615002
#576×C972
#358597