Finite Group Databasebeta

← #27156514

Subgroups — #27156514

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#20261572C211yesmaximal, characteristic, complemented
2H(5,𝔽2)×D97C211yesmaximal, characteristic, complemented
3#20261544C213yesmaximal, complemented
4#20261543C211yesmaximal, characteristic, complemented
5#20261541C213yesmaximal, complemented
6#20261540C211yesmaximal, characteristic, complemented
7#20261539C213yesmaximal, complemented
8#20261538C211yesmaximal, characteristic, complemented
9#20261537C213yesmaximal, complemented
10#20261534C213yesmaximal, complemented
11#20261532C213yesmaximal, complemented
12C2×#15690837C213yesmaximal, complemented
13AGL(1,ℤ/8)×D97C213yesmaximal, complemented
14#20261500C211yesmaximal, characteristic, complemented
15#577×C97C211yesmaximal, characteristic, complemented
16#15690858—21no—
17#15690858C2213yescomplemented
18#15690856—21no—
19#15690856—23no—
20#15690856C2211yescharacteristic, complemented
21#15690856C2211yescharacteristic, complemented
22#15690856C2213yescomplemented
23#15690856C2213yescomplemented
24#15690855—23no—
25#15690855C2213yescomplemented
26C2×#13095512—21no—
27C2×#13095512C2213yescomplemented
28#41×D97—23no—
29#41×D97—21no—
30#41×D97C2211yescharacteristic
31#41×D97C2213yescomplemented
32#41×D97C2213yescomplemented
33C2×D4×D97—23no—
34C2×D4×D97C2213yescomplemented
35#15690842C2213yescomplemented
36#15690842C2213yescomplemented
37#15690841C2213yescomplemented
38#15690839C2211yescharacteristic, complemented
39#15690839C2213yescomplemented
40#15690838C2213yescomplemented
41#15690838C2213yescomplemented
42#15690837C2216yescomplemented
43#15690837C2213yescomplemented
44#15690837C2216yescomplemented
45#15690837C2216yescomplemented
46#15690837C2213yescomplemented
47#15690836C2213yescomplemented
48#15690835C2213yescomplemented
49#15690835C2213yescomplemented
50#15690834C2213yescomplemented
51#15690834C2213yescomplemented
52#15690833C2216yescomplemented
53#15690832C2213yescomplemented
54#15690831C2216yescomplemented
55#15690830C2213yescomplemented
56#15690830C2213yescomplemented
57C2×#13095498C2213yescomplemented
58SD16×D97C2216yescomplemented
59C2×D776C2213yescomplemented
60C2×#13095496C2213yescomplemented
61D8×D97C2216yescomplemented
62#15690818C2213yescomplemented
63#15690817C2211yescharacteristic, complemented
64#15690816C2213yescomplemented
65C2×#13095494C2213yescomplemented
66M16×D97C2213yescomplemented
672−1+4×C97C2211yescharacteristic, complemented
68H(5,𝔽2)×C97C2211yescharacteristic, complemented
69#137×C97C2213yescomplemented
70AGL(1,ℤ/8)×C97C2213yescomplemented
71#135×C97C2213yescomplemented
72C2×SD16×C97C2213yescomplemented
73#131×C97C2211yescharacteristic, complemented
74C23×D97—23no—
75#13095512—21no—
76#13095512—21no—
77#13095512—23no—
78#13095512—23no—
79#13095512—23no—
80#13095512D411yescharacteristic
81#13095512C2313yes—
82#13095512C2313yes—
83#13095512C2316yescomplemented
84#13095511—26no—
85#13095511—23no—
86#13095511—23no—
87#13095511—23no—
88#13095511—23no—
89#13095511C2313yes—
90#13095511C2313yes—
91#13095511C2313yescomplemented
92#13095510—23no—
93#13095510—23no—
94#13095510—23no—
95#13095510D413yes—
96#13095510C2313yes—
97#13095510C2311yescharacteristic
98Q8×D97—23no—
99Q8×D97D411yescharacteristic
100Q8×D97C2311yescharacteristic
101Q8×D97C2313yescomplemented
102C2×D388—23no—
103C2×D388—23no—
104C2×D388—23no—
105C2×D388C2313yes—
106C2×D388C2313yescomplemented
107C2×C22⋊D97—26no—
108C2×C22⋊D97—23no—
109D4×D97—23no—
110D4×D97—23no—
111D4×D97—23no—
112D4×D97—23no—
113D4×D97—26no—
114D4×D97—26no—
115D4×D97—23no—
116D4×D97—23no—
117D4×D97—23no—
118D4×D97D413yes—
119D4×D97C2316yes—
120D4×D97C2313yes—
121D4×D97C2313yescomplemented
122C2×C4×D97—23no—
123C2×C4×D97—23no—
124C2×C4×D97—23no—
125C2×C4×D97C2313yes—
126Dic388C2311yescharacteristic, complemented
127#13095501C2311yescharacteristic, complemented
128#13095501C2313yescomplemented
129#13095500C2313yescomplemented
130#13095500C2313yescomplemented
131#13095499C2313yescomplemented
132#13095499C2311yescharacteristic, complemented
133#13095498C2316yescomplemented
134#13095498C2313yescomplemented
135#13095498C2313yescomplemented
136D776C2316yescomplemented
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—23no—
150#41×C97C2313yes—
151#41×C97C2311yescharacteristic
152#41×C97C2313yescomplemented
153C2×Q8×C97—21no—
154C2×Q8×C97C2313yescomplemented
155C2×D4×C97—23no—
156C2×D4×C97C2313yes—
157Q16×C97C2313yescomplemented
158SD16×C97C2316yescomplemented
159SD16×C97C2313yescomplemented
160SD16×C97C2311yescharacteristic, complemented
161D8×C97C2313yescomplemented
162M16×C97C2313yescomplemented
163C2×C8×C97C2313yescomplemented
164C23×C97—23no—
165C22×D97—23no—
166C22×D97—23no—
167C22×D97—23no—
168C22×D97—23no—
169C22×D97—26no—
170C22×D97—86no—
171C22×D97C2×D413yes—
172Q8×C97—23no—
173Q8×C97C2×D411yescharacteristic
174Q8×C97C2413yes—
175Q8×C97C2411yescharacteristic
176D4×C97—23no—
177D4×C97—23no—
178D4×C97—23no—
179D4×C97C2×D413yes—
180D4×C97C2411yescharacteristic
181D4×C97C2413yes—
182C2×C4×C97—23no—
183C2×C4×C97—23no—
184C2×C4×C97—23no—
185C2×C4×C97C2413yes—
186C22⋊D97—23no—
187C22⋊D97—23no—
188C22⋊D97—23no—
189C22⋊D97—21no—
190C22⋊D97—26no—
191C22⋊D97—23no—
192C22⋊D97—23no—
193C22⋊D97C2×D416yes—
194C2×Dic97—21no—
195C2×Dic97—23no—
196C2×Dic97C2×D413yes—
197D388—23no—
198D388—23no—
199D388—26no—
200D388—23no—
201D388—23no—
202D388C2×D413yes—
203D388C2413yes—
204D388C2413yes—
205C4×D97—21no—
206C4×D97—23no—
207C4×D97—23no—
208C4×D97—23no—
209C4×D97—23no—
210C4×D97—23no—
211C4×D97—26no—
212C4×D97C2×D413yes—
213C4×D97C2×D413yes—
214C4×D97C2413yes—
215C4×D97C2411yescharacteristic
216Dic194—23no—
217Dic194—21no—
218Dic194C2×D413yes—
219Dic194C2411yescharacteristic
220Dic194C2411yescharacteristic
221C8×C97C2411yescharacteristic
222C8×C97C2413yes—
223#11715895C2413yes—
224#11715895C2411yescharacteristic
225C22×C97—21no—
226C22×C97—23no—
227C22×C97—83no—
228C22×C97C22×D413yes—
229C2×D97—23no—
230C2×D97—23no—
231C2×D97—43no—
232C2×D97—46no—
233C2×D97—83no—
234C2×D97—86no—
235C2×D97—83no—
236C2×D97—83no—
237C2×D97C22×D411yescharacteristic
238C2×D97C22×D413yes—
239C4×C97—21no—
240C4×C97—23no—
241C4×C97C22×D413yes—
242C4×C97C2511yescharacteristic
243Dic97—21no—
244Dic97—21no—
245Dic97C22×D413yes—
246Dic97C22×D411yescharacteristic
247C2×C97—23no—
248C2×C97—43no—
249C2×C97—41no—
250C2×C97C23×D411yescharacteristic
251D97—21no—
252D97—23no—
253D97—43no—
254D97—43no—
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#577D9711yescharacteristic, complemented
263#576—971no—
264#576—973no—
265#576—973no—
266#576—971no—
267#575—973no—
268#575—973no—
269C2×AGL(1,ℤ/8)—973no—
270C2×AGL(1,ℤ/8)—973no—
271#568—971no—
2722−1+4—973no—
2732−1+4—1941no—
2742−1+4C2×D9711yescharacteristic, complemented
275H(5,𝔽2)—973no—
276H(5,𝔽2)—971no—
277H(5,𝔽2)—971no—
278H(5,𝔽2)—973no—
279H(5,𝔽2)—973no—
280H(5,𝔽2)—1943no—
281H(5,𝔽2)—1943no—
282H(5,𝔽2)—1941no—
283H(5,𝔽2)C2×D9711yescharacteristic, complemented
284C2×#41—971no—
285C2×#41—973no—
286C2×#41—973no—
287C2×#41—973no—
288C2×#41—1941no—
289C2×#41—1941no—
290C2×#41—1943no—
291C22×D4—973no—
292C22×D4—1943no—
293#137—973no—
294#137—973no—
295#137—973no—
296#137—971no—
297#137—973no—
298#137C2×D9713yescomplemented
299AGL(1,ℤ/8)—973no—
300AGL(1,ℤ/8)—973no—
301AGL(1,ℤ/8)—973no—
302AGL(1,ℤ/8)—976no—
303AGL(1,ℤ/8)—973no—
304AGL(1,ℤ/8)—976no—
305AGL(1,ℤ/8)—976no—
306AGL(1,ℤ/8)—973no—
307AGL(1,ℤ/8)—973no—
308AGL(1,ℤ/8)—973no—
309AGL(1,ℤ/8)—973no—
310AGL(1,ℤ/8)—973no—
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×SD16C2×D9713yescomplemented
321C2×D8—973no—
322C2×D8—976no—
323C2×D8—973no—
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—23no—
333#41—971no—
334#41—973no—
335#41—973no—
336#41—973no—
337#41—973no—
338#41—976no—
339#41—973no—
340#41—971no—
341#41—973no—
342#41—973no—
343#41—1943no—
344#41—1943no—
345#41—1946no—
346#41—1943no—
347#41—1943no—
348#41—1943no—
349#41—1943no—
350#41—1943no—
351#41—1943no—
352#41—1943no—
353#41—1943no—
354#41—1941no—
355#41—1941no—
356#41C22×D9713yes—
357#41C22×D9711yescharacteristic
358#41C22×D9713yescomplemented
359C2×Q8—21no—
360C2×Q8—971no—
361C2×Q8—971no—
362C2×Q8—973no—
363C2×Q8—1943no—
364C2×Q8C22×D9713yescomplemented
365C2×D4—23no—
366C2×D4—973no—
367C2×D4—976no—
368C2×D4—973no—
369C2×D4—973no—
370C2×D4—973no—
371C2×D4—973no—
372C2×D4—1943no—
373C2×D4—1943no—
374C2×D4—1943no—
375C2×D4—1943no—
376C2×D4—1943no—
377C2×D4—1943no—
378C2×D4—1946no—
379C2×D4—1943no—
380C2×D4—1946no—
381C2×D4—1943no—
382C2×D4—1943no—
383C2×D4—1946no—
384C2×D4—1943no—
385C2×D4—1943no—
386C2×D4C22×D9713yes—
387C22×C4—973no—
388C22×C4—1943no—
389C22×C4—1943no—
390C22×C4—1943no—
391Q16—971no—
392Q16—973no—
393Q16—971no—
394Q16C22×D9713yescomplemented
395SD16—976no—
396SD16—973no—
397SD16—973no—
398SD16—973no—
399SD16—971no—
400SD16—973no—
401SD16—973no—
402SD16C22×D9716yescomplemented
403SD16C22×D9713yescomplemented
404SD16C22×D9711yescharacteristic, complemented
405D8—976no—
406D8—973no—
407D8—973no—
408D8—976no—
409D8—973no—
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—7766no—
422C23—973no—
423C23—1943no—
424C23—1946no—
425C23—1943no—
426C23—1943no—
427C23—1943no—
428Q8—23no—
429Q8—973no—
430Q8—971no—
431Q8—971no—
432Q8—1943no—
433Q8—1941no—
434Q8D4×D9711yescharacteristic
435Q8C23×D9713yes—
436Q8C23×D9711yescharacteristic
437D4—23no—
438D4—23no—
439D4—23no—
440D4—976no—
441D4—973no—
442D4—973no—
443D4—973no—
444D4—1943no—
445D4—1943no—
446D4—1946no—
447D4—1943no—
448D4—1946no—
449D4—1943no—
450D4—1943no—
451D4—1941no—
452D4—1943no—
453D4—1943no—
454D4—1943no—
455D4—1943no—
456D4D4×D9713yes—
457D4C23×D9711yescharacteristic
458D4C23×D9713yes—
459C2×C4—23no—
460C2×C4—23no—
461C2×C4—23no—
462C2×C4—973no—
463C2×C4—973no—
464C2×C4—973no—
465C2×C4—973no—
466C2×C4—971no—
467C2×C4—1943no—
468C2×C4—1943no—
469C2×C4—1946no—
470C2×C4—1941no—
471C2×C4—1943no—
472C2×C4—1941no—
473C2×C4—1943no—
474C2×C4—1943no—
475C2×C4—1943no—
476C2×C4C23×D9713yes—
477C8—971no—
478C8—973no—
479C8C23×D9713yes—
480C8C23×D9711yescharacteristic
481C22—21no—
482C22—23no—
483C22—83no—
484C22—7763no—
485C22—7763no—
486C22—7766no—
487C22—7763no—
488C22—971no—
489C22—973no—
490C22—3883no—
491C22—3886no—
492C22—1943no—
493C22—1943no—
494C22C2×D4×D9713yes—
495C4—21no—
496C4—23no—
497C4—971no—
498C4—973no—
499C4—1941no—
500C4—1941no—
501C4C2×D4×D9713yes—
502C4C24×D9711yescharacteristic
503C2—23no—
504C2—43no—
505C2—41no—
506C2—3883no—
507C2—3883no—
508C2—1943no—
509C2—1941no—
510C2C22×D4×D9711yescharacteristic
subgroupG/H
centerC2C22×D4×D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
Fitting#577×C97C2
socleC2×C97C23×D4
hypercenter#577D97
hypocenterC97#3585
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ #41 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #20261572C2semidirect
2. H(5,𝔽2)×D97C2semidirect
3. #20261544C2semidirect
4. #20261543C2semidirect
5. #20261541C2semidirect
6. #20261540C2semidirect
7. #20261539C2semidirect
8. #20261538C2semidirect
9. #20261537C2semidirect
10. #20261534C2semidirect
11. #20261532C2semidirect
12. C2×#15690837C2semidirect
13. AGL(1,ℤ/8)×D97C2semidirect
14. #20261500C2semidirect
15. #577×C97C2semidirect
17. #15690858C22semidirect
20. #15690856C22semidirect
21. #15690856C22semidirect
22. #15690856C22semidirect
23. #15690856C22semidirect
25. #15690855C22semidirect
27. C2×#13095512C22semidirect
31. #41×D97C22semidirect
32. #41×D97C22semidirect
34. C2×D4×D97C22semidirect
35. #15690842C22semidirect
36. #15690842C22semidirect
37. #15690841C22semidirect
38. #15690839C22semidirect
39. #15690839C22semidirect
40. #15690838C22semidirect
41. #15690838C22semidirect
42. #15690837C22semidirect
43. #15690837C22semidirect
44. #15690837C22semidirect
45. #15690837C22semidirect
46. #15690837C22semidirect
47. #15690836C22semidirect
48. #15690835C22semidirect
49. #15690835C22semidirect
50. #15690834C22semidirect
51. #15690834C22semidirect
52. #15690833C22semidirect
53. #15690832C22semidirect
54. #15690831C22semidirect
55. #15690830C22semidirect
56. #15690830C22semidirect
57. C2×#13095498C22semidirect
58. SD16×D97C22semidirect
59. C2×D776C22semidirect
60. C2×#13095496C22semidirect
61. D8×D97C22semidirect
62. #15690818C22semidirect
63. #15690817C22semidirect
64. #15690816C22semidirect
65. C2×#13095494C22semidirect
66. M16×D97C22semidirect
67. 2−1+4×C97C22semidirect
68. H(5,𝔽2)×C97C22semidirect
69. #137×C97C22semidirect
70. AGL(1,ℤ/8)×C97C22semidirect
71. #135×C97C22semidirect
72. C2×SD16×C97C22semidirect
73. #131×C97C22semidirect
83. #13095512C23semidirect
91. #13095511C23semidirect
101. Q8×D97C23semidirect
106. C2×D388C23semidirect
121. D4×D97C23semidirect
126. Dic388C23semidirect
127. #13095501C23semidirect
128. #13095501C23semidirect
129. #13095500C23semidirect
130. #13095500C23semidirect
131. #13095499C23semidirect
132. #13095499C23semidirect
133. #13095498C23semidirect
134. #13095498C23semidirect
135. #13095498C23semidirect
136. D776C23semidirect
137. D776C23semidirect
138. #13095496C23semidirect
139. #13095496C23semidirect
140. #13095496C23semidirect
141. #13095495C23semidirect
144. #13095494C23semidirect
145. #13095494C23semidirect
146. C2×#11715895C23semidirect
147. C8×D97C23semidirect
152. #41×C97C23semidirect
154. C2×Q8×C97C23semidirect
157. Q16×C97C23semidirect
158. SD16×C97C23semidirect
159. SD16×C97C23semidirect
160. SD16×C97C23semidirect
161. D8×C97C23semidirect
162. M16×C97C23semidirect
163. C2×C8×C97C23semidirect
256. C97#3585semidirect
262. #577D97semidirect
274. 2−1+4C2×D97semidirect
283. H(5,𝔽2)C2×D97semidirect
298. #137C2×D97semidirect
311. AGL(1,ℤ/8)C2×D97semidirect
317. #135C2×D97semidirect
320. C2×SD16C2×D97semidirect
327. #131C2×D97semidirect
358. #41C22×D97semidirect
364. C2×Q8C22×D97semidirect
394. Q16C22×D97semidirect
402. SD16C22×D97semidirect
403. SD16C22×D97semidirect
404. SD16C22×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
#202615722
H(5,𝔽2)×D972
#202615442
#202615432
#202615412
#202615402
#202615392
#202615382
#202615372
#202615342
#202615322
C2×#156908372
AGL(1,ℤ/8)×D972
#202615002
#577×C972
#358597