Finite Group Databasebeta

← #27185910

Subgroups — #27185910

Order 12928.

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#20266014C211yesmaximal, characteristic, complemented
2H(5,𝔽2)×D101C211yesmaximal, characteristic, complemented
3#20265986C213yesmaximal, complemented
4#20265985C211yesmaximal, characteristic, complemented
5#20265983C213yesmaximal, complemented
6#20265982C211yesmaximal, characteristic, complemented
7#20265981C213yesmaximal, complemented
8#20265980C211yesmaximal, characteristic, complemented
9#20265979C213yesmaximal, complemented
10#20265976C213yesmaximal, complemented
11#20265974C213yesmaximal, complemented
12C2×#15692282C213yesmaximal, complemented
13AGL(1,ℤ/8)×D101C213yesmaximal, complemented
14#20265942C211yesmaximal, characteristic, complemented
15#577×C101C211yesmaximal, characteristic, complemented
16#15692303—21no—
17#15692303C2213yescomplemented
18#15692301—21no—
19#15692301—23no—
20#15692301C2211yescharacteristic, complemented
21#15692301C2211yescharacteristic, complemented
22#15692301C2213yescomplemented
23#15692301C2213yescomplemented
24#15692300—23no—
25#15692300C2213yescomplemented
26C2×#13108075—21no—
27C2×#13108075C2213yescomplemented
28#41×D101—23no—
29#41×D101—21no—
30#41×D101C2211yescharacteristic
31#41×D101C2213yescomplemented
32#41×D101C2213yescomplemented
33C2×D4×D101—23no—
34C2×D4×D101C2213yescomplemented
35#15692287C2213yescomplemented
36#15692287C2213yescomplemented
37#15692286C2213yescomplemented
38#15692284C2211yescharacteristic, complemented
39#15692284C2213yescomplemented
40#15692283C2213yescomplemented
41#15692283C2213yescomplemented
42#15692282C2216yescomplemented
43#15692282C2213yescomplemented
44#15692282C2216yescomplemented
45#15692282C2216yescomplemented
46#15692282C2213yescomplemented
47#15692281C2213yescomplemented
48#15692280C2213yescomplemented
49#15692280C2213yescomplemented
50#15692279C2213yescomplemented
51#15692279C2213yescomplemented
52#15692278C2216yescomplemented
53#15692277C2213yescomplemented
54#15692276C2216yescomplemented
55#15692275C2213yescomplemented
56#15692275C2213yescomplemented
57C2×#13108061C2213yescomplemented
58SD16×D101C2216yescomplemented
59C2×D808C2213yescomplemented
60C2×#13108059C2213yescomplemented
61D8×D101C2216yescomplemented
62#15692263C2213yescomplemented
63#15692262C2211yescharacteristic, complemented
64#15692261C2213yescomplemented
65C2×#13108057C2213yescomplemented
66M16×D101C2213yescomplemented
672−1+4×C101C2211yescharacteristic, complemented
68H(5,𝔽2)×C101C2211yescharacteristic, complemented
69#137×C101C2213yescomplemented
70AGL(1,ℤ/8)×C101C2213yescomplemented
71#135×C101C2213yescomplemented
72C2×SD16×C101C2213yescomplemented
73#131×C101C2211yescharacteristic, complemented
74C23×D101—23no—
75#13108075—21no—
76#13108075—21no—
77#13108075—23no—
78#13108075—23no—
79#13108075—23no—
80#13108075D411yescharacteristic
81#13108075C2313yes—
82#13108075C2313yes—
83#13108075C2316yescomplemented
84#13108074—26no—
85#13108074—23no—
86#13108074—23no—
87#13108074—23no—
88#13108074—23no—
89#13108074C2313yes—
90#13108074C2313yes—
91#13108074C2313yescomplemented
92#13108073—23no—
93#13108073—23no—
94#13108073—23no—
95#13108073D413yes—
96#13108073C2311yescharacteristic
97#13108073C2313yes—
98Q8×D101—23no—
99Q8×D101D411yescharacteristic
100Q8×D101C2311yescharacteristic
101Q8×D101C2313yescomplemented
102C2×D404—23no—
103C2×D404—23no—
104C2×D404—23no—
105C2×D404C2313yes—
106C2×D404C2313yescomplemented
107C2×C22⋊D101—26no—
108C2×C22⋊D101—23no—
109D4×D101—23no—
110D4×D101—23no—
111D4×D101—23no—
112D4×D101—23no—
113D4×D101—26no—
114D4×D101—26no—
115D4×D101—23no—
116D4×D101—23no—
117D4×D101—23no—
118D4×D101D413yes—
119D4×D101C2313yes—
120D4×D101C2316yes—
121D4×D101C2313yescomplemented
122C2×C4×D101—23no—
123C2×C4×D101—23no—
124C2×C4×D101—23no—
125C2×C4×D101C2313yes—
126Dic404C2311yescharacteristic, complemented
127#13108064C2311yescharacteristic, complemented
128#13108064C2313yescomplemented
129#13108063C2313yescomplemented
130#13108063C2313yescomplemented
131#13108062C2313yescomplemented
132#13108062C2311yescharacteristic, complemented
133#13108061C2316yescomplemented
134#13108061C2313yescomplemented
135#13108061C2313yescomplemented
136D808C2316yescomplemented
137D808C2313yescomplemented
138#13108059C2313yescomplemented
139#13108059C2316yescomplemented
140#13108059C2313yescomplemented
141#13108058C2313yescomplemented
142#13108057C2313yes—
143#13108057C2311yescharacteristic
144#13108057C2313yescomplemented
145#13108057C2313yescomplemented
146C2×#11717579C2313yescomplemented
147C8×D101C2316yescomplemented
148#41×C101—21no—
149#41×C101—23no—
150#41×C101C2313yes—
151#41×C101C2311yescharacteristic
152#41×C101C2313yescomplemented
153C2×Q8×C101—21no—
154C2×Q8×C101C2313yescomplemented
155C2×D4×C101—23no—
156C2×D4×C101C2313yes—
157Q16×C101C2313yescomplemented
158SD16×C101C2313yescomplemented
159SD16×C101C2316yescomplemented
160SD16×C101C2311yescharacteristic, complemented
161D8×C101C2313yescomplemented
162M16×C101C2313yescomplemented
163C2×C8×C101C2313yescomplemented
164C23×C101—23no—
165C22×D101—23no—
166C22×D101—23no—
167C22×D101—23no—
168C22×D101—23no—
169C22×D101—26no—
170C22×D101—86no—
171C22×D101C2×D413yes—
172Q8×C101—23no—
173Q8×C101C2×D411yescharacteristic
174Q8×C101C2413yes—
175Q8×C101C2411yescharacteristic
176D4×C101—23no—
177D4×C101—23no—
178D4×C101—23no—
179D4×C101C2×D413yes—
180D4×C101C2411yescharacteristic
181D4×C101C2413yes—
182C2×C4×C101—23no—
183C2×C4×C101—23no—
184C2×C4×C101—23no—
185C2×C4×C101C2413yes—
186C22⋊D101—23no—
187C22⋊D101—23no—
188C22⋊D101—23no—
189C22⋊D101—21no—
190C22⋊D101—26no—
191C22⋊D101—23no—
192C22⋊D101—23no—
193C22⋊D101C2×D416yes—
194C2×Dic101—21no—
195C2×Dic101—23no—
196C2×Dic101C2×D413yes—
197D404—23no—
198D404—23no—
199D404—26no—
200D404—23no—
201D404—23no—
202D404C2×D413yes—
203D404C2413yes—
204D404C2413yes—
205C4×D101—21no—
206C4×D101—23no—
207C4×D101—23no—
208C4×D101—23no—
209C4×D101—23no—
210C4×D101—23no—
211C4×D101—26no—
212C4×D101C2×D413yes—
213C4×D101C2×D413yes—
214C4×D101C2413yes—
215C4×D101C2411yescharacteristic
216Dic202—23no—
217Dic202—21no—
218Dic202C2×D413yes—
219Dic202C2411yescharacteristic
220Dic202C2411yescharacteristic
221C8×C101C2411yescharacteristic
222C8×C101C2413yes—
223#11717579C2413yes—
224#11717579C2411yescharacteristic
225C22×C101—21no—
226C22×C101—23no—
227C22×C101—83no—
228C22×C101C22×D413yes—
229C2×D101—23no—
230C2×D101—23no—
231C2×D101—43no—
232C2×D101—46no—
233C2×D101—83no—
234C2×D101—86no—
235C2×D101—83no—
236C2×D101—83no—
237C2×D101C22×D411yescharacteristic
238C2×D101C22×D413yes—
239C4×C101—21no—
240C4×C101—23no—
241C4×C101C22×D413yes—
242C4×C101C2511yescharacteristic
243Dic101—21no—
244Dic101—21no—
245Dic101C22×D413yes—
246Dic101C22×D411yescharacteristic
247C2×C101—23no—
248C2×C101—43no—
249C2×C101—41no—
250C2×C101C23×D411yescharacteristic
251D101—21no—
252D101—23no—
253D101—43no—
254D101—43no—
255#3585—1011nomaximal, Sylow
256C101#358511yescharacteristic, complemented, Sylow
257#585—1011no—
258C2×H(5,𝔽2)—1011no—
259#577—1011no—
260#577—1013no—
261#577—1013no—
262#577D10111yescharacteristic, complemented
263#576—1011no—
264#576—1013no—
265#576—1013no—
266#576—1011no—
267#575—1013no—
268#575—1013no—
269C2×AGL(1,ℤ/8)—1013no—
270C2×AGL(1,ℤ/8)—1013no—
271#568—1011no—
2722−1+4—1013no—
2732−1+4—2021no—
2742−1+4C2×D10111yescharacteristic, complemented
275H(5,𝔽2)—1013no—
276H(5,𝔽2)—1011no—
277H(5,𝔽2)—1011no—
278H(5,𝔽2)—1013no—
279H(5,𝔽2)—1013no—
280H(5,𝔽2)—2023no—
281H(5,𝔽2)—2023no—
282H(5,𝔽2)—2021no—
283H(5,𝔽2)C2×D10111yescharacteristic, complemented
284C2×#41—1011no—
285C2×#41—1013no—
286C2×#41—1013no—
287C2×#41—1013no—
288C2×#41—2021no—
289C2×#41—2021no—
290C2×#41—2023no—
291C22×D4—1013no—
292C22×D4—2023no—
293#137—1013no—
294#137—1013no—
295#137—1013no—
296#137—1011no—
297#137—1013no—
298#137C2×D10113yescomplemented
299AGL(1,ℤ/8)—1013no—
300AGL(1,ℤ/8)—1013no—
301AGL(1,ℤ/8)—1013no—
302AGL(1,ℤ/8)—1016no—
303AGL(1,ℤ/8)—1013no—
304AGL(1,ℤ/8)—1016no—
305AGL(1,ℤ/8)—1016no—
306AGL(1,ℤ/8)—1013no—
307AGL(1,ℤ/8)—1013no—
308AGL(1,ℤ/8)—1013no—
309AGL(1,ℤ/8)—1013no—
310AGL(1,ℤ/8)—1013no—
311AGL(1,ℤ/8)C2×D10113yescomplemented
312#135—1013no—
313#135—1013no—
314#135—1016no—
315#135—1016no—
316#135—1013no—
317#135C2×D10113yescomplemented
318C2×SD16—1013no—
319C2×SD16—1016no—
320C2×SD16C2×D10113yescomplemented
321C2×D8—1013no—
322C2×D8—1016no—
323C2×D8—1013no—
324#131—1011no—
325#131—1013no—
326#131—1013no—
327#131C2×D10111yescharacteristic, complemented
328C2×M16—1013no—
329C2×M16—1013no—
330C24—2023no—
331#41—21no—
332#41—23no—
333#41—1011no—
334#41—1013no—
335#41—1013no—
336#41—1013no—
337#41—1013no—
338#41—1016no—
339#41—1013no—
340#41—1011no—
341#41—1013no—
342#41—1013no—
343#41—2023no—
344#41—2023no—
345#41—2026no—
346#41—2023no—
347#41—2023no—
348#41—2023no—
349#41—2023no—
350#41—2023no—
351#41—2023no—
352#41—2023no—
353#41—2023no—
354#41—2021no—
355#41—2021no—
356#41C22×D10113yes—
357#41C22×D10111yescharacteristic
358#41C22×D10113yescomplemented
359C2×Q8—21no—
360C2×Q8—1011no—
361C2×Q8—1011no—
362C2×Q8—1013no—
363C2×Q8—2023no—
364C2×Q8C22×D10113yescomplemented
365C2×D4—23no—
366C2×D4—1013no—
367C2×D4—1016no—
368C2×D4—1013no—
369C2×D4—1013no—
370C2×D4—1013no—
371C2×D4—1013no—
372C2×D4—2023no—
373C2×D4—2023no—
374C2×D4—2023no—
375C2×D4—2023no—
376C2×D4—2023no—
377C2×D4—2023no—
378C2×D4—2026no—
379C2×D4—2023no—
380C2×D4—2026no—
381C2×D4—2023no—
382C2×D4—2023no—
383C2×D4—2026no—
384C2×D4—2023no—
385C2×D4—2023no—
386C2×D4C22×D10113yes—
387C22×C4—1013no—
388C22×C4—2023no—
389C22×C4—2023no—
390C22×C4—2023no—
391Q16—1011no—
392Q16—1013no—
393Q16—1011no—
394Q16C22×D10113yescomplemented
395SD16—1016no—
396SD16—1013no—
397SD16—1013no—
398SD16—1013no—
399SD16—1011no—
400SD16—1013no—
401SD16—1013no—
402SD16C22×D10116yescomplemented
403SD16C22×D10113yescomplemented
404SD16C22×D10111yescharacteristic, complemented
405D8—1016no—
406D8—1013no—
407D8—1013no—
408D8—1016no—
409D8—1013no—
410D8C22×D10113yescomplemented
411M16—1013no—
412M16—1013no—
413M16—1011no—
414M16—1013no—
415M16—1013no—
416M16C22×D10113yescomplemented
417C2×C8—1013no—
418C2×C8—1016no—
419C2×C8C22×D10113yescomplemented
420C23—23no—
421C23—8086no—
422C23—1013no—
423C23—2023no—
424C23—2026no—
425C23—2023no—
426C23—2023no—
427C23—2023no—
428Q8—23no—
429Q8—1013no—
430Q8—1011no—
431Q8—1011no—
432Q8—2023no—
433Q8—2021no—
434Q8D4×D10111yescharacteristic
435Q8C23×D10113yes—
436Q8C23×D10111yescharacteristic
437D4—23no—
438D4—23no—
439D4—23no—
440D4—1016no—
441D4—1013no—
442D4—1013no—
443D4—1013no—
444D4—2023no—
445D4—2023no—
446D4—2026no—
447D4—2023no—
448D4—2026no—
449D4—2023no—
450D4—2023no—
451D4—2021no—
452D4—2023no—
453D4—2023no—
454D4—2023no—
455D4—2023no—
456D4D4×D10113yes—
457D4C23×D10111yescharacteristic
458D4C23×D10113yes—
459C2×C4—23no—
460C2×C4—23no—
461C2×C4—23no—
462C2×C4—1013no—
463C2×C4—1013no—
464C2×C4—1013no—
465C2×C4—1013no—
466C2×C4—1011no—
467C2×C4—2023no—
468C2×C4—2023no—
469C2×C4—2026no—
470C2×C4—2021no—
471C2×C4—2023no—
472C2×C4—2021no—
473C2×C4—2023no—
474C2×C4—2023no—
475C2×C4—2023no—
476C2×C4C23×D10113yes—
477C8—1011no—
478C8—1013no—
479C8C23×D10113yes—
480C8C23×D10111yescharacteristic
481C22—21no—
482C22—23no—
483C22—83no—
484C22—8083no—
485C22—8083no—
486C22—8086no—
487C22—8083no—
488C22—1011no—
489C22—1013no—
490C22—4043no—
491C22—4046no—
492C22—2023no—
493C22—2023no—
494C22C2×D4×D10113yes—
495C4—21no—
496C4—23no—
497C4—1011no—
498C4—1013no—
499C4—2021no—
500C4—2021no—
501C4C2×D4×D10113yes—
502C4C24×D10111yescharacteristic
503C2—23no—
504C2—43no—
505C2—41no—
506C2—4043no—
507C2—4043no—
508C2—2023no—
509C2—2021no—
510C2C22×D4×D10111yescharacteristic
subgroupG/H
centerC2C22×D4×D101
commutator, cosocleC4×C101C25
FrattiniC4C24×D101
Fitting#577×C101C2
socleC2×C101C23×D4
hypercenter#577D101
hypocenterC101#3585
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ #41 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #20266014C2semidirect
2. H(5,𝔽2)×D101C2semidirect
3. #20265986C2semidirect
4. #20265985C2semidirect
5. #20265983C2semidirect
6. #20265982C2semidirect
7. #20265981C2semidirect
8. #20265980C2semidirect
9. #20265979C2semidirect
10. #20265976C2semidirect
11. #20265974C2semidirect
12. C2×#15692282C2semidirect
13. AGL(1,ℤ/8)×D101C2semidirect
14. #20265942C2semidirect
15. #577×C101C2semidirect
17. #15692303C22semidirect
20. #15692301C22semidirect
21. #15692301C22semidirect
22. #15692301C22semidirect
23. #15692301C22semidirect
25. #15692300C22semidirect
27. C2×#13108075C22semidirect
31. #41×D101C22semidirect
32. #41×D101C22semidirect
34. C2×D4×D101C22semidirect
35. #15692287C22semidirect
36. #15692287C22semidirect
37. #15692286C22semidirect
38. #15692284C22semidirect
39. #15692284C22semidirect
40. #15692283C22semidirect
41. #15692283C22semidirect
42. #15692282C22semidirect
43. #15692282C22semidirect
44. #15692282C22semidirect
45. #15692282C22semidirect
46. #15692282C22semidirect
47. #15692281C22semidirect
48. #15692280C22semidirect
49. #15692280C22semidirect
50. #15692279C22semidirect
51. #15692279C22semidirect
52. #15692278C22semidirect
53. #15692277C22semidirect
54. #15692276C22semidirect
55. #15692275C22semidirect
56. #15692275C22semidirect
57. C2×#13108061C22semidirect
58. SD16×D101C22semidirect
59. C2×D808C22semidirect
60. C2×#13108059C22semidirect
61. D8×D101C22semidirect
62. #15692263C22semidirect
63. #15692262C22semidirect
64. #15692261C22semidirect
65. C2×#13108057C22semidirect
66. M16×D101C22semidirect
67. 2−1+4×C101C22semidirect
68. H(5,𝔽2)×C101C22semidirect
69. #137×C101C22semidirect
70. AGL(1,ℤ/8)×C101C22semidirect
71. #135×C101C22semidirect
72. C2×SD16×C101C22semidirect
73. #131×C101C22semidirect
83. #13108075C23semidirect
91. #13108074C23semidirect
101. Q8×D101C23semidirect
106. C2×D404C23semidirect
121. D4×D101C23semidirect
126. Dic404C23semidirect
127. #13108064C23semidirect
128. #13108064C23semidirect
129. #13108063C23semidirect
130. #13108063C23semidirect
131. #13108062C23semidirect
132. #13108062C23semidirect
133. #13108061C23semidirect
134. #13108061C23semidirect
135. #13108061C23semidirect
136. D808C23semidirect
137. D808C23semidirect
138. #13108059C23semidirect
139. #13108059C23semidirect
140. #13108059C23semidirect
141. #13108058C23semidirect
144. #13108057C23semidirect
145. #13108057C23semidirect
146. C2×#11717579C23semidirect
147. C8×D101C23semidirect
152. #41×C101C23semidirect
154. C2×Q8×C101C23semidirect
157. Q16×C101C23semidirect
158. SD16×C101C23semidirect
159. SD16×C101C23semidirect
160. SD16×C101C23semidirect
161. D8×C101C23semidirect
162. M16×C101C23semidirect
163. C2×C8×C101C23semidirect
256. C101#3585semidirect
262. #577D101semidirect
274. 2−1+4C2×D101semidirect
283. H(5,𝔽2)C2×D101semidirect
298. #137C2×D101semidirect
311. AGL(1,ℤ/8)C2×D101semidirect
317. #135C2×D101semidirect
320. C2×SD16C2×D101semidirect
327. #131C2×D101semidirect
358. #41C22×D101semidirect
364. C2×Q8C22×D101semidirect
394. Q16C22×D101semidirect
402. SD16C22×D101semidirect
403. SD16C22×D101semidirect
404. SD16C22×D101semidirect
410. D8C22×D101semidirect
416. M16C22×D101semidirect
419. C2×C8C22×D101semidirect

Maximal subgroups

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

subgroupindex
#202660142
H(5,𝔽2)×D1012
#202659862
#202659852
#202659832
#202659822
#202659812
#202659802
#202659792
#202659762
#202659742
C2×#156922822
AGL(1,ℤ/8)×D1012
#202659422
#577×C1012
#3585101