Finite Group Databasebeta

← #27185907

Subgroups — #27185907

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#20266012C211yesmaximal, characteristic, complemented
2H(5,𝔽2)×D101C211yesmaximal, characteristic, complemented
3#20265989C211yesmaximal, characteristic, complemented
4#20265988C211yesmaximal, characteristic, complemented
5#20265987C213yesmaximal, complemented
6#20265984C213yesmaximal, complemented
7#20265981C213yesmaximal, complemented
8#20265980C211yesmaximal, characteristic, complemented
9#20265978C213yesmaximal, complemented
10#20265974C213yesmaximal, complemented
11#20265973C213yesmaximal, complemented
12C2×#15692279C213yesmaximal, complemented
13AGL(1,ℤ/8)×D101C213yesmaximal, complemented
14#20265942C211yesmaximal, characteristic, complemented
15#576×C101C211yesmaximal, characteristic, complemented
16#15692302—21no—
17#15692302C2211yescharacteristic, complemented
18#15692302C2213yescomplemented
19#15692301—21no—
20#15692301C2211yescharacteristic, complemented
21#15692301C2213yescomplemented
22#15692300—23no—
23#15692300—23no—
24#15692300C2213yescomplemented
25#15692300C2213yescomplemented
26C2×#13108073—23no—
27C2×#13108073C2213yescomplemented
28#41×D101—21no—
29#41×D101—21no—
30#41×D101C2211yescharacteristic
31#41×D101C2213yescomplemented
32#41×D101C2213yescomplemented
33C2×D4×D101—23no—
34C2×D4×D101C2213yescomplemented
35#15692288C2213yescomplemented
36#15692287C2211yescharacteristic, complemented
37#15692287C2213yescomplemented
38#15692285C2213yescomplemented
39#15692284C2213yescomplemented
40#15692284C2213yescomplemented
41#15692283C2213yescomplemented
42#15692282C2213yescomplemented
43#15692282C2213yescomplemented
44#15692281C2213yescomplemented
45#15692281C2213yescomplemented
46#15692280C2213yescomplemented
47#15692279C2216yescomplemented
48#15692279C2213yescomplemented
49#15692279C2216yescomplemented
50#15692279C2216yescomplemented
51#15692279C2213yescomplemented
52#15692277C2213yescomplemented
53#15692276C2216yescomplemented
54#15692275C2213yescomplemented
55#15692275C2213yescomplemented
56#15692274C2216yescomplemented
57C2×#13108063C2213yescomplemented
58C2×#13108062C2213yescomplemented
59SD16×D101C2216yescomplemented
60C2×#13108059C2213yescomplemented
61D8×D101C2216yescomplemented
62#15692263C2213yescomplemented
63#15692262C2211yescharacteristic, complemented
64#15692261C2213yescomplemented
65C2×#13108057C2213yescomplemented
66M16×D101C2213yescomplemented
67H(5,𝔽2)×C101C2211yescharacteristic, complemented
68H(5,𝔽2)×C101C2211yescharacteristic, complemented
69AGL(1,ℤ/8)×C101C2213yescomplemented
70AGL(1,ℤ/8)×C101C2213yescomplemented
71#135×C101C2213yescomplemented
72C2×D8×C101C2213yescomplemented
73#131×C101C2211yescharacteristic, complemented
74C23×D101—23no—
75#13108075—21no—
76#13108075D411yescharacteristic
77#13108075C2311yescharacteristic
78#13108075C2313yes—
79#13108074—23no—
80#13108074—23no—
81#13108074—23no—
82#13108074—23no—
83#13108074C2313yes—
84#13108074C2313yes—
85#13108074C2313yescomplemented
86#13108073—23no—
87#13108073—26no—
88#13108073—23no—
89#13108073—23no—
90#13108073—23no—
91#13108073—23no—
92#13108073—23no—
93#13108073—23no—
94#13108073D413yes—
95#13108073C2313yes—
96#13108073C2313yes—
97#13108073C2316yescomplemented
98C2×Dic202—23no—
99C2×Dic202C2313yescomplemented
100Q8×D101—21no—
101Q8×D101D411yescharacteristic
102Q8×D101C2311yescharacteristic
103Q8×D101C2313yescomplemented
104C2×D404—23no—
105C2×D404C2313yes—
106C2×C22⋊D101—23no—
107C2×C22⋊D101—26no—
108C2×C22⋊D101—26no—
109C2×C22⋊D101—23no—
110D4×D101—23no—
111D4×D101—23no—
112D4×D101—23no—
113D4×D101—26no—
114D4×D101—23no—
115D4×D101—23no—
116D4×D101—23no—
117D4×D101—23no—
118D4×D101D413yes—
119D4×D101C2313yes—
120D4×D101C2316yes—
121D4×D101C2313yescomplemented
122C22×Dic101—23no—
123C2×C4×D101—23no—
124C2×C4×D101—23no—
125C2×C4×D101C2313yes—
126Dic404C2313yescomplemented
127#13108064C2311yescharacteristic, complemented
128#13108064C2313yescomplemented
129#13108063C2316yescomplemented
130#13108063C2313yescomplemented
131#13108063C2311yescharacteristic, complemented
132#13108062C2316yescomplemented
133#13108062C2313yescomplemented
134#13108062C2313yescomplemented
135#13108061C2313yescomplemented
136#13108061C2311yescharacteristic, complemented
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—21no—
150#41×C101C2313yes—
151#41×C101C2311yescharacteristic
152#41×C101C2313yescomplemented
153C2×D4×C101—23no—
154C2×D4×C101—23no—
155C2×D4×C101C2313yes—
156C2×D4×C101C2313yescomplemented
157Q16×C101C2311yescharacteristic, complemented
158SD16×C101C2313yescomplemented
159SD16×C101C2313yescomplemented
160D8×C101C2313yescomplemented
161D8×C101C2316yescomplemented
162M16×C101C2313yescomplemented
163C2×C8×C101C2313yescomplemented
164C23×C101—23no—
165C23×C101—23no—
166C22×D101—23no—
167C22×D101—23no—
168C22×D101—23no—
169C22×D101—26no—
170C22×D101—86no—
171C22×D101C2×D413yes—
172Q8×C101C2×D411yescharacteristic
173Q8×C101C2411yescharacteristic
174Q8×C101C2411yescharacteristic
175D4×C101—23no—
176D4×C101—23no—
177D4×C101—23no—
178D4×C101—23no—
179D4×C101C2×D413yes—
180D4×C101C2413yes—
181D4×C101C2413yes—
182C2×C4×C101—23no—
183C2×C4×C101—23no—
184C2×C4×C101C2413yes—
185C22⋊D101—23no—
186C22⋊D101—23no—
187C22⋊D101—23no—
188C22⋊D101—23no—
189C22⋊D101—26no—
190C22⋊D101—26no—
191C22⋊D101—23no—
192C22⋊D101—23no—
193C22⋊D101C2×D416yes—
194C2×Dic101—23no—
195C2×Dic101—23no—
196C2×Dic101—23no—
197C2×Dic101—26no—
198C2×Dic101C2×D413yes—
199D404—23no—
200D404—21no—
201D404—23no—
202D404C2×D413yes—
203D404C2411yescharacteristic
204D404C2413yes—
205C4×D101—21no—
206C4×D101—23no—
207C4×D101—23no—
208C4×D101—23no—
209C4×D101—21no—
210C4×D101—23no—
211C4×D101C2×D413yes—
212C4×D101C2×D413yes—
213C4×D101C2413yes—
214C4×D101C2411yescharacteristic
215Dic202—23no—
216Dic202—21no—
217Dic202—23no—
218Dic202C2×D413yes—
219Dic202C2413yes—
220Dic202C2411yescharacteristic
221C8×C101C2411yescharacteristic
222C8×C101C2413yes—
223#11717579C2413yes—
224#11717579C2411yescharacteristic
225C22×C101—23no—
226C22×C101—23no—
227C22×C101—83no—
228C22×C101—83no—
229C22×C101C22×D413yes—
230C2×D101—21no—
231C2×D101—23no—
232C2×D101—43no—
233C2×D101—46no—
234C2×D101—83no—
235C2×D101—86no—
236C2×D101—83no—
237C2×D101C22×D411yescharacteristic
238C2×D101C22×D413yes—
239C4×C101—21no—
240C4×C101—21no—
241C4×C101C22×D413yes—
242C4×C101C2511yescharacteristic
243Dic101—23no—
244Dic101—21no—
245Dic101C22×D413yes—
246Dic101C22×D411yescharacteristic
247C2×C101—23no—
248C2×C101—43no—
249C2×C101—43no—
250C2×C101C23×D411yescharacteristic
251D101—21no—
252D101—23no—
253D101—43no—
254D101—41no—
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#577—1011no—
263#576—1011no—
264#576—1013no—
265#576—1013no—
266#576D10111yescharacteristic, complemented
267#575—1013no—
268#575—1013no—
269C2×AGL(1,ℤ/8)—1013no—
270C2×AGL(1,ℤ/8)—1013no—
271#568—1011no—
2722−1+4—1011no—
2732−1+4—1013no—
2742−1+4—2021no—
275H(5,𝔽2)—1013no—
276H(5,𝔽2)—1011no—
277H(5,𝔽2)—1013no—
278H(5,𝔽2)—1013no—
279H(5,𝔽2)—2023no—
280H(5,𝔽2)—2023no—
281H(5,𝔽2)—2021no—
282H(5,𝔽2)C2×D10111yescharacteristic, complemented
283H(5,𝔽2)C2×D10111yescharacteristic, complemented
284C2×#41—1011no—
285C2×#41—1013no—
286C2×#41—1013no—
287C2×#41—1013no—
288C2×#41—2023no—
289C2×#41—2021no—
290C2×#41—2021no—
291C22×D4—1013no—
292C22×D4—2023no—
293#137—1013no—
294#137—1011no—
295#137—1013no—
296#137—1013no—
297#137—1013no—
298#137—1013no—
299AGL(1,ℤ/8)—1013no—
300AGL(1,ℤ/8)—1013no—
301AGL(1,ℤ/8)—1013no—
302AGL(1,ℤ/8)—1013no—
303AGL(1,ℤ/8)—1013no—
304AGL(1,ℤ/8)—1016no—
305AGL(1,ℤ/8)—1013no—
306AGL(1,ℤ/8)—1016no—
307AGL(1,ℤ/8)—1016no—
308AGL(1,ℤ/8)—1013no—
309AGL(1,ℤ/8)—1013no—
310AGL(1,ℤ/8)C2×D10113yescomplemented
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×SD16—1013no—
321C2×D8—1013no—
322C2×D8—1016no—
323C2×D8C2×D10113yescomplemented
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—21no—
333#41—1011no—
334#41—1013no—
335#41—1013no—
336#41—1013no—
337#41—1011no—
338#41—1013no—
339#41—1013no—
340#41—1016no—
341#41—1013no—
342#41—1013no—
343#41—2023no—
344#41—2023no—
345#41—2023no—
346#41—2023no—
347#41—2023no—
348#41—2023no—
349#41—2023no—
350#41—2023no—
351#41—2026no—
352#41—2023no—
353#41—2023no—
354#41—2021no—
355#41—2023no—
356#41C22×D10113yes—
357#41C22×D10111yescharacteristic
358#41C22×D10113yescomplemented
359C2×Q8—1011no—
360C2×Q8—1011no—
361C2×Q8—1013no—
362C2×Q8—1013no—
363C2×Q8—2023no—
364C2×Q8—2021no—
365C2×D4—23no—
366C2×D4—23no—
367C2×D4—1013no—
368C2×D4—1016no—
369C2×D4—1013no—
370C2×D4—1013no—
371C2×D4—1013no—
372C2×D4—2023no—
373C2×D4—2023no—
374C2×D4—2023no—
375C2×D4—2023no—
376C2×D4—2026no—
377C2×D4—2023no—
378C2×D4—2023no—
379C2×D4—2023no—
380C2×D4—2023no—
381C2×D4—2026no—
382C2×D4—2026no—
383C2×D4—2023no—
384C2×D4—2023no—
385C2×D4C22×D10113yes—
386C2×D4C22×D10113yescomplemented
387C22×C4—1013no—
388C22×C4—2023no—
389C22×C4—2023no—
390C22×C4—2023no—
391Q16—1013no—
392Q16—1011no—
393Q16—1013no—
394Q16C22×D10111yescharacteristic, complemented
395SD16—1013no—
396SD16—1011no—
397SD16—1016no—
398SD16—1013no—
399SD16—1013no—
400SD16—1016no—
401SD16—1013no—
402SD16—1011no—
403SD16C22×D10113yescomplemented
404SD16C22×D10113yescomplemented
405D8—1013no—
406D8—1016no—
407D8—1013no—
408D8—1013no—
409D8C22×D10116yescomplemented
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—23no—
422C23—8086no—
423C23—1013no—
424C23—2023no—
425C23—2026no—
426C23—2023no—
427C23—2023no—
428Q8—1013no—
429Q8—1013no—
430Q8—1011no—
431Q8—2023no—
432Q8—2021no—
433Q8—2023no—
434Q8D4×D10111yescharacteristic
435Q8C23×D10111yescharacteristic
436Q8C23×D10111yescharacteristic
437D4—23no—
438D4—23no—
439D4—23no—
440D4—23no—
441D4—1016no—
442D4—1013no—
443D4—1011no—
444D4—1013no—
445D4—2023no—
446D4—2021no—
447D4—2023no—
448D4—2026no—
449D4—2023no—
450D4—2023no—
451D4—2023no—
452D4—2026no—
453D4—2023no—
454D4—2023no—
455D4—2023no—
456D4D4×D10113yes—
457D4C23×D10113yes—
458D4C23×D10113yes—
459C2×C4—23no—
460C2×C4—23no—
461C2×C4—1013no—
462C2×C4—1013no—
463C2×C4—1013no—
464C2×C4—1013no—
465C2×C4—1011no—
466C2×C4—2021no—
467C2×C4—2023no—
468C2×C4—2021no—
469C2×C4—2023no—
470C2×C4—2023no—
471C2×C4—2023no—
472C2×C4—2026no—
473C2×C4—2023no—
474C2×C4—2023no—
475C2×C4—2023no—
476C2×C4C23×D10113yes—
477C8—1011no—
478C8—1013no—
479C8C23×D10113yes—
480C8C23×D10111yescharacteristic
481C22—23no—
482C22—23no—
483C22—83no—
484C22—83no—
485C22—8083no—
486C22—8086no—
487C22—8083no—
488C22—1011no—
489C22—1013no—
490C22—4043no—
491C22—4046no—
492C22—2021no—
493C22—2023no—
494C22C2×D4×D10113yes—
495C4—21no—
496C4—21no—
497C4—1011no—
498C4—1013no—
499C4—2021no—
500C4—2023no—
501C4C2×D4×D10113yes—
502C4C24×D10111yescharacteristic
503C2—23no—
504C2—43no—
505C2—43no—
506C2—4043no—
507C2—4041no—
508C2—2023no—
509C2—2021no—
510C2C22×D4×D10111yescharacteristic
subgroupG/H
centerC2C22×D4×D101
commutator, cosocleC4×C101C25
FrattiniC4C24×D101
Fitting#576×C101C2
socleC2×C101C23×D4
hypercenter#576D101
hypocenterC101#3585
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ #41 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #20266012C2semidirect
2. H(5,𝔽2)×D101C2semidirect
3. #20265989C2semidirect
4. #20265988C2semidirect
5. #20265987C2semidirect
6. #20265984C2semidirect
7. #20265981C2semidirect
8. #20265980C2semidirect
9. #20265978C2semidirect
10. #20265974C2semidirect
11. #20265973C2semidirect
12. C2×#15692279C2semidirect
13. AGL(1,ℤ/8)×D101C2semidirect
14. #20265942C2semidirect
15. #576×C101C2semidirect
17. #15692302C22semidirect
18. #15692302C22semidirect
20. #15692301C22semidirect
21. #15692301C22semidirect
24. #15692300C22semidirect
25. #15692300C22semidirect
27. C2×#13108073C22semidirect
31. #41×D101C22semidirect
32. #41×D101C22semidirect
34. C2×D4×D101C22semidirect
35. #15692288C22semidirect
36. #15692287C22semidirect
37. #15692287C22semidirect
38. #15692285C22semidirect
39. #15692284C22semidirect
40. #15692284C22semidirect
41. #15692283C22semidirect
42. #15692282C22semidirect
43. #15692282C22semidirect
44. #15692281C22semidirect
45. #15692281C22semidirect
46. #15692280C22semidirect
47. #15692279C22semidirect
48. #15692279C22semidirect
49. #15692279C22semidirect
50. #15692279C22semidirect
51. #15692279C22semidirect
52. #15692277C22semidirect
53. #15692276C22semidirect
54. #15692275C22semidirect
55. #15692275C22semidirect
56. #15692274C22semidirect
57. C2×#13108063C22semidirect
58. C2×#13108062C22semidirect
59. SD16×D101C22semidirect
60. C2×#13108059C22semidirect
61. D8×D101C22semidirect
62. #15692263C22semidirect
63. #15692262C22semidirect
64. #15692261C22semidirect
65. C2×#13108057C22semidirect
66. M16×D101C22semidirect
67. H(5,𝔽2)×C101C22semidirect
68. H(5,𝔽2)×C101C22semidirect
69. AGL(1,ℤ/8)×C101C22semidirect
70. AGL(1,ℤ/8)×C101C22semidirect
71. #135×C101C22semidirect
72. C2×D8×C101C22semidirect
73. #131×C101C22semidirect
85. #13108074C23semidirect
97. #13108073C23semidirect
99. C2×Dic202C23semidirect
103. Q8×D101C23semidirect
121. D4×D101C23semidirect
126. Dic404C23semidirect
127. #13108064C23semidirect
128. #13108064C23semidirect
129. #13108063C23semidirect
130. #13108063C23semidirect
131. #13108063C23semidirect
132. #13108062C23semidirect
133. #13108062C23semidirect
134. #13108062C23semidirect
135. #13108061C23semidirect
136. #13108061C23semidirect
137. D808C23semidirect
138. #13108059C23semidirect
139. #13108059C23semidirect
140. #13108059C23semidirect
141. #13108058C23semidirect
144. #13108057C23semidirect
145. #13108057C23semidirect
146. C2×#11717579C23semidirect
147. C8×D101C23semidirect
152. #41×C101C23semidirect
156. C2×D4×C101C23semidirect
157. Q16×C101C23semidirect
158. SD16×C101C23semidirect
159. SD16×C101C23semidirect
160. D8×C101C23semidirect
161. D8×C101C23semidirect
162. M16×C101C23semidirect
163. C2×C8×C101C23semidirect
256. C101#3585semidirect
266. #576D101semidirect
282. H(5,𝔽2)C2×D101semidirect
283. H(5,𝔽2)C2×D101semidirect
310. AGL(1,ℤ/8)C2×D101semidirect
311. AGL(1,ℤ/8)C2×D101semidirect
317. #135C2×D101semidirect
323. C2×D8C2×D101semidirect
327. #131C2×D101semidirect
358. #41C22×D101semidirect
386. C2×D4C22×D101semidirect
394. Q16C22×D101semidirect
403. SD16C22×D101semidirect
404. SD16C22×D101semidirect
409. D8C22×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
#202660122
H(5,𝔽2)×D1012
#202659892
#202659882
#202659872
#202659842
#202659812
#202659802
#202659782
#202659742
#202659732
C2×#156922792
AGL(1,ℤ/8)×D1012
#202659422
#576×C1012
#3585101