Finite Group Databasebeta

← #36368338

Subgroups — #36368338

Order 22144.

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#26012326C211yesmaximal, characteristic, complemented
2H(5,𝔽2)×D173C211yesmaximal, characteristic, complemented
3#26012303C211yesmaximal, characteristic, complemented
4#26012302C211yesmaximal, characteristic, complemented
5#26012301C213yesmaximal, complemented
6#26012298C213yesmaximal, complemented
7#26012295C213yesmaximal, complemented
8#26012294C211yesmaximal, characteristic, complemented
9#26012292C213yesmaximal, complemented
10#26012288C213yesmaximal, complemented
11#26012287C213yesmaximal, complemented
12C2×#19149703C213yesmaximal, complemented
13AGL(1,ℤ/8)×D173C213yesmaximal, complemented
14#26012256C211yesmaximal, characteristic, complemented
15#576×C173C211yesmaximal, characteristic, complemented
16#19149726—21no—
17#19149726C2211yescharacteristic, complemented
18#19149726C2213yescomplemented
19#19149725—21no—
20#19149725C2211yescharacteristic, complemented
21#19149725C2213yescomplemented
22#19149724—23no—
23#19149724—23no—
24#19149724C2213yescomplemented
25#19149724C2213yescomplemented
26C2×#14583825—23no—
27C2×#14583825C2213yescomplemented
28#41×D173—21no—
29#41×D173—21no—
30#41×D173C2211yescharacteristic
31#41×D173C2213yescomplemented
32#41×D173C2213yescomplemented
33C2×D4×D173—23no—
34C2×D4×D173C2213yescomplemented
35#19149712C2213yescomplemented
36#19149711C2211yescharacteristic, complemented
37#19149711C2213yescomplemented
38#19149709C2213yescomplemented
39#19149708C2213yescomplemented
40#19149708C2213yescomplemented
41#19149707C2213yescomplemented
42#19149706C2213yescomplemented
43#19149706C2213yescomplemented
44#19149705C2213yescomplemented
45#19149705C2213yescomplemented
46#19149704C2213yescomplemented
47#19149703C2216yescomplemented
48#19149703C2213yescomplemented
49#19149703C2216yescomplemented
50#19149703C2216yescomplemented
51#19149703C2213yescomplemented
52#19149701C2213yescomplemented
53#19149700C2216yescomplemented
54#19149699C2213yescomplemented
55#19149699C2213yescomplemented
56#19149698C2216yescomplemented
57C2×#14583815C2213yescomplemented
58C2×#14583814C2213yescomplemented
59SD16×D173C2216yescomplemented
60C2×#14583811C2213yescomplemented
61D8×D173C2216yescomplemented
62#19149687C2213yescomplemented
63#19149686C2211yescharacteristic, complemented
64#19149685C2213yescomplemented
65C2×#14583809C2213yescomplemented
66M16×D173C2213yescomplemented
67H(5,𝔽2)×C173C2211yescharacteristic, complemented
68H(5,𝔽2)×C173C2211yescharacteristic, complemented
69AGL(1,ℤ/8)×C173C2213yescomplemented
70AGL(1,ℤ/8)×C173C2213yescomplemented
71#135×C173C2213yescomplemented
72C2×D8×C173C2213yescomplemented
73#131×C173C2211yescharacteristic, complemented
74C23×D173—23no—
75#14583827—21no—
76#14583827D411yescharacteristic
77#14583827C2313yes—
78#14583827C2311yescharacteristic
79#14583826—23no—
80#14583826—23no—
81#14583826—23no—
82#14583826—23no—
83#14583826C2313yes—
84#14583826C2313yes—
85#14583826C2313yescomplemented
86#14583825—23no—
87#14583825—23no—
88#14583825—26no—
89#14583825—23no—
90#14583825—23no—
91#14583825—23no—
92#14583825—23no—
93#14583825—23no—
94#14583825D413yes—
95#14583825C2313yes—
96#14583825C2313yes—
97#14583825C2316yescomplemented
98C2×Dic346—23no—
99C2×Dic346C2313yescomplemented
100Q8×D173—21no—
101Q8×D173D411yescharacteristic
102Q8×D173C2311yescharacteristic
103Q8×D173C2313yescomplemented
104C2×D692—23no—
105C2×D692C2313yes—
106C2×C22⋊D173—23no—
107C2×C22⋊D173—26no—
108C2×C22⋊D173—26no—
109C2×C22⋊D173—23no—
110D4×D173—23no—
111D4×D173—23no—
112D4×D173—23no—
113D4×D173—23no—
114D4×D173—26no—
115D4×D173—23no—
116D4×D173—23no—
117D4×D173—23no—
118D4×D173D413yes—
119D4×D173C2316yes—
120D4×D173C2313yes—
121D4×D173C2313yescomplemented
122C22×Dic173—23no—
123C2×C4×D173—23no—
124C2×C4×D173—23no—
125C2×C4×D173C2313yes—
126Dic692C2313yescomplemented
127#14583816C2311yescharacteristic, complemented
128#14583816C2313yescomplemented
129#14583815C2316yescomplemented
130#14583815C2313yescomplemented
131#14583815C2311yescharacteristic, complemented
132#14583814C2316yescomplemented
133#14583814C2313yescomplemented
134#14583814C2313yescomplemented
135#14583813C2313yescomplemented
136#14583813C2311yescharacteristic, complemented
137D1384C2313yescomplemented
138#14583811C2313yescomplemented
139#14583811C2316yescomplemented
140#14583811C2313yescomplemented
141#14583810C2313yescomplemented
142#14583809C2313yes—
143#14583809C2311yescharacteristic
144#14583809C2313yescomplemented
145#14583809C2313yescomplemented
146C2×#13063436C2313yescomplemented
147C8×D173C2316yescomplemented
148#41×C173—21no—
149#41×C173—21no—
150#41×C173C2313yes—
151#41×C173C2311yescharacteristic
152#41×C173C2313yescomplemented
153C2×D4×C173—23no—
154C2×D4×C173—23no—
155C2×D4×C173C2313yes—
156C2×D4×C173C2313yescomplemented
157Q16×C173C2311yescharacteristic, complemented
158SD16×C173C2313yescomplemented
159SD16×C173C2313yescomplemented
160D8×C173C2313yescomplemented
161D8×C173C2316yescomplemented
162M16×C173C2313yescomplemented
163C2×C8×C173C2313yescomplemented
164C22×D173—23no—
165C22×D173—23no—
166C22×D173—23no—
167C22×D173—26no—
168C22×D173—86no—
169C22×D173C2×D413yes—
170Dic346—23no—
171Dic346—21no—
172Dic346—23no—
173Dic346C2×D413yes—
174Dic346C2413yes—
175Dic346C2411yescharacteristic
176D692—23no—
177D692—21no—
178D692—23no—
179D692C2×D413yes—
180D692C2411yescharacteristic
181D692C2413yes—
182C22⋊D173—23no—
183C22⋊D173—23no—
184C22⋊D173—23no—
185C22⋊D173—23no—
186C22⋊D173—26no—
187C22⋊D173—23no—
188C22⋊D173—26no—
189C22⋊D173—23no—
190C22⋊D173C2×D416yes—
191C2×Dic173—23no—
192C2×Dic173—23no—
193C2×Dic173—23no—
194C2×Dic173—26no—
195C2×Dic173C2×D413yes—
196C4×D173—21no—
197C4×D173—23no—
198C4×D173—23no—
199C4×D173—23no—
200C4×D173—21no—
201C4×D173—23no—
202C4×D173C2×D413yes—
203C4×D173C2×D413yes—
204C4×D173C2411yescharacteristic
205C4×D173C2413yes—
206#13063436C2411yescharacteristic
207#13063436C2413yes—
208C23×C173—23no—
209C23×C173—23no—
210Q8×C173C2×D411yescharacteristic
211Q8×C173C2411yescharacteristic
212Q8×C173C2411yescharacteristic
213D4×C173—23no—
214D4×C173—23no—
215D4×C173—23no—
216D4×C173—23no—
217D4×C173C2×D413yes—
218D4×C173C2413yes—
219D4×C173C2413yes—
220C2×C4×C173—23no—
221C2×C4×C173—23no—
222C2×C4×C173C2413yes—
223C8×C173C2411yescharacteristic
224C8×C173C2413yes—
225C22×C173—23no—
226C22×C173—23no—
227C22×C173—83no—
228C22×C173—83no—
229C22×C173C22×D413yes—
230C2×D173—21no—
231C2×D173—23no—
232C2×D173—43no—
233C2×D173—46no—
234C2×D173—83no—
235C2×D173—86no—
236C2×D173—83no—
237C2×D173C22×D411yescharacteristic
238C2×D173C22×D413yes—
239C4×C173—21no—
240C4×C173—21no—
241C4×C173C22×D413yes—
242C4×C173C2511yescharacteristic
243Dic173—21no—
244Dic173—23no—
245Dic173C22×D411yescharacteristic
246Dic173C22×D413yes—
247C2×C173—23no—
248C2×C173—43no—
249C2×C173—43no—
250C2×C173C23×D411yescharacteristic
251D173—21no—
252D173—23no—
253D173—43no—
254D173—41no—
255C173#358511yescharacteristic, complemented, Sylow
256#3585—1731nomaximal, Sylow
257#585—1731no—
258C2×H(5,𝔽2)—1731no—
259#577—1731no—
260#577—1733no—
261#577—1733no—
262#577—1731no—
263#576—1731no—
264#576—1733no—
265#576—1733no—
266#576D17311yescharacteristic, complemented
267#575—1733no—
268#575—1733no—
269C2×AGL(1,ℤ/8)—1733no—
270C2×AGL(1,ℤ/8)—1733no—
271#568—1731no—
2722−1+4—3461no—
2732−1+4—1731no—
2742−1+4—1733no—
275H(5,𝔽2)—3463no—
276H(5,𝔽2)—3463no—
277H(5,𝔽2)—3461no—
278H(5,𝔽2)—1733no—
279H(5,𝔽2)—1731no—
280H(5,𝔽2)—1733no—
281H(5,𝔽2)—1733no—
282H(5,𝔽2)C2×D17311yescharacteristic, complemented
283H(5,𝔽2)C2×D17311yescharacteristic, complemented
284C2×#41—3463no—
285C2×#41—3461no—
286C2×#41—3461no—
287C2×#41—1733no—
288C2×#41—1733no—
289C2×#41—1733no—
290C2×#41—1731no—
291C22×D4—3463no—
292C22×D4—1733no—
293#137—1733no—
294#137—1731no—
295#137—1733no—
296#137—1733no—
297#137—1733no—
298#137—1733no—
299AGL(1,ℤ/8)—1733no—
300AGL(1,ℤ/8)—1733no—
301AGL(1,ℤ/8)—1733no—
302AGL(1,ℤ/8)—1733no—
303AGL(1,ℤ/8)—1733no—
304AGL(1,ℤ/8)—1736no—
305AGL(1,ℤ/8)—1733no—
306AGL(1,ℤ/8)—1736no—
307AGL(1,ℤ/8)—1736no—
308AGL(1,ℤ/8)—1733no—
309AGL(1,ℤ/8)—1733no—
310AGL(1,ℤ/8)C2×D17313yescomplemented
311AGL(1,ℤ/8)C2×D17313yescomplemented
312#135—1733no—
313#135—1733no—
314#135—1736no—
315#135—1736no—
316#135—1733no—
317#135C2×D17313yescomplemented
318C2×SD16—1733no—
319C2×SD16—1736no—
320C2×SD16—1733no—
321C2×D8—1733no—
322C2×D8—1736no—
323C2×D8C2×D17313yescomplemented
324#131—1731no—
325#131—1733no—
326#131—1733no—
327#131C2×D17311yescharacteristic, complemented
328C2×M16—1733no—
329C2×M16—1733no—
330C24—3463no—
331#41—21no—
332#41—21no—
333#41—3463no—
334#41—3463no—
335#41—3463no—
336#41—3463no—
337#41—3463no—
338#41—3463no—
339#41—3466no—
340#41—3461no—
341#41—3463no—
342#41—3463no—
343#41—3463no—
344#41—3463no—
345#41—3463no—
346#41—1733no—
347#41—1731no—
348#41—1733no—
349#41—1733no—
350#41—1736no—
351#41—1733no—
352#41—1733no—
353#41—1731no—
354#41—1733no—
355#41—1733no—
356#41C22×D17313yes—
357#41C22×D17311yescharacteristic
358#41C22×D17313yescomplemented
359C2×Q8—3463no—
360C2×Q8—3461no—
361C2×Q8—1731no—
362C2×Q8—1733no—
363C2×Q8—1733no—
364C2×Q8—1731no—
365C2×D4—23no—
366C2×D4—23no—
367C2×D4—3463no—
368C2×D4—3463no—
369C2×D4—3463no—
370C2×D4—3466no—
371C2×D4—3463no—
372C2×D4—3463no—
373C2×D4—3463no—
374C2×D4—3466no—
375C2×D4—3466no—
376C2×D4—3463no—
377C2×D4—3463no—
378C2×D4—3463no—
379C2×D4—3463no—
380C2×D4—1736no—
381C2×D4—1733no—
382C2×D4—1733no—
383C2×D4—1733no—
384C2×D4—1733no—
385C2×D4C22×D17313yes—
386C2×D4C22×D17313yescomplemented
387C22×C4—3463no—
388C22×C4—3463no—
389C22×C4—3463no—
390C22×C4—1733no—
391Q16—1733no—
392Q16—1731no—
393Q16—1733no—
394Q16C22×D17311yescharacteristic, complemented
395SD16—1733no—
396SD16—1731no—
397SD16—1736no—
398SD16—1733no—
399SD16—1733no—
400SD16—1736no—
401SD16—1733no—
402SD16—1731no—
403SD16C22×D17313yescomplemented
404SD16C22×D17313yescomplemented
405D8—1733no—
406D8—1736no—
407D8—1733no—
408D8—1733no—
409D8C22×D17316yescomplemented
410D8C22×D17313yescomplemented
411M16—1733no—
412M16—1733no—
413M16—1731no—
414M16—1733no—
415M16—1733no—
416M16C22×D17313yescomplemented
417C2×C8—1733no—
418C2×C8—1736no—
419C2×C8C22×D17313yescomplemented
420C23—23no—
421C23—23no—
422C23—3463no—
423C23—3466no—
424C23—3463no—
425C23—3463no—
426C23—13846no—
427C23—1733no—
428Q8—3463no—
429Q8—3461no—
430Q8—3463no—
431Q8—1733no—
432Q8—1731no—
433Q8—1733no—
434Q8D4×D17311yescharacteristic
435Q8C23×D17311yescharacteristic
436Q8C23×D17311yescharacteristic
437D4—23no—
438D4—23no—
439D4—23no—
440D4—23no—
441D4—3463no—
442D4—3461no—
443D4—3463no—
444D4—3463no—
445D4—3466no—
446D4—3463no—
447D4—3463no—
448D4—3463no—
449D4—3466no—
450D4—3463no—
451D4—3463no—
452D4—1736no—
453D4—1733no—
454D4—1731no—
455D4—1733no—
456D4D4×D17313yes—
457D4C23×D17313yes—
458D4C23×D17313yes—
459C2×C4—23no—
460C2×C4—23no—
461C2×C4—3461no—
462C2×C4—3463no—
463C2×C4—3461no—
464C2×C4—3463no—
465C2×C4—3463no—
466C2×C4—3463no—
467C2×C4—3463no—
468C2×C4—3466no—
469C2×C4—3463no—
470C2×C4—3463no—
471C2×C4—1733no—
472C2×C4—1733no—
473C2×C4—1733no—
474C2×C4—1733no—
475C2×C4—1731no—
476C2×C4C23×D17313yes—
477C8—1731no—
478C8—1733no—
479C8C23×D17313yes—
480C8C23×D17311yescharacteristic
481C22—23no—
482C22—23no—
483C22—83no—
484C22—83no—
485C22—3461no—
486C22—3463no—
487C22—13843no—
488C22—13846no—
489C22—13843no—
490C22—1731no—
491C22—1733no—
492C22—6923no—
493C22—6926no—
494C22C2×D4×D17313yes—
495C4—21no—
496C4—21no—
497C4—3461no—
498C4—3463no—
499C4—1731no—
500C4—1733no—
501C4C2×D4×D17313yes—
502C4C24×D17311yescharacteristic
503C2—23no—
504C2—43no—
505C2—43no—
506C2—3463no—
507C2—3461no—
508C2—6921no—
509C2—6923no—
510C2C22×D4×D17311yescharacteristic
subgroupG/H
centerC2C22×D4×D173
commutator, cosocleC4×C173C25
FrattiniC4C24×D173
Fitting#576×C173C2
socleC2×C173C23×D4
hypercenter#576D173
hypocenterC173#3585
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ #41 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #26012326C2semidirect
2. H(5,𝔽2)×D173C2semidirect
3. #26012303C2semidirect
4. #26012302C2semidirect
5. #26012301C2semidirect
6. #26012298C2semidirect
7. #26012295C2semidirect
8. #26012294C2semidirect
9. #26012292C2semidirect
10. #26012288C2semidirect
11. #26012287C2semidirect
12. C2×#19149703C2semidirect
13. AGL(1,ℤ/8)×D173C2semidirect
14. #26012256C2semidirect
15. #576×C173C2semidirect
17. #19149726C22semidirect
18. #19149726C22semidirect
20. #19149725C22semidirect
21. #19149725C22semidirect
24. #19149724C22semidirect
25. #19149724C22semidirect
27. C2×#14583825C22semidirect
31. #41×D173C22semidirect
32. #41×D173C22semidirect
34. C2×D4×D173C22semidirect
35. #19149712C22semidirect
36. #19149711C22semidirect
37. #19149711C22semidirect
38. #19149709C22semidirect
39. #19149708C22semidirect
40. #19149708C22semidirect
41. #19149707C22semidirect
42. #19149706C22semidirect
43. #19149706C22semidirect
44. #19149705C22semidirect
45. #19149705C22semidirect
46. #19149704C22semidirect
47. #19149703C22semidirect
48. #19149703C22semidirect
49. #19149703C22semidirect
50. #19149703C22semidirect
51. #19149703C22semidirect
52. #19149701C22semidirect
53. #19149700C22semidirect
54. #19149699C22semidirect
55. #19149699C22semidirect
56. #19149698C22semidirect
57. C2×#14583815C22semidirect
58. C2×#14583814C22semidirect
59. SD16×D173C22semidirect
60. C2×#14583811C22semidirect
61. D8×D173C22semidirect
62. #19149687C22semidirect
63. #19149686C22semidirect
64. #19149685C22semidirect
65. C2×#14583809C22semidirect
66. M16×D173C22semidirect
67. H(5,𝔽2)×C173C22semidirect
68. H(5,𝔽2)×C173C22semidirect
69. AGL(1,ℤ/8)×C173C22semidirect
70. AGL(1,ℤ/8)×C173C22semidirect
71. #135×C173C22semidirect
72. C2×D8×C173C22semidirect
73. #131×C173C22semidirect
85. #14583826C23semidirect
97. #14583825C23semidirect
99. C2×Dic346C23semidirect
103. Q8×D173C23semidirect
121. D4×D173C23semidirect
126. Dic692C23semidirect
127. #14583816C23semidirect
128. #14583816C23semidirect
129. #14583815C23semidirect
130. #14583815C23semidirect
131. #14583815C23semidirect
132. #14583814C23semidirect
133. #14583814C23semidirect
134. #14583814C23semidirect
135. #14583813C23semidirect
136. #14583813C23semidirect
137. D1384C23semidirect
138. #14583811C23semidirect
139. #14583811C23semidirect
140. #14583811C23semidirect
141. #14583810C23semidirect
144. #14583809C23semidirect
145. #14583809C23semidirect
146. C2×#13063436C23semidirect
147. C8×D173C23semidirect
152. #41×C173C23semidirect
156. C2×D4×C173C23semidirect
157. Q16×C173C23semidirect
158. SD16×C173C23semidirect
159. SD16×C173C23semidirect
160. D8×C173C23semidirect
161. D8×C173C23semidirect
162. M16×C173C23semidirect
163. C2×C8×C173C23semidirect
255. C173#3585semidirect
266. #576D173semidirect
282. H(5,𝔽2)C2×D173semidirect
283. H(5,𝔽2)C2×D173semidirect
310. AGL(1,ℤ/8)C2×D173semidirect
311. AGL(1,ℤ/8)C2×D173semidirect
317. #135C2×D173semidirect
323. C2×D8C2×D173semidirect
327. #131C2×D173semidirect
358. #41C22×D173semidirect
386. C2×D4C22×D173semidirect
394. Q16C22×D173semidirect
403. SD16C22×D173semidirect
404. SD16C22×D173semidirect
409. D8C22×D173semidirect
410. D8C22×D173semidirect
416. M16C22×D173semidirect
419. C2×C8C22×D173semidirect

Maximal subgroups

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

subgroupindex
#260123262
H(5,𝔽2)×D1732
#260123032
#260123022
#260123012
#260122982
#260122952
#260122942
#260122922
#260122882
#260122872
C2×#191497032
AGL(1,ℤ/8)×D1732
#260122562
#576×C1732
#3585173