Finite Group Databasebeta

← #37548506

Subgroups — #37548506

Order 23168.

1412 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042116C211yesmaximal, characteristic, complemented
22−1+4×D181C211yesmaximal, characteristic, complemented
3#26042095C213yesmaximal, complemented
4#26042094C211yesmaximal, characteristic, complemented
5#26042092C213yesmaximal, complemented
6#26042091C211yesmaximal, characteristic, complemented
7#26042088C213yesmaximal, complemented
8#26042087C211yesmaximal, characteristic, complemented
9#26042086C213yesmaximal, complemented
10#26042078C213yesmaximal, complemented
11#26042076C213yesmaximal, complemented
12C2×#19153848C213yesmaximal, complemented
13#137×D181C213yesmaximal, complemented
14#26042044C211yesmaximal, characteristic, complemented
15#578×C181C211yesmaximal, characteristic, complemented
16#19153864—21no—
17#19153864—21no—
18#19153864C2213yescomplemented
19#19153864C2213yescomplemented
20#19153863—23no—
21#19153863C2211yescharacteristic, complemented
22#19153863C2213yescomplemented
23#19153862—23no—
24#19153862C2211yescharacteristic, complemented
25#19153862C2213yescomplemented
26C2×#15668450—21no—
27C2×#15668450C2213yes—
28#41×D181—23no—
29#41×D181—23no—
30#41×D181C2213yes—
31#41×D181C2211yescharacteristic
32#41×D181C2213yescomplemented
33C2×Q8×D181—21no—
34C2×Q8×D181C2213yescomplemented
35#19153849C2213yescomplemented
36#19153848C2216yescomplemented
37#19153848C2213yescomplemented
38#19153848C2216yescomplemented
39#19153848C2216yescomplemented
40#19153848C2213yescomplemented
41#19153847C2213yescomplemented
42#19153847C2213yescomplemented
43#19153846C2213yescomplemented
44#19153845C2213yescomplemented
45#19153845C2213yescomplemented
46#19153844C2213yescomplemented
47#19153843C2213yescomplemented
48#19153843C2213yescomplemented
49#19153841C2213yescomplemented
50#19153840C2211yescharacteristic, complemented
51#19153840C2213yescomplemented
52#19153839C2216yescomplemented
53#19153838C2213yescomplemented
54#19153837C2216yescomplemented
55#19153836C2213yescomplemented
56#19153836C2213yescomplemented
57C2×#15668439C2213yescomplemented
58Q16×D181C2216yescomplemented
59C2×#15668438C2213yescomplemented
60C2×#15668436C2213yescomplemented
61SD16×D181C2216yescomplemented
62#19153824C2213yescomplemented
63#19153823C2211yescharacteristic, complemented
64#19153822C2213yescomplemented
65C2×#15668432C2213yescomplemented
66M16×D181C2213yescomplemented
672−1+4×C181C2211yescharacteristic, complemented
682−1+4×C181C2211yescharacteristic, complemented
69#137×C181C2213yescomplemented
70#137×C181C2213yescomplemented
71#135×C181C2213yescomplemented
72C2×Q16×C181C2213yescomplemented
73#131×C181C2211yescharacteristic, complemented
74#15668450—23no—
75#15668450—23no—
76#15668450—23no—
77#15668450—23no—
78#15668450—21no—
79#15668450D411yescharacteristic
80#15668450C2316yes—
81#15668450C2313yes—
82#15668450C2313yes—
83#15668449—23no—
84#15668449—23no—
85#15668449—26no—
86#15668449—26no—
87#15668449—23no—
88#15668449—23no—
89#15668449C2313yes—
90#15668449C2313yes—
91#15668449C2313yes—
92#15668448—26no—
93#15668448—23no—
94#15668448—23no—
95#15668448—23no—
96#15668448D413yes—
97#15668448C2313yes—
98#15668448C2311yescharacteristic
99C2×Dic362—23no—
100C2×Dic362—23no—
101C2×Dic362C2313yes—
102Q8×D181—23no—
103Q8×D181—23no—
104Q8×D181—23no—
105Q8×D181—23no—
106Q8×D181—21no—
107Q8×D181D411yescharacteristic
108Q8×D181C2316yes—
109Q8×D181C2313yes—
110Q8×D181C2313yes—
111C2×D724—23no—
112C2×D724—23no—
113C2×D724C2313yes—
114D4×D181—23no—
115D4×D181—23no—
116D4×D181—26no—
117D4×D181—23no—
118D4×D181D413yes—
119D4×D181C2313yes—
120D4×D181C2311yescharacteristic
121C2×C4×D181—23no—
122C2×C4×D181—23no—
123C2×C4×D181—23no—
124C2×C4×D181—23no—
125C2×C4×D181C2313yes—
126Dic724C2313yes—
127#15668439C2313yes—
128#15668439C2316yes—
129#15668439C2313yes—
130#15668438C2316yes—
131#15668438C2313yes—
132#15668438C2311yescharacteristic
133#15668437C2313yes—
134#15668437C2311yescharacteristic
135#15668436C2316yes—
136#15668436C2313yes—
137#15668436C2313yes—
138D1448C2313yes—
139#15668434C2311yescharacteristic
140#15668434C2313yes—
141#15668433C2313yes—
142#15668432C2313yes—
143#15668432C2311yescharacteristic
144#15668432C2313yes—
145#15668432C2313yes—
146C2×#13089678C2313yes—
147C8×D181C2316yes—
148#41×C181—23no—
149#41×C181—23no—
150#41×C181C2313yes—
151#41×C181C2313yes—
152#41×C181C2311yescharacteristic
153C2×Q8×C181—21no—
154C2×Q8×C181—21no—
155C2×Q8×C181C2313yes—
156C2×Q8×C181C2313yes—
157Q16×C181C2313yes—
158Q16×C181C2316yes—
159SD16×C181C2313yes—
160SD16×C181C2313yes—
161D8×C181C2311yescharacteristic
162M16×C181C2313yes—
163C2×C8×C181C2313yes—
164C22×D181—23no—
165C22×D181—21no—
166C22×D181—23no—
167C22×D181C2×D413yes—
168Dic362—23no—
169Dic362—23no—
170Dic362—26no—
171Dic362—23no—
172Dic362C2×D413yes—
173Dic362C2413yes—
174Dic362C2411yescharacteristic
175D724—23no—
176D724—26no—
177D724—23no—
178D724—23no—
179D724C2×D413yes—
180D724C2411yescharacteristic
181D724C2413yes—
182C22⋊D181—23no—
183C22⋊D181—23no—
184C22⋊D181—21no—
185C22⋊D181—23no—
186C22⋊D181—23no—
187C22⋊D181—21no—
188C22⋊D181C2×D416yes—
189C2×Dic181—23no—
190C2×Dic181—21no—
191C2×Dic181—23no—
192C2×Dic181C2×D413yes—
193C4×D181—23no—
194C4×D181—23no—
195C4×D181—26no—
196C4×D181—23no—
197C4×D181—23no—
198C4×D181—23no—
199C4×D181—26no—
200C4×D181—23no—
201C4×D181C2×D413yes—
202C4×D181C2×D413yes—
203C4×D181C2413yes—
204C4×D181C2411yescharacteristic
205#13089678C2413yes—
206#13089678C2411yescharacteristic
207Q8×C181—23no—
208Q8×C181—23no—
209Q8×C181C2×D411yescharacteristic
210Q8×C181C2413yes—
211Q8×C181C2413yes—
212D4×C181—23no—
213D4×C181—23no—
214D4×C181C2×D413yes—
215D4×C181C2411yescharacteristic
216D4×C181C2411yescharacteristic
217C2×C4×C181—23no—
218C2×C4×C181—23no—
219C2×C4×C181—23no—
220C2×C4×C181—23no—
221C2×C4×C181C2413yes—
222C8×C181C2411yescharacteristic
223C8×C181C2413yes—
224C22×C181—21no—
225C22×C181—21no—
226C22×C181C22×D413yes—
227C2×D181—21no—
228C2×D181—23no—
229C2×D181—43no—
230C2×D181—42no—
231C2×D181—83no—
232C2×D181—83no—
233C2×D181C22×D411yescharacteristic
234C2×D181C22×D413yes—
235C4×C181—23no—
236C4×C181—23no—
237C4×C181C22×D413yes—
238C4×C181C2511yescharacteristic
239Dic181—21no—
240Dic181—23no—
241Dic181C22×D411yescharacteristic
242Dic181C22×D413yes—
243C2×C181—23no—
244C2×C181—41no—
245C2×C181—41no—
246C2×C181C23×D411yescharacteristic
247D181—21no—
248D181—23no—
249D181—41no—
250D181—43no—
251C181#358611yescharacteristic, complemented, Sylow
252#3586—1811nomaximal, Sylow
253#585—1811no—
254C2×2−1+4—1811no—
255#578—1811no—
256#578—1813no—
257#578—1813no—
258#578D18111yescharacteristic, complemented
259#577—1811no—
260#577—1813no—
261#577—1813no—
262#577—1811no—
263#575—1813no—
264#575—1813no—
265C2×#137—1813no—
266C2×#137—1813no—
267#568—1811no—
2682−1+4—3621no—
2692−1+4—3621no—
2702−1+4—3623no—
2712−1+4—1813no—
2722−1+4—1811no—
2732−1+4—1813no—
2742−1+4—1813no—
2752−1+4C2×D18111yescharacteristic, complemented
2762−1+4C2×D18111yescharacteristic, complemented
277H(5,𝔽2)—3623no—
278H(5,𝔽2)—1813no—
279H(5,𝔽2)—1811no—
280C2×#41—3621no—
281C2×#41—3623no—
282C2×#41—3623no—
283C2×#41—1813no—
284C2×#41—1813no—
285C2×#41—1813no—
286C2×#41—1811no—
287C22×Q8—3621no—
288C22×Q8—1813no—
289#137—1813no—
290#137—1813no—
291#137—1813no—
292#137—1816no—
293#137—1813no—
294#137—1816no—
295#137—1816no—
296#137—1813no—
297#137—1813no—
298#137—1813no—
299#137—1813no—
300#137C2×D18113yescomplemented
301#137C2×D18113yescomplemented
302AGL(1,ℤ/8)—1813no—
303AGL(1,ℤ/8)—1813no—
304AGL(1,ℤ/8)—1813no—
305AGL(1,ℤ/8)—1811no—
306AGL(1,ℤ/8)—1813no—
307AGL(1,ℤ/8)—1813no—
308#135—1813no—
309#135—1813no—
310#135—1816no—
311#135—1816no—
312#135—1813no—
313#135C2×D18113yescomplemented
314C2×Q16—1813no—
315C2×Q16—1816no—
316C2×Q16C2×D18113yescomplemented
317C2×SD16—1813no—
318C2×SD16—1816no—
319C2×SD16—1813no—
320#131—1811no—
321#131—1813no—
322#131—1813no—
323#131C2×D18111yescharacteristic, complemented
324C2×M16—1813no—
325C2×M16—1813no—
326#41—23no—
327#41—23no—
328#41—3623no—
329#41—3626no—
330#41—3626no—
331#41—3623no—
332#41—3623no—
333#41—3623no—
334#41—3626no—
335#41—3621no—
336#41—3623no—
337#41—3623no—
338#41—3623no—
339#41—3623no—
340#41—3623no—
341#41—3623no—
342#41—3623no—
343#41—1813no—
344#41—1816no—
345#41—1813no—
346#41—1813no—
347#41—1811no—
348#41—1813no—
349#41—1813no—
350#41—1811no—
351#41—1813no—
352#41—1813no—
353#41C22×D18113yes—
354#41C22×D18113yes—
355#41C22×D18111yescharacteristic
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—3623no—
359C2×Q8—3623no—
360C2×Q8—3623no—
361C2×Q8—3623no—
362C2×Q8—3621no—
363C2×Q8—3623no—
364C2×Q8—3623no—
365C2×Q8—1816no—
366C2×Q8—1813no—
367C2×Q8—1813no—
368C2×Q8—1813no—
369C2×Q8—1811no—
370C2×Q8C22×D18113yes—
371C2×Q8C22×D18113yes—
372C2×D4—3623no—
373C2×D4—3623no—
374C2×D4—3623no—
375C2×D4—3623no—
376C2×D4—3626no—
377C2×D4—3623no—
378C2×D4—1813no—
379C2×D4—1811no—
380C2×D4—1813no—
381C2×D4—1813no—
382C22×C4—3623no—
383C22×C4—3623no—
384C22×C4—3623no—
385C22×C4—3623no—
386C22×C4—1813no—
387Q16—1813no—
388Q16—1816no—
389Q16—1813no—
390Q16—1813no—
391Q16C22×D18116yes—
392Q16C22×D18113yes—
393SD16—1813no—
394SD16—1816no—
395SD16—1813no—
396SD16—1813no—
397SD16—1811no—
398SD16—1816no—
399SD16—1813no—
400SD16—1811no—
401SD16C22×D18113yes—
402SD16C22×D18113yes—
403D8—1811no—
404D8—1813no—
405D8—1813no—
406D8C22×D18111yescharacteristic
407M16—1813no—
408M16—1813no—
409M16—1811no—
410M16—1813no—
411M16—1813no—
412M16C22×D18113yes—
413C2×C8—1813no—
414C2×C8—1816no—
415C2×C8C22×D18113yes—
416C23—3623no—
417C23—3621no—
418C23—3623no—
419C23—1813no—
420Q8—23no—
421Q8—23no—
422Q8—3623no—
423Q8—3626no—
424Q8—3623no—
425Q8—3623no—
426Q8—1813no—
427Q8—1811no—
428Q8—1813no—
429Q8D4×D18111yescharacteristic
430Q8C23×D18113yes—
431Q8C23×D18113yes—
432D4—23no—
433D4—23no—
434D4—3623no—
435D4—3626no—
436D4—3623no—
437D4—3621no—
438D4—3623no—
439D4—3623no—
440D4—3623no—
441D4—3623no—
442D4—3621no—
443D4—3623no—
444D4—1816no—
445D4—1813no—
446D4—1811no—
447D4—1813no—
448D4D4×D18113yes—
449D4C23×D18111yescharacteristic
450D4C23×D18111yescharacteristic
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—3623no—
456C2×C4—3623no—
457C2×C4—3626no—
458C2×C4—3623no—
459C2×C4—3626no—
460C2×C4—3623no—
461C2×C4—3623no—
462C2×C4—3621no—
463C2×C4—3623no—
464C2×C4—3623no—
465C2×C4—3623no—
466C2×C4—1813no—
467C2×C4—1813no—
468C2×C4—1813no—
469C2×C4—1813no—
470C2×C4—1811no—
471C2×C4C23×D18113yes—
472C8—1811no—
473C8—1813no—
474C8C23×D18113yes—
475C8C23×D18111yescharacteristic
476C22—21no—
477C22—21no—
478C22—3621no—
479C22—3623no—
480C22—14483no—
481C22—14483no—
482C22—1811no—
483C22—1813no—
484C22—7243no—
485C22—7242no—
486C22C2×D4×D18113yes—
487C4—23no—
488C4—23no—
489C4—3621no—
490C4—3623no—
491C4—1811no—
492C4—1813no—
493C4C2×D4×D18113yes—
494C4C24×D18111yescharacteristic
495C2—23no—
496C2—41no—
497C2—41no—
498C2—3623no—
499C2—3621no—
500C2—7243no—
501C2—7241no—
502C2C22×D4×D18111yescharacteristic
subgroupG/H
centerC2C22×D4×D181
commutator, cosocleC4×C181C25
FrattiniC4C24×D181
Fitting#578×C181C2
socleC2×C181C23×D4
hypercenter#578D181
hypocenterC181#3586
LCSG ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ #41 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #26042116C2semidirect
2. 2−1+4×D181C2semidirect
3. #26042095C2semidirect
4. #26042094C2semidirect
5. #26042092C2semidirect
6. #26042091C2semidirect
7. #26042088C2semidirect
8. #26042087C2semidirect
9. #26042086C2semidirect
10. #26042078C2semidirect
11. #26042076C2semidirect
12. C2×#19153848C2semidirect
13. #137×D181C2semidirect
14. #26042044C2semidirect
15. #578×C181C2semidirect
18. #19153864C22semidirect
19. #19153864C22semidirect
21. #19153863C22semidirect
22. #19153863C22semidirect
24. #19153862C22semidirect
25. #19153862C22semidirect
32. #41×D181C22semidirect
34. C2×Q8×D181C22semidirect
35. #19153849C22semidirect
36. #19153848C22semidirect
37. #19153848C22semidirect
38. #19153848C22semidirect
39. #19153848C22semidirect
40. #19153848C22semidirect
41. #19153847C22semidirect
42. #19153847C22semidirect
43. #19153846C22semidirect
44. #19153845C22semidirect
45. #19153845C22semidirect
46. #19153844C22semidirect
47. #19153843C22semidirect
48. #19153843C22semidirect
49. #19153841C22semidirect
50. #19153840C22semidirect
51. #19153840C22semidirect
52. #19153839C22semidirect
53. #19153838C22semidirect
54. #19153837C22semidirect
55. #19153836C22semidirect
56. #19153836C22semidirect
57. C2×#15668439C22semidirect
58. Q16×D181C22semidirect
59. C2×#15668438C22semidirect
60. C2×#15668436C22semidirect
61. SD16×D181C22semidirect
62. #19153824C22semidirect
63. #19153823C22semidirect
64. #19153822C22semidirect
65. C2×#15668432C22semidirect
66. M16×D181C22semidirect
67. 2−1+4×C181C22semidirect
68. 2−1+4×C181C22semidirect
69. #137×C181C22semidirect
70. #137×C181C22semidirect
71. #135×C181C22semidirect
72. C2×Q16×C181C22semidirect
73. #131×C181C22semidirect
251. C181#3586semidirect
258. #578D181semidirect
275. 2−1+4C2×D181semidirect
276. 2−1+4C2×D181semidirect
300. #137C2×D181semidirect
301. #137C2×D181semidirect
313. #135C2×D181semidirect
316. C2×Q16C2×D181semidirect
323. #131C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260421162
2−1+4×D1812
#260420952
#260420942
#260420922
#260420912
#260420882
#260420872
#260420862
#260420782
#260420762
C2×#191538482
#137×D1812
#260420442
#578×C1812
#3586181