Finite Group Databasebeta

← #46825657

Subgroups — #46825657

Order 34432.

1424 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771442C211yesmaximal, characteristic, complemented
2#31771440C211yesmaximal, characteristic, complemented
3#31771421C213yesmaximal, complemented
4#31771420C211yesmaximal, characteristic, complemented
5#31771417C211yesmaximal, characteristic, complemented
6#31771412C213yesmaximal, complemented
7#31771411C213yesmaximal, complemented
8#31771409C213yesmaximal, complemented
9#31771408C211yesmaximal, characteristic, complemented
10#31771404C213yesmaximal, complemented
11#31771401C213yesmaximal, complemented
12C2×#22575474C213yesmaximal, complemented
13#135×D269C213yesmaximal, complemented
14#131×D269C211yesmaximal, characteristic, complemented
15#577×C269C211yesmaximal, characteristic, complemented
16#22575501—21no—
17#22575501C2213yescomplemented
18#22575500—21no—
19#22575500C2211yescharacteristic, complemented
20#22575500C2213yescomplemented
21#22575499—23no—
22#22575499C2211yescharacteristic, complemented
23#22575499C2213yescomplemented
24#22575498—23no—
25#22575498C2213yescomplemented
26C2×#16864403—21no—
27C2×#16864403C2213yescomplemented
28C2×#16864401—23no—
29C2×#16864401C2213yescomplemented
30#41×D269—21no—
31#41×D269—23no—
32#41×D269C2211yescharacteristic
33#41×D269C2213yescomplemented
34#41×D269C2213yescomplemented
35#22575486C2213yescomplemented
36#22575485C2216yescomplemented
37#22575484C2213yescomplemented
38#22575483C2213yescomplemented
39#22575482C2216yescomplemented
40#22575481C2213yescomplemented
41#22575480C2216yescomplemented
42#22575479C2213yescomplemented
43#22575478C2213yescomplemented
44#22575477C2216yescomplemented
45#22575476C2213yescomplemented
46#22575475C2213yescomplemented
47#22575474C2216yescomplemented
48#22575474C2216yescomplemented
49#22575474C2216yescomplemented
50#22575474C2211yescharacteristic, complemented
51#22575474C2213yescomplemented
52#22575473C2213yescomplemented
53#22575473C2213yescomplemented
54#22575472C2213yescomplemented
55C2×#16864392C2213yescomplemented
56Q16×D269C2213yescomplemented
57C2×#16864391C2213yescomplemented
58SD16×D269C2213yescomplemented
59SD16×D269C2213yescomplemented
60C2×#16864387C2213yescomplemented
61D8×D269C2213yescomplemented
62#22575461C2213yescomplemented
63#22575460C2211yescharacteristic, complemented
64#22575459C2213yescomplemented
65M16×D269C2213yescomplemented
66C2×C8×D269C2213yescomplemented
672−1+4×C269C2211yescharacteristic, complemented
68H(5,𝔽2)×C269C2211yescharacteristic, complemented
69#137×C269C2213yescomplemented
70AGL(1,ℤ/8)×C269C2213yescomplemented
71#135×C269C2213yescomplemented
72C2×SD16×C269C2213yescomplemented
73#131×C269C2211yescharacteristic, complemented
74#16864403—23no—
75#16864403—23no—
76#16864403—23no—
77#16864403—21no—
78#16864403D411yescharacteristic
79#16864403C2313yes—
80#16864403C2316yes—
81#16864403C2311yescharacteristic
82#16864402—23no—
83#16864402—23no—
84#16864402—26no—
85#16864402—23no—
86#16864402—23no—
87#16864402C2313yes—
88#16864402C2313yes—
89#16864402C2313yes—
90#16864401—23no—
91#16864401—26no—
92#16864401—23no—
93#16864401—23no—
94#16864401—23no—
95#16864401—23no—
96#16864401—23no—
97#16864401D413yes—
98#16864401C2311yescharacteristic
99#16864401C2313yes—
100#16864401C2316yes—
101C2×Dic538—23no—
102C2×Dic538C2313yes—
103Q8×D269—23no—
104Q8×D269—21no—
105Q8×D269D411yescharacteristic
106Q8×D269C2313yes—
107Q8×D269C2313yes—
108C2×D1076—23no—
109C2×D1076—23no—
110C2×D1076C2313yes—
111C2×C22⋊D269—23no—
112C2×C22⋊D269—26no—
113D4×D269—23no—
114D4×D269—23no—
115D4×D269—23no—
116D4×D269—26no—
117D4×D269—23no—
118D4×D269D413yes—
119D4×D269C2313yes—
120D4×D269C2313yes—
121C22×Dic269—23no—
122C2×C4×D269—23no—
123C2×C4×D269—23no—
124C2×C4×D269—23no—
125C2×C4×D269C2313yes—
126Dic1076C2313yes—
127#16864392C2311yescharacteristic
128#16864392C2313yes—
129#16864392C2316yes—
130#16864391C2311yescharacteristic
131#16864391C2313yes—
132#16864391C2316yes—
133#16864390C2313yes—
134#16864390C2313yes—
135#16864389C2313yes—
136#16864389C2313yes—
137D2152C2313yes—
138#16864387C2311yescharacteristic
139#16864387C2313yes—
140#16864387C2316yes—
141#16864386C2313yes—
142#16864385C2316yes—
143C2×#14524130C2313yes—
144C8×D269C2311yescharacteristic
145C8×D269C2313yes—
146C8×D269C2313yes—
147C8×D269C2313yes—
148#41×C269—21no—
149#41×C269—23no—
150#41×C269C2313yes—
151#41×C269C2313yes—
152#41×C269C2311yescharacteristic
153C2×Q8×C269—21no—
154C2×Q8×C269C2313yes—
155C2×D4×C269—23no—
156C2×D4×C269C2313yes—
157Q16×C269C2313yes—
158SD16×C269C2311yescharacteristic
159SD16×C269C2313yes—
160SD16×C269C2316yes—
161D8×C269C2313yes—
162M16×C269C2313yes—
163C2×C8×C269C2313yes—
164C22×D269—23no—
165C22×D269—23no—
166C22×D269—23no—
167C22×D269C2×D413yes—
168Dic538—23no—
169Dic538—23no—
170Dic538—23no—
171Dic538C2×D413yes—
172Dic538C2411yescharacteristic
173Dic538C2413yes—
174D1076—21no—
175D1076—26no—
176D1076—23no—
177D1076—23no—
178D1076C2×D413yes—
179D1076C2413yes—
180D1076C2411yescharacteristic
181C22⋊D269—23no—
182C22⋊D269—23no—
183C22⋊D269—21no—
184C22⋊D269—26no—
185C22⋊D269—23no—
186C22⋊D269—23no—
187C22⋊D269—23no—
188C22⋊D269C2×D416yes—
189C2×Dic269—23no—
190C2×Dic269—21no—
191C2×Dic269—23no—
192C2×Dic269—26no—
193C2×Dic269C2×D413yes—
194C4×D269—23no—
195C4×D269—23no—
196C4×D269—23no—
197C4×D269—23no—
198C4×D269—21no—
199C4×D269—23no—
200C4×D269—26no—
201C4×D269C2×D413yes—
202C4×D269C2×D413yes—
203C4×D269C2411yescharacteristic
204C4×D269C2413yes—
205#14524130C2411yescharacteristic
206#14524130C2413yes—
207C23×C269—23no—
208Q8×C269—23no—
209Q8×C269C2×D411yescharacteristic
210Q8×C269C2413yes—
211Q8×C269C2411yescharacteristic
212D4×C269—23no—
213D4×C269—23no—
214D4×C269—23no—
215D4×C269C2×D413yes—
216D4×C269C2411yescharacteristic
217D4×C269C2413yes—
218C2×C4×C269—23no—
219C2×C4×C269—23no—
220C2×C4×C269—23no—
221C2×C4×C269C2413yes—
222C8×C269C2411yescharacteristic
223C8×C269C2413yes—
224C22×C269—21no—
225C22×C269—23no—
226C22×C269—83no—
227C22×C269C22×D413yes—
228C2×D269—21no—
229C2×D269—23no—
230C2×D269—43no—
231C2×D269—83no—
232C2×D269—83no—
233C2×D269—83no—
234C2×D269C22×D413yes—
235C2×D269C22×D411yescharacteristic
236C4×C269—21no—
237C4×C269—23no—
238C4×C269C22×D413yes—
239C4×C269C2511yescharacteristic
240Dic269—21no—
241Dic269—23no—
242Dic269C22×D411yescharacteristic
243Dic269C22×D413yes—
244C2×C269—23no—
245C2×C269—41no—
246C2×C269—43no—
247C2×C269C23×D411yescharacteristic
248D269—21no—
249D269—23no—
250D269—41no—
251D269—43no—
252C269#358411yescharacteristic, complemented, Sylow
253#3584—2691nomaximal, Sylow
254#585—2691no—
255#585—2691no—
256#578—2691no—
257#578—2693no—
258#577—2693no—
259#577—2693no—
260#577—2691no—
261#577D26911yescharacteristic, complemented
262#576—2691no—
263#576—2693no—
264#575—2693no—
265#575—2693no—
266C2×#135—2693no—
267C2×#135—2693no—
268C2×#131—2691no—
2692−1+4—2693no—
2702−1+4—2691no—
2712−1+4—2693no—
2722−1+4—5381no—
2732−1+4—5381no—
2742−1+4C2×D26911yescharacteristic, complemented
275H(5,𝔽2)—2693no—
276H(5,𝔽2)—2691no—
277H(5,𝔽2)—2693no—
278H(5,𝔽2)—5383no—
279H(5,𝔽2)—5383no—
280H(5,𝔽2)C2×D26911yescharacteristic, complemented
281C2×#41—2691no—
282C2×#41—2693no—
283C2×#41—2693no—
284C2×#41—2693no—
285C2×#41—2693no—
286C2×#41—5383no—
287C2×#41—5381no—
288C2×#41—5381no—
289C2×#41—5383no—
290#137—2693no—
291#137—2693no—
292#137—2696no—
293#137—2696no—
294#137—2693no—
295#137C2×D26913yescomplemented
296AGL(1,ℤ/8)—2693no—
297AGL(1,ℤ/8)—2693no—
298AGL(1,ℤ/8)—2696no—
299AGL(1,ℤ/8)—2696no—
300AGL(1,ℤ/8)—2693no—
301AGL(1,ℤ/8)C2×D26913yescomplemented
302#135—2693no—
303#135—2693no—
304#135—2693no—
305#135—2691no—
306#135—2693no—
307#135—2696no—
308#135—2696no—
309#135—2696no—
310#135—2693no—
311#135—2693no—
312#135C2×D26913yescomplemented
313C2×Q16—2693no—
314C2×Q16—2693no—
315C2×SD16—2693no—
316C2×SD16—2693no—
317C2×SD16—2693no—
318C2×SD16C2×D26913yescomplemented
319C2×D8—2693no—
320C2×D8—2693no—
321#131—2691no—
322#131—2693no—
323#131—2693no—
324#131C2×D26911yescharacteristic, complemented
325C2×M16—2693no—
326C22×C8—2693no—
327#41—21no—
328#41—23no—
329#41—2691no—
330#41—2693no—
331#41—2693no—
332#41—2691no—
333#41—2693no—
334#41—2696no—
335#41—2693no—
336#41—2691no—
337#41—2696no—
338#41—2693no—
339#41—2693no—
340#41—5383no—
341#41—5383no—
342#41—5386no—
343#41—5383no—
344#41—5383no—
345#41—5383no—
346#41—5383no—
347#41—5383no—
348#41—5383no—
349#41—5383no—
350#41—5383no—
351#41—5386no—
352#41—5383no—
353#41—5383no—
354#41—5381no—
355#41—5383no—
356#41C22×D26913yes—
357#41C22×D26913yes—
358#41C22×D26911yescharacteristic
359C2×Q8—21no—
360C2×Q8—2691no—
361C2×Q8—2693no—
362C2×Q8—2693no—
363C2×Q8—2693no—
364C2×Q8—5383no—
365C2×Q8—5383no—
366C2×Q8—5381no—
367C2×Q8C22×D26913yes—
368C2×D4—23no—
369C2×D4—2693no—
370C2×D4—2693no—
371C2×D4—2693no—
372C2×D4—2693no—
373C2×D4—5383no—
374C2×D4—5383no—
375C2×D4—5386no—
376C2×D4—5383no—
377C2×D4—5383no—
378C2×D4—5383no—
379C2×D4—5386no—
380C2×D4—5383no—
381C2×D4—5383no—
382C2×D4C22×D26913yes—
383C22×C4—2693no—
384C22×C4—5383no—
385C22×C4—5383no—
386C22×C4—5383no—
387C22×C4—5383no—
388Q16—2691no—
389Q16—2693no—
390Q16—2696no—
391Q16—2693no—
392Q16C22×D26913yes—
393SD16—2693no—
394SD16—2693no—
395SD16—2693no—
396SD16—2693no—
397SD16—2691no—
398SD16—2693no—
399SD16—2696no—
400SD16C22×D26911yescharacteristic
401SD16C22×D26913yes—
402SD16C22×D26916yes—
403D8—2693no—
404D8—2691no—
405D8—2696no—
406D8—2693no—
407D8C22×D26913yes—
408M16—2693no—
409M16—2696no—
410M16C22×D26913yes—
411C2×C8—2693no—
412C2×C8—2691no—
413C2×C8—2693no—
414C2×C8—2693no—
415C2×C8—2693no—
416C2×C8C22×D26913yes—
417C23—23no—
418C23—2693no—
419C23—5383no—
420C23—5383no—
421C23—5383no—
422Q8—23no—
423Q8—2693no—
424Q8—2691no—
425Q8—2693no—
426Q8—5383no—
427Q8—5383no—
428Q8—5383no—
429Q8D4×D26911yescharacteristic
430Q8C23×D26911yescharacteristic
431Q8C23×D26913yes—
432D4—23no—
433D4—23no—
434D4—23no—
435D4—2696no—
436D4—2693no—
437D4—2693no—
438D4—2691no—
439D4—5381no—
440D4—5383no—
441D4—5386no—
442D4—5383no—
443D4—5383no—
444D4—5381no—
445D4—5383no—
446D4—5386no—
447D4—5383no—
448D4—5383no—
449D4—5383no—
450D4D4×D26913yes—
451D4C23×D26913yes—
452D4C23×D26911yescharacteristic
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—23no—
456C2×C4—2693no—
457C2×C4—2693no—
458C2×C4—2693no—
459C2×C4—2691no—
460C2×C4—2693no—
461C2×C4—5381no—
462C2×C4—5383no—
463C2×C4—5386no—
464C2×C4—5383no—
465C2×C4—5383no—
466C2×C4—5381no—
467C2×C4—5383no—
468C2×C4—5386no—
469C2×C4—5383no—
470C2×C4—5383no—
471C2×C4—5383no—
472C2×C4C23×D26913yes—
473C8—2691no—
474C8—2693no—
475C8C23×D26911yescharacteristic
476C8C23×D26913yes—
477C22—21no—
478C22—23no—
479C22—83no—
480C22—2691no—
481C22—2693no—
482C22—5383no—
483C22—5381no—
484C22—10763no—
485C22—21523no—
486C22—21523no—
487C22—21523no—
488C22C2×D4×D26913yes—
489C4—21no—
490C4—23no—
491C4—2691no—
492C4—2693no—
493C4—5381no—
494C4—5383no—
495C4C2×D4×D26913yes—
496C4C24×D26911yescharacteristic
497C2—23no—
498C2—43no—
499C2—41no—
500C2—5381no—
501C2—5383no—
502C2—10763no—
503C2—10761no—
504C2C22×D4×D26911yescharacteristic
subgroupG/H
centerC2C22×D4×D269
commutator, cosocleC4×C269C25
FrattiniC4C24×D269
Fitting#577×C269C2
socleC2×C269C23×D4
hypercenter#577D269
hypocenterC269#3584
LCSG ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ #41 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #31771442C2semidirect
2. #31771440C2semidirect
3. #31771421C2semidirect
4. #31771420C2semidirect
5. #31771417C2semidirect
6. #31771412C2semidirect
7. #31771411C2semidirect
8. #31771409C2semidirect
9. #31771408C2semidirect
10. #31771404C2semidirect
11. #31771401C2semidirect
12. C2×#22575474C2semidirect
13. #135×D269C2semidirect
14. #131×D269C2semidirect
15. #577×C269C2semidirect
17. #22575501C22semidirect
19. #22575500C22semidirect
20. #22575500C22semidirect
22. #22575499C22semidirect
23. #22575499C22semidirect
25. #22575498C22semidirect
27. C2×#16864403C22semidirect
29. C2×#16864401C22semidirect
33. #41×D269C22semidirect
34. #41×D269C22semidirect
35. #22575486C22semidirect
36. #22575485C22semidirect
37. #22575484C22semidirect
38. #22575483C22semidirect
39. #22575482C22semidirect
40. #22575481C22semidirect
41. #22575480C22semidirect
42. #22575479C22semidirect
43. #22575478C22semidirect
44. #22575477C22semidirect
45. #22575476C22semidirect
46. #22575475C22semidirect
47. #22575474C22semidirect
48. #22575474C22semidirect
49. #22575474C22semidirect
50. #22575474C22semidirect
51. #22575474C22semidirect
52. #22575473C22semidirect
53. #22575473C22semidirect
54. #22575472C22semidirect
55. C2×#16864392C22semidirect
56. Q16×D269C22semidirect
57. C2×#16864391C22semidirect
58. SD16×D269C22semidirect
59. SD16×D269C22semidirect
60. C2×#16864387C22semidirect
61. D8×D269C22semidirect
62. #22575461C22semidirect
63. #22575460C22semidirect
64. #22575459C22semidirect
65. M16×D269C22semidirect
66. C2×C8×D269C22semidirect
67. 2−1+4×C269C22semidirect
68. H(5,𝔽2)×C269C22semidirect
69. #137×C269C22semidirect
70. AGL(1,ℤ/8)×C269C22semidirect
71. #135×C269C22semidirect
72. C2×SD16×C269C22semidirect
73. #131×C269C22semidirect
252. C269#3584semidirect
261. #577D269semidirect
274. 2−1+4C2×D269semidirect
280. H(5,𝔽2)C2×D269semidirect
295. #137C2×D269semidirect
301. AGL(1,ℤ/8)C2×D269semidirect
312. #135C2×D269semidirect
318. C2×SD16C2×D269semidirect
324. #131C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714422
#317714402
#317714212
#317714202
#317714172
#317714122
#317714112
#317714092
#317714082
#317714042
#317714012
C2×#225754742
#135×D2692
#131×D2692
#577×C2692
#3584269