Finite Group Databasebeta

← #46825667

Subgroups — #46825667

Order 34432.

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#31771440C211yesmaximal, characteristic, complemented
22−1+4×D269C211yesmaximal, characteristic, complemented
3#31771422C211yesmaximal, characteristic, complemented
4#31771421C213yesmaximal, complemented
5#31771420C211yesmaximal, characteristic, complemented
6#31771419C213yesmaximal, complemented
7#31771416C211yesmaximal, characteristic, complemented
8#31771415C213yesmaximal, complemented
9#31771411C213yesmaximal, complemented
10#31771402C213yesmaximal, complemented
11#31771401C213yesmaximal, complemented
12C2×#22575482C213yesmaximal, complemented
13#137×D269C213yesmaximal, complemented
14#31771370C211yesmaximal, characteristic, complemented
15#577×C269C211yesmaximal, characteristic, complemented
16#22575501—21no—
17#22575501C2213yescomplemented
18#22575500—23no—
19#22575500—21no—
20#22575500C2211yescharacteristic, complemented
21#22575500C2211yescharacteristic, complemented
22#22575500C2213yescomplemented
23#22575500C2213yescomplemented
24#22575498—23no—
25#22575498C2213yescomplemented
26C2×#16864401—23no—
27C2×#16864401C2213yes—
28#41×D269—21no—
29#41×D269—23no—
30#41×D269C2213yes—
31#41×D269C2211yescharacteristic
32#41×D269C2213yescomplemented
33C2×Q8×D269—21no—
34C2×Q8×D269C2213yescomplemented
35#22575486C2213yescomplemented
36#22575486C2213yescomplemented
37#22575485C2213yescomplemented
38#22575485C2213yescomplemented
39#22575484C2213yescomplemented
40#22575483C2213yescomplemented
41#22575483C2213yescomplemented
42#22575482C2216yescomplemented
43#22575482C2213yescomplemented
44#22575482C2216yescomplemented
45#22575482C2216yescomplemented
46#22575482C2213yescomplemented
47#22575480C2211yescharacteristic, complemented
48#22575480C2213yescomplemented
49#22575479C2213yescomplemented
50#22575477C2213yescomplemented
51#22575477C2213yescomplemented
52#22575475C2213yescomplemented
53#22575474C2216yescomplemented
54#22575473C2213yescomplemented
55#22575473C2213yescomplemented
56#22575472C2216yescomplemented
57C2×Dic1076C2213yescomplemented
58C2×#16864392C2213yescomplemented
59Q16×D269C2216yescomplemented
60C2×#16864390C2213yescomplemented
61SD16×D269C2216yescomplemented
62#22575461C2213yescomplemented
63#22575460C2211yescharacteristic, complemented
64#22575459C2213yescomplemented
65C2×#16864385C2213yescomplemented
66M16×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#16864403D411yescharacteristic
76#16864403C2313yes—
77#16864403C2311yescharacteristic
78#16864402—23no—
79#16864402—23no—
80#16864402—26no—
81#16864402—23no—
82#16864402—23no—
83#16864402C2313yes—
84#16864402C2313yes—
85#16864402C2313yes—
86#16864401—23no—
87#16864401—26no—
88#16864401—26no—
89#16864401—23no—
90#16864401—23no—
91#16864401—23no—
92#16864401—23no—
93#16864401—23no—
94#16864401—23no—
95#16864401D413yes—
96#16864401C2313yes—
97#16864401C2316yes—
98#16864401C2313yes—
99C2×Dic538—23no—
100C2×Dic538—23no—
101C2×Dic538—23no—
102C2×Dic538C2313yes—
103C2×Dic538C2313yes—
104Q8×D269—23no—
105Q8×D269—23no—
106Q8×D269—23no—
107Q8×D269—21no—
108Q8×D269—21no—
109Q8×D269D411yescharacteristic
110Q8×D269C2316yes—
111Q8×D269C2313yes—
112Q8×D269C2313yes—
113C2×C22⋊D269—23no—
114C2×C22⋊D269—26no—
115D4×D269—23no—
116D4×D269—23no—
117D4×D269—23no—
118D4×D269D413yes—
119D4×D269C2313yes—
120D4×D269C2311yescharacteristic
121C22×Dic269—23no—
122C2×C4×D269—23no—
123C2×C4×D269—23no—
124C2×C4×D269—23no—
125C2×C4×D269C2313yes—
126Dic1076C2316yes—
127Dic1076C2313yes—
128#16864392C2313yes—
129#16864392C2316yes—
130#16864392C2313yes—
131#16864391C2313yes—
132#16864391C2313yes—
133#16864390C2316yes—
134#16864390C2313yes—
135#16864390C2313yes—
136#16864389C2313yes—
137#16864389C2311yescharacteristic
138D2152C2311yescharacteristic
139#16864387C2311yescharacteristic
140#16864387C2313yes—
141#16864386C2313yes—
142#16864385C2313yes—
143#16864385C2311yescharacteristic
144#16864385C2313yes—
145#16864385C2313yes—
146C2×#14524130C2313yes—
147C8×D269C2316yes—
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×C269C2316yes—
159SD16×C269C2313yes—
160SD16×C269C2311yescharacteristic
161D8×C269C2313yes—
162M16×C269C2313yes—
163C2×C8×C269C2313yes—
164C22×D269—21no—
165C22×D269—23no—
166C22×D269C2×D413yes—
167Dic538—23no—
168Dic538—23no—
169Dic538—23no—
170Dic538—26no—
171Dic538—23no—
172Dic538C2×D413yes—
173Dic538C2413yes—
174Dic538C2413yes—
175D1076—21no—
176D1076—23no—
177D1076C2×D413yes—
178D1076C2411yescharacteristic
179D1076C2411yescharacteristic
180C22⋊D269—23no—
181C22⋊D269—23no—
182C22⋊D269—23no—
183C22⋊D269—26no—
184C22⋊D269—23no—
185C22⋊D269—23no—
186C22⋊D269—21no—
187C22⋊D269C2×D416yes—
188C2×Dic269—23no—
189C2×Dic269—23no—
190C2×Dic269—23no—
191C2×Dic269—23no—
192C2×Dic269—26no—
193C2×Dic269C2×D413yes—
194C4×D269—23no—
195C4×D269—23no—
196C4×D269—26no—
197C4×D269—23no—
198C4×D269—23no—
199C4×D269—21no—
200C4×D269—23no—
201C4×D269C2×D413yes—
202C4×D269C2×D413yes—
203C4×D269C2413yes—
204C4×D269C2411yescharacteristic
205#14524130C2413yes—
206#14524130C2411yescharacteristic
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—21no—
230C2×D269—43no—
231C2×D269—42no—
232C2×D269—83no—
233C2×D269C22×D411yescharacteristic
234C2×D269C22×D413yes—
235C4×C269—21no—
236C4×C269—23no—
237C4×C269C22×D413yes—
238C4×C269C2511yescharacteristic
239Dic269—23no—
240Dic269—23no—
241Dic269C22×D411yescharacteristic
242Dic269C22×D413yes—
243C2×C269—23no—
244C2×C269—41no—
245C2×C269—43no—
246C2×C269C23×D411yescharacteristic
247D269—21no—
248D269—23no—
249D269—41no—
250D269—41no—
251C269#358611yescharacteristic, complemented, Sylow
252#3586—2691nomaximal, Sylow
253#585—2691no—
254C2×2−1+4—2691no—
255#578—2691no—
256#578—2693no—
257#578—2693no—
258#578—2691no—
259#577—2691no—
260#577—2693no—
261#577—2693no—
262#577D26911yescharacteristic, complemented
263#575—2693no—
264#575—2693no—
265C2×#137—2693no—
266C2×#137—2693no—
267#568—2691no—
2682−1+4—2693no—
2692−1+4—2693no—
2702−1+4—2691no—
2712−1+4—2691no—
2722−1+4—2693no—
2732−1+4—5381no—
2742−1+4—5381no—
2752−1+4—5383no—
2762−1+4C2×D26911yescharacteristic, complemented
277H(5,𝔽2)—2693no—
278H(5,𝔽2)—5383no—
279H(5,𝔽2)C2×D26911yescharacteristic, complemented
280C2×#41—2691no—
281C2×#41—2693no—
282C2×#41—2693no—
283C2×#41—2693no—
284C2×#41—5383no—
285C2×#41—5383no—
286C2×#41—5381no—
287C22×Q8—2693no—
288C22×Q8—5381no—
289#137—2693no—
290#137—2693no—
291#137—2693no—
292#137—2693no—
293#137—2693no—
294#137—2696no—
295#137—2693no—
296#137—2696no—
297#137—2696no—
298#137—2693no—
299#137—2693no—
300#137—2693no—
301#137C2×D26913yescomplemented
302AGL(1,ℤ/8)—2693no—
303AGL(1,ℤ/8)—2691no—
304AGL(1,ℤ/8)—2693no—
305AGL(1,ℤ/8)—2693no—
306AGL(1,ℤ/8)—2693no—
307AGL(1,ℤ/8)C2×D26913yescomplemented
308#135—2693no—
309#135—2693no—
310#135—2696no—
311#135—2696no—
312#135—2693no—
313#135C2×D26913yescomplemented
314C2×Q16—2693no—
315C2×Q16—2696no—
316C2×Q16—2693no—
317C2×SD16—2693no—
318C2×SD16—2696no—
319C2×SD16C2×D26913yescomplemented
320#131—2691no—
321#131—2693no—
322#131—2693no—
323#131C2×D26911yescharacteristic, complemented
324C2×M16—2693no—
325C2×M16—2693no—
326#41—21no—
327#41—23no—
328#41—2691no—
329#41—2693no—
330#41—2693no—
331#41—2693no—
332#41—2691no—
333#41—2693no—
334#41—2696no—
335#41—2693no—
336#41—2693no—
337#41—2693no—
338#41—5383no—
339#41—5383no—
340#41—5383no—
341#41—5386no—
342#41—5383no—
343#41—5383no—
344#41—5383no—
345#41—5383no—
346#41—5383no—
347#41—5386no—
348#41—5386no—
349#41—5383no—
350#41—5383no—
351#41—5383no—
352#41—5383no—
353#41C22×D26913yes—
354#41C22×D26913yes—
355#41C22×D26911yescharacteristic
356C2×Q8—21no—
357C2×Q8—2691no—
358C2×Q8—2696no—
359C2×Q8—2693no—
360C2×Q8—2693no—
361C2×Q8—2693no—
362C2×Q8—2693no—
363C2×Q8—5383no—
364C2×Q8—5383no—
365C2×Q8—5383no—
366C2×Q8—5383no—
367C2×Q8—5383no—
368C2×Q8—5383no—
369C2×Q8—5381no—
370C2×Q8—5381no—
371C2×Q8C22×D26913yes—
372C2×D4—23no—
373C2×D4—2693no—
374C2×D4—2693no—
375C2×D4—2691no—
376C2×D4—5383no—
377C2×D4—5383no—
378C2×D4—5383no—
379C2×D4—5386no—
380C2×D4—5383no—
381C2×D4C22×D26913yes—
382C22×C4—2693no—
383C22×C4—5383no—
384C22×C4—5383no—
385C22×C4—5383no—
386C22×C4—5383no—
387Q16—2693no—
388Q16—2693no—
389Q16—2696no—
390Q16—2696no—
391Q16—2693no—
392Q16C22×D26913yes—
393SD16—2693no—
394SD16—2691no—
395SD16—2693no—
396SD16—2696no—
397SD16—2693no—
398SD16—2693no—
399SD16—2693no—
400SD16C22×D26916yes—
401SD16C22×D26913yes—
402SD16C22×D26911yescharacteristic
403D8—2693no—
404D8—2691no—
405D8—2691no—
406D8C22×D26913yes—
407M16—2693no—
408M16—2693no—
409M16—2691no—
410M16—2693no—
411M16—2693no—
412M16C22×D26913yes—
413C2×C8—2693no—
414C2×C8—2696no—
415C2×C8C22×D26913yes—
416C23—23no—
417C23—2693no—
418C23—5381no—
419C23—5383no—
420Q8—23no—
421Q8—2693no—
422Q8—2693no—
423Q8—2693no—
424Q8—5383no—
425Q8—5386no—
426Q8—5383no—
427Q8—5383no—
428Q8—5383no—
429Q8D4×D26911yescharacteristic
430Q8C23×D26913yes—
431Q8C23×D26911yescharacteristic
432D4—23no—
433D4—23no—
434D4—23no—
435D4—2696no—
436D4—2693no—
437D4—2691no—
438D4—2691no—
439D4—5381no—
440D4—5383no—
441D4—5383no—
442D4—5381no—
443D4—5383no—
444D4—5386no—
445D4—5383no—
446D4—5383no—
447D4—5383no—
448D4D4×D26913yes—
449D4C23×D26911yescharacteristic
450D4C23×D26913yes—
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—2693no—
455C2×C4—2693no—
456C2×C4—2693no—
457C2×C4—2693no—
458C2×C4—2691no—
459C2×C4—5381no—
460C2×C4—5383no—
461C2×C4—5383no—
462C2×C4—5383no—
463C2×C4—5386no—
464C2×C4—5383no—
465C2×C4—5386no—
466C2×C4—5383no—
467C2×C4—5383no—
468C2×C4—5383no—
469C2×C4—5383no—
470C2×C4—5383no—
471C2×C4C23×D26913yes—
472C8—2691no—
473C8—2693no—
474C8C23×D26913yes—
475C8C23×D26911yescharacteristic
476C22—21no—
477C22—23no—
478C22—83no—
479C22—2691no—
480C22—2693no—
481C22—5381no—
482C22—5381no—
483C22—10763no—
484C22—10762no—
485C22—21523no—
486C22C2×D4×D26913yes—
487C4—21no—
488C4—23no—
489C4—2691no—
490C4—2693no—
491C4—5383no—
492C4—5383no—
493C4C2×D4×D26913yes—
494C4C24×D26911yescharacteristic
495C2—23no—
496C2—43no—
497C2—41no—
498C2—5381no—
499C2—5383no—
500C2—10761no—
501C2—10761no—
502C2C22×D4×D26911yescharacteristic
subgroupG/H
centerC2C22×D4×D269
commutator, cosocleC4×C269C25
FrattiniC4C24×D269
Fitting#577×C269C2
socleC2×C269C23×D4
hypercenter#577D269
hypocenterC269#3586
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. #31771440C2semidirect
2. 2−1+4×D269C2semidirect
3. #31771422C2semidirect
4. #31771421C2semidirect
5. #31771420C2semidirect
6. #31771419C2semidirect
7. #31771416C2semidirect
8. #31771415C2semidirect
9. #31771411C2semidirect
10. #31771402C2semidirect
11. #31771401C2semidirect
12. C2×#22575482C2semidirect
13. #137×D269C2semidirect
14. #31771370C2semidirect
15. #577×C269C2semidirect
17. #22575501C22semidirect
20. #22575500C22semidirect
21. #22575500C22semidirect
22. #22575500C22semidirect
23. #22575500C22semidirect
25. #22575498C22semidirect
32. #41×D269C22semidirect
34. C2×Q8×D269C22semidirect
35. #22575486C22semidirect
36. #22575486C22semidirect
37. #22575485C22semidirect
38. #22575485C22semidirect
39. #22575484C22semidirect
40. #22575483C22semidirect
41. #22575483C22semidirect
42. #22575482C22semidirect
43. #22575482C22semidirect
44. #22575482C22semidirect
45. #22575482C22semidirect
46. #22575482C22semidirect
47. #22575480C22semidirect
48. #22575480C22semidirect
49. #22575479C22semidirect
50. #22575477C22semidirect
51. #22575477C22semidirect
52. #22575475C22semidirect
53. #22575474C22semidirect
54. #22575473C22semidirect
55. #22575473C22semidirect
56. #22575472C22semidirect
57. C2×Dic1076C22semidirect
58. C2×#16864392C22semidirect
59. Q16×D269C22semidirect
60. C2×#16864390C22semidirect
61. SD16×D269C22semidirect
62. #22575461C22semidirect
63. #22575460C22semidirect
64. #22575459C22semidirect
65. C2×#16864385C22semidirect
66. M16×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
251. C269#3586semidirect
262. #577D269semidirect
276. 2−1+4C2×D269semidirect
279. H(5,𝔽2)C2×D269semidirect
301. #137C2×D269semidirect
307. AGL(1,ℤ/8)C2×D269semidirect
313. #135C2×D269semidirect
319. C2×SD16C2×D269semidirect
323. #131C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714402
2−1+4×D2692
#317714222
#317714212
#317714202
#317714192
#317714162
#317714152
#317714112
#317714022
#317714012
C2×#225754822
#137×D2692
#317713702
#577×C2692
#3586269