Finite Group Databasebeta

← #6344601

Subgroups — #6344601

Order 512.

1789 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60343C211yesmaximal, characteristic, complemented
2#60325C211yesmaximal, characteristic, complemented
3C4×#577C211yesmaximal, characteristic
4#17898C213yesmaximal, complemented
5#17896C213yesmaximal, complemented
6#17894C213yesmaximal, complemented
7#17893C213yesmaximal, complemented
8#17892C213yesmaximal, complemented
9#17891C213yesmaximal, complemented
10#17890C213yesmaximal, complemented
11C2×#1592C213yesmaximal, complemented
12#17789C211yesmaximal, characteristic, complemented
13#17439C211yesmaximal, characteristic, complemented
14#17438C211yesmaximal, characteristic, complemented
15#17316C211yesmaximal, characteristic, complemented
16C2×#577C411yescharacteristic
17C4×2−1+4C2211yescharacteristic
18C4×H(5,𝔽2)C2211yescharacteristic
19#2991—21no—
20#2990—23no—
21#2989C2213yes—
22#2989C2213yescomplemented
23#2988C2211yescharacteristic
24#2988C2211yescharacteristic
25#2984—21no—
26#2983C2213yescomplemented
27#2975C2213yes—
28#2974—23no—
29#2971—21no—
30#2970—23no—
31#2965C2213yes—
32#2965C2213yescomplemented
33C8×#41C2211yescharacteristic
34C8×#41C2211yescharacteristic
35Q8×M16—21no—
36C2×#446C2213yescomplemented
37#2952C413yes—
38#2949C413yes—
39#2947C413yes—
40#2946C413yes—
41C4×#137C2213yes—
42C4×AGL(1,ℤ/8)C2213yes—
43C4×#135C2213yes—
44C2×C4×SD16C2213yes—
45D4×M16—23no—
46C2×#436C2213yes—
47#2912C411yescharacteristic
48#2898C411yescharacteristic
49#2897C411yescharacteristic
50C4×#131C2211yescharacteristic
51#1594C2213yescomplemented
52#1594C2213yescomplemented
53#1593C2213yescomplemented
54#1593C2213yescomplemented
55#1592C2213yescomplemented
56#1592C2216yescomplemented
57#1592C2216yescomplemented
58#1592C2216yescomplemented
59#1592C2211yescharacteristic, complemented
60#1592C2213yescomplemented
61#1592C2213yescomplemented
62#1591C2216yescomplemented
63#1590C2216yescomplemented
64#1590C2216yescomplemented
65#1589C2216yescomplemented
66#1565C2213yescomplemented
67#1565C2213yescomplemented
68#1564C2213yescomplemented
69C2×#335C2213yescomplemented
70#1479C2213yescomplemented
71#1478C2213yes—
72#1477C2213yes—
73#1477C2213yescomplemented
74#1476C2213yes—
75#1476C2213yescomplemented
76C2×Q8⋊C8C2213yescomplemented
77C2×D4⋊C8C2213yes—
78#1450C2213yescomplemented
79C8×M16C2213yescomplemented
80C8×M16C2213yescomplemented
81C2×#322C2213yescomplemented
82C2×2−1+4C2×C411yescharacteristic
83C2×H(5,𝔽2)C2×C411yescharacteristic
84#577—28no—
85C2×#137C2×C413yes—
86C2×AGL(1,ℤ/8)C2×C413yes—
87C2×#135C2×C413yes—
88C22×SD16C2×C413yes—
89C2×#131C2×C411yescharacteristic
90C2×#131C2×C411yescharacteristic
91C22×M16—23no—
92#520—21no—
93#520—23no—
94#520C2×C413yes—
95#520C2×C413yes—
96#520C2×C411yescharacteristic
97#520C2×C411yescharacteristic
98#519—21no—
99#519C2×C413yes—
100#518—23no—
101#518C2×C413yes—
102C4×#41—21no—
103C4×#41—23no—
104C4×#41C2313yes—
105C4×#41C2313yes—
106C4×#41C2311yescharacteristic
107C2×C4×Q8—21no—
108C2×C4×Q8C2313yes—
109C2×C4×D4—23no—
110C2×C4×D4C2313yes—
111#446—23no—
112#446—23no—
113#446—23no—
114#446—23no—
115#446C2313yes—
116#446C2316yes—
117#446C2313yes—
118#446C2313yes—
119#446C2311yescharacteristic
120C8×Q8—21no—
121C8×Q8—21no—
122C8×Q8—21no—
123C8×Q8D411yescharacteristic
124C8×Q8D411yescharacteristic
125#442C2×C416yes—
126#441C2×C416yes—
127#440C2×C416yes—
128#440C2×C416yes—
129C4×Q16C2×C413yes—
130C4×Q16C2313yes—
131C4×SD16C2×C416yes—
132C4×SD16C2×C416yes—
133C4×SD16C2×C413yes—
134C4×SD16C2×C413yes—
135C4×SD16C2316yes—
136C4×SD16C2313yes—
137C4×SD16C2311yescharacteristic
138C4×D8C2×C413yes—
139C4×D8C2313yes—
140#436—23no—
141#436—23no—
142#436—23no—
143#436—23no—
144#436C2313yes—
145#436C2316yes—
146#436C2313yes—
147#436C2313yes—
148#436C2311yescharacteristic
149#435—23no—
150#435—23no—
151#435—23no—
152#435—23no—
153#435—26no—
154#435—26no—
155C8×D4—23no—
156C8×D4—23no—
157C8×D4—23no—
158C8×D4D413yes—
159C8×D4D413yes—
160#432—23no—
161#432—23no—
162#432—26no—
163#431—23no—
164#431—23no—
165#431C2313yes—
166#431C2313yes—
167#431C2313yes—
168#431C2313yes—
169#428C2×C413yes—
170#428C2×C413yes—
171#427C2×C413yes—
172C2×#106C2×C413yes—
173#423—23no—
174#423—23no—
175C2×C4⋊C8—23no—
176C2×C4⋊C8C2313yes—
177C2×C4⋊C8C2313yes—
178#419C2×C413yes—
179#418C2×C413yes—
180#417C2×C413yes—
181#417C2×C413yes—
182#416C2×C413yes—
183#416C2×C413yes—
184C2×#103C2×C413yes—
185C2×#102C2×C413yes—
186#407—26no—
187C2×C22⋊C8—23no—
188#405C2×C413yes—
189#405C2×C413yes—
190C4×M16—23no—
191C4×M16—23no—
192C4×M16C2313yes—
193C2×#97—23no—
194C2×C4×C8C2313yes—
195C2×C4×C8C2313yes—
196C2×C4×C8C2313yes—
197#335C2313yes—
198#335C2316yes—
199#335C2311yescharacteristic
200#334C2313yes—
201#334C2313yes—
202Q8⋊C8C2313yes—
203Q8⋊C8C2316yes—
204Q8⋊C8C2313yes—
205Q8⋊C8C2313yes—
206Q8⋊C8C2311yescharacteristic
207D4⋊C8C2313yes—
208D4⋊C8C2316yes—
209D4⋊C8C2313yes—
210D4⋊C8C2313yes—
211D4⋊C8C2311yescharacteristic
212#322C2311yescharacteristic
213#322C2313yes—
214#322C2313yes—
215#322C2313yes—
216C82C2316yes—
2172−1+4—24no—
2182−1+4—24no—
219H(5,𝔽2)—24no—
220H(5,𝔽2)—24no—
221C2×#41—21no—
222C2×#41—23no—
223C2×#41C22×C413yes—
224C2×#41C22×C413yes—
225C2×#41C22×C411yescharacteristic
226C22×Q8—21no—
227C22×Q8C22×C413yes—
228C22×D4—23no—
229C22×D4C22×C413yes—
230C23×C4—23no—
231#137—212no—
232AGL(1,ℤ/8)—212no—
233#135—212no—
234C2×Q16C22×C413yes—
235C2×SD16—212no—
236C2×SD16C22×C413yes—
237C2×SD16C22×C416yes—
238C2×SD16C22×C411yescharacteristic
239C2×D8C22×C413yes—
240#131—24no—
241#131M1618yescomplemented
242C2×M16—23no—
243C2×M16—21no—
244C2×M16—26no—
245C2×M16—23no—
246C2×M16—23no—
247C2×M16—86no—
248C2×M16C22×C413yes—
249C2×M16C22×C413yes—
250C22×C8—23no—
251C22×C8—23no—
252C22×C8—23no—
253C22×C8C22×C413yes—
254C22×C8C22×C413yes—
255C22×C8C2×D413yes—
256C22×C8C2×D413yes—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—23no—
260C4×Q8—23no—
261C4×Q8—23no—
262C4×Q8—23no—
263C4×Q8C22×C416yes—
264C4×Q8C22×C413yes—
265C4×Q8C22×C413yes—
266C4×Q8C2×D411yescharacteristic
267C4×Q8#4111yescharacteristic
268C4×Q8C2413yes—
269C4×Q8C2411yescharacteristic
270C4×D4—23no—
271C4×D4—23no—
272C4×D4—23no—
273C4×D4—23no—
274C4×D4—23no—
275C4×D4—23no—
276C4×D4—23no—
277C4×D4—23no—
278C4×D4—26no—
279C4×D4—26no—
280C4×D4—23no—
281C4×D4—23no—
282C4×D4C22×C416yes—
283C4×D4C22×C413yes—
284C4×D4C22×C413yes—
285C4×D4C2×D413yes—
286C4×D4#4113yes—
287C4×D4C2413yes—
288C4×D4C2411yescharacteristic
289#117—23no—
290#117—23no—
291#117—26no—
292#117—23no—
293#117—23no—
294#117C22×C413yes—
295#117C22×C413yes—
296#117C22×C413yes—
297C2×C4⋊C4—23no—
298C2×C4⋊C4—23no—
299C2×C4⋊C4—23no—
300C2×C4⋊C4C22×C413yes—
301C2×C4⋊C4C22×C413yes—
302C2×C22⋊C4—23no—
303C2×C22⋊C4—26no—
304C2×C42—23no—
305C2×C42—23no—
306C2×C42—23no—
307C2×C42C2413yes—
308#107C22×C413yes—
309#107C22×C413yes—
310#106C22×C413yes—
311#106C22×C416yes—
312#106C22×C411yescharacteristic
313C4⋊C8—23no—
314C4⋊C8—26no—
315C4⋊C8—23no—
316C4⋊C8—21no—
317C4⋊C8—23no—
318C4⋊C8—23no—
319C4⋊C8—23no—
320C4⋊C8—23no—
321C4⋊C8C2×D413yes—
322C4⋊C8C2×D413yes—
323C4⋊C8C2411yescharacteristic
324C4⋊C8C2413yes—
325C4⋊C8C2411yescharacteristic
326C4⋊C8C2413yes—
327#103C22×C413yes—
328#103C22×C416yes—
329#103C22×C411yescharacteristic
330#103C22×C413yes—
331#103C22×C413yes—
332#102C22×C413yes—
333#102C22×C416yes—
334#102C22×C411yescharacteristic
335#102C22×C413yes—
336#102C22×C413yes—
337C22⋊C8—23no—
338C22⋊C8—23no—
339C22⋊C8—23no—
340C22⋊C8—23no—
341C22⋊C8—26no—
342C22⋊C8—23no—
343C22⋊C8—21no—
344C22⋊C8—23no—
345C22⋊C8C2×D416yes—
346#97—23no—
347#97—26no—
348#97—21no—
349#97—23no—
350#97—23no—
351#97C22×C416yes—
352C4×C8—23no—
353C4×C8—23no—
354C4×C8—23no—
355C4×C8C22×C413yes—
356C4×C8C22×C413yes—
357C4×C8C2×D413yes—
358C4×C8C2×D413yes—
359C4×C8C2413yes—
360C4×C8C2411yescharacteristic
361C4×C8C2413yes—
362C4×C8C2411yescharacteristic
363C4×C8C2413yes—
364C4×C8C2411yescharacteristic
365C24—23no—
366#41—26no—
367#41—26no—
368#41—26no—
369#41—26no—
370#41—44no—
371#41—412no—
372#41C2×M1614yes—
373#41#13114yes—
374C2×Q8—23no—
375C2×Q8—26no—
376C2×Q8—26no—
377C2×Q8—44no—
378C2×Q8C4×D411yescharacteristic
379C2×Q8C23×C413yes—
380C2×Q8C23×C411yescharacteristic
381C2×D4—23no—
382C2×D4—23no—
383C2×D4—23no—
384C2×D4—26no—
385C2×D4—26no—
386C2×D4—412no—
387C2×D4C4×D413yes—
388C2×D4C23×C411yescharacteristic
389C2×D4C23×C413yes—
390C22×C4—23no—
391C22×C4—23no—
392C22×C4—23no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4—21no—
396C22×C4—23no—
397C22×C4—23no—
398C22×C4—23no—
399C22×C4—26no—
400C22×C4—83no—
401C22×C4—83no—
402C22×C4C23×C413yes—
403C22×C4C22×D413yes—
404C22×C4C2×#4113yes—
405Q16—26no—
406SD16—26no—
407SD16—212no—
408SD16—22no—
409D8—26no—
410M16—26no—
411M16—42no—
412M16—46no—
413M16—83no—
414M16—83no—
415M16—86no—
416M16C2×M16112yescomplemented
417C2×C8—26no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—23no—
421C2×C8—23no—
422C2×C8—21no—
423C2×C8—23no—
424C2×C8—43no—
425C2×C8—43no—
426C2×C8—83no—
427C2×C8—83no—
428C2×C8—83no—
429C2×C8C2×M16112yescomplemented
430C2×C8C23×C411yescharacteristic
431C2×C8C23×C413yes—
432C2×C8C23×C411yescharacteristic
433C2×C8C23×C413yes—
434C2×C8C22×D411yescharacteristic
435C2×C8C22×D411yescharacteristic
436C2×C8C22×D413yes—
437C2×C8C22×D413yes—
438C4⋊C4—23no—
439C4⋊C4—26no—
440C4⋊C4—23no—
441C4⋊C4—23no—
442C4⋊C4—23no—
443C4⋊C4—23no—
444C4⋊C4—21no—
445C4⋊C4C4×D413yes—
446C4⋊C4C4×D413yes—
447C4⋊C4C23×C413yes—
448C4⋊C4C23×C411yescharacteristic
449C4⋊C4C23×C413yes—
450C4⋊C4C23×C411yescharacteristic
451C22⋊C4—23no—
452C22⋊C4—23no—
453C22⋊C4—23no—
454C22⋊C4—26no—
455C22⋊C4—23no—
456C22⋊C4—23no—
457C22⋊C4—21no—
458C22⋊C4C4×D416yes—
459C42—23no—
460C42—23no—
461C42—26no—
462C42—23no—
463C42—23no—
464C42—21no—
465C42—23no—
466C42C23×C413yes—
467C42C22×D413yes—
468C42C2×#4113yes—
469C42C2511yescharacteristic
470C23—21no—
471C23—23no—
472C23—46no—
473C23—83no—
474C23—83no—
475C23C2×C4×D413yes—
476Q8—23no—
477Q8—21no—
478Q8—23no—
479Q8—21no—
480Q8—46no—
481Q8#43614yes—
482D4—21no—
483D4—23no—
484D4—23no—
485D4—21no—
486D4—46no—
487D4—46no—
488D4—46no—
489D4#436112yes—
490C2×C4—23no—
491C2×C4—23no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—23no—
495C2×C4—21no—
496C2×C4—23no—
497C2×C4—23no—
498C2×C4—43no—
499C2×C4—46no—
500C2×C4—43no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—43no—
504C2×C4—46no—
505C2×C4—46no—
506C2×C4—83no—
507C2×C4—83no—
508C2×C4—86no—
509C2×C4—83no—
510C2×C4—83no—
511C2×C4C2×C4×D413yes—
512C2×C4C2×C4×D413yes—
513C2×C4C2×C4×D413yes—
514C2×C4C22×M1616yes—
515C2×C4C2×#13116yes—
516C2×C4C24×C411yescharacteristic
517C2×C4C23×D411yescharacteristic
518C2×C4C22×#4111yescharacteristic
519C8—21no—
520C8—23no—
521C8—21no—
522C8—23no—
523C8—21no—
524C8—23no—
525C8—43no—
526C8—41no—
527C8—43no—
528C8—43no—
529C8—41no—
530C8—41no—
531C8C22×M1612yes—
532C8C22×M1616yes—
533C22—23no—
534C22—23no—
535C22—43no—
536C22—41no—
537C22—41no—
538C22—43no—
539C22—41no—
540C22—43no—
541C22—166no—
542C22C2×#43616yes—
543C22C22×C4×D411yescharacteristic
544C4—23no—
545C4—21no—
546C4—23no—
547C4—41no—
548C4—43no—
549C4—41no—
550C4—43no—
551C4—41no—
552C4—43no—
553C4C2×#43616yes—
554C4C23×M1611yescharacteristic
555C4C22×#13111yescharacteristic
556C4#358511yescharacteristic
557C4#358611yescharacteristic
558C2—26no—
559C2—81no—
560C2—83no—
561C2C22×#43611yescharacteristic
562C2#6017411yescharacteristic
563C2#6018411yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radicalC2×#577C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C8×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60343C2semidirect
2. #60325C2semidirect
4. #17898C2semidirect
5. #17896C2semidirect
6. #17894C2semidirect
7. #17893C2semidirect
8. #17892C2semidirect
9. #17891C2semidirect
10. #17890C2semidirect
11. C2×#1592C2semidirect
12. #17789C2semidirect
13. #17439C2semidirect
14. #17438C2semidirect
15. #17316C2semidirect
22. #2989C22semidirect
26. #2983C22semidirect
32. #2965C22semidirect
36. C2×#446C22semidirect
51. #1594C22semidirect
52. #1594C22semidirect
53. #1593C22semidirect
54. #1593C22semidirect
55. #1592C22semidirect
56. #1592C22semidirect
57. #1592C22semidirect
58. #1592C22semidirect
59. #1592C22semidirect
60. #1592C22semidirect
61. #1592C22semidirect
62. #1591C22semidirect
63. #1590C22semidirect
64. #1590C22semidirect
65. #1589C22semidirect
66. #1565C22semidirect
67. #1565C22semidirect
68. #1564C22semidirect
69. C2×#335C22semidirect
70. #1479C22semidirect
73. #1477C22semidirect
75. #1476C22semidirect
76. C2×Q8⋊C8C22semidirect
78. #1450C22semidirect
79. C8×M16C22semidirect
80. C8×M16C22semidirect
81. C2×#322C22semidirect
241. #131M16semidirect
416. M16C2×M16semidirect
429. C2×C8C2×M16semidirect

Maximal subgroups

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

subgroupindex
#603432
#603252
C4×#5772
#178982
#178962
#178942
#178932
#178922
#178912
#178902
C2×#15922
#177892
#174392
#174382
#173162