Finite Group Databasebeta

← #6345565

Subgroups — #6345565

Order 512.

1732 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60940C211yesmaximal, characteristic
2#60236C211yesmaximal, characteristic, complemented
3#60232C211yesmaximal, characteristic, complemented
4#18091C214yesmaximal, complemented
5#18084C214yesmaximal, complemented
6#18001C214yesmaximal, complemented
7#17994C214yesmaximal, complemented
8#17950C214yesmaximal, complemented
9#17922C214yesmaximal, complemented
10#17803C211yesmaximal, characteristic, complemented
11#17500C211yesmaximal, characteristic, complemented
12C2×#1484C211yesmaximal, characteristic, complemented
13#17327C211yesmaximal, characteristic, complemented
14C2×#532C2211yescharacteristic
15C2×#532C2211yescharacteristic
16#3245C414yes—
17#3226C412yes—
18#3226C412yes—
19#3205C412yes—
20#3205C412yes—
21#3186C414yes—
22#3169C2214yes—
23#3100C2214yes—
24#3100C2212yes—
25#3100C2212yes—
26#2981—22no—
27#2981—22no—
28#2981C2214yes—
29#2980—22no—
30#2979C2214yescomplemented
31#2979C2214yescomplemented
32#2975C2212yescomplemented
33#2974—22no—
34#2972C2212yescomplemented
35#2935C2214yes—
36D4×M16—22no—
37D4×M16C2214yescomplemented
38#2931—22no—
39#2920C2211yescharacteristic
40#2920C2211yescharacteristic
41C2×#431C2211yescharacteristic
42C2×#431C2211yescharacteristic
43#2880—22no—
44#2879—22no—
45C2×C4×M16C2211yescharacteristic
46#1594C2218yescomplemented
47#1589C2218yescomplemented
48#1586C2218yescomplemented
49#1586C2218yescomplemented
50#1581C2218yescomplemented
51#1581C2218yescomplemented
52#1580C2218yescomplemented
53C8×D8C2218yescomplemented
54#1572C2212yescomplemented
55#1572C2212yescomplemented
56#1571C2212yescomplemented
57#1570C2212yescomplemented
58#1564C2214yescomplemented
59#1492C2214yes—
60#1490C2214yescomplemented
61#1484C2212yes—
62#1484C2212yes—
63#1484C2212yescomplemented
64#1484C2212yescomplemented
65#1484C2212yescomplemented
66#1484C2212yescomplemented
67C2×D4⋊C8C2214yescomplemented
68#1453C2214yescomplemented
69#1451C2212yescomplemented
70#1451C2212yescomplemented
71C8×M16C2212yescomplemented
72C8×M16C2212yescomplemented
73C22×#41—21no—
74C22×#41—21no—
75C2×#131—24no—
76C22×M16—21no—
77C22×M16—24no—
78#532—21no—
79#532—21no—
80#532—21no—
81#532—21no—
82#532—21no—
83#532—21no—
84#532—21no—
85#532—21no—
86#532C2×C412yes—
87#532C2×C412yes—
88#532C2×C412yes—
89#532C2×C412yes—
90#532C2312yes—
91#532C2312yes—
92#532C2312yes—
93#532C2312yes—
94C2×C4⋊Q8C2×C411yescharacteristic
95C2×C4⋊Q8C2×C411yescharacteristic
96C2×#127C2×C411yescharacteristic
97C2×#127C2×C411yescharacteristic
98C2×#124—21no—
99C2×#124—21no—
100C2×#121—21no—
101C2×#121—21no—
102C2×#121C2×C412yes—
103C2×#121C2×C412yes—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C22×C42C2311yescharacteristic
109#500C2×C414yes—
110#493C2×C414yes—
111#486C2×C414yes—
112#486C2×C414yes—
113#480C2×C418yes—
114#480C2×C414yes—
115#480C2×C414yes—
116#474C2×C418yes—
117#474C2×C414yes—
118#474C2×C414yes—
119#466C2×C418yes—
120#466C2×C418yes—
121#459C2×C418yes—
122#459C2×C414yes—
123#459C2×C414yes—
124#447C2×C418yes—
125#447C2×C414yes—
126#447C2×C414yes—
127C4×D8C2318yes—
128C4×D8C2318yes—
129#436—24no—
130#436—24no—
131#436—24no—
132#436C2318yes—
133#436C2318yes—
134#435—28no—
135#435—28no—
136#435—28no—
137C8×D4—24no—
138C8×D4—24no—
139C8×D4—24no—
140C8×D4C2318yes—
141C8×D4C2318yes—
142#432—21no—
143#432D412yes—
144#432D412yes—
145#432C2312yes—
146#431—21no—
147#431—21no—
148#431—21no—
149#431C2×C412yes—
150#431C2×C412yes—
151#431C2312yes—
152#431C2312yes—
153#431C2311yescharacteristic
154#431C2311yescharacteristic
155#431C2311yescharacteristic
156#431C2311yescharacteristic
157#431C2312yes—
158#431C2312yes—
159#431C2312yes—
160#423D412yes—
161#423C2312yes—
162C2×C4⋊C8C2×C412yes—
163C2×C4⋊C8C2312yes—
164#408—24no—
165#408—28no—
166#408—28no—
167#408—28no—
168#407—21no—
169#407—24no—
170#407—24no—
171#407D412yes—
172C2×C22⋊C8—21no—
173C2×C22⋊C8—24no—
174C2×C22⋊C8C2×C412yes—
175C4×M16—21no—
176C4×M16—21no—
177C4×M16—21no—
178C4×M16C2311yescharacteristic
179C4×M16C2311yescharacteristic
180C4×M16C2312yes—
181C4×M16C2312yes—
182C2×#97—21no—
183C2×C4×C8—21no—
184C2×C4×C8C2312yes—
185C2×C4×C8C2312yes—
186#335C2314yes—
187#335C2314yes—
188#334C2314yes—
189#334C2314yes—
190D4⋊C8C2314yes—
191D4⋊C8C2318yes—
192D4⋊C8C2314yes—
193D4⋊C8C2318yes—
194D4⋊C8C2314yes—
195D4⋊C8C2314yes—
196#322C2314yes—
197#322C2314yes—
198#322C2314yes—
199C82C2314yes—
200C2×#41—22no—
201C2×#41—22no—
202C2×#41—22no—
203C2×#41—22no—
204C2×#41—22no—
205C2×#41—22no—
206C2×#41—22no—
207C2×#41—22no—
208C2×#41—84no—
209C2×#41—84no—
210C22×Q8—21no—
211C22×Q8—21no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4C22×C412yes—
219C22×D4C22×C412yes—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4C22×C411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4#4111yescharacteristic
225C2×D8C22×C418yes—
226C2×D8C22×C418yes—
227#131—88no—
228C2×M16—21no—
229C2×M16—22no—
230C2×M16—22no—
231C2×M16—21no—
232C2×M16—28no—
233C2×M16—28no—
234C2×M16—44no—
235C2×M16—88no—
236C2×M16C2×D414yes—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—28no—
240C22×C8—28no—
241C22×C8C22×C414yes—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8C22×C412yes—
247C4⋊Q8C22×C412yes—
248C4⋊Q8C22×C412yes—
249C4⋊Q8C22×C412yes—
250#127—21no—
251#127—21no—
252#127—21no—
253#127—21no—
254#127C22×C412yes—
255#127C22×C412yes—
256#127C22×C412yes—
257#127C22×C412yes—
258#124—22no—
259#124—22no—
260#124—22no—
261#124—22no—
262#124—22no—
263#124—22no—
264#124—22no—
265#124—22no—
266#121—22no—
267#121—22no—
268#121—22no—
269#121—22no—
270#121—22no—
271#121—22no—
272#121—22no—
273#121—22no—
274#121—22no—
275#121—22no—
276#121—22no—
277#121—22no—
278#121C22×C414yes—
279#121C22×C414yes—
280#121C22×C414yes—
281#121C22×C414yes—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4C22×C414yes—
295C4×D4C22×C414yes—
296C4×D4C2414yes—
297C4×D4C2414yes—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4C22×C412yes—
301C2×C4⋊C4C22×C412yes—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C42C22×C411yescharacteristic
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42#4111yescharacteristic
313C2×C42#4111yescharacteristic
314C2×C42#4111yescharacteristic
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318#107C22×C418yes—
319#107C22×C418yes—
320C4⋊C8—22no—
321C4⋊C8—22no—
322C4⋊C8C22×C412yes—
323C4⋊C8C22×C412yes—
324C4⋊C8C22×C414yes—
325C4⋊C8C2×D414yes—
326C4⋊C8C2×D414yes—
327C4⋊C8C2412yes—
328C4⋊C8C2412yes—
329C4⋊C8C2414yes—
330#102C22×C418yes—
331#102C22×C418yes—
332#102C22×C418yes—
333#102C22×C418yes—
334C22⋊C8—21no—
335C22⋊C8—21no—
336C22⋊C8—22no—
337C22⋊C8—22no—
338C22⋊C8—22no—
339C22⋊C8—28no—
340C22⋊C8—28no—
341C22⋊C8—28no—
342C22⋊C8—28no—
343C22⋊C8C22×C412yes—
344C22⋊C8C22×C412yes—
345C22⋊C8C22×C414yes—
346C22⋊C8C2×D414yes—
347C22⋊C8C2×D414yes—
348#97—22no—
349#97—22no—
350#97—21no—
351#97—21no—
352#97C2×D414yes—
353C4×C8—21no—
354C4×C8—22no—
355C4×C8—21no—
356C4×C8—22no—
357C4×C8—22no—
358C4×C8C22×C414yes—
359C4×C8C2412yes—
360C4×C8C2412yes—
361C4×C8C2412yes—
362C4×C8C2412yes—
363C4×C8C2414yes—
364C24—22no—
365C24—22no—
366C24C4×D411yescharacteristic
367#41—48no—
368#41—84no—
369#41—84no—
370#41—84no—
371#41—82no—
372#41—84no—
373#41—82no—
374#41—84no—
375#41—84no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—82no—
381C2×Q8—82no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—82no—
399C2×D4—82no—
400C2×D4—82no—
401C2×D4—82no—
402C2×D4—84no—
403C2×D4—84no—
404C2×D4C23×C414yes—
405C2×D4C23×C414yes—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—48no—
416C22×C4—84no—
417C22×C4C4×D411yescharacteristic
418C22×C4C4×D411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4C4×D411yescharacteristic
421C22×C4C4×D411yescharacteristic
422C22×C4C4×D411yescharacteristic
423C22×C4C4×D411yescharacteristic
424C22×C4C4×D411yescharacteristic
425C22×C4C4×D411yescharacteristic
426C22×C4C23×C411yescharacteristic
427C22×C4C23×C411yescharacteristic
428C22×C4C23×C411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435D8—48no—
436D8—48no—
437M16—24no—
438M16—28no—
439M16—44no—
440M16—42no—
441M16—44no—
442M16—88no—
443M16—88no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—44no—
450C2×C8—44no—
451C2×C8—44no—
452C2×C8—88no—
453C2×C8—88no—
454C2×C8C23×C414yes—
455C2×C8C23×C414yes—
456C2×C8C22×D418yes—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4C23×C414yes—
464C4⋊C4C23×C414yes—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C42C4×D411yescharacteristic
474C42C4×D411yescharacteristic
475C42C4×D411yescharacteristic
476C42C4×D411yescharacteristic
477C42C4×D411yescharacteristic
478C42C4×D411yescharacteristic
479C42C23×C411yescharacteristic
480C42C23×C411yescharacteristic
481C42C23×C411yescharacteristic
482C42C22×D411yescharacteristic
483C42C22×D411yescharacteristic
484C42C22×D411yescharacteristic
485C42C2×#4111yescharacteristic
486C42C2×#4111yescharacteristic
487C42C2×#4111yescharacteristic
488C42C2511yescharacteristic
489C23—22no—
490C23—24no—
491C23—24no—
492C23—41no—
493C23—41no—
494C23—44no—
495C23—44no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23C2×C4×D411yescharacteristic
501C23C2×C4×D411yescharacteristic
502C23C2×C4×D411yescharacteristic
503Q8—44no—
504Q8—82no—
505Q8—82no—
506Q8—82no—
507D4—28no—
508D4—24no—
509D4—24no—
510D4—44no—
511D4—48no—
512D4—44no—
513D4—44no—
514D4—44no—
515D4—82no—
516D4—82no—
517D4—82no—
518D4—82no—
519D4—82no—
520D4—82no—
521D4—84no—
522D4—84no—
523D4—84no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—84no—
549C2×C4—82no—
550C2×C4C2×C4×D411yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C2×C4×D411yescharacteristic
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C2×C4×D411yescharacteristic
555C2×C4C2×C4×D411yescharacteristic
556C2×C4C2×C4×D411yescharacteristic
557C2×C4C2×C4×D411yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4#57614yes—
560C2×C4C24×C411yescharacteristic
561C2×C4C23×D411yescharacteristic
562C2×C4C22×#4111yescharacteristic
563C8—24no—
564C8—24no—
565C8—24no—
566C8—28no—
567C8—42no—
568C8—42no—
569C8—42no—
570C8—44no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—82no—
586C22—164no—
587C22—164no—
588C22#294812yes—
589C22C22×C4×D411yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—44no—
599C4—44no—
600C4#294812yes—
601C4#295612yes—
602C4#357312yes—
603C4C2×#57612yes—
604C2—22no—
605C2—41no—
606C2—41no—
607C2—84no—
608C2—84no—
609C2#6014211yescharacteristic
610C2C2×#294811yescharacteristic
611C2C2×#295611yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#60940C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #2920 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60236C2semidirect
3. #60232C2semidirect
4. #18091C2semidirect
5. #18084C2semidirect
6. #18001C2semidirect
7. #17994C2semidirect
8. #17950C2semidirect
9. #17922C2semidirect
10. #17803C2semidirect
11. #17500C2semidirect
12. C2×#1484C2semidirect
13. #17327C2semidirect
30. #2979C22semidirect
31. #2979C22semidirect
32. #2975C22semidirect
34. #2972C22semidirect
37. D4×M16C22semidirect
46. #1594C22semidirect
47. #1589C22semidirect
48. #1586C22semidirect
49. #1586C22semidirect
50. #1581C22semidirect
51. #1581C22semidirect
52. #1580C22semidirect
53. C8×D8C22semidirect
54. #1572C22semidirect
55. #1572C22semidirect
56. #1571C22semidirect
57. #1570C22semidirect
58. #1564C22semidirect
60. #1490C22semidirect
63. #1484C22semidirect
64. #1484C22semidirect
65. #1484C22semidirect
66. #1484C22semidirect
67. C2×D4⋊C8C22semidirect
68. #1453C22semidirect
69. #1451C22semidirect
70. #1451C22semidirect
71. C8×M16C22semidirect
72. C8×M16C22semidirect

Maximal subgroups

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

subgroupindex
#609402
#602362
#602322
#180912
#180842
#180012
#179942
#179502
#179222
#178032
#175002
C2×#14842
#173272