Finite Group Databasebeta

← #6345659

Subgroups — #6345659

Order 512.

1705 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60940C211yesmaximal, characteristic
2#60237C211yesmaximal, characteristic, complemented
3#60232C211yesmaximal, characteristic, complemented
4#18026C218yesmaximal, complemented
5#18001C214yesmaximal, complemented
6#17994C214yesmaximal, complemented
7#17989C214yesmaximal, complemented
8#17970C214yesmaximal, complemented
9#17816C211yesmaximal, characteristic, complemented
10#17478C211yesmaximal, characteristic, complemented
11C2×#1496C211yesmaximal, characteristic, complemented
12#17326C211yesmaximal, characteristic, complemented
13C2×#532C2211yescharacteristic
14C2×#532C2211yescharacteristic
15#3245C414yes—
16#3226C412yes—
17#3226C412yes—
18#3205C412yes—
19#3205C412yes—
20#3186C414yes—
21#3169C2214yes—
22#3100C2212yes—
23#3100C2212yes—
24#3100C2214yes—
25#2981—22no—
26#2981C2214yescomplemented
27#2981C2214yescomplemented
28#2980—22no—
29#2980—22no—
30#2979C2214yes—
31#2974C2214yescomplemented
32#2973—22no—
33#2935C2214yes—
34D4×M16—22no—
35D4×M16C2214yescomplemented
36#2932—22no—
37#2920C2211yescharacteristic
38#2920C2211yescharacteristic
39C2×#432C2211yescharacteristic
40C2×#431C2211yescharacteristic
41C2×#431C2211yescharacteristic
42#2880—22no—
43#2879—22no—
44#1594C2218yescomplemented
45#1589C2218yescomplemented
46#1586C2218yescomplemented
47#1586C2218yescomplemented
48#1581C2218yescomplemented
49#1581C2218yescomplemented
50#1580C2218yescomplemented
51C8×D8C2218yescomplemented
52#1572C2214yescomplemented
53#1571C2212yescomplemented
54#1570C2212yescomplemented
55#1568C2214yescomplemented
56#1496C2214yes—
57#1496C2212yescomplemented
58#1496C2212yescomplemented
59#1496C2212yescomplemented
60#1496C2212yescomplemented
61#1492C2214yes—
62#1478C2214yescomplemented
63C2×D4⋊C8C2214yescomplemented
64#1454C2214yescomplemented
65#1452C2212yescomplemented
66#1451C2212yescomplemented
67#1450C2212yescomplemented
68C8×M16C2212yescomplemented
69C22×#41—21no—
70C22×#41—21no—
71C2×#131—24no—
72C22×M16—24no—
73#532—21no—
74#532—21no—
75#532—21no—
76#532—21no—
77#532—21no—
78#532—21no—
79#532—21no—
80#532—21no—
81#532C2×C412yes—
82#532C2×C412yes—
83#532C2×C412yes—
84#532C2×C412yes—
85#532C2312yes—
86#532C2312yes—
87#532C2312yes—
88#532C2312yes—
89C2×C4⋊Q8C2×C411yescharacteristic
90C2×C4⋊Q8C2×C411yescharacteristic
91C2×#127C2×C411yescharacteristic
92C2×#127C2×C411yescharacteristic
93C2×#124—21no—
94C2×#124—21no—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121C2×C412yes—
98C2×#121C2×C412yes—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4C2312yes—
102C2×C4×D4C2312yes—
103C22×C42C2311yescharacteristic
104#500C2×C414yes—
105#493C2×C414yes—
106#486C2×C414yes—
107#486C2×C414yes—
108#480C2×C414yes—
109#480C2×C414yes—
110#480C2×C418yes—
111#474C2×C414yes—
112#474C2×C414yes—
113#474C2×C418yes—
114#466C2×C418yes—
115#466C2×C418yes—
116#459C2×C414yes—
117#459C2×C414yes—
118#459C2×C418yes—
119#447C2×C414yes—
120#447C2×C414yes—
121#447C2×C418yes—
122C4×D8C2318yes—
123C4×D8C2318yes—
124#436—24no—
125#436C2318yes—
126#436C2318yes—
127#435—28no—
128#435—28no—
129#435—28no—
130#435—28no—
131#435—28no—
132C8×D4—24no—
133C8×D4C2318yes—
134C8×D4C2318yes—
135#432—21no—
136#432—21no—
137#432—21no—
138#432D412yes—
139#432D412yes—
140#432C2312yes—
141#432C2312yes—
142#432C2312yes—
143#431—21no—
144#431—21no—
145#431—21no—
146#431C2×C412yes—
147#431C2×C412yes—
148#431C2312yes—
149#431C2311yescharacteristic
150#431C2311yescharacteristic
151#431C2312yes—
152#431C2311yescharacteristic
153#431C2311yescharacteristic
154#431C2312yes—
155#431C2312yes—
156#431C2312yes—
157#423D412yes—
158#423C2312yes—
159C2×C4⋊C8—21no—
160C2×C4⋊C8C2×C412yes—
161C2×C4⋊C8C2312yes—
162#408—24no—
163#408—28no—
164#408—28no—
165#408—28no—
166#407—21no—
167#407—24no—
168#407—24no—
169#407D412yes—
170C2×C22⋊C8—21no—
171C2×C22⋊C8—21no—
172C2×C22⋊C8—24no—
173C2×C22⋊C8C2×C412yes—
174C4×M16—21no—
175C4×M16C2312yes—
176C2×#97—21no—
177C2×C4×C8C2312yes—
178C2×C4×C8C2312yes—
179#335C2318yes—
180#334C2318yes—
181D4⋊C8C2314yes—
182D4⋊C8C2318yes—
183D4⋊C8C2314yes—
184D4⋊C8C2314yes—
185D4⋊C8C2314yes—
186D4⋊C8C2318yes—
187#322C2314yes—
188#322C2314yes—
189#322C2314yes—
190C82C2314yes—
191C2×#41—22no—
192C2×#41—22no—
193C2×#41—22no—
194C2×#41—22no—
195C2×#41—22no—
196C2×#41—22no—
197C2×#41—22no—
198C2×#41—22no—
199C2×#41—84no—
200C2×#41—84no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—21no—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4C22×C412yes—
210C22×D4C22×C412yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4C22×C411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4#4111yescharacteristic
216C2×D8C22×C418yes—
217C2×D8C22×C418yes—
218#131—88no—
219C2×M16—22no—
220C2×M16—28no—
221C2×M16—28no—
222C2×M16—88no—
223C2×M16C2×D414yes—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—28no—
227C22×C8—28no—
228C22×C8C22×C414yes—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8C22×C412yes—
234C4⋊Q8C22×C412yes—
235C4⋊Q8C22×C412yes—
236C4⋊Q8C22×C412yes—
237#127—21no—
238#127—21no—
239#127—21no—
240#127—21no—
241#127C22×C412yes—
242#127C22×C412yes—
243#127C22×C412yes—
244#127C22×C412yes—
245#124—22no—
246#124—22no—
247#124—22no—
248#124—22no—
249#124—22no—
250#124—22no—
251#124—22no—
252#124—22no—
253#121—22no—
254#121—22no—
255#121—22no—
256#121—22no—
257#121—22no—
258#121—22no—
259#121—22no—
260#121—22no—
261#121—22no—
262#121—22no—
263#121—22no—
264#121—22no—
265#121C22×C414yes—
266#121C22×C414yes—
267#121C22×C414yes—
268#121C22×C414yes—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4C22×C414yes—
282C4×D4C22×C414yes—
283C4×D4C2414yes—
284C4×D4C2414yes—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4C22×C412yes—
288C2×C4⋊C4C22×C412yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42#4111yescharacteristic
300C2×C42#4111yescharacteristic
301C2×C42#4111yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305#107C22×C418yes—
306#107C22×C418yes—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8C22×C412yes—
313C4⋊C8C22×C412yes—
314C4⋊C8C22×C414yes—
315C4⋊C8C2×D414yes—
316C4⋊C8C2×D414yes—
317C4⋊C8C2414yes—
318C4⋊C8C2414yes—
319#102C22×C418yes—
320#102C22×C418yes—
321#102C22×C418yes—
322#102C22×C418yes—
323C22⋊C8—22no—
324C22⋊C8—22no—
325C22⋊C8—22no—
326C22⋊C8—22no—
327C22⋊C8—22no—
328C22⋊C8—21no—
329C22⋊C8—21no—
330C22⋊C8—28no—
331C22⋊C8—28no—
332C22⋊C8—28no—
333C22⋊C8—28no—
334C22⋊C8C22×C412yes—
335C22⋊C8C22×C412yes—
336C22⋊C8C22×C414yes—
337C22⋊C8C2×D414yes—
338C22⋊C8C2×D414yes—
339#97—22no—
340#97—22no—
341#97—22no—
342#97C2×D414yes—
343C4×C8—22no—
344C4×C8—22no—
345C4×C8C22×C414yes—
346C4×C8C2412yes—
347C4×C8C2412yes—
348C4×C8C2412yes—
349C4×C8C2412yes—
350C4×C8C2414yes—
351C24—22no—
352C24—22no—
353C24C4×D411yescharacteristic
354#41—48no—
355#41—84no—
356#41—84no—
357#41—82no—
358#41—82no—
359#41—84no—
360#41—84no—
361#41—84no—
362#41—84no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—82no—
368C2×Q8—82no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—82no—
386C2×D4—82no—
387C2×D4—82no—
388C2×D4—84no—
389C2×D4—82no—
390C2×D4—84no—
391C2×D4C23×C414yes—
392C2×D4C23×C414yes—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—42no—
401C22×C4—48no—
402C22×C4—84no—
403C22×C4—81no—
404C22×C4C4×D411yescharacteristic
405C22×C4C4×D411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4C4×D411yescharacteristic
409C22×C4C4×D411yescharacteristic
410C22×C4C4×D411yescharacteristic
411C22×C4C4×D411yescharacteristic
412C22×C4C4×D411yescharacteristic
413C22×C4C23×C411yescharacteristic
414C22×C4C23×C411yescharacteristic
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422D8—48no—
423D8—48no—
424M16—28no—
425M16—44no—
426M16—88no—
427M16—88no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—44no—
433C2×C8—44no—
434C2×C8—88no—
435C2×C8—88no—
436C2×C8—82no—
437C2×C8C23×C414yes—
438C2×C8C23×C414yes—
439C2×C8C22×D418yes—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4C23×C414yes—
447C4⋊C4C23×C414yes—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C42C4×D411yescharacteristic
457C42C4×D411yescharacteristic
458C42C4×D411yescharacteristic
459C42C4×D411yescharacteristic
460C42C4×D411yescharacteristic
461C42C4×D411yescharacteristic
462C42C23×C411yescharacteristic
463C42C23×C411yescharacteristic
464C42C23×C411yescharacteristic
465C42C22×D411yescharacteristic
466C42C22×D411yescharacteristic
467C42C22×D411yescharacteristic
468C42C2×#4111yescharacteristic
469C42C2×#4111yescharacteristic
470C42C2×#4111yescharacteristic
471C42C2511yescharacteristic
472C23—22no—
473C23—24no—
474C23—24no—
475C23—41no—
476C23—41no—
477C23—44no—
478C23—44no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23C2×C4×D411yescharacteristic
484C23C2×C4×D411yescharacteristic
485C23C2×C4×D411yescharacteristic
486Q8—44no—
487Q8—82no—
488Q8—82no—
489Q8—82no—
490D4—24no—
491D4—24no—
492D4—28no—
493D4—44no—
494D4—48no—
495D4—44no—
496D4—82no—
497D4—82no—
498D4—82no—
499D4—82no—
500D4—82no—
501D4—82no—
502D4—82no—
503D4—82no—
504D4—84no—
505D4—84no—
506D4—84no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—48no—
528C2×C4—44no—
529C2×C4—48no—
530C2×C4—44no—
531C2×C4—84no—
532C2×C4—82no—
533C2×C4C2×C4×D411yescharacteristic
534C2×C4C2×C4×D411yescharacteristic
535C2×C4C2×C4×D411yescharacteristic
536C2×C4C2×C4×D411yescharacteristic
537C2×C4C2×C4×D411yescharacteristic
538C2×C4C2×C4×D411yescharacteristic
539C2×C4C2×C4×D411yescharacteristic
540C2×C4C2×C4×D411yescharacteristic
541C2×C4C2×C4×D411yescharacteristic
542C2×C4#57614yes—
543C2×C4C24×C411yescharacteristic
544C2×C4C23×D411yescharacteristic
545C2×C4C22×#4111yescharacteristic
546C8—24no—
547C8—24no—
548C8—28no—
549C8—42no—
550C8—42no—
551C8—44no—
552C8—44no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—82no—
568C22—164no—
569C22—164no—
570C22#294812yes—
571C22C22×C4×D411yescharacteristic
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—44no—
582C4—44no—
583C4#294812yes—
584C4#357312yes—
585C4C2×#57612yes—
586C2—22no—
587C2—41no—
588C2—41no—
589C2—84no—
590C2—84no—
591C2#6014311yescharacteristic
592C2C2×#294811yescharacteristic
593C2#6018211yescharacteristic
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. #60237C2semidirect
3. #60232C2semidirect
4. #18026C2semidirect
5. #18001C2semidirect
6. #17994C2semidirect
7. #17989C2semidirect
8. #17970C2semidirect
9. #17816C2semidirect
10. #17478C2semidirect
11. C2×#1496C2semidirect
12. #17326C2semidirect
26. #2981C22semidirect
27. #2981C22semidirect
31. #2974C22semidirect
35. D4×M16C22semidirect
44. #1594C22semidirect
45. #1589C22semidirect
46. #1586C22semidirect
47. #1586C22semidirect
48. #1581C22semidirect
49. #1581C22semidirect
50. #1580C22semidirect
51. C8×D8C22semidirect
52. #1572C22semidirect
53. #1571C22semidirect
54. #1570C22semidirect
55. #1568C22semidirect
57. #1496C22semidirect
58. #1496C22semidirect
59. #1496C22semidirect
60. #1496C22semidirect
62. #1478C22semidirect
63. C2×D4⋊C8C22semidirect
64. #1454C22semidirect
65. #1452C22semidirect
66. #1451C22semidirect
67. #1450C22semidirect
68. C8×M16C22semidirect

Maximal subgroups

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

subgroupindex
#609402
#602372
#602322
#180262
#180012
#179942
#179892
#179702
#178162
#174782
C2×#14962
#173262