Finite Group Databasebeta

← #6345543

Subgroups — #6345543

Order 512.

1709 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#60236C211yesmaximal, characteristic, complemented
4#18073C214yesmaximal, complemented
5#18066C214yesmaximal, complemented
6#18026C218yesmaximal, complemented
7#17950C214yesmaximal, complemented
8#17922C214yesmaximal, complemented
9C2×#1572C211yesmaximal, characteristic, complemented
10#17495C211yesmaximal, characteristic, complemented
11#17463C211yesmaximal, characteristic, complemented
12#17324C211yesmaximal, characteristic, complemented
13C2×#532C2211yescharacteristic
14C2×#532C2211yescharacteristic
15#3245C414yes—
16#3226C412yes—
17#3226C412yes—
18#3205C412yes—
19#3205C412yes—
20#3186C414yes—
21#3169C2212yes—
22#3169C2212yes—
23#3100C2214yes—
24#3100C2214yes—
25#2981—22no—
26#2981C2214yes—
27#2980—22no—
28#2979C2214yes—
29#2975C2212yescomplemented
30#2974—22no—
31#2974C2214yescomplemented
32#2973—22no—
33#2972C2212yescomplemented
34#2935C2214yescomplemented
35#2935C2214yescomplemented
36#2932—22no—
37#2931—22no—
38#2920C2211yescharacteristic
39#2920C2211yescharacteristic
40C2×#432C2211yescharacteristic
41C2×#431C2211yescharacteristic
42C2×#431C2211yescharacteristic
43#2880—22no—
44#2880—22no—
45#1594C2218yescomplemented
46#1589C2218yescomplemented
47#1586C2218yescomplemented
48#1586C2218yescomplemented
49#1581C2218yescomplemented
50#1581C2218yescomplemented
51#1580C2218yescomplemented
52C8×D8C2218yescomplemented
53#1572C2212yescomplemented
54#1572C2212yescomplemented
55#1572C2214yescomplemented
56C2×#335C2212yescomplemented
57C2×#334C2212yescomplemented
58#1496C2214yes—
59#1490C2214yescomplemented
60#1484C2212yes—
61#1484C2212yes—
62#1480C2214yescomplemented
63#1480C2214yescomplemented
64#1478C2214yescomplemented
65#1454C2214yescomplemented
66#1453C2214yescomplemented
67#1451C2212yescomplemented
68C8×M16C2212yescomplemented
69C22×#41—21no—
70C22×#41—21no—
71C2×#131—24no—
72C2×#131—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×C412yes—
105#500C2×C412yes—
106#493C2×C412yes—
107#493C2×C412yes—
108#486C2×C412yes—
109#486C2×C412yes—
110#486C2×C412yes—
111#486C2×C412yes—
112#480C2×C418yes—
113#480C2×C418yes—
114#474C2×C418yes—
115#474C2×C418yes—
116#466C2×C418yes—
117#466C2×C418yes—
118#459C2×C418yes—
119#459C2×C418yes—
120#447C2×C418yes—
121#447C2×C418yes—
122C4×D8C2318yes—
123C4×D8C2318yes—
124#436—24no—
125#436—24no—
126#436C2318yes—
127#436C2318yes—
128#435—28no—
129#435—28no—
130#435—28no—
131#435—28no—
132C8×D4—24no—
133C8×D4—24no—
134C8×D4C2318yes—
135C8×D4C2318yes—
136#432—21no—
137#432—21no—
138#432—21no—
139#432—21no—
140#432D412yes—
141#432D412yes—
142#432C2312yes—
143#432C2312yes—
144#431—21no—
145#431—21no—
146#431C2×C412yes—
147#431C2×C412yes—
148#431C2311yescharacteristic
149#431C2311yescharacteristic
150#431C2312yes—
151#431C2312yes—
152#431C2312yes—
153#431C2312yes—
154#431C2312yes—
155#431C2312yes—
156#431C2312yes—
157#423—21no—
158#423D412yes—
159C2×C4⋊C8C2×C412yes—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2312yes—
162#408—24no—
163#408—24no—
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⋊C8—24no—
174C2×C22⋊C8C2×C412yes—
175C4×M16C2312yes—
176C4×M16C2312yes—
177C2×#97—21no—
178C2×C4×C8—21no—
179C2×C4×C8C2311yescharacteristic
180C2×C4×C8C2311yescharacteristic
181#335C2314yes—
182#335C2314yes—
183#334C2314yes—
184#334C2314yes—
185D4⋊C8C2318yes—
186D4⋊C8C2318yes—
187D4⋊C8C2318yes—
188D4⋊C8C2318yes—
189#322C2314yes—
190#322C2314yes—
191#322C2314yes—
192C82C2314yes—
193C2×#41—22no—
194C2×#41—22no—
195C2×#41—22no—
196C2×#41—22no—
197C2×#41—22no—
198C2×#41—22no—
199C2×#41—22no—
200C2×#41—22no—
201C2×#41—84no—
202C2×#41—84no—
203C22×Q8—21no—
204C22×Q8—21no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4C22×C412yes—
212C22×D4C22×C412yes—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4C22×C411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4#4111yescharacteristic
218C2×D8C22×C418yes—
219C2×D8C22×C418yes—
220#131—88no—
221#131—88no—
222C2×M16—22no—
223C2×M16—28no—
224C2×M16—28no—
225C2×M16C2×D414yes—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—28no—
229C22×C8—28no—
230C22×C8C22×C412yes—
231C22×C8C22×C412yes—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8C22×C412yes—
237C4⋊Q8C22×C412yes—
238C4⋊Q8C22×C412yes—
239C4⋊Q8C22×C412yes—
240#127—21no—
241#127—21no—
242#127—21no—
243#127—21no—
244#127C22×C412yes—
245#127C22×C412yes—
246#127C22×C412yes—
247#127C22×C412yes—
248#124—22no—
249#124—22no—
250#124—22no—
251#124—22no—
252#124—22no—
253#124—22no—
254#124—22no—
255#124—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#121—22no—
266#121—22no—
267#121—22no—
268#121C22×C414yes—
269#121C22×C414yes—
270#121C22×C414yes—
271#121C22×C414yes—
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×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4C22×C414yes—
285C4×D4C22×C414yes—
286C4×D4C2414yes—
287C4×D4C2414yes—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4C22×C412yes—
291C2×C4⋊C4C22×C412yes—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42#4111yescharacteristic
303C2×C42#4111yescharacteristic
304C2×C42#4111yescharacteristic
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308#107C22×C418yes—
309#107C22×C418yes—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314C4⋊C8C22×C414yes—
315C4⋊C8C22×C414yes—
316C4⋊C8C2×D414yes—
317C4⋊C8C2×D414yes—
318C4⋊C8C2412yes—
319C4⋊C8C2412yes—
320C4⋊C8C2414yes—
321#102C22×C418yes—
322#102C22×C418yes—
323#102C22×C418yes—
324#102C22×C418yes—
325C22⋊C8—21no—
326C22⋊C8—22no—
327C22⋊C8—21no—
328C22⋊C8—22no—
329C22⋊C8—22no—
330C22⋊C8—22no—
331C22⋊C8—22no—
332C22⋊C8—28no—
333C22⋊C8—28no—
334C22⋊C8—28no—
335C22⋊C8—28no—
336C22⋊C8C22×C414yes—
337C22⋊C8C22×C414yes—
338C22⋊C8C2×D414yes—
339C22⋊C8C2×D414yes—
340#97—22no—
341#97—22no—
342#97—22no—
343#97C2×D414yes—
344C4×C8—21no—
345C4×C8—22no—
346C4×C8—21no—
347C4×C8C22×C412yes—
348C4×C8C22×C412yes—
349C4×C8C2414yes—
350C4×C8C2412yes—
351C4×C8C2412yes—
352C4×C8C2414yes—
353C24—22no—
354C24—22no—
355C24C4×D411yescharacteristic
356#41—48no—
357#41—48no—
358#41—84no—
359#41—84no—
360#41—84no—
361#41—84no—
362#41—84no—
363#41—84no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—82no—
369C2×Q8—82no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—24no—
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—22no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—82no—
387C2×D4—82no—
388C2×D4—82no—
389C2×D4—82no—
390C2×D4—84no—
391C2×D4—84no—
392C2×D4C23×C414yes—
393C2×D4C23×C414yes—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—42no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—81no—
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×C4C4×D411yescharacteristic
414C22×C4C23×C411yescharacteristic
415C22×C4C23×C411yescharacteristic
416C22×C4C23×C411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423D8—48no—
424D8—48no—
425M16—28no—
426M16—88no—
427M16—88no—
428M16—82no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—44no—
436C2×C8—44no—
437C2×C8—88no—
438C2×C8—88no—
439C2×C8C23×C414yes—
440C2×C8C23×C414yes—
441C2×C8C22×D418yes—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4C23×C414yes—
449C4⋊C4C23×C414yes—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C42C4×D411yescharacteristic
459C42C4×D411yescharacteristic
460C42C4×D411yescharacteristic
461C42C4×D411yescharacteristic
462C42C4×D411yescharacteristic
463C42C4×D411yescharacteristic
464C42C23×C411yescharacteristic
465C42C23×C411yescharacteristic
466C42C23×C411yescharacteristic
467C42C22×D411yescharacteristic
468C42C22×D411yescharacteristic
469C42C22×D411yescharacteristic
470C42C2×#4111yescharacteristic
471C42C2×#4111yescharacteristic
472C42C2×#4111yescharacteristic
473C42C2511yescharacteristic
474C23—22no—
475C23—24no—
476C23—24no—
477C23—41no—
478C23—41no—
479C23—44no—
480C23—44no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23C2×C4×D411yescharacteristic
486C23C2×C4×D411yescharacteristic
487C23C2×C4×D411yescharacteristic
488Q8—44no—
489Q8—44no—
490Q8—82no—
491Q8—82no—
492D4—28no—
493D4—28no—
494D4—44no—
495D4—44no—
496D4—48no—
497D4—48no—
498D4—44no—
499D4—44no—
500D4—82no—
501D4—82no—
502D4—82no—
503D4—82no—
504D4—82no—
505D4—82no—
506D4—84no—
507D4—84no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
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—41no—
527C2×C4—48no—
528C2×C4—48no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4—82no—
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—44no—
550C8—42no—
551C8—44no—
552C8—42no—
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#6014211yescharacteristic
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. #60236C2semidirect
4. #18073C2semidirect
5. #18066C2semidirect
6. #18026C2semidirect
7. #17950C2semidirect
8. #17922C2semidirect
9. C2×#1572C2semidirect
10. #17495C2semidirect
11. #17463C2semidirect
12. #17324C2semidirect
29. #2975C22semidirect
31. #2974C22semidirect
33. #2972C22semidirect
34. #2935C22semidirect
35. #2935C22semidirect
45. #1594C22semidirect
46. #1589C22semidirect
47. #1586C22semidirect
48. #1586C22semidirect
49. #1581C22semidirect
50. #1581C22semidirect
51. #1580C22semidirect
52. C8×D8C22semidirect
53. #1572C22semidirect
54. #1572C22semidirect
55. #1572C22semidirect
56. C2×#335C22semidirect
57. C2×#334C22semidirect
59. #1490C22semidirect
62. #1480C22semidirect
63. #1480C22semidirect
64. #1478C22semidirect
65. #1454C22semidirect
66. #1453C22semidirect
67. #1451C22semidirect
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
#602362
#180732
#180662
#180262
#179502
#179222
C2×#15722
#174952
#174632
#173242