Finite Group Databasebeta

← #6508802

Subgroups — #6508802

Order 512.

2370 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic, complemented
2C2×#2972C211yesmaximal, characteristic, complemented
3#35000C211yesmaximal, characteristic
4C2×#2487C211yesmaximal, characteristic, complemented
5#28097C214yesmaximal, complemented
6#27658C214yesmaximal, complemented
7#25238C218yesmaximal, complemented
8#24603C214yesmaximal, complemented
9#24553C214yesmaximal, complemented
10#21938C211yesmaximal, characteristic, complemented
11#21758C211yesmaximal, characteristic, complemented
12C2×#1824C211yesmaximal, characteristic, complemented
13C2×#534—21no—
14C22×#121C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#481—28no—
18C2×#481C2214yes—
19C2×#481C2214yescomplemented
20#2972—28no—
21C2×C8×D4C2214yes—
22C2×C8×D4C2214yescomplemented
23C2×#431C2212yescomplemented
24C22×#106C2212yescomplemented
25C22×#102C2212yescomplemented
26C22×C22⋊C8C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C22×C22⋊C8C2212yescomplemented
29#2854C2212yes—
30#2847C2212yes—
31#2618—21no—
32#2617—21no—
33#2543—22no—
34#2534—22no—
35#2487—22no—
36#2487—22no—
37#2487C2214yes—
38#2487C2214yescomplemented
39#2486C2212yes—
40#2416C2214yes—
41#2408C2212yes—
42C2×#400—21no—
43C2×#399—21no—
44#2308—21no—
45C2×#389C2212yescomplemented
46C2×#386—21no—
47C2×#385—21no—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2212yescomplemented
51#2088C2218yescomplemented
52#2077C2218yescomplemented
53#2065C2218yescomplemented
54#2034C2218yescomplemented
55#1941C2218yescomplemented
56#1934C2218yescomplemented
57#1893C2214yescomplemented
58#1893C2218yescomplemented
59#1874C2214yes—
60#1853C2214yescomplemented
61#1853C2214yescomplemented
62#1824C2214yescomplemented
63#1824C2214yescomplemented
64#1823C2212yescomplemented
65#1807C2218yescomplemented
66#1807C2214yescomplemented
67#1786C2212yescomplemented
68C2×#340C2214yescomplemented
69C23×D4—21no—
70C23×D4D411yescharacteristic
71C23×D4C2311yescharacteristic
72C24×C4—21no—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C23×C8C2314yes—
76C23×C8C2312yes—
77C23×C8C2312yes—
78#534—48no—
79C2×#123—22no—
80C2×#123—22no—
81C2×#121—24no—
82C2×#121—21no—
83C2×#121—21no—
84C2×#121—24no—
85C2×#121—24no—
86C2×#121C2312yes—
87C2×#121C2312yes—
88C2×#121C2314yes—
89C2×#120—22no—
90C2×#120—22no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4D412yes—
94C2×C4×D4D412yes—
95C2×C4×D4C2312yes—
96C2×C4×D4C2312yes—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—22no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—22no—
114C22×C22⋊C4D411yescharacteristic
115C22×C22⋊C4D411yescharacteristic
116C22×C42C2311yescharacteristic
117#481—216no—
118#481—216no—
119C8×D4—216no—
120C8×D4—216no—
121#431—28no—
122C2×#106—28no—
123C2×#106C2318yes—
124C2×#106C2318yes—
125C2×C4⋊C8C2314yes—
126C2×#102—28no—
127C2×#102C2318yes—
128C2×#102C2318yes—
129C2×C22⋊C8—24no—
130C2×C22⋊C8—28no—
131C2×C22⋊C8D418yescomplemented
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C4×C8C2314yes—
138#400—24no—
139#400—24no—
140#399—24no—
141#399—24no—
142#398—22no—
143#397—22no—
144#396—24no—
145#394—22no—
146#394—24no—
147#392—22no—
148#390C2314yes—
149#389—24no—
150#389C2318yes—
151#389C2314yes—
152#388—24no—
153#386—24no—
154#386—24no—
155#386—22no—
156#386—22no—
157#386—24no—
158#386—24no—
159#385—24no—
160#385—24no—
161#381—22no—
162#380—21no—
163#380—21no—
164#380—21no—
165#380—24no—
166#380—21no—
167#380C2314yes—
168#380C2312yes—
169#380C2312yes—
170#380C2318yes—
171#380C2312yes—
172#380C2312yes—
173#379—22no—
174C2×#95—21no—
175C2×#95—22no—
176C2×#95—22no—
177C2×#95—21no—
178#340C2318yes—
179#340C2318yes—
180#340C2318yes—
181#340C2318yes—
182C25—21no—
183C25C2×D411yescharacteristic
184C25C2×D411yescharacteristic
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—21no—
192C22×D4—22no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4SD1618yes—
198C22×D4C2×D412yes—
199C22×D4C2×D412yes—
200C22×D4C2412yes—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—84no—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D412yes—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D414yes—
221C23×C4#4114yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C22×C8—24no—
227C22×C8—28no—
228C22×C8—44no—
229C22×C8—48no—
230C22×C8—44no—
231C22×C8C2×D418yes—
232C22×C8C2414yes—
233C22×C8C2414yes—
234#123—48no—
235#123—48no—
236#121—24no—
237#121—24no—
238#121—24no—
239#121—24no—
240#121—416no—
241#121—44no—
242#121—44no—
243#121—48no—
244#120—48no—
245#120—48no—
246C4×D4—28no—
247C4×D4—28no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—84no—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2414yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—84no—
298C2×C22⋊C4C2×D412yes—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D412yes—
301C2×C22⋊C4C2×D412yes—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42C2×D412yes—
305C2×C42C2412yes—
306#106—216no—
307#106—48no—
308C4⋊C8—28no—
309#102—216no—
310#102—48no—
311C22⋊C8—24no—
312C22⋊C8—24no—
313C22⋊C8—28no—
314C22⋊C8C2×D418yes—
315C22⋊C8C2×D418yes—
316C4×C8—28no—
317#95—28no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—28no—
323#95—24no—
324#95—24no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—82no—
336C24—82no—
337C24#12114yes—
338C24C2×SD1614yes—
339C24C2×SD1614yes—
340C24#13514yes—
341C24C22×D412yes—
342C24C22×D411yescharacteristic
343C24C22×D411yescharacteristic
344C24C22×D411yescharacteristic
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—232no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—28no—
353C2×D4—48no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—48no—
361C2×D4—42no—
362C2×D4—84no—
363C2×D4—82no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—816no—
367C2×D4C2×SD1618yes—
368C2×D4C2×SD1618yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—28no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4—816no—
411C22×C4#12114yes—
412C22×C4#12114yes—
413C22×C4#12114yes—
414C22×C4C2×SD1614yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2511yescharacteristic
428C2×C8—28no—
429C2×C8—24no—
430C2×C8—28no—
431C2×C8—48no—
432C2×C8—88no—
433C2×C8—84no—
434C2×C8C22×D418yes—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—48no—
449C4⋊C4—82no—
450C4⋊C4—88no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—48no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—48no—
463C22⋊C4—84no—
464C22⋊C4—84no—
465C22⋊C4—88no—
466C42—24no—
467C42—22no—
468C42—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—22no—
477C23—24no—
478C23—22no—
479C23—28no—
480C23—44no—
481C23—44no—
482C23—48no—
483C23—48no—
484C23—48no—
485C23—44no—
486C23—42no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—82no—
491C23—84no—
492C23—82no—
493C23—84no—
494C23—84no—
495C23—84no—
496C23—164no—
497C23—164no—
498C23#465116yes—
499C23C2×#12112yes—
500C23C2×#12112yes—
501C23C2×#12112yes—
502C23C22×SD1614yes—
503C23C22×SD1612yes—
504C23C22×SD1612yes—
505C23C22×SD1612yes—
506C23C2×#13512yes—
507C23C2×#13512yes—
508C23C2×#13512yes—
509C23C23×D411yescharacteristic
510D4—216no—
511D4—216no—
512D4—44no—
513D4—44no—
514D4—84no—
515D4—88no—
516D4—88no—
517D4—168no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—88no—
555C2×C4—88no—
556C2×C4—88no—
557C2×C4—88no—
558C2×C4—84no—
559C2×C4—164no—
560C2×C4C2×#12112yes—
561C2×C4C2×#12112yes—
562C2×C4C2×#12112yes—
563C2×C4C22×SD1614yes—
564C2×C4C23×D411yescharacteristic
565C2×C4C22×#4111yescharacteristic
566C2×C4#58514yes—
567C8—28no—
568C8—44no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—28no—
582C22—24no—
583C22—416no—
584C22—44no—
585C22—42no—
586C22—42no—
587C22—88no—
588C22—84no—
589C22—84no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—168no—
594C22C2×#46514yes—
595C22C2×#46514yes—
596C22C2×#46514yes—
597C22C22×#12111yescharacteristic
598C22#348512yes—
599C22C23×SD1611yescharacteristic
600C22C22×#13511yescharacteristic
601C22#358512yes—
602C4—24no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—42no—
607C4—42no—
608C4—84no—
609C4—84no—
610C4—82no—
611C4—84no—
612C4#348512yes—
613C4#349112yes—
614C2—24no—
615C2—24no—
616C2—24no—
617C2—44no—
618C2—162no—
619C2#3544711yescharacteristic
620C2C22×#46511yescharacteristic
621C2#6045411yescharacteristic
622C2#6169712yes—
623C2#6169912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35000C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#481C2semidirect
2. C2×#2972C2semidirect
4. C2×#2487C2semidirect
5. #28097C2semidirect
6. #27658C2semidirect
7. #25238C2semidirect
8. #24603C2semidirect
9. #24553C2semidirect
10. #21938C2semidirect
11. #21758C2semidirect
12. C2×#1824C2semidirect
19. C2×#481C22semidirect
22. C2×C8×D4C22semidirect
23. C2×#431C22semidirect
24. C22×#106C22semidirect
25. C22×#102C22semidirect
28. C22×C22⋊C8C22semidirect
38. #2487C22semidirect
45. C2×#389C22semidirect
50. C2×#380C22semidirect
51. #2088C22semidirect
52. #2077C22semidirect
53. #2065C22semidirect
54. #2034C22semidirect
55. #1941C22semidirect
56. #1934C22semidirect
57. #1893C22semidirect
58. #1893C22semidirect
60. #1853C22semidirect
61. #1853C22semidirect
62. #1824C22semidirect
63. #1824C22semidirect
64. #1823C22semidirect
65. #1807C22semidirect
66. #1807C22semidirect
67. #1786C22semidirect
68. C2×#340C22semidirect
131. C2×C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4812
C2×#29722
#350002
C2×#24872
#280972
#276582
#252382
#246032
#245532
#219382
#217582
C2×#18242