Finite Group Databasebeta

← #206428

Subgroups — #206428

Order 512.

1057 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60700C211yesmaximal, characteristic
2#21856C211yesmaximal, characteristic, complemented
3#21815C211yesmaximal, characteristic, complemented
4#20472C211yesmaximal, characteristic, complemented
5#17796C211yesmaximal, characteristic, complemented
6#17489C211yesmaximal, characteristic, complemented
7#17485C211yesmaximal, characteristic, complemented
8#9854C211yesmaximal, characteristic, complemented
9#9852C211yesmaximal, characteristic, complemented
10#9851C211yesmaximal, characteristic, complemented
11#9849C211yesmaximal, characteristic, complemented
12#9847C211yesmaximal, characteristic, complemented
13#9846C211yesmaximal, characteristic, complemented
14#9844C211yesmaximal, characteristic, complemented
15#9843C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3184C411yescharacteristic, complemented
19#3184C411yescharacteristic, complemented
20#3182C411yescharacteristic, complemented
21#3182C411yescharacteristic, complemented
22#3180—22no—
23#3171—22no—
24#3169C2211yescharacteristic
25#3169C2211yescharacteristic
26#3167C2211yescharacteristic
27#3167C2211yescharacteristic
28#3165C411yescharacteristic, complemented
29#3151C411yescharacteristic, complemented
30#3141C411yescharacteristic, complemented
31#3138C411yescharacteristic, complemented
32#3055—22no—
33C4×AGL(1,ℤ/8)—22no—
34#2920C2211yescharacteristic
35C2×#431C2211yescharacteristic
36C2×#423C2211yescharacteristic
37C2×C4×M16C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1994C2211yescharacteristic, complemented
40#1994C2211yescharacteristic, complemented
41#1980—22no—
42#1962C2211yescharacteristic
43#1962C2211yescharacteristic
44#1962C2211yescharacteristic
45#1962C2211yescharacteristic
46#1962C2211yescharacteristic, complemented
47#1962C2211yescharacteristic, complemented
48#1871—22no—
49#1806—22no—
50#1789—22no—
51#1771—22no—
52#1753—22no—
53#1744C2211yescharacteristic, complemented
54#1744C2211yescharacteristic, complemented
55C4×#97C2211yescharacteristic, complemented
56C42×C8C2211yescharacteristic, complemented
57#1569—22no—
58#1567C2211yescharacteristic, complemented
59#1567C2211yescharacteristic, complemented
60#1566C2211yescharacteristic, complemented
61#1566C2211yescharacteristic, complemented
62#1565—22no—
63#1493C2211yescharacteristic, complemented
64#1493C2211yescharacteristic, complemented
65#1492C2211yescharacteristic
66#1492C2211yescharacteristic
67#1492C2211yescharacteristic
68#1492C2211yescharacteristic
69#1492C2211yescharacteristic, complemented
70#1492C2211yescharacteristic, complemented
71#1488—22no—
72#1486—22no—
73#1482—22no—
74#1480—22no—
75C22×#41—21no—
76C2×AGL(1,ℤ/8)—22no—
77C22×M16—21no—
78C22×M16D412yes—
79#533C2×C411yescharacteristic
80#533C2×C411yescharacteristic
81#533C2311yescharacteristic
82#532—21no—
83#532—21no—
84#532—21no—
85#532—21no—
86#532—21no—
87#532—21no—
88#532C2×C411yescharacteristic
89#532C2×C411yescharacteristic
90#532C2×C411yescharacteristic
91#532C2×C411yescharacteristic
92#532C2311yescharacteristic
93#532C2311yescharacteristic
94#532C2311yescharacteristic
95C2×C4⋊Q8C2×C411yescharacteristic
96C2×#127C2×C411yescharacteristic
97#529—21no—
98#529—21no—
99#529—21no—
100#529—21no—
101#529—21no—
102#529—21no—
103C2×#124—21no—
104#525—21no—
105#525—21no—
106#525—21no—
107C2×#121—21no—
108C2×#121—21no—
109C4×#41—21no—
110C4×#41—21no—
111C4×#41—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×#117—21no—
115C22×C42C2311yescharacteristic
116#501—22no—
117#500C2×C411yescharacteristic
118#500C2×C411yescharacteristic
119#498C2×C411yescharacteristic
120#498C2×C411yescharacteristic
121#496—22no—
122#493C2×C411yescharacteristic
123#493C2×C411yescharacteristic
124#492C2×C411yescharacteristic
125#492C2×C411yescharacteristic
126#490—22no—
127#489—22no—
128#487C2×C411yescharacteristic
129#487C2×C411yescharacteristic
130#486C2×C411yescharacteristic
131#486C2×C411yescharacteristic
132#486C2×C411yescharacteristic
133#486C2×C411yescharacteristic
134#486C2×C411yescharacteristic
135#486C2×C411yescharacteristic
136#469—22no—
137#469—22no—
138#468—22no—
139#468—22no—
140#466—22no—
141#466—22no—
142#465—22no—
143#465—22no—
144#442—22no—
145#442—22no—
146#440—22no—
147#440—22no—
148C4×SD16—22no—
149C4×SD16—22no—
150C4×D8—22no—
151C4×D8—22no—
152#432—21no—
153#432—21no—
154#431—21no—
155#431—21no—
156#431—21no—
157#431—21no—
158#431D412yes—
159#431D412yes—
160#428—22no—
161#423—21no—
162#423—21no—
163#423—21no—
164#423C2311yescharacteristic
165#423C2311yescharacteristic
166#423C2311yescharacteristic
167#423C2311yescharacteristic
168C2×C4⋊C8—21no—
169C2×C4⋊C8—21no—
170C2×C4⋊C8—21no—
171C2×#104—22no—
172C2×#104—22no—
173C2×#104—22no—
174C2×#104—22no—
175C2×#104—22no—
176C2×#104—22no—
177#418—22no—
178#417—22no—
179#407—21no—
180C2×C22⋊C8—21no—
181C4×M16—21no—
182C4×M16C2×C411yescharacteristic
183C4×M16C2×C411yescharacteristic
184C4×M16C2×C411yescharacteristic
185C4×M16C2×C411yescharacteristic
186C4×M16D412yes—
187C4×M16D412yes—
188C2×#97—22no—
189C2×#97D412yes—
190C2×C4×C8—22no—
191C2×C4×C8C2311yescharacteristic
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2311yescharacteristic
195C4×C4⋊C4—21no—
196C4×C4⋊C4D412yes—
197C4×C22⋊C4—21no—
198C4×C22⋊C4D412yes—
199#376—21no—
200C43C2311yescharacteristic
201C43C2311yescharacteristic
202#339—22no—
203#339—22no—
204#336—22no—
205#336—22no—
206#335—22no—
207#335—22no—
208#334—22no—
209#334—22no—
210Q8⋊C8—22no—
211Q8⋊C8—22no—
212D4⋊C8—22no—
213D4⋊C8—22no—
214D4⋊C8—22no—
215D4⋊C8—22no—
216D4⋊C8—22no—
217D4⋊C8—22no—
218C2×#41—21no—
219C2×#41—21no—
220C2×#41—21no—
221C2×#41—21no—
222C2×#41—21no—
223C2×#41—21no—
224C2×#41—21no—
225C2×#41—41no—
226C2×#41—41no—
227C2×#41—82no—
228C2×#41—82no—
229C22×Q8—21no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—41no—
234C23×C4—41no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4#4111yescharacteristic
238AGL(1,ℤ/8)—48no—
239C2×SD16—22no—
240C2×SD16—22no—
241C2×D8—22no—
242C2×D8—22no—
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—44no—
248C2×M16—41no—
249C2×M16—82no—
250C2×M16C22⋊C412yes—
251C2×M16C22⋊C412yes—
252C2×M16C22⋊C412yes—
253C2×M16C22⋊C412yes—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—41no—
257C22×C8C2×D412yes—
258C22×C8C2×D412yes—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8C22×C411yescharacteristic
263C4⋊Q8C22×C411yescharacteristic
264C4⋊Q8C22×C411yescharacteristic
265C4⋊Q8C22×C411yescharacteristic
266C4⋊Q8C22×C411yescharacteristic
267#127—21no—
268#127—21no—
269#127—21no—
270#127C22×C411yescharacteristic
271#127C22×C411yescharacteristic
272#127C22×C411yescharacteristic
273#126—41no—
274#126—41no—
275#125—21no—
276#125—21no—
277#125—21no—
278#124—21no—
279#124—21no—
280#124—21no—
281#124—21no—
282#124—21no—
283#124—21no—
284#124—21no—
285#124—41no—
286#124—41no—
287#123—41no—
288#123—41no—
289#123—41no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296#121—21no—
297#121—21no—
298#121—21no—
299#121—21no—
300#121—21no—
301#121—21no—
302#121—21no—
303#121—21no—
304#121—21no—
305#121—21no—
306#121—21no—
307#121—21no—
308#121—21no—
309#121—21no—
310#121—41no—
311#121—41no—
312#121—41no—
313#121—41no—
314#120—41no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—41no—
336C4×D4—41no—
337C4×D4—41no—
338C4×D4—41no—
339C4×D4—41no—
340C4×D4—41no—
341C4×D4—41no—
342C4×D4—41no—
343C4×D4—41no—
344C4×D4—41no—
345#117—21no—
346#117—21no—
347#117—21no—
348#117—44no—
349#117—41no—
350#117—41no—
351#117—41no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—41no—
357C2×C22⋊C4—41no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—41no—
363C2×C42C22×C411yescharacteristic
364C2×C42C22×C411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42#4111yescharacteristic
374C2×C42#4111yescharacteristic
375C2×C42#4111yescharacteristic
376C2×C42C2411yescharacteristic
377#107—22no—
378#107—22no—
379#106—22no—
380#106—22no—
381C4⋊C8—22no—
382C4⋊C8—22no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—21no—
391C4⋊C8—41no—
392C4⋊C8—41no—
393#104—82no—
394#104—82no—
395#104—82no—
396#104—82no—
397#104—82no—
398#104—82no—
399#103—22no—
400#103—22no—
401#102—22no—
402#102—22no—
403#102—22no—
404#102—22no—
405#102—22no—
406#102—22no—
407C22⋊C8—22no—
408C22⋊C8—22no—
409C22⋊C8—41no—
410C22⋊C8—41no—
411#97—44no—
412#97—41no—
413#97C22⋊C412yes—
414#97C22⋊C412yes—
415#97C22⋊C412yes—
416#97C22⋊C412yes—
417C4×C8—21no—
418C4×C8—21no—
419C4×C8—44no—
420C4×C8—41no—
421C4×C8C22×C411yescharacteristic
422C4×C8C22×C411yescharacteristic
423C4×C8C22×C411yescharacteristic
424C4×C8C22×C411yescharacteristic
425C4×C8C2×D412yes—
426C4×C8C2×D412yes—
427C4×C8C2×D412yes—
428C4×C8C2×D412yes—
429#95—22no—
430#95—22no—
431C24—41no—
432C24#12111yescharacteristic
433#41—44no—
434#41—42no—
435#41—42no—
436#41—42no—
437#41—42no—
438#41—42no—
439#41—42no—
440#41—82no—
441#41—82no—
442#41—82no—
443#41—84no—
444#41—82no—
445#41—82no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—41no—
452C2×Q8—82no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—44no—
463C2×D4—41no—
464C2×D4—41no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—82no—
474C2×D4—82no—
475C2×D4—84no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4C2×C22⋊C411yescharacteristic
493C22×C4C2×C22⋊C411yescharacteristic
494C22×C4C4×D411yescharacteristic
495C22×C4C4×D411yescharacteristic
496C22×C4C4×D411yescharacteristic
497C22×C4C4×D411yescharacteristic
498C22×C4#12111yescharacteristic
499C22×C4#12111yescharacteristic
500C22×C4#12111yescharacteristic
501C22×C4#12111yescharacteristic
502C22×C4#12111yescharacteristic
503C22×C4#12411yescharacteristic
504C22×C4#12411yescharacteristic
505C22×C4#12711yescharacteristic
506C22×C4#12711yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C2×#4111yescharacteristic
510SD16—44no—
511SD16—44no—
512D8—44no—
513D8—44no—
514M16—24no—
515M16—24no—
516M16—42no—
517M16—42no—
518M16—42no—
519M16—42no—
520M16—42no—
521M16—42no—
522M16—82no—
523C2×C8—24no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—42no—
527C2×C8—42no—
528C2×C8—42no—
529C2×C8—41no—
530C2×C8—41no—
531C2×C8—41no—
532C2×C8—41no—
533C2×C8—82no—
534C2×C8C2×C22⋊C412yes—
535C2×C8C2×C22⋊C412yes—
536C2×C8C2×C22⋊C412yes—
537C2×C8C2×C22⋊C412yes—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—44no—
549C4⋊C4—41no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C4⋊C4—41no—
553C4⋊C4—41no—
554C4⋊C4—41no—
555C4⋊C4—41no—
556C4⋊C4—41no—
557C22⋊C4—44no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4—41no—
561C22⋊C4—41no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—41no—
566C22⋊C4—41no—
567C22⋊C4—41no—
568C22⋊C4—41no—
569C22⋊C4—41no—
570C22⋊C4—41no—
571C22⋊C4—41no—
572C22⋊C4—41no—
573C22⋊C4—41no—
574C22⋊C4—41no—
575C42—24no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—42no—
581C42—42no—
582C42—42no—
583C42—41no—
584C42—41no—
585C42—41no—
586C42C2×C22⋊C411yescharacteristic
587C42C2×C22⋊C411yescharacteristic
588C42C4×D411yescharacteristic
589C42C4×D411yescharacteristic
590C42C4×D411yescharacteristic
591C42C4×D411yescharacteristic
592C42#12111yescharacteristic
593C42#12111yescharacteristic
594C42#12411yescharacteristic
595C42#12411yescharacteristic
596C42#12711yescharacteristic
597C42#12711yescharacteristic
598C42C23×C411yescharacteristic
599C42C22×D411yescharacteristic
600C42C22×D411yescharacteristic
601C42C2×#4111yescharacteristic
602C23—22no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23—41no—
607C23—41no—
608C23—82no—
609C23—82no—
610C23—82no—
611C23#39011yescharacteristic
612C23#39011yescharacteristic
613C23C2×#12111yescharacteristic
614Q8—42no—
615Q8—42no—
616Q8—42no—
617Q8—42no—
618Q8—42no—
619Q8—82no—
620D4—42no—
621D4—42no—
622D4—42no—
623D4—42no—
624D4—42no—
625D4—42no—
626D4—42no—
627D4—42no—
628D4—42no—
629D4—82no—
630D4—82no—
631D4—82no—
632D4—82no—
633D4—82no—
634D4—82no—
635D4—82no—
636D4—82no—
637D4—82no—
638D4—82no—
639C2×C4—22no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—24no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—82no—
670C2×C4—82no—
671C2×C4—82no—
672C2×C4—82no—
673C2×C4—82no—
674C2×C4—81no—
675C2×C4—81no—
676C2×C4—81no—
677C2×C4—81no—
678C2×C4—81no—
679C2×C4—81no—
680C2×C4#39011yescharacteristic
681C2×C4#39011yescharacteristic
682C2×C4#39011yescharacteristic
683C2×C4#39011yescharacteristic
684C2×C4#39011yescharacteristic
685C2×C4#39011yescharacteristic
686C2×C4C22×C22⋊C411yescharacteristic
687C2×C4C2×C4×D411yescharacteristic
688C2×C4C2×C4×D411yescharacteristic
689C2×C4C2×#12111yescharacteristic
690C2×C4C2×#12411yescharacteristic
691C2×C4C2×#12711yescharacteristic
692C2×C4#57612yes—
693C2×C4#57712yes—
694C8—22no—
695C8—22no—
696C8—22no—
697C8—42no—
698C8—42no—
699C8—81no—
700C8—81no—
701C8—81no—
702C8—81no—
703C8#42114yes—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—21no—
711C22—21no—
712C22—82no—
713C22—81no—
714C22—81no—
715C22—81no—
716C22—81no—
717C22—81no—
718C22—81no—
719C22—81no—
720C22—81no—
721C22—81no—
722C22—164no—
723C22C2×#39011yescharacteristic
724C22#306212yes—
725C4—21no—
726C4—21no—
727C4—21no—
728C4—21no—
729C4—21no—
730C4—21no—
731C4—21no—
732C4—21no—
733C4—44no—
734C4—82no—
735C4—81no—
736C4—81no—
737C4—81no—
738C4—81no—
739C4—81no—
740C4C2×#42112yes—
741C4#294612yes—
742C4#295612yes—
743C4#315411yescharacteristic
744C4#315511yescharacteristic
745C2—22no—
746C2—41no—
747C2—41no—
748C2—161no—
749C2—161no—
750C2—161no—
751C2#2120911yescharacteristic
752C2#2123511yescharacteristic
753C2#2124111yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60700C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21856C2semidirect
3. #21815C2semidirect
4. #20472C2semidirect
5. #17796C2semidirect
6. #17489C2semidirect
7. #17485C2semidirect
8. #9854C2semidirect
9. #9852C2semidirect
10. #9851C2semidirect
11. #9849C2semidirect
12. #9847C2semidirect
13. #9846C2semidirect
14. #9844C2semidirect
15. #9843C2semidirect
18. #3184C4semidirect
19. #3184C4semidirect
20. #3182C4semidirect
21. #3182C4semidirect
28. #3165C4semidirect
29. #3151C4semidirect
30. #3141C4semidirect
31. #3138C4semidirect
39. #1994C22semidirect
40. #1994C22semidirect
46. #1962C22semidirect
47. #1962C22semidirect
53. #1744C22semidirect
54. #1744C22semidirect
55. C4×#97C22semidirect
56. C42×C8C22semidirect
58. #1567C22semidirect
59. #1567C22semidirect
60. #1566C22semidirect
61. #1566C22semidirect
63. #1493C22semidirect
64. #1493C22semidirect
69. #1492C22semidirect
70. #1492C22semidirect

Maximal subgroups

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

subgroupindex
#607002
#218562
#218152
#204722
#177962
#174892
#174852
#98542
#98522
#98512
#98492
#98472
#98462
#98442
#98432