Finite Group Databasebeta

← #6344738

Subgroups — #6344738

Order 512.

1750 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60558C211yesmaximal, characteristic
2#60268C211yesmaximal, characteristic, complemented
3C2×#2975C211yesmaximal, characteristic, complemented
4#17946C218yesmaximal, complemented
5#17935C212yesmaximal, complemented
6#17933C212yesmaximal, complemented
7#17930C212yesmaximal, complemented
8#17928C212yesmaximal, complemented
9#17893C214yesmaximal, complemented
10#17875C214yesmaximal, complemented
11#17809C211yesmaximal, characteristic, complemented
12#17458C211yesmaximal, characteristic, complemented
13C2×#1486C211yesmaximal, characteristic, complemented
14#17326C211yesmaximal, characteristic, complemented
15#3444C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17#3114C414yes—
18#3110C412yes—
19#3109C412yes—
20#3105C2214yes—
21#3101C2212yes—
22#3100C2212yes—
23#3092C412yes—
24#3077C412yes—
25#3036C412yes—
26#3003C412yes—
27#2994—22no—
28#2993C2214yescomplemented
29#2991—21no—
30#2990—21no—
31#2989C2212yescomplemented
32#2988C2212yescomplemented
33#2975—21no—
34#2975—21no—
35#2975—22no—
36#2975C2214yes—
37#2975C2212yes—
38#2975C2212yes—
39#2960C2212yescomplemented
40C4×AGL(1,ℤ/8)C2214yes—
41#2933—22no—
42#2931C2212yescomplemented
43C2×#436—21no—
44C2×#436—21no—
45C2×#436C2212yes—
46C2×#436C2212yes—
47#2925—21no—
48#2920C2211yescharacteristic
49C2×#431C2211yescharacteristic
50C2×#431C2211yescharacteristic
51C2×#431C2211yescharacteristic
52#2906—21no—
53C2×#423C2211yescharacteristic
54C2×#407—22no—
55#1594C2218yescomplemented
56#1592C2218yescomplemented
57#1590C2218yescomplemented
58#1589C2218yescomplemented
59#1588C2218yescomplemented
60#1586C2218yescomplemented
61#1583C2218yescomplemented
62#1581C2218yescomplemented
63#1574C2214yescomplemented
64#1571C2212yescomplemented
65#1570C2212yescomplemented
66#1565C2214yescomplemented
67#1486C2212yes—
68#1486C2212yes—
69#1486C2214yes—
70#1482C2214yescomplemented
71#1481C2211yescharacteristic, complemented
72#1481C2211yescharacteristic, complemented
73#1480C2211yescharacteristic, complemented
74#1480C2211yescharacteristic, complemented
75#1476C2214yescomplemented
76C2×D4⋊C8C2214yes—
77#1454C2214yescomplemented
78#1452C2212yescomplemented
79#1451C2212yescomplemented
80#1450C2212yescomplemented
81C8×M16C2212yescomplemented
82C23×D4C2×C411yescharacteristic
83C24×C4—21no—
84C2×AGL(1,ℤ/8)C2×C414yes—
85C2×#131—22no—
86C22×M16—21no—
87C22×M16—22no—
88C22×M16—22no—
89#533C2×C411yescharacteristic
90#533C2×C411yescharacteristic
91#533C2311yescharacteristic
92#532C2×C411yescharacteristic
93#532C2×C411yescharacteristic
94#532C2311yescharacteristic
95#529—21no—
96#529—21no—
97#529—21no—
98#529—21no—
99#529C2×C412yes—
100#529C2312yes—
101#525—21no—
102#525—21no—
103#525C2×C412yes—
104C4×#41—21no—
105C4×#41—21no—
106C4×#41C2312yes—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4C2×C412yes—
114C2×C4×D4C2×C412yes—
115C2×C4×D4C2×C412yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C22×C4⋊C4C2×C411yescharacteristic
120C22×C22⋊C4—21no—
121C22×C42C2311yescharacteristic
122#484C2×C414yes—
123#483C2×C414yes—
124#481C2×C414yes—
125#480C2×C414yes—
126#478C2×C418yes—
127#476C2×C414yes—
128#474C2×C414yes—
129#463C2×C418yes—
130#461C2×C414yes—
131#459C2×C414yes—
132#450C2×C414yes—
133#450C2×C414yes—
134#447C2×C414yes—
135#447C2×C414yes—
136#446—22no—
137#446C2314yescomplemented
138C8×Q8—22no—
139C8×Q8C2314yescomplemented
140#442C2×C418yes—
141#440C2×C418yes—
142C4×SD16C2318yes—
143C4×D8C2318yes—
144#436—22no—
145#436—22no—
146#436—24no—
147#436—22no—
148#436—22no—
149#436—24no—
150#436—24no—
151#436—22no—
152#436—22no—
153#436—22no—
154#436—22no—
155#436—22no—
156#436C2314yes—
157#436C2314yes—
158#436C2314yes—
159#436C2314yes—
160#436C2314yescomplemented
161#435—24no—
162#435—24no—
163C8×D4—24no—
164C8×D4—22no—
165C8×D4C2314yescomplemented
166#433—24no—
167#432—22no—
168#432C2×C412yes—
169#432C2×C412yes—
170#431—21no—
171#431—21no—
172#431—21no—
173#431—21no—
174#431—22no—
175#431D411yescharacteristic
176#431D411yescharacteristic
177#431D412yes—
178#431C2312yes—
179#431C2312yes—
180#431C2311yescharacteristic
181#431C2311yescharacteristic
182#431C2312yes—
183#431C2312yes—
184#431C2312yes—
185#431C2311yescharacteristic
186#431C2311yescharacteristic
187#431C2312yes—
188#428C2×C414yes—
189#424—22no—
190#424—24no—
191#423—21no—
192#423—21no—
193#423—22no—
194#423C2×C412yes—
195#423C2312yes—
196#423C2312yes—
197C2×C4⋊C8—21no—
198C2×C4⋊C8—21no—
199C2×C4⋊C8—22no—
200C2×C4⋊C8D411yescharacteristic
201C2×C4⋊C8D411yescharacteristic
202C2×C4⋊C8C2312yes—
203C2×C4⋊C8C2311yescharacteristic
204C2×C4⋊C8C2311yescharacteristic
205#418C2×C414yes—
206#417C2×C414yes—
207#408—22no—
208#407—24no—
209#407—24no—
210#407—24no—
211#407—24no—
212#407C2×C412yes—
213C2×C22⋊C8—21no—
214C2×C22⋊C8—21no—
215C2×C22⋊C8—22no—
216C2×C22⋊C8—22no—
217C2×C22⋊C8D412yes—
218#405—22no—
219C4×M16C2312yes—
220C2×C4×C8—21no—
221C2×C4×C8C2312yes—
222C2×C4×C8C2312yes—
223#335C2318yescomplemented
224#334C2318yescomplemented
225Q8⋊C8C2314yescomplemented
226Q8⋊C8C2314yescomplemented
227D4⋊C8C2318yes—
228D4⋊C8C2318yes—
229D4⋊C8C2314yescomplemented
230D4⋊C8C2314yescomplemented
231#322C2314yescomplemented
232#322C2314yescomplemented
233#322C2314yescomplemented
234C82C2314yescomplemented
235C25—21no—
236C2×#41—21no—
237C2×#41—21no—
238C2×#41C22×C412yes—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—44no—
243C22×D4C22×C412yes—
244C22×D4C22×C412yes—
245C22×D4C22×C412yes—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—82no—
254C23×C4—82no—
255C23×C4C22×C411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4#4111yescharacteristic
258AGL(1,ℤ/8)—48no—
259C2×SD16C22×C418yes—
260C2×D8C22×C418yes—
261#131—84no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—84no—
270C2×M16—84no—
271C2×M16—82no—
272C2×M16C22×C414yes—
273C22×C8—21no—
274C22×C8—21no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—24no—
278C22×C8C2×D412yes—
279C22×C8C2×D412yes—
280C4⋊Q8C22×C411yescharacteristic
281C4⋊Q8C22×C411yescharacteristic
282C4⋊Q8C22×C411yescharacteristic
283#127C22×C411yescharacteristic
284#126—22no—
285#126—22no—
286#125—21no—
287#125—21no—
288#125C22×C412yes—
289#124—21no—
290#124—21no—
291#124C22×C412yes—
292#123—22no—
293#123—22no—
294#123—22no—
295C22⋊Q8—21no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8C22×C412yes—
300C22⋊Q8C22×C412yes—
301#121—21no—
302#121—21no—
303#121—21no—
304#121—21no—
305#121C22×C412yes—
306#121C22×C412yes—
307#120—22no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8C22×C412yes—
313C4×Q8C2412yes—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—21no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4C22×C414yes—
337C4×D4C22×C414yes—
338C4×D4C22×C414yes—
339C4×D4C22×C412yes—
340C4×D4C2414yes—
341C4×D4C2412yes—
342#117—21no—
343#117—21no—
344#117—22no—
345#117—22no—
346#117—22no—
347#117C22×C412yes—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4C22×C412yes—
352C2×C4⋊C4C22×C412yes—
353C2×C4⋊C4C22×C412yes—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C42—22no—
361C2×C42C22×C411yescharacteristic
362C2×C42C22×C411yescharacteristic
363C2×C42C22×C411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42#4111yescharacteristic
368C2×C42#4111yescharacteristic
369C2×C42#4111yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373#107C22×C418yes—
374#106C22×C418yes—
375C4⋊C8—24no—
376C4⋊C8—24no—
377C4⋊C8—22no—
378C4⋊C8—22no—
379C4⋊C8—22no—
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—21no—
384C4⋊C8—22no—
385C4⋊C8C22×C414yes—
386C4⋊C8C22×C414yes—
387C4⋊C8C2×D411yescharacteristic
388C4⋊C8C2×D411yescharacteristic
389C4⋊C8C2×D412yes—
390C4⋊C8C2×D411yescharacteristic
391C4⋊C8C2×D411yescharacteristic
392C4⋊C8C2×D412yes—
393C4⋊C8C2414yes—
394C4⋊C8C2412yes—
395C4⋊C8C2412yes—
396#103C22×C418yes—
397#102C22×C418yes—
398#102C22×C418yes—
399#102C22×C418yes—
400C22⋊C8—22no—
401C22⋊C8—22no—
402C22⋊C8—21no—
403C22⋊C8—21no—
404C22⋊C8—22no—
405C22⋊C8—24no—
406C22⋊C8—24no—
407C22⋊C8—24no—
408C22⋊C8—24no—
409C22⋊C8—24no—
410C22⋊C8—24no—
411C22⋊C8C22×C414yes—
412C22⋊C8C22×C414yes—
413C22⋊C8C2×D412yes—
414C22⋊C8C2×D412yes—
415C22⋊C8C2×D412yes—
416C22⋊C8C2×D412yes—
417#97—24no—
418#97C22×C414yes—
419C4×C8—24no—
420C4×C8—22no—
421C4×C8—22no—
422C4×C8—22no—
423C4×C8C2×D412yes—
424C4×C8C2×D412yes—
425C4×C8C2412yes—
426C4×C8C2412yes—
427C4×C8C2414yes—
428C4×C8C2412yes—
429C4×C8C2412yes—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—44no—
434C24—82no—
435C24—82no—
436C24C4×D411yescharacteristic
437#41—28no—
438#41—42no—
439#41—42no—
440#41—82no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C23×C412yes—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4C23×C414yes—
460C2×D4C23×C412yes—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—84no—
483C22×C4—81no—
484C22×C4—82no—
485C22×C4C4×D411yescharacteristic
486C22×C4C4×D411yescharacteristic
487C22×C4C4×D411yescharacteristic
488C22×C4C4×D411yescharacteristic
489C22×C4C4×D411yescharacteristic
490C22×C4C4×D411yescharacteristic
491C22×C4C4×D411yescharacteristic
492C22×C4C4×D411yescharacteristic
493C22×C4C4×D411yescharacteristic
494C22×C4C23×C411yescharacteristic
495C22×C4C23×C411yescharacteristic
496C22×C4C23×C411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C2×#4111yescharacteristic
501C22×C4C2×#4111yescharacteristic
502C22×C4C2×#4111yescharacteristic
503SD16—48no—
504D8—48no—
505M16—28no—
506M16—44no—
507M16—84no—
508M16—84no—
509M16—84no—
510M16—84no—
511M16—84no—
512M16—82no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—24no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—42no—
521C2×C8—42no—
522C2×C8—84no—
523C2×C8—82no—
524C2×C8—82no—
525C2×C8C23×C418yes—
526C2×C8C22×D412yes—
527C2×C8C22×D412yes—
528C2×C8C22×D412yes—
529C2×C8C22×D412yes—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4C23×C412yes—
544C4⋊C4C23×C412yes—
545C4⋊C4C23×C414yes—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—24no—
558C22⋊C4—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42C4×D411yescharacteristic
563C42C4×D411yescharacteristic
564C42C4×D411yescharacteristic
565C42C4×D411yescharacteristic
566C42C4×D411yescharacteristic
567C42C4×D411yescharacteristic
568C42C23×C411yescharacteristic
569C42C23×C411yescharacteristic
570C42C23×C411yescharacteristic
571C42C22×D411yescharacteristic
572C42C22×D411yescharacteristic
573C42C22×D411yescharacteristic
574C42C2×#4111yescharacteristic
575C42C2×#4111yescharacteristic
576C42C2×#4111yescharacteristic
577C42C2511yescharacteristic
578C23—22no—
579C23—22no—
580C23—22no—
581C23—24no—
582C23—41no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—82no—
589C23—82no—
590C23—82no—
591C23—82no—
592C23—816no—
593C23C2×C4×D411yescharacteristic
594C23C2×C4×D411yescharacteristic
595C23C2×C4×D411yescharacteristic
596Q8—24no—
597Q8—42no—
598Q8—42no—
599D4—24no—
600D4—24no—
601D4—24no—
602D4—44no—
603D4—44no—
604D4—42no—
605D4—44no—
606D4—44no—
607D4—44no—
608D4—42no—
609D4—82no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—48no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4C2×C4×D411yescharacteristic
645C2×C4C2×C4×D411yescharacteristic
646C2×C4C2×C4×D411yescharacteristic
647C2×C4C2×C4×D411yescharacteristic
648C2×C4C2×C4×D411yescharacteristic
649C2×C4C2×C4×D411yescharacteristic
650C2×C4C2×C4×D411yescharacteristic
651C2×C4C2×C4×D411yescharacteristic
652C2×C4C2×C4×D411yescharacteristic
653C2×C4#56814yes—
654C2×C4#57612yes—
655C2×C4#57712yes—
656C2×C4C24×C411yescharacteristic
657C2×C4C23×D411yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C8—28no—
660C8—22no—
661C8—22no—
662C8—22no—
663C8—22no—
664C8—42no—
665C8—42no—
666C8—42no—
667C8—44no—
668C8—42no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—44no—
677C22—42no—
678C22—44no—
679C22—42no—
680C22—44no—
681C22—44no—
682C22—42no—
683C22—88no—
684C22—88no—
685C22—88no—
686C22#293012yes—
687C22#294612yes—
688C22C22×C4×D411yescharacteristic
689C4—21no—
690C4—21no—
691C4—21no—
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—42no—
698C4—42no—
699C4—44no—
700C4—42no—
701C4#293012yes—
702C4#294612yes—
703C4C2×#56812yes—
704C4C2×#57611yescharacteristic
705C4C2×#57711yescharacteristic
706C2—22no—
707C2—22no—
708C2—41no—
709C2—82no—
710C2—84no—
711C2C2×#293011yescharacteristic
712C2C2×#294611yescharacteristic
713C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#60558C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#431 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60268C2semidirect
3. C2×#2975C2semidirect
4. #17946C2semidirect
5. #17935C2semidirect
6. #17933C2semidirect
7. #17930C2semidirect
8. #17928C2semidirect
9. #17893C2semidirect
10. #17875C2semidirect
11. #17809C2semidirect
12. #17458C2semidirect
13. C2×#1486C2semidirect
14. #17326C2semidirect
28. #2993C22semidirect
31. #2989C22semidirect
32. #2988C22semidirect
39. #2960C22semidirect
42. #2931C22semidirect
55. #1594C22semidirect
56. #1592C22semidirect
57. #1590C22semidirect
58. #1589C22semidirect
59. #1588C22semidirect
60. #1586C22semidirect
61. #1583C22semidirect
62. #1581C22semidirect
63. #1574C22semidirect
64. #1571C22semidirect
65. #1570C22semidirect
66. #1565C22semidirect
70. #1482C22semidirect
71. #1481C22semidirect
72. #1481C22semidirect
73. #1480C22semidirect
74. #1480C22semidirect
75. #1476C22semidirect
77. #1454C22semidirect
78. #1452C22semidirect
79. #1451C22semidirect
80. #1450C22semidirect
81. C8×M16C22semidirect
137. #446C23semidirect
139. C8×Q8C23semidirect
160. #436C23semidirect
165. C8×D4C23semidirect
223. #335C23semidirect
224. #334C23semidirect
225. Q8⋊C8C23semidirect
226. Q8⋊C8C23semidirect
229. D4⋊C8C23semidirect
230. D4⋊C8C23semidirect
231. #322C23semidirect
232. #322C23semidirect
233. #322C23semidirect
234. C82C23semidirect

Maximal subgroups

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

subgroupindex
#605582
#602682
C2×#29752
#179462
#179352
#179332
#179302
#179282
#178932
#178752
#178092
#174582
C2×#14862
#173262