Finite Group Databasebeta

← #6345471

Subgroups — #6345471

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60951C211yesmaximal, characteristic
2#60268C211yesmaximal, characteristic, complemented
3C2×#2979C211yesmaximal, characteristic, complemented
4#18080C218yesmaximal, complemented
5#18025C214yesmaximal, complemented
6#18015C214yesmaximal, complemented
7#17935C212yesmaximal, complemented
8#17930C212yesmaximal, complemented
9#17905C212yesmaximal, complemented
10#17900C212yesmaximal, complemented
11#17809C211yesmaximal, characteristic, complemented
12#17485C211yesmaximal, characteristic, complemented
13#17458C211yesmaximal, characteristic, complemented
14#17325C211yesmaximal, characteristic, complemented
15#3444C2211yescharacteristic
16C2×#532C2211yescharacteristic
17#3254C412yes—
18#3253C412yes—
19#3238C412yes—
20#3235C412yes—
21#3217C412yes—
22#3214C412yes—
23#3194C414yes—
24#3171C2214yes—
25#3106C2214yes—
26#3101C2212yes—
27#3100C2212yes—
28#2994C2214yescomplemented
29#2993—22no—
30#2991C2212yescomplemented
31#2990C2212yescomplemented
32#2989—21no—
33#2988—21no—
34#2979—21no—
35#2979—21no—
36#2979—22no—
37#2979C2212yes—
38#2979C2212yes—
39#2979C2214yes—
40#2960C2212yes—
41#2933—22no—
42#2931C2212yes—
43C2×#436—21no—
44C2×#436C2212yes—
45C2×C8×D4—21no—
46C2×C8×D4C2212yes—
47#2925—21no—
48#2920C2211yescharacteristic
49C2×#431C2211yescharacteristic
50C2×#431C2211yescharacteristic
51C2×#431C2211yescharacteristic
52#2906—21no—
53C2×#423C2211yescharacteristic
54C2×#408—22no—
55#1594C2218yescomplemented
56#1590C2218yescomplemented
57#1586C2218yescomplemented
58#1585C2218yescomplemented
59#1583C2218yescomplemented
60#1581C2218yescomplemented
61SD16⋊C8C2218yescomplemented
62C8×D8C2218yescomplemented
63#1574C2214yescomplemented
64#1571C2212yes—
65#1570C2212yes—
66#1565C2214yescomplemented
67#1492C2214yes—
68#1486C2212yes—
69#1486C2212yes—
70#1482C2214yescomplemented
71#1481C2211yescharacteristic
72#1481C2211yescharacteristic
73#1480C2211yescharacteristic
74#1480C2211yescharacteristic
75#1480C2214yes—
76#1476C2214yescomplemented
77#1453C2214yescomplemented
78#1452C2212yes—
79#1452C2212yescomplemented
80#1450C2212yes—
81#1450C2212yescomplemented
82C22×#41—21no—
83C2×#131—22no—
84C22×M16—21no—
85C22×M16—22no—
86C23×C8—22no—
87#533C2×C411yescharacteristic
88#533C2×C411yescharacteristic
89#533C2311yescharacteristic
90#532—21no—
91#532—21no—
92#532—21no—
93#532—21no—
94#532C2×C411yescharacteristic
95#532C2×C412yes—
96#532C2×C412yes—
97#532C2×C411yescharacteristic
98#532C2311yescharacteristic
99#532C2312yes—
100#532C2312yes—
101C2×C4⋊Q8C2×C411yescharacteristic
102C2×#127C2×C411yescharacteristic
103#529—21no—
104#529—21no—
105#529—21no—
106#529—21no—
107#529C2×C412yes—
108#529C2312yes—
109C2×#124—21no—
110#525—21no—
111#525—21no—
112#525C2×C412yes—
113C2×#121—21no—
114C2×#121C2×C412yes—
115C4×#41—21no—
116C4×#41—21no—
117C4×#41C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4C2312yes—
120C22×C42C2311yescharacteristic
121#501C2×C414yes—
122#496C2×C414yes—
123#490C2×C414yes—
124#489C2×C414yes—
125#483C2×C414yes—
126#482C2×C418yes—
127#480C2×C414yes—
128#477C2×C414yes—
129#476C2×C414yes—
130#476C2×C414yes—
131#474C2×C414yes—
132#469C2×C418yes—
133#468C2×C418yes—
134#461C2×C414yes—
135#460C2×C414yes—
136#459C2×C414yes—
137#459C2×C414yes—
138#450C2×C414yes—
139#449C2×C418yes—
140#447C2×C414yes—
141#446—22no—
142#446C2314yes—
143C8×Q8—22no—
144C8×Q8C2314yes—
145C4×SD16C2318yes—
146C4×D8C2318yes—
147#436—22no—
148#436—22no—
149#436—24no—
150#436—24no—
151#436—22no—
152#436—22no—
153#436—22no—
154#436C2314yes—
155#436C2314yes—
156#436C2314yes—
157#435—24no—
158#435—24no—
159C8×D4—22no—
160C8×D4—22no—
161C8×D4—24no—
162C8×D4—24no—
163C8×D4—22no—
164C8×D4—22no—
165C8×D4—22no—
166C8×D4C2314yes—
167C8×D4C2314yes—
168C8×D4C2314yes—
169#433—24no—
170#432—22no—
171#432C2×C412yes—
172#432C2×C412yes—
173#431—21no—
174#431—21no—
175#431—21no—
176#431—21no—
177#431—22no—
178#431D411yescharacteristic
179#431D411yescharacteristic
180#431D412yes—
181#431C2312yes—
182#431C2312yes—
183#431C2311yescharacteristic
184#431C2311yescharacteristic
185#431C2312yes—
186#431C2312yes—
187#431C2312yes—
188#431C2311yescharacteristic
189#431C2311yescharacteristic
190#431C2312yes—
191#424—22no—
192#424—24no—
193#423—21no—
194#423—21no—
195#423—22no—
196#423C2×C412yes—
197#423C2312yes—
198#423C2312yes—
199C2×C4⋊C8—21no—
200C2×C4⋊C8—21no—
201C2×C4⋊C8—22no—
202C2×C4⋊C8D411yescharacteristic
203C2×C4⋊C8D411yescharacteristic
204C2×C4⋊C8C2312yes—
205C2×C4⋊C8C2311yescharacteristic
206C2×C4⋊C8C2311yescharacteristic
207#408—22no—
208#408—24no—
209#408—24no—
210#408—24no—
211#408—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#335C2318yes—
224#334C2318yes—
225Q8⋊C8C2314yes—
226Q8⋊C8C2314yes—
227D4⋊C8C2314yes—
228D4⋊C8C2318yes—
229D4⋊C8C2318yes—
230D4⋊C8C2314yes—
231#322C2314yes—
232#322C2314yes—
233#322C2314yes—
234C82C2314yes—
235C2×#41—21no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—21no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—82no—
242C2×#41—82no—
243C2×#41C22×C412yes—
244C22×Q8—21no—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C22×C412yes—
249C23×C4—22no—
250C23×C4C22×C411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4#4111yescharacteristic
253C2×SD16C22×C418yes—
254C2×D8C22×C418yes—
255#131—84no—
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—24no—
259C2×M16—24no—
260C2×M16—24no—
261C2×M16—84no—
262C2×M16—82no—
263C2×M16C22×C414yes—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—24no—
269C22×C8—24no—
270C22×C8—24no—
271C22×C8—84no—
272C22×C8C2×D412yes—
273C22×C8C2×D412yes—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8C22×C411yescharacteristic
277C4⋊Q8C22×C412yes—
278C4⋊Q8C22×C412yes—
279C4⋊Q8C22×C411yescharacteristic
280C4⋊Q8C22×C411yescharacteristic
281#127—21no—
282#127—21no—
283#127C22×C412yes—
284#127C22×C412yes—
285#127C22×C411yescharacteristic
286#126—22no—
287#126—22no—
288#125—21no—
289#125—21no—
290#125C22×C412yes—
291#124—22no—
292#124—22no—
293#124—21no—
294#124—22no—
295#124—22no—
296#124—21no—
297#124C22×C412yes—
298#123—22no—
299#123—22no—
300#123—22no—
301C22⋊Q8—21no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304C22⋊Q8—21no—
305C22⋊Q8C22×C412yes—
306C22⋊Q8C22×C412yes—
307#121—22no—
308#121—22no—
309#121—22no—
310#121—22no—
311#121—21no—
312#121—21no—
313#121—21no—
314#121—22no—
315#121—22no—
316#121—21no—
317#121C22×C414yes—
318#121C22×C414yes—
319#121C22×C412yes—
320#121C22×C412yes—
321#120—22no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8C22×C412yes—
327C4×Q8C2412yes—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4—22no—
342C4×D4—21no—
343C4×D4—22no—
344C4×D4C22×C414yes—
345C4×D4C22×C412yes—
346C4×D4C2414yes—
347C4×D4C2412yes—
348#117—21no—
349#117—21no—
350#117—22no—
351#117—22no—
352#117—22no—
353#117C22×C412yes—
354C2×C4⋊C4—21no—
355C2×C4⋊C4C22×C412yes—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C42—22no—
359C2×C42C22×C411yescharacteristic
360C2×C42C22×C411yescharacteristic
361C2×C42C22×C411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42#4111yescharacteristic
366C2×C42#4111yescharacteristic
367C2×C42#4111yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107C22×C418yes—
372#106C22×C418yes—
373C4⋊C8—24no—
374C4⋊C8—24no—
375C4⋊C8—22no—
376C4⋊C8—22no—
377C4⋊C8—22no—
378C4⋊C8—21no—
379C4⋊C8—21no—
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8—22no—
383C4⋊C8C22×C414yes—
384C4⋊C8C22×C414yes—
385C4⋊C8C2×D411yescharacteristic
386C4⋊C8C2×D411yescharacteristic
387C4⋊C8C2×D412yes—
388C4⋊C8C2×D411yescharacteristic
389C4⋊C8C2×D411yescharacteristic
390C4⋊C8C2×D412yes—
391C4⋊C8C2414yes—
392C4⋊C8C2412yes—
393C4⋊C8C2412yes—
394#103C22×C418yes—
395#102C22×C418yes—
396#102C22×C418yes—
397#102C22×C418yes—
398C22⋊C8—22no—
399C22⋊C8—22no—
400C22⋊C8—21no—
401C22⋊C8—21no—
402C22⋊C8—22no—
403C22⋊C8—24no—
404C22⋊C8—24no—
405C22⋊C8—24no—
406C22⋊C8—24no—
407C22⋊C8—24no—
408C22⋊C8—24no—
409C22⋊C8C22×C414yes—
410C22⋊C8C22×C414yes—
411C22⋊C8C2×D412yes—
412C22⋊C8C2×D412yes—
413C22⋊C8C2×D412yes—
414C22⋊C8C2×D412yes—
415#97—24no—
416#97C22×C414yes—
417C4×C8—24no—
418C4×C8—22no—
419C4×C8—22no—
420C4×C8—22no—
421C4×C8C2×D412yes—
422C4×C8C2×D412yes—
423C4×C8C2412yes—
424C4×C8C2412yes—
425C4×C8C2414yes—
426C4×C8C2412yes—
427C4×C8C2412yes—
428C24—22no—
429C24C4×D411yescharacteristic
430#41—44no—
431#41—44no—
432#41—42no—
433#41—42no—
434#41—84no—
435#41—84no—
436#41—82no—
437#41—82no—
438#41—82no—
439#41—82no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—82no—
445C2×Q8C23×C412yes—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—82no—
461C2×D4—84no—
462C2×D4C23×C412yes—
463C2×D4C23×C414yes—
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—42no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4C4×D411yescharacteristic
477C22×C4C4×D411yescharacteristic
478C22×C4C4×D411yescharacteristic
479C22×C4C4×D411yescharacteristic
480C22×C4C4×D411yescharacteristic
481C22×C4C4×D411yescharacteristic
482C22×C4C4×D411yescharacteristic
483C22×C4C4×D411yescharacteristic
484C22×C4C4×D411yescharacteristic
485C22×C4C23×C411yescharacteristic
486C22×C4C23×C411yescharacteristic
487C22×C4C23×C411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4111yescharacteristic
492C22×C4C2×#4111yescharacteristic
493C22×C4C2×#4111yescharacteristic
494SD16—48no—
495D8—48no—
496M16—44no—
497M16—44no—
498M16—84no—
499M16—84no—
500M16—84no—
501M16—82no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—24no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—44no—
510C2×C8—42no—
511C2×C8—42no—
512C2×C8—84no—
513C2×C8—84no—
514C2×C8—84no—
515C2×C8—82no—
516C2×C8C23×C418yes—
517C2×C8C22×D412yes—
518C2×C8C22×D412yes—
519C2×C8C22×D412yes—
520C2×C8C22×D412yes—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4C23×C412yes—
535C4⋊C4C23×C412yes—
536C4⋊C4C23×C414yes—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—24no—
549C22⋊C4—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42C4×D411yescharacteristic
554C42C4×D411yescharacteristic
555C42C4×D411yescharacteristic
556C42C4×D411yescharacteristic
557C42C4×D411yescharacteristic
558C42C4×D411yescharacteristic
559C42C23×C411yescharacteristic
560C42C23×C411yescharacteristic
561C42C23×C411yescharacteristic
562C42C22×D411yescharacteristic
563C42C22×D411yescharacteristic
564C42C22×D411yescharacteristic
565C42C2×#4111yescharacteristic
566C42C2×#4111yescharacteristic
567C42C2×#4111yescharacteristic
568C42C2511yescharacteristic
569C23—22no—
570C23—22no—
571C23—24no—
572C23—41no—
573C23—41no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23C2×C4×D411yescharacteristic
578C23C2×C4×D411yescharacteristic
579C23C2×C4×D411yescharacteristic
580Q8—24no—
581Q8—42no—
582Q8—42no—
583Q8—82no—
584Q8—82no—
585D4—24no—
586D4—24no—
587D4—24no—
588D4—44no—
589D4—44no—
590D4—42no—
591D4—44no—
592D4—44no—
593D4—42no—
594D4—44no—
595D4—82no—
596D4—82no—
597D4—84no—
598D4—84no—
599D4—82no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—82no—
631C2×C4C2×C4×D411yescharacteristic
632C2×C4C2×C4×D411yescharacteristic
633C2×C4C2×C4×D411yescharacteristic
634C2×C4C2×C4×D411yescharacteristic
635C2×C4C2×C4×D411yescharacteristic
636C2×C4C2×C4×D411yescharacteristic
637C2×C4C2×C4×D411yescharacteristic
638C2×C4C2×C4×D411yescharacteristic
639C2×C4C2×C4×D411yescharacteristic
640C2×C4#57612yes—
641C2×C4#57712yes—
642C2×C4C24×C411yescharacteristic
643C2×C4C23×D411yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C8—28no—
646C8—22no—
647C8—22no—
648C8—22no—
649C8—22no—
650C8—42no—
651C8—42no—
652C8—42no—
653C8—44no—
654C8—42no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—42no—
664C22—44no—
665C22—42no—
666C22—44no—
667C22—44no—
668C22—42no—
669C22—82no—
670C22—88no—
671C22#294612yes—
672C22C22×C4×D411yescharacteristic
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—21no—
680C4—21no—
681C4—42no—
682C4—42no—
683C4—44no—
684C4—42no—
685C4#294612yes—
686C4#295512yes—
687C4#357312yes—
688C4C2×#57611yescharacteristic
689C4C2×#57711yescharacteristic
690C2—22no—
691C2—41no—
692C2—41no—
693C2—82no—
694C2—84no—
695C2#6014211yescharacteristic
696C2C2×#294611yescharacteristic
697C2C2×#295511yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#60951C2
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×#2979C2semidirect
4. #18080C2semidirect
5. #18025C2semidirect
6. #18015C2semidirect
7. #17935C2semidirect
8. #17930C2semidirect
9. #17905C2semidirect
10. #17900C2semidirect
11. #17809C2semidirect
12. #17485C2semidirect
13. #17458C2semidirect
14. #17325C2semidirect
28. #2994C22semidirect
30. #2991C22semidirect
31. #2990C22semidirect
55. #1594C22semidirect
56. #1590C22semidirect
57. #1586C22semidirect
58. #1585C22semidirect
59. #1583C22semidirect
60. #1581C22semidirect
61. SD16⋊C8C22semidirect
62. C8×D8C22semidirect
63. #1574C22semidirect
66. #1565C22semidirect
70. #1482C22semidirect
76. #1476C22semidirect
77. #1453C22semidirect
79. #1452C22semidirect
81. #1450C22semidirect

Maximal subgroups

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

subgroupindex
#609512
#602682
C2×#29792
#180802
#180252
#180152
#179352
#179302
#179052
#179002
#178092
#174852
#174582
#173252