Finite Group Databasebeta

← #122017

Subgroups — #122017

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic
2#10730C212yesmaximal
3C2×#1294C212yesmaximal
4C2×#1280C211yesmaximal, characteristic
5#7557C211yesmaximal, characteristic
6C22×C22⋊Q8—21no—
7C23×C4⋊C4C2211yescharacteristic
8C22×C22⋊C8C2212yes—
9#2659—24no—
10#2659—22no—
11#2659C414yescomplemented
12C2×#398—22no—
13C2×#395—22no—
14C2×#380—21no—
15C2×#380C2212yes—
16C22×#95—21no—
17C22×#95—21no—
18C22×#95C2211yescharacteristic
19C22×#95C2211yescharacteristic
20C2×#340—22no—
21C2×#340—22no—
22#1294—24no—
23#1280—22no—
24C23×Q8—21no—
25C24×C4—21no—
26C24×C4—21no—
27C24×C4D411yescharacteristic
28C24×C4D411yescharacteristic
29C24×C4C2311yescharacteristic
30C23×C8—22no—
31C2×C22⋊Q8—24no—
32C2×C22⋊Q8—24no—
33C2×C22⋊Q8—42no—
34C2×C22⋊Q8—42no—
35C2×C22⋊Q8—42no—
36C2×C22⋊Q8—42no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—42no—
45C22×C4⋊C4—42no—
46C22×C4⋊C4D412yes—
47C22×C4⋊C4D412yes—
48C22×C4⋊C4Q812yescomplemented
49C22×C22⋊C4—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8Q814yescomplemented
56#398—28no—
57#398—28no—
58#395—28no—
59#395—28no—
60#380—24no—
61#380—24no—
62#380—22no—
63#380—22no—
64#380C2×C414yes—
65#380D414yes—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—21no—
73C2×#95—42no—
74C2×#95—42no—
75C2×#95—42no—
76C2×#95—42no—
77C2×#95—42no—
78C2×#95—42no—
79C2×#95C2×C412yes—
80C2×#95D412yes—
81C2×#95Q812yescomplemented
82#340—44no—
83#340—44no—
84C25C2×D411yescharacteristic
85C22×Q8—24no—
86C22×Q8—42no—
87C22×Q8—42no—
88C22×Q8—81no—
89C22×Q8—81no—
90C22×Q8—81no—
91C22×Q8—81no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4—41no—
111C23×C4—41no—
112C23×C4—41no—
113C23×C4—41no—
114C23×C4—41no—
115C23×C4—41no—
116C23×C4—41no—
117C23×C4—41no—
118C23×C4—41no—
119C23×C4—41no—
120C23×C4—41no—
121C23×C4—41no—
122C23×C4—41no—
123C23×C4—41no—
124C23×C4—41no—
125C23×C4—41no—
126C23×C4—41no—
127C23×C4—41no—
128C23×C4—41no—
129C23×C4—41no—
130C23×C4—41no—
131C23×C4—41no—
132C23×C4—41no—
133C23×C4—41no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C412yes—
146C23×C4D812yes—
147C23×C4SD1612yes—
148C23×C4SD1612yes—
149C23×C4Q1612yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×Q812yes—
155C23×C4C2×Q811yescharacteristic
156C23×C4C2×Q811yescharacteristic
157C23×C4#4112yes—
158C23×C4#4111yescharacteristic
159C23×C4#4111yescharacteristic
160C22×C8—24no—
161C22×C8—42no—
162C22×C8—42no—
163C22×C8—42no—
164C22×C8—42no—
165C22×C8—42no—
166C22×C8—42no—
167C22⋊Q8—48no—
168C22⋊Q8—48no—
169C22⋊Q8—48no—
170C22⋊Q8—48no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4C22⋊C414yes—
199C2×C4⋊C4C22⋊C414yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C22⋊C8—24no—
207C22⋊C8—24no—
208C22⋊C8—42no—
209C22⋊C8—42no—
210#95—28no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—48no—
217#95—48no—
218#95—42no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24C2×C22⋊C411yescharacteristic
228C24#12011yescharacteristic
229C24#12011yescharacteristic
230C24C22⋊Q811yescharacteristic
231C24C22⋊Q811yescharacteristic
232C24#12411yescharacteristic
233C24C4⋊Q811yescharacteristic
234C24C2×D811yescharacteristic
235C24C2×SD1611yescharacteristic
236C24C2×SD1611yescharacteristic
237C24C2×Q1611yescharacteristic
238C24AGL(1,ℤ/8)11yescharacteristic
239C24AGL(1,ℤ/8)11yescharacteristic
240C24#13711yescharacteristic
241C24#13711yescharacteristic
242C2×Q8—48no—
243C2×Q8—84no—
244C2×Q8—84no—
245C2×Q8—84no—
246C2×Q8—84no—
247C2×Q8—84no—
248C2×Q8—84no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4#10214yes—
400C22×C4#10314yes—
401C22×C4C2×C22⋊C412yes—
402C22×C4C4×Q811yescharacteristic
403C22×C4C4×Q811yescharacteristic
404C22×C4C4×Q812yes—
405C22×C4#12012yes—
406C22×C4C22⋊Q812yes—
407C22×C4C22⋊Q812yes—
408C22×C4C22⋊Q812yes—
409C22×C4C22⋊Q812yes—
410C22×C4C22⋊Q812yes—
411C22×C4#12412yes—
412C22×C4#12412yes—
413C22×C4#12411yescharacteristic
414C22×C4C4⋊Q812yes—
415C22×C4C4⋊Q812yes—
416C22×C4C4⋊Q811yescharacteristic
417C2×C8—44no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—82no—
421C2×C8—82no—
422C2×C8—82no—
423C2×C8—82no—
424C4⋊C4—28no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—48no—
431C4⋊C4—416no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23#37911yescharacteristic
497C23#39111yescharacteristic
498C23#39111yescharacteristic
499C23#39311yescharacteristic
500C23#39311yescharacteristic
501C23#39511yescharacteristic
502C23#39511yescharacteristic
503C23C2×#10211yescharacteristic
504C23C2×#10311yescharacteristic
505C23#41811yescharacteristic
506C23#41911yescharacteristic
507C23#44711yescharacteristic
508C23#44711yescharacteristic
509C23#44811yescharacteristic
510C23#44811yescharacteristic
511C23#45011yescharacteristic
512C23#45011yescharacteristic
513C23#45111yescharacteristic
514C23#45111yescharacteristic
515C23#45311yescharacteristic
516C23#45411yescharacteristic
517C23#45511yescharacteristic
518C23#45611yescharacteristic
519C23B(4,𝔽2)11yescharacteristic
520C23B(4,𝔽2)11yescharacteristic
521C23#45811yescharacteristic
522C23#45811yescharacteristic
523C23#47411yescharacteristic
524C23#47411yescharacteristic
525C23#47511yescharacteristic
526C23#47511yescharacteristic
527C23#47611yescharacteristic
528C23#47611yescharacteristic
529C23#47711yescharacteristic
530C23#47711yescharacteristic
531C23#48611yescharacteristic
532C23#48711yescharacteristic
533C23#48911yescharacteristic
534C23#48911yescharacteristic
535C23#49811yescharacteristic
536C23#50011yescharacteristic
537C23#50111yescharacteristic
538C23#50111yescharacteristic
539Q8—816no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4#39112yes—
629C2×C4#39112yes—
630C2×C4#39312yes—
631C2×C4#39512yes—
632C8—84no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22—21no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—21no—
689C22#144611yescharacteristic
690C22#159511yescharacteristic
691C22#159611yescharacteristic
692C22#159811yescharacteristic
693C22#159911yescharacteristic
694C22#160111yescharacteristic
695C22#160211yescharacteristic
696C22#160411yescharacteristic
697C22#160511yescharacteristic
698C22#160711yescharacteristic
699C22#160811yescharacteristic
700C22#161011yescharacteristic
701C22#161111yescharacteristic
702C22#161311yescharacteristic
703C22#161411yescharacteristic
704C22#161611yescharacteristic
705C22#161711yescharacteristic
706C22#178611yescharacteristic
707C22#178711yescharacteristic
708C22#179011yescharacteristic
709C22#179011yescharacteristic
710C22#179211yescharacteristic
711C22#179311yescharacteristic
712C22#198711yescharacteristic
713C22#198811yescharacteristic
714C22#199011yescharacteristic
715C22#199311yescharacteristic
716C22#202311yescharacteristic
717C22#202311yescharacteristic
718C22#202411yescharacteristic
719C22#202411yescharacteristic
720C22#202711yescharacteristic
721C22#202811yescharacteristic
722C22#202911yescharacteristic
723C22#203011yescharacteristic
724C22#205711yescharacteristic
725C22#205711yescharacteristic
726C22#205811yescharacteristic
727C22#205811yescharacteristic
728C22#206011yescharacteristic
729C22#206111yescharacteristic
730C22#206211yescharacteristic
731C22#206211yescharacteristic
732C4—22no—
733C4—22no—
734C4—22no—
735C4—22no—
736C4—44no—
737C4—44no—
738C4—84no—
739C4—84no—
740C4—84no—
741C4—84no—
742C2—21no—
743C2—21no—
744C2—21no—
745C2—21no—
746C2—21no—
747C2—21no—
748C2—21no—
749C2—21no—
750C2#840311yescharacteristic
751C2#840411yescharacteristic
752C2#840511yescharacteristic
753C2#840611yescharacteristic
754C2#1034011yescharacteristic
755C2#1034111yescharacteristic
756C2#1034211yescharacteristic
757C2#1034311yescharacteristic
758C2#1056711yescharacteristic
759C2#1056811yescharacteristic
760C2#1056911yescharacteristic
761C2#1057011yescharacteristic
762C2#1138411yescharacteristic
763C2#1139011yescharacteristic
764C2#1139211yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2659C4semidirect
48. C22×C4⋊C4Q8semidirect
55. C2×C22⋊C8Q8semidirect
81. C2×#95Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
#107302
C2×#12942
C2×#12802
#75572