Finite Group Databasebeta

← #467021

Subgroups — #467021

Order 512.

1592 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic, complemented
3C2×#2043C211yesmaximal, characteristic, complemented
4#14697C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11067C212yesmaximal, complemented
8#11010C211yesmaximal, characteristic, complemented
9#10970C212yesmaximal
10#10970C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C23×C42C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2212yes—
20#2857—24no—
21#2857—24no—
22C2×#391C2212yes—
23C2×#389—21no—
24C2×#389—21no—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#385—21no—
28C2×#384C2211yescharacteristic
29C2×#380—21no—
30C2×#380—21no—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33#2100—24no—
34#2043—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1436—21no—
43#1436—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C22⋊#95—21no—
47C23×Q8C2312yes—
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8D414yes—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—42no—
62C2×C22⋊Q8—42no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—42no—
81C22×C42—42no—
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—42no—
100C2×C22⋊C8—42no—
101#391—24no—
102#391—24no—
103#389—28no—
104#389—28no—
105#389—22no—
106#389—22no—
107#389—42no—
108#385—44no—
109#384—22no—
110#384D414yes—
111#380—28no—
112#380—28no—
113#380—22no—
114#380—22no—
115#380—42no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126#340—24no—
127#340—24no—
128#340—42no—
129#340—42no—
130#336—22no—
131#336—22no—
132#336—42no—
133C25C2×D411yescharacteristic
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8C2×D414yes—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—41no—
168C23×C4—41no—
169C23×C4—41no—
170C23×C4—41no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4SD1614yes—
176C23×C4Q1614yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4#4114yes—
183C23×C4C2411yescharacteristic
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C4⋊Q8—48no—
199C4⋊Q8—48no—
200C22⋊Q8—48no—
201C22⋊Q8—48no—
202C22⋊Q8—48no—
203C22⋊Q8—48no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4#4112yes—
248C2×C4⋊C4#4112yes—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—48no—
267C2×C42—48no—
268C2×C42—41no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274#103—48no—
275#103—48no—
276#103—48no—
277#103—48no—
278C22⋊C8—48no—
279#95—24no—
280#95—28no—
281#95—28no—
282#95—42no—
283#95—42no—
284#95—42no—
285#95—42no—
286#95—42no—
287#95—42no—
288#95—42no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24#12412yes—
302C24C2×SD1612yes—
303C24C2×Q1612yes—
304C24C22×D411yescharacteristic
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4#12012yes—
432C22×C4#12012yes—
433C22×C4#12412yes—
434C22×C4C2×SD1612yes—
435C22×C4C2×SD1612yes—
436C22×C4C2×Q1612yes—
437C22×C4C2×Q1612yes—
438C22×C4#13712yes—
439C22×C4#13712yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4112yes—
444C22×C4C2×#4111yescharacteristic
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4H(5,𝔽2)11yescharacteristic
448C22×C4H(5,𝔽2)12yes—
449C22×C42−1+411yescharacteristic
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C22⋊C4—48no—
486C22⋊C4—416no—
487C22⋊C4—84no—
488C42—24no—
489C42—24no—
490C42—44no—
491C42—44no—
492C42—42no—
493C42—42no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23#48714yes—
529C23C2×#12011yescharacteristic
530C23#52511yescharacteristic
531C23C2×#12411yescharacteristic
532C23#53411yescharacteristic
533C23#53411yescharacteristic
534C23#53511yescharacteristic
535C23#53711yescharacteristic
536C23C22×SD1611yescharacteristic
537C23C22×Q1611yescharacteristic
538C23C2×#13711yescharacteristic
539C23C2×#13711yescharacteristic
540C23#57511yescharacteristic
541C23#57511yescharacteristic
542C23#57511yescharacteristic
543C23#57511yescharacteristic
544Q8—44no—
545Q8—44no—
546Q8—44no—
547Q8—44no—
548Q8—82no—
549Q8—82no—
550Q8—82no—
551Q8—82no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—82no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4#44814yes—
634C2×C4#45114yes—
635C2×C4#53412yes—
636C2×C4#53512yes—
637C2×C4#53812yes—
638C2×C4#53812yes—
639C2×C4#53912yes—
640C2×C4#54012yes—
641C2×C4#54611yescharacteristic
642C2×C4#54612yes—
643C2×C4#54612yes—
644C2×C4#54711yescharacteristic
645C2×C4#54811yescharacteristic
646C2×C4#55011yescharacteristic
647C2×C4#55312yes—
648C2×C4#55312yes—
649C2×C4#56012yes—
650C2×C4#56212yes—
651C8—82no—
652C8—82no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22#240711yescharacteristic
669C22#243511yescharacteristic
670C22#259111yescharacteristic
671C22#261311yescharacteristic
672C22#263511yescharacteristic
673C22#263911yescharacteristic
674C22#281311yescharacteristic
675C22C2×#44811yescharacteristic
676C22C2×#45111yescharacteristic
677C22#300511yescharacteristic
678C22#300511yescharacteristic
679C22#310111yescharacteristic
680C22#310411yescharacteristic
681C22#311611yescharacteristic
682C22#311611yescharacteristic
683C22C2×#48711yescharacteristic
684C22#313511yescharacteristic
685C22#313911yescharacteristic
686C22#314211yescharacteristic
687C22#318711yescharacteristic
688C22#318711yescharacteristic
689C22#318911yescharacteristic
690C22#318911yescharacteristic
691C22#319611yescharacteristic
692C22#319611yescharacteristic
693C22#319611yescharacteristic
694C22#319611yescharacteristic
695C22#320611yescharacteristic
696C22#320911yescharacteristic
697C22#321911yescharacteristic
698C22#321911yescharacteristic
699C22#324611yescharacteristic
700C22#324911yescharacteristic
701C22#325211yescharacteristic
702C22#325211yescharacteristic
703C4—24no—
704C4—24no—
705C4—42no—
706C4—42no—
707C4—42no—
708C4—84no—
709C4—82no—
710C4—82no—
711C4—84no—
712C4—161no—
713C4—161no—
714C4#268812yes—
715C4#270812yes—
716C4#271112yes—
717C4#271412yes—
718C2—44no—
719C2#1655711yescharacteristic
720C2#2415111yescharacteristic
721C2#2415511yescharacteristic
722C2#2585611yescharacteristic
723C2#2586211yescharacteristic
724C2#2932911yescharacteristic
725C2#2933811yescharacteristic
726C2#2959311yescharacteristic
727C2#2959711yescharacteristic
728C2#3010611yescharacteristic
729C2#3010911yescharacteristic
730C2#3018711yescharacteristic
731C2#3020211yescharacteristic
732C2#3272211yescharacteristic
733C2#3272411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2100C2semidirect
3. C2×#2043C2semidirect
5. #11371C2semidirect
6. #11190C2semidirect
7. #11067C2semidirect
8. #11010C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#21002
C2×#20432
#146972
#113712
#111902
#110672
#110102
#109702
#107122