Finite Group Databasebeta

← #397188

Subgroups — #397188

Order 512.

1908 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2043C211yesmaximal, characteristic, complemented
3C2×#1892C212yesmaximal
4C2×#1892C212yesmaximal
5C22×#336C211yesmaximal, characteristic
6#14697C211yesmaximal, characteristic
7#11267C211yesmaximal, characteristic, complemented
8#11207C211yesmaximal, characteristic, complemented
9#11190C212yesmaximal, complemented
10#11067C212yesmaximal, complemented
11#11010C211yesmaximal, characteristic, complemented
12C24×C8C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C23×C42C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20#2857—28no—
21C2×#391C2212yes—
22C2×#389—21no—
23C2×#389—21no—
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#385—21no—
27C2×#384C2211yescharacteristic
28C2×#380—21no—
29C2×#380—21no—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32#2043—24no—
33#1892—28no—
34#1892—28no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336—24no—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43#1436—21no—
44#1436—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47C22⋊#95—21no—
48C23×Q8C2312yes—
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8—24no—
53C23×C8—24no—
54C23×C8—24no—
55C23×C8—24no—
56C23×C8—24no—
57C23×C8—42no—
58C23×C8—42no—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—42no—
65C2×C22⋊Q8—24no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8D418yes—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C42—24no—
87C22×C42—22no—
88C22×C42—22no—
89C22×C42—42no—
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—42no—
103C2×#103—42no—
104C2×#103—42no—
105C2×#103—42no—
106#391—24no—
107#391—24no—
108#389—28no—
109#389—28no—
110#389—22no—
111#389—22no—
112#389—42no—
113#385—44no—
114#384—24no—
115#380—28no—
116#380—28no—
117#380—22no—
118#380—24no—
119#380D414yes—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#336—24no—
135#336—22no—
136#336D414yes—
137C25C2×D411yescharacteristic
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×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8C2×D414yes—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4SD1614yes—
177C23×C4Q1614yes—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D414yes—
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4#4114yes—
186C23×C4#4112yes—
187C23×C4C2411yescharacteristic
188C22×C8—216no—
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—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—48no—
202C22×C8—48no—
203C22×C8—48no—
204C22×C8—48no—
205C22×C8—48no—
206C22×C8—48no—
207C22×C8C2×D414yes—
208C22×C8#4114yes—
209C4⋊Q8—48no—
210C4⋊Q8—48no—
211C22⋊Q8—216no—
212C22⋊Q8—216no—
213C22⋊Q8—48no—
214C22⋊Q8—48no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—22no—
277C2×C42—21no—
278C2×C42—22no—
279C2×C42—48no—
280C2×C42—48no—
281C2×C42—41no—
282C2×C42—41no—
283C2×C42C2×D412yes—
284C2×C42#4112yes—
285#103—48no—
286#103—48no—
287#103—48no—
288#103—48no—
289#95—28no—
290#95—28no—
291#95—24no—
292#95—42no—
293#95—42no—
294#95—42no—
295#95—42no—
296#95—42no—
297#95—42no—
298#95—42no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24#12012yes—
308C24#12112yes—
309C24#12412yes—
310C24C2×SD1612yes—
311C24C2×Q1612yes—
312C24#13712yes—
313C24#13712yes—
314C24C22×D411yescharacteristic
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
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—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
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—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
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—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4#12012yes—
438C22×C4#12118yes—
439C22×C4#12112yes—
440C22×C4#12112yes—
441C22×C4#12112yes—
442C22×C4#12318yes—
443C22×C4#12412yes—
444C22×C4C2×SD1612yes—
445C22×C4C2×SD1612yes—
446C22×C4C2×Q1612yes—
447C22×C4C2×Q1612yes—
448C22×C4#13512yes—
449C22×C4#13512yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C2×#4112yes—
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4112yes—
456C22×C4C2×#4111yescharacteristic
457C22×C4H(5,𝔽2)11yescharacteristic
458C22×C4H(5,𝔽2)12yes—
459C22×C42−1+411yescharacteristic
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—432no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4—416no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C22⋊C4—84no—
499C42—24no—
500C42—24no—
501C42—44no—
502C42—44no—
503C42—42no—
504C42—42no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23#39414yes—
534C23#44814yes—
535C23#45114yes—
536C23#46114yes—
537C23#46214yes—
538C23#48714yes—
539C23#49114yes—
540C23C2×#12011yescharacteristic
541C23C2×#12111yescharacteristic
542C23C2×#12111yescharacteristic
543C23C2×#12411yescharacteristic
544C23#53211yescharacteristic
545C23#53411yescharacteristic
546C23#53711yescharacteristic
547C23C22×SD1611yescharacteristic
548C23C22×Q1611yescharacteristic
549C23C2×#13511yescharacteristic
550C23C2×#13511yescharacteristic
551C23C2×#13711yescharacteristic
552C23C2×#13711yescharacteristic
553C23#57511yescharacteristic
554C23#57511yescharacteristic
555Q8—44no—
556Q8—44no—
557Q8—44no—
558Q8—44no—
559Q8—82no—
560Q8—82no—
561Q8—82no—
562Q8—82no—
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—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—416no—
592C2×C4—416no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—82no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—81no—
627C2×C4—82no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4#39414yes—
644C2×C4#46514yes—
645C2×C4#46714yes—
646C2×C4C2×#12112yes—
647C2×C4C2×#12312yes—
648C2×C4#52512yes—
649C2×C4#52912yes—
650C2×C4#53812yes—
651C2×C4#53912yes—
652C2×C4D4211yescharacteristic
653C2×C4D4212yes—
654C2×C4#54612yes—
655C2×C4#54612yes—
656C2×C4#54612yes—
657C2×C4#54711yescharacteristic
658C2×C4#54712yes—
659C2×C4#54811yescharacteristic
660C2×C4#55312yes—
661C2×C4#55311yescharacteristic
662C8—44no—
663C8—44no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22#204318yescomplemented
679C22C2×#39411yescharacteristic
680C22#241011yescharacteristic
681C22#241611yescharacteristic
682C22#243011yescharacteristic
683C22#254311yescharacteristic
684C22#263911yescharacteristic
685C22#285011yescharacteristic
686C22C2×#44811yescharacteristic
687C22C2×#45111yescharacteristic
688C22#300211yescharacteristic
689C22#300211yescharacteristic
690C22C2×#46111yescharacteristic
691C22C2×#46211yescharacteristic
692C22#303511yescharacteristic
693C22#303511yescharacteristic
694C22C2×#46511yescharacteristic
695C22C2×#46711yescharacteristic
696C22#305611yescharacteristic
697C22#305611yescharacteristic
698C22C2×#48711yescharacteristic
699C22#313111yescharacteristic
700C22C2×#49111yescharacteristic
701C22#313511yescharacteristic
702C22#316711yescharacteristic
703C22#317011yescharacteristic
704C22#318111yescharacteristic
705C22#318111yescharacteristic
706C22#318711yescharacteristic
707C22#318911yescharacteristic
708C22#319011yescharacteristic
709C22#319011yescharacteristic
710C22#324611yescharacteristic
711C22#324811yescharacteristic
712C22#324811yescharacteristic
713C22#324911yescharacteristic
714C4—24no—
715C4—24no—
716C4—24no—
717C4—42no—
718C4—42no—
719C4—84no—
720C4—82no—
721C4—82no—
722C4—84no—
723C4—161no—
724C4—161no—
725C4#242412yes—
726C4#242512yes—
727C4#244312yes—
728C4#244412yes—
729C2—28no—
730C2#1474711yescharacteristic
731C2C2×#204311yescharacteristic
732C2#2350411yescharacteristic
733C2#2438011yescharacteristic
734C2#2438111yescharacteristic
735C2#2475111yescharacteristic
736C2#2475211yescharacteristic
737C2#2549711yescharacteristic
738C2#2549811yescharacteristic
739C2#2857111yescharacteristic
740C2#2857511yescharacteristic
741C2#3018711yescharacteristic
742C2#3019011yescharacteristic
743C2#3304311yescharacteristic
744C2#3305311yescharacteristic
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×#2043C2semidirect
7. #11267C2semidirect
8. #11207C2semidirect
9. #11190C2semidirect
10. #11067C2semidirect
11. #11010C2semidirect
678. C22#2043semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#20432
C2×#18922
C22×#3362
#146972
#112672
#112072
#111902
#110672
#110102