Finite Group Databasebeta

← #464121

Subgroups — #464121

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#2043C211yesmaximal, characteristic, complemented
4C2×#2036C212yesmaximal
5#14697C211yesmaximal, characteristic
6#11357C211yesmaximal, characteristic, complemented
7#11124C212yesmaximal, complemented
8#11067C212yesmaximal, complemented
9#11010C211yesmaximal, characteristic, complemented
10#10970C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C23×C42C2211yescharacteristic
15C22×#107C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2212yes—
21#2857—24no—
22#2857—24no—
23C2×#391C2212yes—
24C2×#389—21no—
25C2×#389—21no—
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#385—21no—
29C2×#384C2211yescharacteristic
30C2×#380—21no—
31C2×#380—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34#2083—24no—
35#2043—24no—
36#2036—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1436—21no—
44#1436—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47C22⋊#95—21no—
48C23×Q8C2312yes—
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8D414yes—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—42no—
63C2×C22⋊Q8—42no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C42—22no—
80C22×C42—22no—
81C22×C42—42no—
82C22×C42—42no—
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×#107—24no—
88C2×#107—24no—
89C2×#107—42no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103—42no—
101C2×#103—42no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—42no—
105#391—24no—
106#391—24no—
107#389—28no—
108#389—28no—
109#389—22no—
110#389—22no—
111#389—42no—
112#385—44no—
113#384—22no—
114#384D414yes—
115#380—28no—
116#380—28no—
117#380—22no—
118#380—42no—
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—42no—
133#336—22no—
134#336—22no—
135#336—42no—
136C25C2×D411yescharacteristic
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×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8C2×D414yes—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4—41no—
178C23×C4Q1614yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4#4112yes—
186C23×C4C2411yescharacteristic
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—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8#4114yes—
201C4⋊Q8—48no—
202C4⋊Q8—48no—
203C22⋊Q8—48no—
204C22⋊Q8—48no—
205C22⋊Q8—48no—
206C22⋊Q8—48no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—48no—
270C2×C42—48no—
271C2×C42—41no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277#107—48no—
278#103—48no—
279#103—48no—
280#103—48no—
281#103—48no—
282C22⋊C8—48no—
283#95—28no—
284#95—28no—
285#95—24no—
286#95—42no—
287#95—42no—
288#95—42no—
289#95—42no—
290#95—42no—
291#95—42no—
292#95—42no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24#12112yes—
306C24C2×Q1612yes—
307C24#13712yes—
308C24C22×D411yescharacteristic
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
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—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
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—28no—
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—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—42no—
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—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4#12012yes—
436C22×C4#12012yes—
437C22×C4#12112yes—
438C22×C4C2×Q1612yes—
439C22×C4C2×Q1612yes—
440C22×C4#13512yes—
441C22×C4AGL(1,ℤ/8)12yes—
442C22×C4#13712yes—
443C22×C4#13712yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2×#4112yes—
448C22×C4C2×#4111yescharacteristic
449C22×C4H(5,𝔽2)12yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)11yescharacteristic
452C22×C4H(5,𝔽2)11yescharacteristic
453C22×C42−1+412yes—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—48no—
474C4⋊C4—416no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—42no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—84no—
492C42—24no—
493C42—24no—
494C42—44no—
495C42—44no—
496C42—42no—
497C42—42no—
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—22no—
513C23—22no—
514C23—22no—
515C23—22no—
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—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23#46214yes—
533C23C2×#12011yescharacteristic
534C23C2×#12111yescharacteristic
535C23#53211yescharacteristic
536C23#53411yescharacteristic
537C23#53411yescharacteristic
538C23#53511yescharacteristic
539C23#53611yescharacteristic
540C23C22×Q1611yescharacteristic
541C23C2×#13511yescharacteristic
542C23C2×AGL(1,ℤ/8)11yescharacteristic
543C23C2×#13711yescharacteristic
544C23C2×#13711yescharacteristic
545C23C2×#13711yescharacteristic
546C23#57511yescharacteristic
547C23#57511yescharacteristic
548Q8—44no—
549Q8—44no—
550Q8—44no—
551Q8—44no—
552Q8—82no—
553Q8—82no—
554Q8—82no—
555Q8—82no—
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—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—416no—
578C2×C4—416no—
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—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—81no—
610C2×C4—82no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—82no—
615C2×C4—81no—
616C2×C4—82no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
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—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4#44914yes—
638C2×C4#45114yes—
639C2×C4#53412yes—
640C2×C4#53512yes—
641C2×C4#53712yes—
642C2×C4#53812yes—
643C2×C4#53912yes—
644C2×C4#54012yes—
645C2×C4D4211yescharacteristic
646C2×C4D4211yescharacteristic
647C2×C4#54612yes—
648C2×C4#54611yescharacteristic
649C2×C4#54611yescharacteristic
650C2×C4#54712yes—
651C2×C4#54812yes—
652C2×C4#55312yes—
653C2×C4#56212yes—
654C2×C4#56212yes—
655C8—44no—
656C8—82no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22#240811yescharacteristic
673C22#243211yescharacteristic
674C22#254311yescharacteristic
675C22#261311yescharacteristic
676C22#261811yescharacteristic
677C22#266111yescharacteristic
678C22#281611yescharacteristic
679C22C2×#45111yescharacteristic
680C22C2×#44911yescharacteristic
681C22#300411yescharacteristic
682C22#300511yescharacteristic
683C22C2×#46211yescharacteristic
684C22#303511yescharacteristic
685C22#303611yescharacteristic
686C22#303911yescharacteristic
687C22#317011yescharacteristic
688C22#317211yescharacteristic
689C22#318111yescharacteristic
690C22#318211yescharacteristic
691C22#318911yescharacteristic
692C22#318911yescharacteristic
693C22#319011yescharacteristic
694C22#319011yescharacteristic
695C22#319311yescharacteristic
696C22#319311yescharacteristic
697C22#319611yescharacteristic
698C22#319611yescharacteristic
699C22#320911yescharacteristic
700C22#321211yescharacteristic
701C22#321411yescharacteristic
702C22#321811yescharacteristic
703C22#323011yescharacteristic
704C22#323111yescharacteristic
705C22#323611yescharacteristic
706C22#323711yescharacteristic
707C4—24no—
708C4—24no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—84no—
713C4—84no—
714C4—82no—
715C4—82no—
716C4—161no—
717C4—161no—
718C4#267112yes—
719C4#267612yes—
720C4#270812yes—
721C4#271112yes—
722C2—44no—
723C2#1650111yescharacteristic
724C2#2421011yescharacteristic
725C2#2421211yescharacteristic
726C2#2567411yescharacteristic
727C2#2567711yescharacteristic
728C2#2857511yescharacteristic
729C2#2857711yescharacteristic
730C2#2959511yescharacteristic
731C2#2959711yescharacteristic
732C2#2975711yescharacteristic
733C2#2975911yescharacteristic
734C2#3031811yescharacteristic
735C2#3032511yescharacteristic
736C2#3277411yescharacteristic
737C2#3277711yescharacteristic
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×#2083C2semidirect
3. C2×#2043C2semidirect
6. #11357C2semidirect
7. #11124C2semidirect
8. #11067C2semidirect
9. #11010C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#20832
C2×#20432
C2×#20362
#146972
#113572
#111242
#110672
#110102
#109702
#107122