Finite Group Databasebeta

← #366970

Subgroups — #366970

Order 512.

2102 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#459C211yesmaximal, characteristic
2C2×#1970C212yesmaximal, complemented
3C2×#1957C211yesmaximal, characteristic, complemented
4C2×#1947C211yesmaximal, characteristic, complemented
5C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
6C2×C4×#102C211yesmaximal, characteristic, complemented
7#14154C211yesmaximal, characteristic, complemented
8#13906C211yesmaximal, characteristic, complemented
9#11123C212yesmaximal, complemented
10#11022C212yesmaximal, complemented
11#10925C212yesmaximal, complemented
12C23×D8C2212yes—
13C22×#127C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#459—24no—
16C2×#459—24no—
17C2×#459C418yescomplemented
18C22×#107C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#102C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×#102C2212yes—
25C22×C4×C8C2212yes—
26C22×C4×C8C2211yescharacteristic
27C2×#390—22no—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#384C2211yescharacteristic
32C2×#380—22no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1970—28no—
36#1957—24no—
37#1947—24no—
38C8×C4⋊C4—24no—
39C4×#102—24no—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#336C2212yes—
44#1439—22no—
45#1438—22no—
46C4⋊#95C2212yes—
47C23×D4—21no—
48C23×D4—21no—
49C23×D4C2312yes—
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C22×D8—24no—
53C22×D8—24no—
54C22×D8—24no—
55C22×D8—24no—
56C22×D8—42no—
57C22×D8—42no—
58C22×D8C2×C418yes—
59C23×C8C2311yescharacteristic
60C23×C8C2312yes—
61C23×C8C2311yescharacteristic
62C2×#127—22no—
63C2×#127—22no—
64C2×#127—22no—
65C2×#127—22no—
66C2×#127—22no—
67C2×#127C2×C414yes—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4C2×C414yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—24no—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—24no—
85C22×C42—21no—
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88#459—232no—
89C2×#107—24no—
90C2×#107—24no—
91C2×#107—42no—
92C2×C4⋊C8C2×C414yes—
93C2×C4⋊C8D414yes—
94C2×C4⋊C8D414yes—
95C2×#102—22no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—42no—
107C2×#102—42no—
108C2×#102C2×C418yes—
109C2×#102D414yescomplemented
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8—24no—
113C2×C4×C8—24no—
114C2×C4×C8—42no—
115C2×C4×C8D414yes—
116#390—24no—
117#390—24no—
118#390—24no—
119#390—42no—
120#390—42no—
121#390—42no—
122#390—48no—
123#384—22no—
124#384—22no—
125#380—48no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C2×#95—24no—
131C2×#95—21no—
132C2×#95D412yes—
133#340—24no—
134#340—22no—
135#340—22no—
136#336—24no—
137C25—21no—
138C25—22no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—22no—
165C22×D4—42no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—82no—
172C22×D4—82no—
173C22×D4C22×C414yes—
174C22×D4C22×C412yes—
175C22×D4C2×D412yes—
176C22×D4#4112yes—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4C2×D414yes—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2411yescharacteristic
190C2×D8—216no—
191C2×D8—216no—
192C2×D8—48no—
193C2×D8—48no—
194C2×D8—48no—
195C2×D8—48no—
196C2×D8—48no—
197C22×C8—22no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8C22×C414yes—
212C22×C8C2×D412yes—
213C22×C8C2×D414yes—
214C22×C8C2×D414yes—
215C22×C8C2×D412yes—
216C22×C8C2×D412yes—
217C22×C8#4112yes—
218#127—28no—
219#127—28no—
220#127—44no—
221#127—44no—
222C4×D4—28no—
223C4×D4—28no—
224C4×D4—44no—
225C4×D4—44no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—82no—
253C2×C4⋊C4D818yescomplemented
254C2×C4⋊C4C22×C412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4#4112yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—84no—
267C2×C22⋊C4—84no—
268C2×C22⋊C4—84no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—42no—
276C2×C42—44no—
277C2×C42C22×C412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42#4112yes—
282C2×C42#4112yes—
283#107—48no—
284C4⋊C8C22⋊C4116yescomplemented
285#102—28no—
286#102—216no—
287#102—28no—
288#102—48no—
289#102—48no—
290C4×C8—28no—
291C4×C8—48no—
292#95—24no—
293#95—42no—
294#95—48no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—24no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—44no—
314C24—44no—
315C24—44no—
316C24—44no—
317C24C22×D411yescharacteristic
318C2×D4—22no—
319C2×D4—24no—
320C2×D4—22no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—24no—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—41no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—41no—
348C2×D4—42no—
349C2×D4—41no—
350C2×D4—42no—
351C2×D4—41no—
352C2×D4—42no—
353C2×D4—41no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—416no—
376C2×D4—88no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4C4×D414yes—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—48no—
408C22×C4—416no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—88no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4C4×D412yes—
418C22×C4C4×D412yes—
419C22×C4#12112yes—
420C22×C4#12112yes—
421C22×C4#12712yes—
422C22×C4#12712yes—
423C22×C4C2×D818yes—
424C22×C4C2×D814yes—
425C22×C4#13514yes—
426C22×C4C23×C411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4H(5,𝔽2)11yescharacteristic
437C22×C4H(5,𝔽2)11yescharacteristic
438C22×C4H(5,𝔽2)11yescharacteristic
439C22×C42−1+411yescharacteristic
440D8—432no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8C2×C22⋊C418yes—
461C2×C8#12114yes—
462C4⋊C4—216no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C22⋊C4—48no—
482C22⋊C4—816no—
483C22⋊C4—816no—
484C22⋊C4—816no—
485C42—42no—
486C42—42no—
487C42—42no—
488C42—42no—
489C42—84no—
490C42C2×C22⋊C414yes—
491C42C4×D414yes—
492C42#12114yes—
493C42#12414yes—
494C23—24no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—48no—
510C23—48no—
511C23—44no—
512C23—82no—
513C23—82no—
514C23—81no—
515C23—81no—
516C23—81no—
517C23—81no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23—82no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23—82no—
538C23—82no—
539C23—84no—
540C23C2×C4×D411yescharacteristic
541C23C2×#12111yescharacteristic
542C23C2×#12711yescharacteristic
543C23D4211yescharacteristic
544C23#54611yescharacteristic
545C23#54811yescharacteristic
546C23#55011yescharacteristic
547C23C22×D812yes—
548C23C2×#13512yes—
549C23#57612yes—
550C23#57712yes—
551D4—24no—
552D4—24no—
553D4—42no—
554D4—44no—
555D4—44no—
556D4—42no—
557D4—44no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—44no—
563D4—816no—
564D4—82no—
565D4—82no—
566D4—816no—
567D4—82no—
568D4—82no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—82no—
595C2×C4—816no—
596C2×C4—84no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—84no—
600C2×C4—82no—
601C2×C4#39014yes—
602C2×C4#39014yes—
603C2×C4C4×D818yes—
604C2×C4#46618yes—
605C2×C4#49314yes—
606C2×C4#49514yes—
607C2×C4C22×C22⋊C411yescharacteristic
608C2×C4C2×C4×D411yescharacteristic
609C2×C4C2×C4×D411yescharacteristic
610C2×C4C4×#4111yescharacteristic
611C2×C4#51811yescharacteristic
612C2×C4#52011yescharacteristic
613C2×C4C2×#12111yescharacteristic
614C2×C4C2×#12111yescharacteristic
615C2×C4#52511yescharacteristic
616C2×C4C2×#12411yescharacteristic
617C2×C4#52911yescharacteristic
618C2×C4#53211yescharacteristic
619C2×C4#53511yescharacteristic
620C2×C4#53711yescharacteristic
621C2×C4#54011yescharacteristic
622C2×C4#54211yescharacteristic
623C2×C4D4211yescharacteristic
624C2×C4D4211yescharacteristic
625C2×C4#54611yescharacteristic
626C2×C4#54611yescharacteristic
627C2×C4#54711yescharacteristic
628C2×C4#54811yescharacteristic
629C2×C4#54811yescharacteristic
630C2×C4#55011yescharacteristic
631C2×C4#55011yescharacteristic
632C2×C4#55311yescharacteristic
633C2×C4#55311yescharacteristic
634C2×C4#55511yescharacteristic
635C8—28no—
636C8—24no—
637C8—42no—
638C22—44no—
639C22—48no—
640C22—84no—
641C22—82no—
642C22—82no—
643C22—84no—
644C22—84no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—88no—
650C22—84no—
651C22—88no—
652C22—84no—
653C22—84no—
654C22—164no—
655C22—162no—
656C22—162no—
657C22—164no—
658C22—162no—
659C22—162no—
660C22C2×#39011yescharacteristic
661C22#236111yescharacteristic
662C22#236311yescharacteristic
663C22#242611yescharacteristic
664C22#242811yescharacteristic
665C22#254011yescharacteristic
666C22#254111yescharacteristic
667C22C2×C4×D812yes—
668C22#294612yes—
669C22C2×#46612yes—
670C22#306212yes—
671C22C2×#49311yescharacteristic
672C22C2×#49511yescharacteristic
673C22#315411yescharacteristic
674C22#315511yescharacteristic
675C22D4×D812yes—
676C22#328412yes—
677C22#329412yes—
678C22#330512yes—
679C22#335812yes—
680C22#336112yes—
681C22#339112yes—
682C22#339412yes—
683C4—24no—
684C4—48no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—162no—
689C4#197018yescomplemented
690C4C4×#12111yescharacteristic
691C4#232411yescharacteristic
692C4D4×C22⋊C411yescharacteristic
693C4#234111yescharacteristic
694C4#241411yescharacteristic
695C4#241611yescharacteristic
696C4#249011yescharacteristic
697C4#249111yescharacteristic
698C2—84no—
699C2—164no—
700C2#1399811yescharacteristic
701C2C2×#197011yescharacteristic
702C2#2123511yescharacteristic
703C2#2299512yes—
704C2#2309312yes—
705C2#2529212yes—
706C2#2538412yes—
707C2#2839711yescharacteristic
708C2#2839911yescharacteristic
709C2#2845811yescharacteristic
710C2#2846011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1970C2semidirect
3. C2×#1957C2semidirect
4. C2×#1947C2semidirect
5. C2×C8×C4⋊C4C2semidirect
6. C2×C4×#102C2semidirect
7. #14154C2semidirect
8. #13906C2semidirect
9. #11123C2semidirect
10. #11022C2semidirect
11. #10925C2semidirect
17. C2×#459C4semidirect
109. C2×#102D4semidirect
253. C2×C4⋊C4D8semidirect
284. C4⋊C8C22⋊C4semidirect
689. C4#1970semidirect

Maximal subgroups

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

subgroupindex
C22×#4592
C2×#19702
C2×#19572
C2×#19472
C2×C8×C4⋊C42
C2×C4×#1022
#141542
#139062
#111232
#110222
#109252