Finite Group Databasebeta

← #392162

Subgroups — #392162

Order 512.

3966 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#41C211yesmaximal, characteristic
2C2×#1805C212yesmaximal, complemented
3C2×#1804C212yesmaximal, complemented
4C2×#1797C211yesmaximal, characteristic, complemented
5C2×#1788C212yesmaximal
6#14682C211yesmaximal, characteristic, complemented
7#10789C212yesmaximal, complemented
8#10788C212yesmaximal, complemented
9#10731C212yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C2×C4×#41—24no—
12C2×C4×#41—28no—
13C2×C4×#41—28no—
14C2×C4×#41C418yescomplemented
15C22×C4×Q8C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4—21no—
18C22×C4×D4—21no—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117—21no—
22C22×#117C2211yescharacteristic
23C23×C42—21no—
24C23×C42C2211yescharacteristic
25C22×C4⋊C8C2212yes—
26C22×#103C2214yes—
27C22×#102C2214yes—
28C22×C22⋊C8C2212yes—
29C2×#382C2212yes—
30C2×#380C2212yes—
31#1805—28no—
32#1804—28no—
33#1797—24no—
34#1788—28no—
35C2×#340C2214yes—
36C2×#340C2214yes—
37C2×#336C2212yes—
38C4⋊#95C2212yes—
39#1434—21no—
40#1434—21no—
41C22⋊#95—21no—
42C22⋊#95—21no—
43C22×#41—24no—
44C22×#41—24no—
45C22×#41—24no—
46C22×#41—24no—
47C22×#41—22no—
48C22×#41—22no—
49C22×#41—22no—
50C22×#41C2×C414yes—
51C22×#41D414yes—
52C22×#41Q814yes—
53C23×Q8C2311yescharacteristic
54C23×D4—21no—
55C23×D4C2311yescharacteristic
56C24×C4—21no—
57C24×C4—21no—
58C24×C4—21no—
59C24×C4D411yescharacteristic
60C24×C4D411yescharacteristic
61C24×C4C2311yescharacteristic
62C23×C8C2314yes—
63C4×#41—264no—
64C4×#41—232no—
65C4×#41—232no—
66C2×C4×Q8—24no—
67C2×C4×Q8—24no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—24no—
71C2×C4×Q8C2×C414yes—
72C2×C4×Q8C2×C414yes—
73C2×C4×Q8D418yes—
74C2×C4×D4—216no—
75C2×C4×D4—216no—
76C2×C4×D4—24no—
77C2×C4×D4—28no—
78C2×C4×D4—28no—
79C2×C4×D4—24no—
80C2×C4×D4—22no—
81C2×C4×D4—24no—
82C2×C4×D4—22no—
83C2×C4×D4C2×C414yes—
84C2×C4×D4C2×C414yes—
85C2×C4×D4D418yes—
86C2×#117—216no—
87C2×#117—22no—
88C2×#117—24no—
89C2×#117—24no—
90C2×#117—28no—
91C2×#117C2×C414yes—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—22no—
101C22×C42—216no—
102C22×C42—21no—
103C22×C42—21no—
104C22×C42—21no—
105C22×C42—28no—
106C22×C42—24no—
107C22×C42—24no—
108C22×C42C2×C414yes—
109C22×C42D411yescharacteristic
110C22×C42D411yescharacteristic
111C22×C42D414yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C2×C4⋊C8—24no—
115C2×C4⋊C8—24no—
116C2×C4⋊C8—24no—
117C2×#103—28no—
118C2×#103—28no—
119C2×#103—28no—
120C2×#102—28no—
121C2×#102—28no—
122C2×#102—28no—
123C2×C22⋊C8—24no—
124C2×C22⋊C8—24no—
125C2×C22⋊C8—24no—
126#382—24no—
127#382—24no—
128#380—24no—
129#380—24no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135#340—28no—
136#340—28no—
137#336—24no—
138C25—21no—
139C25C2×D411yescharacteristic
140C2×#41—216no—
141C2×#41—216no—
142C2×#41—216no—
143C2×#41—216no—
144C2×#41—216no—
145C2×#41—216no—
146C2×#41—28no—
147C2×#41—28no—
148C2×#41—216no—
149C2×#41—216no—
150C2×#41—28no—
151C2×#41—28no—
152C2×#41—216no—
153C2×#41—216no—
154C2×#41—28no—
155C2×#41—28no—
156C2×#41—28no—
157C2×#41C4⋊C4116yes—
158C22×Q8—21no—
159C22×Q8—22no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8C22×C412yes—
164C22×Q8C2×D414yes—
165C22×Q8C2×D412yes—
166C22×Q8C2×Q812yes—
167C22×Q8#4114yes—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—22no—
174C22×D4—21no—
175C22×D4—22no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4C22×C412yes—
179C22×D4C2×D414yes—
180C22×D4C2×D412yes—
181C22×D4C2×Q812yes—
182C22×D4#4114yes—
183C23×C4—28no—
184C23×C4—28no—
185C23×C4—21no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—28no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C22⋊C414yes—
211C23×C4C22⋊C414yes—
212C23×C4C22×C412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D412yes—
219C23×C4C2×D412yes—
220C23×C4C2×Q812yes—
221C23×C4#4112yes—
222C23×C4#4112yes—
223C23×C4C2411yescharacteristic
224C22×C8—24no—
225C22×C8—24no—
226C22×C8—24no—
227C22×C8—24no—
228C22×C8—24no—
229C22×C8—24no—
230C22×C8—24no—
231C4×Q8—216no—
232C4×Q8—28no—
233C4×Q8—28no—
234C4×Q8—28no—
235C4×Q8—216no—
236C4×Q8C22⋊C4116yescomplemented
237C4×D4—264no—
238C4×D4—216no—
239C4×D4—264no—
240C4×D4—28no—
241C4×D4—28no—
242C4×D4—28no—
243C4×D4—216no—
244C4×D4C22⋊C4116yescomplemented
245#117—216no—
246#117—264no—
247#117—28no—
248#117—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C42—28no—
277C2×C42—28no—
278C2×C42—264no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—216no—
290C2×C42C22⋊C414yes—
291C2×C42C22⋊C414yes—
292C2×C42C22×C412yes—
293C2×C42C22×C412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D414yes—
296C2×C42C2×D414yes—
297C2×C42C2×D412yes—
298C4⋊C8—48no—
299#103—416no—
300#102—416no—
301C22⋊C8—48no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307C24—22no—
308C24—22no—
309C24—24no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—24no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24C2×C22⋊C412yes—
323C24#12112yes—
324C24C22⋊Q812yes—
325C24C22×D411yescharacteristic
326#41—264no—
327#41—264no—
328#41—232no—
329#41—232no—
330#41—232no—
331#41—232no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—22no—
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8C2×C22⋊C414yes—
346C2×Q8C2×C4⋊C414yes—
347C2×Q8#11714yes—
348C2×Q8#12118yes—
349C2×Q8C22⋊Q818yes—
350C2×D4—216no—
351C2×D4—232no—
352C2×D4—232no—
353C2×D4—216no—
354C2×D4—216no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—22no—
360C2×D4—24no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4C2×C22⋊C414yes—
369C2×D4C2×C4⋊C414yes—
370C2×D4#11714yes—
371C2×D4#12118yes—
372C2×D4C22⋊Q818yes—
373C22×C4—216no—
374C22×C4—232no—
375C22×C4—232no—
376C22×C4—216no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—216no—
416C22×C4—28no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—416no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—432no—
425C22×C4—416no—
426C22×C4—432no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4C2×C22⋊C412yes—
440C22×C4C2×C22⋊C412yes—
441C22×C4C2×C22⋊C412yes—
442C22×C4C2×C22⋊C412yes—
443C22×C4C2×C22⋊C412yes—
444C22×C4C2×C4⋊C414yes—
445C22×C4#11714yes—
446C22×C4#12014yes—
447C22×C4#12014yes—
448C22×C4#12114yes—
449C22×C4#12112yes—
450C22×C4C22⋊Q814yes—
451C22×C4C22⋊Q812yes—
452C22×C4#13514yes—
453C22×C4C23×C411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×Q811yescharacteristic
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4111yescharacteristic
463C2×C8—44no—
464C2×C8—44no—
465C2×C8—44no—
466C2×C8—44no—
467C2×C8—44no—
468C2×C8—44no—
469C2×C8—44no—
470C4⋊C4—216no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—216no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4C2×C22⋊C414yes—
481C4⋊C4C2×C22⋊C414yes—
482C22⋊C4—216no—
483C22⋊C4—216no—
484C22⋊C4—216no—
485C22⋊C4—88no—
486C42—216no—
487C42—216no—
488C42—22no—
489C42—22no—
490C42—24no—
491C42—28no—
492C42—216no—
493C42—28no—
494C42C2×C22⋊C414yes—
495C42C2×C22⋊C414yes—
496C42#12018yes—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—24no—
502C23—28no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—24no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—48no—
518C23—44no—
519C23—48no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—48no—
526C23—44no—
527C23—48no—
528C23—44no—
529C23—44no—
530C23#38014yes—
531C23C22×C22⋊C411yescharacteristic
532C23C2×#12012yes—
533C23C2×#12111yescharacteristic
534C23C2×C22⋊Q811yescharacteristic
535C23#52512yes—
536C23C2×#13512yes—
537C23#57512yes—
538C23#57512yes—
539C23#57512yes—
540Q8—28no—
541Q8—28no—
542Q8—24no—
543Q8—24no—
544Q8—24no—
545Q8#38018yes—
546D4—264no—
547D4—28no—
548D4—28no—
549D4—24no—
550D4—24no—
551D4—24no—
552D4#38018yes—
553C2×C4—264no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—416no—
584C2×C4—432no—
585C2×C4—416no—
586C2×C4—432no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4#37914yes—
602C2×C4#37914yes—
603C2×C4#38014yes—
604C2×C4#41614yes—
605C2×C4C22×C22⋊C411yescharacteristic
606C2×C4C22×C22⋊C411yescharacteristic
607C2×C4C22×C4⋊C411yescharacteristic
608C2×C4C2×#11711yescharacteristic
609C2×C4C2×#11711yescharacteristic
610C2×C4C2×#11711yescharacteristic
611C2×C4C2×#12012yes—
612C2×C4C2×#12111yescharacteristic
613C2×C4C2×#12112yes—
614C2×C4C2×#12112yes—
615C2×C4C2×C22⋊Q811yescharacteristic
616C2×C4C2×C22⋊Q812yes—
617C2×C4C2×C22⋊Q812yes—
618C2×C4#52512yes—
619C2×C4#52512yes—
620C2×C4#52512yes—
621C2×C4#52512yes—
622C2×C4#52512yes—
623C8—84no—
624C22—24no—
625C22—28no—
626C22—24no—
627C22—24no—
628C22—42no—
629C22—44no—
630C22—42no—
631C22—44no—
632C22—42no—
633C22—44no—
634C22—42no—
635C22—44no—
636C22—48no—
637C22—416no—
638C22—44no—
639C22—44no—
640C22—48no—
641C22—416no—
642C22C2×#37911yescharacteristic
643C22C2×#38011yescharacteristic
644C22#228011yescharacteristic
645C22#284712yes—
646C22#284812yes—
647C22C2×#41611yescharacteristic
648C22#289611yescharacteristic
649C22#289611yescharacteristic
650C22#289611yescharacteristic
651C22#300214yes—
652C22#300514yes—
653C22#303512yes—
654C22#304012yes—
655C22#307612yes—
656C22#308012yes—
657C22#310214yes—
658C22#311114yes—
659C4—24no—
660C4—28no—
661C4—24no—
662C4—24no—
663C4—24no—
664C4—44no—
665C4—42no—
666C4—44no—
667C4—164no—
668C4C2×#38011yescharacteristic
669C4C2×#38011yescharacteristic
670C4#228011yescharacteristic
671C4#228011yescharacteristic
672C4#284712yes—
673C4#284812yes—
674C2—44no—
675C2—48no—
676C2#1468011yescharacteristic
677C2#2059711yescharacteristic
678C2#2060011yescharacteristic
679C2#2064511yescharacteristic
680C2#2064911yescharacteristic
681C2#2073611yescharacteristic
682C2#2074211yescharacteristic
683C2#3291014yes—
684C2#3296114yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×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×#1805C2semidirect
3. C2×#1804C2semidirect
4. C2×#1797C2semidirect
6. #14682C2semidirect
7. #10789C2semidirect
8. #10788C2semidirect
9. #10731C2semidirect
14. C2×C4×#41C4semidirect
236. C4×Q8C22⋊C4semidirect
244. C4×D4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#412
C2×#18052
C2×#18042
C2×#17972
C2×#17882
#146822
#107892
#107882
#107312