Finite Group Databasebeta

← #393047

Subgroups — #393047

Order 512.

2192 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2620C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic, complemented
3C2×#1983C212yesmaximal, complemented
4C2×#1875C211yesmaximal, characteristic
5C2×#1873C211yesmaximal, characteristic, complemented
6C2×#1874C211yesmaximal, characteristic
7C2×#1872C211yesmaximal, characteristic, complemented
8C2×#1798C211yesmaximal, characteristic, complemented
9#10932C212yesmaximal, complemented
10#10931C212yesmaximal, complemented
11#10849C212yesmaximal, complemented
12C22×#124C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C22×C4⋊C8C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2620—24no—
26#2620C418yescomplemented
27C2×#396—22no—
28C2×#394—22no—
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#384C2211yescharacteristic
32#2274—28no—
33C2×C4×C4⋊C4—22no—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4—22no—
37#1983—28no—
38#1875—24no—
39#1874—24no—
40#1873—24no—
41#1872—24no—
42#1798—24no—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#336C2212yes—
48C23×Q8C2311yescharacteristic
49C23×D4—21no—
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×#124—22no—
56C2×#124—24no—
57C2×#124—24no—
58C2×#124C2×C414yescomplemented
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2×C414yes—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121—22no—
68C2×#121—22no—
69C2×#121—22no—
70C2×#121C2×C414yes—
71C2×#117—22no—
72C2×#117—24no—
73C2×#117—24no—
74C2×#117—24no—
75C2×#117—24no—
76C2×#117C2×C414yes—
77C2×#117D414yes—
78C2×#117D414yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4D414yes—
86C22×C4⋊C4D412yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—22no—
96C22×C22⋊C4D412yes—
97C22×C42—21no—
98C22×C42—22no—
99C22×C42—22no—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C2×C4⋊C8—24no—
103C2×C4⋊C8—24no—
104C2×C4⋊C8—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#102—24no—
112C2×#102—24no—
113C2×#102—24no—
114C2×#102—24no—
115C2×#102—24no—
116C2×#102—24no—
117C2×C22⋊C8—22no—
118C2×C22⋊C8—22no—
119C2×C22⋊C8—22no—
120C2×C22⋊C8—22no—
121C2×C22⋊C8—22no—
122C2×C22⋊C8—22no—
123#396—216no—
124#394—216no—
125#391—22no—
126#391C2×C414yescomplemented
127#390—22no—
128#390C2×C414yescomplemented
129#384—22no—
130#384C2×C414yescomplemented
131C4×C4⋊C4—216no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4D418yescomplemented
136C4×C22⋊C4—216no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—24no—
144#336—24no—
145C25—21no—
146C25C2×D411yescharacteristic
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8C22×C412yes—
157C22×D4—24no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—82no—
170C22×D4—82no—
171C22×D4C22×C412yes—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—82no—
185C23×C4—82no—
186C23×C4C22×C412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4#4112yes—
194C23×C4#4112yes—
195C23×C4C2411yescharacteristic
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210#124—216no—
211#124—48no—
212C22⋊Q8—28no—
213C22⋊Q8—28no—
214C22⋊Q8—44no—
215C22⋊Q8—44no—
216#121—28no—
217#121—28no—
218#121—44no—
219#121—44no—
220#117—216no—
221#117—216no—
222#117—216no—
223#117—216no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—216no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—216no—
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—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—82no—
252C2×C4⋊C4—82no—
253C2×C4⋊C4C22⋊C418yes—
254C2×C4⋊C4C22×C412yes—
255C2×C4⋊C4C22×C412yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4#4114yes—
261C2×C4⋊C4#4114yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—82no—
278C2×C22⋊C4—82no—
279C2×C22⋊C4C22⋊C418yes—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—28no—
286C2×C42—28no—
287C2×C42—82no—
288C2×C42—82no—
289C2×C42—82no—
290C2×C42—82no—
291C2×C42C22×C412yes—
292C2×C42C22×C412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D414yes—
295C2×C42C2×D412yes—
296C2×C42#4114yes—
297C4⋊C8—48no—
298#103—48no—
299#103—48no—
300#102—48no—
301#102—48no—
302C22⋊C8—44no—
303C22⋊C8—44no—
304#95—28no—
305#95—28no—
306#95—28no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—41no—
317C24—41no—
318C24—41no—
319C24—41no—
320C24—41no—
321C24—41no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24C2×C22⋊C412yes—
327C24#12112yes—
328C24#12112yes—
329C24C22×D411yescharacteristic
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—42no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—48no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—41no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—41no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—88no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—88no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—216no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4C2×C22⋊C414yes—
430C22×C4C2×C22⋊C414yes—
431C22×C4C2×C22⋊C412yes—
432C22×C4C4×D414yes—
433C22×C4C4×D414yes—
434C22×C4#12014yes—
435C22×C4#12112yes—
436C22×C4#12112yes—
437C22×C4#12314yes—
438C22×C4#12414yes—
439C22×C4#12714yes—
440C22×C4C23×C411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C2×#4112yes—
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
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—216no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—216no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—48no—
481C4⋊C4—88no—
482C4⋊C4#12118yes—
483C4⋊C4#12118yes—
484C22⋊C4—216no—
485C22⋊C4—216no—
486C22⋊C4—48no—
487C22⋊C4—416no—
488C22⋊C4—48no—
489C22⋊C4—88no—
490C42—216no—
491C42—24no—
492C42—24no—
493C42—88no—
494C42—88no—
495C42C4×D418yes—
496C42#12118yes—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—82no—
525C23—82no—
526C23—81no—
527C23—81no—
528C23—81no—
529C23—81no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—81no—
537C23—81no—
538C23—81no—
539C23—81no—
540C23—82no—
541C23—82no—
542C23#39014yes—
543C23C22×C22⋊C411yescharacteristic
544C23C2×#12011yescharacteristic
545C23C2×#12111yescharacteristic
546C23C2×#12111yescharacteristic
547C23C2×#12311yescharacteristic
548C23#52511yescharacteristic
549C23#52511yescharacteristic
550C23#57612yes—
551C23#57712yes—
552C23#57712yes—
553C23#57812yes—
554Q8—44no—
555Q8—44no—
556Q8—82no—
557Q8—82no—
558D4—44no—
559D4—44no—
560D4—816no—
561D4—82no—
562D4—82no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
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—816no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4#38618yes—
619C2×C4#39014yes—
620C2×C4C2×C4×D412yes—
621C2×C4C2×C4×D411yescharacteristic
622C2×C4C2×C4×D411yescharacteristic
623C2×C4C4×#4112yes—
624C2×C4C2×#12112yes—
625C2×C4C2×#12112yes—
626C2×C4C2×#12112yes—
627C2×C4#52512yes—
628C2×C4#52512yes—
629C2×C4C2×#12411yescharacteristic
630C2×C4#52912yes—
631C2×C4#52912yes—
632C2×C4C2×#12711yescharacteristic
633C2×C4#53212yes—
634C2×C4#53212yes—
635C2×C4#53212yes—
636C8—82no—
637C8—82no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—44no—
642C22—42no—
643C22—42no—
644C22—44no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22—44no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—84no—
653C22—84no—
654C22—162no—
655C22—161no—
656C22—161no—
657C22—161no—
658C22—161no—
659C22—162no—
660C22C2×#38611yescharacteristic
661C22C2×#39011yescharacteristic
662C22#229711yescharacteristic
663C22#284711yescharacteristic
664C22#284711yescharacteristic
665C22#285411yescharacteristic
666C22#285411yescharacteristic
667C22#289712yes—
668C22#289812yes—
669C22#301212yes—
670C22#301312yes—
671C22#304111yescharacteristic
672C22#304211yescharacteristic
673C22#304311yescharacteristic
674C22#304411yescharacteristic
675C22#304512yes—
676C22#304612yes—
677C22#309812yes—
678C22#309912yes—
679C22#311812yes—
680C22#312112yes—
681C22#312412yes—
682C22#312512yes—
683C4—24no—
684C4—28no—
685C4—48no—
686C4—48no—
687C4—44no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4C4×#12112yes—
692C4C4×#12112yes—
693C4#285012yes—
694C4#285612yes—
695C2—44no—
696C2—162no—
697C2#1469811yescharacteristic
698C2#2091512yes—
699C2#2126211yescharacteristic
700C2#2126311yescharacteristic
701C2#2139712yes—
702C2#3291711yescharacteristic
703C2#3291811yescharacteristic
704C2#3291912yes—
705C2#3311512yes—
706C2#3312011yescharacteristic
707C2#3312111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2274C2semidirect
3. C2×#1983C2semidirect
5. C2×#1873C2semidirect
7. C2×#1872C2semidirect
8. C2×#1798C2semidirect
9. #10932C2semidirect
10. #10931C2semidirect
11. #10849C2semidirect
26. #2620C4semidirect
58. C2×#124C2×C4semidirect
126. #391C2×C4semidirect
128. #390C2×C4semidirect
130. #384C2×C4semidirect
135. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26202
C2×#22742
C2×#19832
C2×#18752
C2×#18732
C2×#18742
C2×#18722
C2×#17982
#109322
#109312
#108492