Finite Group Databasebeta

← #393085

Subgroups — #393085

Order 512.

2028 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#464C211yesmaximal, characteristic
2C2×#2276C211yesmaximal, characteristic, complemented
3C2×#1983C211yesmaximal, characteristic
4C2×#1880C212yesmaximal
5C2×#1878C212yesmaximal, complemented
6C2×#1803C211yesmaximal, characteristic, complemented
7#14273C211yesmaximal, characteristic
8#10932C212yesmaximal, complemented
9#10931C212yesmaximal
10#10849C211yesmaximal, characteristic, complemented
11#10833C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C23×SD16C2211yescharacteristic
14C22×#124C2211yescharacteristic
15C22×C4×Q8C2212yes—
16C22×C4×Q8C2211yescharacteristic
17C2×#464—24no—
18C2×#464—24no—
19C2×#464C418yescomplemented
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#103C2211yescharacteristic
25C22×#102C2212yes—
26C22×#102C2211yescharacteristic
27C2×#391C2212yes—
28C2×#390C2212yes—
29C2×#384C2211yescharacteristic
30#2276—24no—
31#2276—24no—
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4—22no—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×#376—21no—
37C2×#376C2211yescharacteristic
38#1983—24no—
39#1880—28no—
40#1878—28no—
41#1803—24no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47#1439—21no—
48#1439—22no—
49#1439—21no—
50C23×Q8C2311yescharacteristic
51C23×Q8C2311yescharacteristic
52C23×D4C2311yescharacteristic
53C22×Q16—24no—
54C22×Q16—22no—
55C22×Q16—22no—
56C22×Q16—24no—
57C22×Q16C2×C414yes—
58C22×SD16—24no—
59C22×SD16—22no—
60C22×SD16—22no—
61C22×SD16—24no—
62C22×SD16C2×C414yes—
63C23×C8C2312yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×#124—22no—
67C2×#124—22no—
68C2×#124—22no—
69C2×#124—22no—
70C2×#124—22no—
71C2×#124C2×C414yes—
72C2×C4×Q8—24no—
73C2×C4×Q8—24no—
74C2×C4×Q8—24no—
75C2×C4×Q8—24no—
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8—24no—
79C2×C4×Q8—22no—
80C2×C4×Q8C2×C414yes—
81C2×C4×Q8D418yes—
82C2×C4×Q8D414yes—
83C2×C4×Q8D414yes—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C42—22no—
96C22×C42—21no—
97C22×C42—22no—
98C22×C42—21no—
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101#464—232no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8C2×C414yes—
108C2×#103—22no—
109C2×#103—22no—
110C2×#103—24no—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—24no—
115C2×#103—24no—
116C2×#103C2×C414yes—
117C2×#102—22no—
118C2×#102—22no—
119C2×#102—24no—
120C2×#102—24no—
121C2×#102—24no—
122C2×#102C2×C414yes—
123#391—24no—
124#391—24no—
125#390—24no—
126#390—24no—
127#384—22no—
128#384—22no—
129C4×C4⋊C4—216no—
130C4×C4⋊C4—28no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4D414yes—
135#376—28no—
136#376—22no—
137#376—22no—
138C2×#95—22no—
139C2×#95—21no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95D411yescharacteristic
143C2×#95D411yescharacteristic
144C2×#95D412yes—
145#340—24no—
146#340—24no—
147#340—24no—
148#336—24no—
149#336—24no—
150C25—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—22no—
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8C22×C412yes—
166C22×Q8C22×C412yes—
167C22×Q8C2×D412yes—
168C22×Q8C2×D412yes—
169C22×Q8C2×D414yes—
170C22×Q8#4114yes—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—22no—
175C22×D4—21no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—21no—
180C22×D4C22×C412yes—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C2×Q16—28no—
192C2×Q16—216no—
193C2×Q16—48no—
194C2×Q16—44no—
195C2×Q16—44no—
196C2×SD16—28no—
197C2×SD16—216no—
198C2×SD16—48no—
199C2×SD16—44no—
200C2×SD16—44no—
201C22×C8—22no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8C22×C412yes—
217C22×C8C22×C412yes—
218#124—28no—
219#124—28no—
220#124—44no—
221#124—44no—
222C4×Q8—216no—
223C4×Q8—216no—
224C4×Q8—216no—
225C4×Q8—216no—
226C4×Q8—28no—
227C4×Q8—28no—
228C4×Q8—28no—
229C4×Q8C22⋊C4116yescomplemented
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—81no—
251C2×C4⋊C4—82no—
252C2×C4⋊C4—81no—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4#4114yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—82no—
268C2×C22⋊C4—82no—
269C2×C22⋊C4—82no—
270C2×C22⋊C4—82no—
271C2×C42—28no—
272C2×C42—28no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—28no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—28no—
282C2×C42—24no—
283C2×C42—81no—
284C2×C42—81no—
285C2×C42—82no—
286C2×C42—82no—
287C2×C42C22×C412yes—
288C2×C42C2×D414yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42#4114yes—
293C2×C42#4112yes—
294C2×C42#4112yes—
295C4⋊C8—28no—
296C4⋊C8—44no—
297#103—28no—
298#103—48no—
299#103—48no—
300#102—28no—
301#102—48no—
302#95—28no—
303#95—24no—
304#95—22no—
305#95—24no—
306#95—22no—
307#95—44no—
308#95—44no—
309C24—22no—
310C24—41no—
311C24—41no—
312C24—42no—
313C24—42no—
314C24—41no—
315C24—42no—
316C24—41no—
317C24—41no—
318C24—42no—
319C24—41no—
320C24C22×D411yescharacteristic
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—41no—
347C2×Q8—42no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—42no—
352C2×Q8C2×C22⋊C414yes—
353C2×Q8C4×D418yes—
354C2×Q8#12118yes—
355C2×Q8#12414yes—
356C2×Q8#12714yes—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—41no—
374C2×D4—42no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4C4×D412yes—
425C22×C4C4×D412yes—
426C22×C4#12012yes—
427C22×C4#12012yes—
428C22×C4#12114yes—
429C22×C4#12114yes—
430C22×C4#12112yes—
431C22×C4#12112yes—
432C22×C4C23×C411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4111yescharacteristic
442Q16—416no—
443SD16—416no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C4⋊C4—232no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—216no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4C2×C22⋊C414yes—
476C4⋊C4#12118yes—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C42—232no—
482C42—24no—
483C42—24no—
484C42—24no—
485C42—24no—
486C42—216no—
487C42—84no—
488C42—88no—
489C42C2×C22⋊C414yes—
490C42C4×D414yes—
491C42#12114yes—
492C42#12314yes—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—82no—
499C23—81no—
500C23—81no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—81no—
506C23—82no—
507C23—81no—
508C23—82no—
509C23—81no—
510C23—82no—
511C23—81no—
512C23—82no—
513C23—81no—
514C23—82no—
515C23—81no—
516C23C2×C4×D411yescharacteristic
517C23C2×#12011yescharacteristic
518C23C2×#12112yes—
519C23C2×#12111yescharacteristic
520C23#52512yes—
521C23#57611yescharacteristic
522C23#57711yescharacteristic
523C23#57712yes—
524C23#57711yescharacteristic
525C23#57811yescharacteristic
526C23#57812yes—
527Q8—24no—
528Q8—28no—
529Q8—24no—
530Q8—24no—
531Q8—28no—
532Q8—42no—
533Q8—42no—
534Q8—44no—
535Q8—81no—
536Q8—82no—
537Q8—81no—
538Q8#39018yes—
539D4—42no—
540D4—42no—
541D4—44no—
542D4—81no—
543D4—82no—
544D4—81no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—82no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4#38614yes—
601C2×C4#38614yes—
602C2×C4C22×C22⋊C411yescharacteristic
603C2×C4C2×C4×D412yes—
604C2×C4C2×C4×D411yescharacteristic
605C2×C4C4×#4112yes—
606C2×C4C2×#12111yescharacteristic
607C2×C4C2×#12112yes—
608C2×C4C2×#12112yes—
609C2×C4C2×#12311yescharacteristic
610C2×C4#52512yes—
611C2×C4#52512yes—
612C2×C4C2×#12411yescharacteristic
613C2×C4#52912yes—
614C2×C4#52912yes—
615C2×C4C2×#12711yescharacteristic
616C2×C4#53211yescharacteristic
617C2×C4#53212yes—
618C2×C4#53212yes—
619C2×C4#53211yescharacteristic
620C8—42no—
621C8—42no—
622C8—82no—
623C22—84no—
624C22—84no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—161no—
629C22—161no—
630C22—161no—
631C22—161no—
632C22—162no—
633C22—162no—
634C22C2×#38611yescharacteristic
635C22C4×#12112yes—
636C22#284712yes—
637C22#285011yescharacteristic
638C22#285611yescharacteristic
639C22#294712yes—
640C22#295111yescharacteristic
641C22#295211yescharacteristic
642C22#301211yescharacteristic
643C22#301311yescharacteristic
644C22#301312yes—
645C22#304312yes—
646C22#304311yescharacteristic
647C22#304412yes—
648C22#304411yescharacteristic
649C22#304512yes—
650C22#304511yescharacteristic
651C22#304612yes—
652C22#304611yescharacteristic
653C22#312112yes—
654C22#312112yes—
655C22#312412yes—
656C22#312512yes—
657C4—28no—
658C4—24no—
659C4—24no—
660C4—44no—
661C4—44no—
662C4—48no—
663C4—44no—
664C4—82no—
665C4—162no—
666C4C2×#39011yescharacteristic
667C4#229711yescharacteristic
668C4C4×#12112yes—
669C4#285011yescharacteristic
670C4#285412yes—
671C4#285611yescharacteristic
672C2—162no—
673C2#1469811yescharacteristic
674C2#2093011yescharacteristic
675C2#2093711yescharacteristic
676C2#2151112yes—
677C2#2151212yes—
678C2#3291812yes—
679C2#3291912yes—
680C2#3303511yescharacteristic
681C2#3303711yescharacteristic
682C2#3316711yescharacteristic
683C2#3316911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2276C2semidirect
5. C2×#1878C2semidirect
6. C2×#1803C2semidirect
8. #10932C2semidirect
10. #10849C2semidirect
11. #10833C2semidirect
19. C2×#464C4semidirect
229. C4×Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4642
C2×#22762
C2×#19832
C2×#18802
C2×#18782
C2×#18032
#142732
#109322
#109312
#108492
#108332