Finite Group Databasebeta

← #393547

Subgroups — #393547

Order 512.

1936 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#482C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic, complemented
3C2×#1874C211yesmaximal, characteristic
4C2×#1825C211yesmaximal, characteristic, complemented
5#14257C211yesmaximal, characteristic
6#10931C212yesmaximal, complemented
7#10931C212yesmaximal, complemented
8#10908C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#482—24no—
17C2×#482—24no—
18C2×#482C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C2×#390C2212yes—
27C2×#380C2211yescharacteristic
28#2274—24no—
29#2274—24no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C22⋊C4—22no—
35#1874—24no—
36#1825—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1439—21no—
45#1439—21no—
46#1438—22no—
47C4⋊#95C2212yes—
48C23×D4—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121C2×C414yes—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—24no—
67C2×#117—24no—
68C2×#117C2×C414yes—
69C2×#117D414yes—
70C2×#117D414yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D414yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4D412yes—
89C22×C42—21no—
90C22×C42—22no—
91C22×C42—22no—
92C22×C42C2312yes—
93#482—232no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107C2×C414yescomplemented
97C2×#106—22no—
98C2×#106—22no—
99C2×#106C2×C414yescomplemented
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102C2×C414yescomplemented
108C2×#102C2×C414yescomplemented
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8C2×C414yes—
115#390—24no—
116#390—24no—
117#380—22no—
118#380—22no—
119C4×C4⋊C4—28no—
120C4×C4⋊C4—28no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C22⋊C4—216no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#336—24no—
137#336—24no—
138C25—21no—
139C25C2×D411yescharacteristic
140C22×D4—24no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—82no—
153C22×D4—82no—
154C22×D4C22×C412yes—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—28no—
163C23×C4—24no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—81no—
167C23×C4—82no—
168C23×C4—81no—
169C23×C4C22×C412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176C23×C4#4112yes—
177C23×C4#4112yes—
178C23×C4C2411yescharacteristic
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8C22×C412yes—
195C22×C8C22×C412yes—
196#121—28no—
197#121—28no—
198#121—44no—
199#121—44no—
200#117—28no—
201#117—28no—
202#117—216no—
203#117—28no—
204#117—28no—
205#117—28no—
206#117C22⋊C4116yescomplemented
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—216no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
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⋊C4C22×C412yes—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4#4114yes—
245C2×C4⋊C4#4114yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—82no—
259C2×C22⋊C4—82no—
260C2×C22⋊C4—82no—
261C2×C22⋊C4—82no—
262C2×C22⋊C4—81no—
263C2×C22⋊C4—81no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—28no—
272C2×C42—28no—
273C2×C42—82no—
274C2×C42—82no—
275C2×C42—82no—
276C2×C42—82no—
277C2×C42#4114yes—
278C2×C42#4114yes—
279#107—28no—
280#106—28no—
281#102—28no—
282#102—28no—
283#102—48no—
284C22⋊C8—28no—
285C22⋊C8—44no—
286#95—28no—
287#95—28no—
288#95—42no—
289#95—82no—
290#95—82no—
291C24—22no—
292C24—22no—
293C24—21no—
294C24—22no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—21no—
299C24—22no—
300C24—41no—
301C24—41no—
302C24—42no—
303C24—42no—
304C24—41no—
305C24—42no—
306C24—41no—
307C24—42no—
308C24—41no—
309C24—41no—
310C24C4×D412yes—
311C24#12012yes—
312C24#12112yes—
313C24C22×D411yescharacteristic
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—48no—
319C2×D4—41no—
320C2×D4—42no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—88no—
334C2×D4—84no—
335C2×D4—84no—
336C2×D4—88no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—216no—
375C22×C4—22no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
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—42no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4—88no—
400C22×C4—88no—
401C22×C4C2×C22⋊C414yes—
402C22×C4C4×D414yes—
403C22×C4C4×D412yes—
404C22×C4#12012yes—
405C22×C4#12114yes—
406C22×C4#12112yes—
407C22×C4#12314yes—
408C22×C4#12314yes—
409C22×C4#12314yes—
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4111yescharacteristic
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C4⋊C4—216no—
438C4⋊C4—28no—
439C4⋊C4—28no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4C2×C22⋊C414yes—
458C4⋊C4C2×C22⋊C414yes—
459C4⋊C4#12318yes—
460C22⋊C4—216no—
461C22⋊C4—216no—
462C22⋊C4—48no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C22⋊C4—84no—
466C42—24no—
467C42—24no—
468C42—24no—
469C42—24no—
470C42—216no—
471C42—88no—
472C42—88no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—81no—
505C23—81no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—81no—
512C23—81no—
513C23—81no—
514C23—81no—
515C23—82no—
516C23—81no—
517C23—81no—
518C23#38614yes—
519C23C2×C4×D411yescharacteristic
520C23C2×#12011yescharacteristic
521C23C2×#12111yescharacteristic
522C23C2×#12312yes—
523C23#52512yes—
524C23#57611yescharacteristic
525C23#57612yes—
526C23#57711yescharacteristic
527C23#57712yes—
528C23#57711yescharacteristic
529C23#57811yescharacteristic
530D4—42no—
531D4—42no—
532D4—44no—
533D4—816no—
534D4—81no—
535D4—81no—
536D4—82no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—816no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4#38614yes—
594C2×C4C22×C22⋊C411yescharacteristic
595C2×C4C2×C4×D411yescharacteristic
596C2×C4C4×#4112yes—
597C2×C4C4×#4112yes—
598C2×C4C2×#12111yescharacteristic
599C2×C4C2×#12311yescharacteristic
600C2×C4C2×#12312yes—
601C2×C4#52511yescharacteristic
602C2×C4#52511yescharacteristic
603C2×C4#52511yescharacteristic
604C2×C4#52511yescharacteristic
605C2×C4#52512yes—
606C2×C4#52912yes—
607C2×C4#52912yes—
608C2×C4#52912yes—
609C2×C4#52912yes—
610C2×C4#53212yes—
611C2×C4#53212yes—
612C8—42no—
613C8—42no—
614C8—82no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—84no—
627C22—84no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—162no—
632C22—162no—
633C22—161no—
634C22—161no—
635C22—161no—
636C22—161no—
637C22C2×#38611yescharacteristic
638C22C4×#12312yes—
639C22#284911yescharacteristic
640C22#284911yescharacteristic
641C22#285412yes—
642C22#294612yes—
643C22#295111yescharacteristic
644C22#295211yescharacteristic
645C22#300811yescharacteristic
646C22#301011yescharacteristic
647C22#301211yescharacteristic
648C22#301311yescharacteristic
649C22#304512yes—
650C22#304511yescharacteristic
651C22#304611yescharacteristic
652C22#309612yes—
653C22#309612yes—
654C22#309812yes—
655C22#309912yes—
656C22#311812yes—
657C22#312112yes—
658C22#312212yes—
659C22#312612yes—
660C4—24no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—48no—
666C4—82no—
667C4—162no—
668C4#229711yescharacteristic
669C4#229711yescharacteristic
670C4C4×#12312yes—
671C4#285311yescharacteristic
672C4#285311yescharacteristic
673C4#285412yes—
674C2—44no—
675C2—162no—
676C2#1470611yescharacteristic
677C2#2093211yescharacteristic
678C2#2093711yescharacteristic
679C2#2158812yes—
680C2#2159012yes—
681C2#3298611yescharacteristic
682C2#3298711yescharacteristic
683C2#3301411yescharacteristic
684C2#3301511yescharacteristic
685C2#3311512yes—
686C2#3311612yes—
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
4. C2×#1825C2semidirect
6. #10931C2semidirect
7. #10931C2semidirect
8. #10908C2semidirect
9. #10907C2semidirect
10. #10730C2semidirect
11. #10713C2semidirect
18. C2×#482C4semidirect
96. C2×#107C2×C4semidirect
99. C2×#106C2×C4semidirect
107. C2×#102C2×C4semidirect
108. C2×#102C2×C4semidirect
206. #117C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4822
C2×#22742
C2×#18742
C2×#18252
#142572
#109312
#109082
#109072
#107302
#107132