Finite Group Databasebeta

← #392780

Subgroups — #392780

Order 512.

2156 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×C8C211yesmaximal, characteristic, complemented
2C2×#1923C212yesmaximal, complemented
3C2×#1869C211yesmaximal, characteristic
4C2×#1869C211yesmaximal, characteristic
5C2×#1850C211yesmaximal, characteristic, complemented
6C4×#386C211yesmaximal, characteristic
7#10921C212yesmaximal, complemented
8#10876C212yesmaximal, complemented
9#10797C212yesmaximal, complemented
10#10714C211yesmaximal, characteristic, complemented
11#10712C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C4×C8—28no—
23C22×C4×C8—24no—
24C22×C4×C8C2212yes—
25C22×C4×C8C2212yes—
26C2×#386C411yescharacteristic
27C2×#386C411yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30#1923—28no—
31#1869—24no—
32#1869—24no—
33#1850—24no—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1437C412yes—
43#1436C412yes—
44C22⋊#95C411yescharacteristic
45C22⋊#95C411yescharacteristic
46C4×#95C2212yes—
47C23×D4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—24no—
57C23×C8—24no—
58C23×C8—24no—
59C23×C8—24no—
60C23×C8D414yes—
61C23×C8D414yes—
62C23×C8C2311yescharacteristic
63C23×C8C2312yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2312yes—
67C23×C8C2311yescharacteristic
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C412yes—
81C22×C42—22no—
82C22×C42—22no—
83C22×C42—24no—
84C22×C42—24no—
85C22×C42C2×C412yes—
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88C22×C42C2312yes—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—24no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—24no—
105C2×C4×C8—232no—
106C2×C4×C8—24no—
107C2×C4×C8—24no—
108C2×C4×C8—24no—
109C2×C4×C8—24no—
110C2×C4×C8—24no—
111C2×C4×C8D418yescomplemented
112#386—24no—
113#386C818yescomplemented
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—24no—
117C4×C22⋊C4—24no—
118C2×#95C2×C412yes—
119C2×#95C2×C412yes—
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C2×#95C2×C412yes—
123C2×#95C2×C412yes—
124C2×#95C2×C412yes—
125#336—24no—
126#336—22no—
127#336—24no—
128#336—22no—
129#336—24no—
130#336—24no—
131#336—24no—
132#336—24no—
133C25C22×C411yescharacteristic
134C25C22×C411yescharacteristic
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4C2×C814yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4C2×C814yes—
165C23×C4M1614yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C412yes—
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C412yes—
173C23×C4C22×C412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4#4112yes—
177C23×C4#4112yes—
178C23×C4#4112yes—
179C23×C4#4112yes—
180C23×C4C2411yescharacteristic
181C22×C8—216no—
182C22×C8—216no—
183C22×C8—216no—
184C22×C8—28no—
185C22×C8—28no—
186C22×C8—28no—
187C22×C8—28no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—28no—
199C22×C8—216no—
200C22×C8—28no—
201C22×C8—22no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—21no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8C2×D414yes—
219C22×C8C2×D412yes—
220C22×C8C2×D414yes—
221C22×C8C2×D414yes—
222C22×C8C2×D412yes—
223C22×C8#4114yes—
224C22×C8#4114yes—
225C4×D4—44no—
226C4×D4—44no—
227C4×D4—44no—
228C4×D4—44no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4C2×C814yes—
249C2×C22⋊C4C2×C818yescomplemented
250C2×C42—28no—
251C2×C42—216no—
252C2×C42—28no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42C2×D414yes—
273C2×C42#4114yes—
274C4⋊C8—48no—
275C22⋊C8—44no—
276C22⋊C8—44no—
277C22⋊C8—44no—
278C22⋊C8—48no—
279C22⋊C8—44no—
280C4×C8—216no—
281C4×C8—216no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95C2×C818yescomplemented
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—21no—
294C24—22no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24—21no—
302C24—21no—
303C24#11712yes—
304C24C4×D412yes—
305C24C4×D412yes—
306C24C22×C812yes—
307C24C22×C812yes—
308C24C2×M1612yes—
309C24#13112yes—
310C24#13112yes—
311C24C23×C411yescharacteristic
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—216no—
322C22×C4—216no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—216no—
352C22×C4—216no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4#11712yes—
401C22×C4C4×D412yes—
402C22×C4C4×D412yes—
403C22×C4C4×D414yes—
404C22×C4C4×D414yes—
405C22×C4#12114yes—
406C22×C4#12314yes—
407C22×C4#12414yes—
408C22×C4#12614yes—
409C22×C4C22×C812yes—
410C22×C4C22×C814yes—
411C22×C4C22×C814yes—
412C22×C4C2×M1614yes—
413C22×C4#13114yes—
414C22×C4#13114yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4112yes—
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4112yes—
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4112yes—
425C2×C8—232no—
426C2×C8—232no—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—42no—
444C2×C8—41no—
445C2×C8—42no—
446C2×C8—41no—
447C2×C8—42no—
448C2×C8—41no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—41no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8#12014yes—
459C2×C8#12118yes—
460C2×C8#12314yes—
461C4⋊C4—44no—
462C22⋊C4—28no—
463C22⋊C4—216no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—48no—
468C42—216no—
469C42—24no—
470C42—24no—
471C42—24no—
472C42—24no—
473C42—42no—
474C42—41no—
475C42—41no—
476C42—42no—
477C42—41no—
478C42—41no—
479C42—42no—
480C42—42no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23—41no—
520C23#38814yes—
521C23#43114yes—
522C23#43314yes—
523C23C8×D414yes—
524C23C8×D414yes—
525C23#43514yes—
526C23#43614yes—
527C23C2×#11711yescharacteristic
528C23C2×C4×D411yescharacteristic
529C23C2×C4×D412yes—
530C23C2×C4×D411yescharacteristic
531C23C4×#4112yes—
532C23C23×C811yescharacteristic
533C23C22×M1611yescharacteristic
534C23C2×#13111yescharacteristic
535C23C2×#13111yescharacteristic
536C23C2×#13112yes—
537C23C2×#13112yes—
538D4—416no—
539C2×C4—232no—
540C2×C4—232no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4#38814yes—
601C2×C4C8×D418yes—
602C2×C4#43518yes—
603C2×C4C2×#12011yescharacteristic
604C2×C4C2×#12112yes—
605C2×C4C2×#12111yescharacteristic
606C2×C4C2×#12311yescharacteristic
607C2×C4C2×#12311yescharacteristic
608C2×C4#52512yes—
609C2×C4#52511yescharacteristic
610C2×C4#52512yes—
611C2×C4#52512yes—
612C2×C4#52512yes—
613C2×C4#52511yescharacteristic
614C2×C4C2×#12411yescharacteristic
615C2×C4C2×#12611yescharacteristic
616C2×C4#52912yes—
617C2×C4#52912yes—
618C2×C4#52912yes—
619C2×C4#52912yes—
620C2×C4#53212yes—
621C8—28no—
622C8—24no—
623C8—24no—
624C8—81no—
625C8—82no—
626C8—81no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—28no—
632C22—28no—
633C22—24no—
634C22—24no—
635C22—28no—
636C22—28no—
637C22—42no—
638C22—42no—
639C22—44no—
640C22—42no—
641C22—44no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—82no—
646C22—81no—
647C22—82no—
648C22#192318yescomplemented
649C22C2×#38811yescharacteristic
650C22#229012yes—
651C22C4×#12011yescharacteristic
652C22C4×#12112yes—
653C22C4×#12311yescharacteristic
654C22C2×#43111yescharacteristic
655C22C2×#43311yescharacteristic
656C22#292212yes—
657C22C2×C8×D411yescharacteristic
658C22C2×C8×D411yescharacteristic
659C22C2×C8×D412yes—
660C22C2×#43511yescharacteristic
661C22C2×#43512yes—
662C22C2×#43611yescharacteristic
663C22#292911yescharacteristic
664C22#292911yescharacteristic
665C22#292912yes—
666C22#292912yes—
667C22#293112yes—
668C22C8×#4112yes—
669C22#296512yes—
670C22#296612yes—
671C22#296612yes—
672C4—24no—
673C4—28no—
674C4—28no—
675C4—42no—
676C4—44no—
677C4—81no—
678C4—81no—
679C4—82no—
680C4#284711yescharacteristic
681C4#284911yescharacteristic
682C4#285012yes—
683C4#285212yes—
684C4#285311yescharacteristic
685C4#285411yescharacteristic
686C2—28no—
687C2—82no—
688C2#1469611yescharacteristic
689C2C2×#192311yescharacteristic
690C2#2104611yescharacteristic
691C2#2110112yes—
692C2#2110312yes—
693C2C8×#12011yescharacteristic
694C2#2144011yescharacteristic
695C2C8×#12112yes—
696C2#2148712yes—
697C2C8×#12311yescharacteristic
698C2#2156311yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C4×C8C2semidirect
2. C2×#1923C2semidirect
5. C2×#1850C2semidirect
7. #10921C2semidirect
8. #10876C2semidirect
9. #10797C2semidirect
10. #10714C2semidirect
11. #10712C2semidirect
111. C2×C4×C8D4semidirect
113. #386C8semidirect
249. C2×C22⋊C4C2×C8semidirect
288. #95C2×C8semidirect
648. C22#1923semidirect

Maximal subgroups

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

subgroupindex
C23×C4×C82
C2×#19232
C2×#18692
C2×#18502
C4×#3862
#109212
#108762
#107972
#107142
#107122