Finite Group Databasebeta

← #375122

Subgroups — #375122

Order 512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2392C211yesmaximal, characteristic
2C2×#1870C212yesmaximal, complemented
3C2×#1869C212yesmaximal, complemented
4C2×#1750C211yesmaximal, characteristic, complemented
5C2×#1729C211yesmaximal, characteristic, complemented
6#11346C212yesmaximal, complemented
7#10964C212yesmaximal, complemented
8#10922C212yesmaximal, complemented
9#10921C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×C4×D4C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×#97C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2392—24no—
23#2392—24no—
24C2×#398C411yescharacteristic
25C2×#397C411yescharacteristic
26C2×#396C412yes—
27C2×#394C411yescharacteristic
28C2×#394C412yes—
29C2×#392C411yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2212yes—
32#1870—28no—
33#1869—28no—
34#1750—24no—
35#1729—24no—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C23×D4C2×C412yes—
44C23×D4C2×C411yescharacteristic
45C24×C4C2312yes—
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48C22×M16—24no—
49C22×M16—24no—
50C22×M16—41no—
51C22×M16—41no—
52C23×C8—22no—
53C23×C8—24no—
54C23×C8—24no—
55C23×C8—41no—
56C23×C8—41no—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—24no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—41no—
70C2×C4×D4—41no—
71C2×C4×D4—41no—
72C2×C4×D4—41no—
73C2×#117—41no—
74C2×#117—41no—
75C2×#117—41no—
76C2×#117—41no—
77C2×#117D414yes—
78C2×#117D414yes—
79C22×C4⋊C4C2×C412yes—
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C22⋊C4C2×C412yes—
85C22×C22⋊C4C2×C412yes—
86C22×C22⋊C4C2×C412yes—
87C22×C22⋊C4C2×C412yes—
88C22×C22⋊C4C2×C412yes—
89C22×C22⋊C4C2×C411yescharacteristic
90C22×C22⋊C4C2×C411yescharacteristic
91C22×C42C2312yes—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—42no—
105C2×C22⋊C8—42no—
106C2×C22⋊C8—42no—
107C2×C22⋊C8—42no—
108C2×C22⋊C8D418yescomplemented
109C2×#97—22no—
110C2×#97—41no—
111C2×#97—41no—
112C2×C4×C8—22no—
113C2×C4×C8—41no—
114C2×C4×C8—41no—
115#398—24no—
116#397—24no—
117#396—28no—
118#394—24no—
119#394—28no—
120#392—24no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C22⋊C4—24no—
124C4×C22⋊C4—24no—
125C2×#95C2×C411yescharacteristic
126C2×#95C2×C411yescharacteristic
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129#336—24no—
130#336—24no—
131#336—24no—
132#336—24no—
133#336—24no—
134#336—24no—
135#336—24no—
136C25C22×C412yes—
137C25C22×C411yescharacteristic
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—22no—
144C22×D4—24no—
145C22×D4—41no—
146C22×D4—41no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4C22×C412yes—
171C23×C4C22×C412yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C412yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D414yes—
181C23×C4#4114yes—
182C23×C4C2411yescharacteristic
183C2×M16—28no—
184C2×M16—48no—
185C2×M16—48no—
186C2×M16—44no—
187C2×M16—44no—
188C22×C8—28no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
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—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—48no—
214C22×C8—48no—
215C22×C8—44no—
216C22×C8—44no—
217C22×C8C2×D414yes—
218C22×C8C2×D414yes—
219C22×C8#4114yes—
220C22×C8#4114yes—
221C4×D4—44no—
222C4×D4—44no—
223C4×D4—44no—
224C4×D4—44no—
225C4×D4—48no—
226C4×D4—48no—
227C4×D4—48no—
228C4×D4—48no—
229#117—44no—
230#117—44no—
231#117—44no—
232#117—44no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4C22⋊C414yes—
247C2×C4⋊C4C22⋊C414yes—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4C22⋊C414yes—
268C2×C22⋊C4C22⋊C414yes—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C22⋊C8—216no—
288C22⋊C8—48no—
289C22⋊C8—48no—
290C22⋊C8—48no—
291C22⋊C8—48no—
292#97—44no—
293C4×C8—44no—
294#95—24no—
295#95—24no—
296#95—22no—
297#95—22no—
298C24—22no—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24C2×C22⋊C412yes—
320C24C4×D414yes—
321C24#13114yes—
322C24C23×C411yescharacteristic
323C2×D4—216no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—44no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—44no—
334C2×D4—48no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
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—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
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—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
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—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
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—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4C2×C22⋊C412yes—
418C22×C4C2×C22⋊C412yes—
419C22×C4C2×C22⋊C412yes—
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C412yes—
422C22×C4C4×D414yes—
423C22×C4#12118yes—
424C22×C4#12414yes—
425C22×C4#12714yes—
426C22×C4#13114yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4H(5,𝔽2)11yescharacteristic
434C22×C4H(5,𝔽2)11yescharacteristic
435C22×C4H(5,𝔽2)11yescharacteristic
436C22×C4H(5,𝔽2)11yescharacteristic
437M16—416no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—416no—
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—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—42no—
488C42—42no—
489C42—42no—
490C42—42no—
491C23—22no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23#39014yes—
547C23C22×C22⋊C411yescharacteristic
548C23C2×C4×D412yes—
549C23C4×#4112yes—
550C23#51811yescharacteristic
551C23#51811yescharacteristic
552C23C2×#13112yes—
553C23C2×#13112yes—
554C23#56812yes—
555C23#56812yes—
556D4—432no—
557D4—416no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4#39014yes—
618C2×C4C2×#12112yes—
619C2×C4#52511yescharacteristic
620C2×C4#52511yescharacteristic
621C2×C4C2×#12411yescharacteristic
622C2×C4C2×#12711yescharacteristic
623C2×C4#53212yes—
624C2×C4#53411yescharacteristic
625C2×C4#53411yescharacteristic
626C2×C4#53411yescharacteristic
627C2×C4#53411yescharacteristic
628C2×C4#54612yes—
629C2×C4#54612yes—
630C2×C4#54612yes—
631C2×C4#54612yes—
632C2×C4#55312yes—
633C2×C4#55312yes—
634C2×C4#55312yes—
635C2×C4#55312yes—
636C8—44no—
637C8—44no—
638C8—44no—
639C8—44no—
640C22—24no—
641C22—28no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—42no—
648C22—42no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22C2×#39011yescharacteristic
665C22#229711yescharacteristic
666C22#230811yescharacteristic
667C22#236412yes—
668C22#236412yes—
669C22#288012yes—
670C22#288012yes—
671C22#292912yes—
672C22#293012yes—
673C22#293312yes—
674C22#293312yes—
675C22#296612yes—
676C22#296712yes—
677C22#296812yes—
678C22#296912yes—
679C22#297612yes—
680C22#297711yescharacteristic
681C22#297811yescharacteristic
682C22#297911yescharacteristic
683C22#298011yescharacteristic
684C22#298112yes—
685C4—24no—
686C4—44no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—82no—
692C4—82no—
693C4#253911yescharacteristic
694C4#253911yescharacteristic
695C4#254312yes—
696C4#254312yes—
697C4#259111yescharacteristic
698C4#259111yescharacteristic
699C2—44no—
700C2—84no—
701C2#1415211yescharacteristic
702C2#2119112yes—
703C2#2138112yes—
704C2#2174411yescharacteristic
705C2#2174511yescharacteristic
706C2#2312212yes—
707C2#2312312yes—
708C2#2312412yes—
709C2#2312512yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC2×#392C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1870C2semidirect
3. C2×#1869C2semidirect
4. C2×#1750C2semidirect
5. C2×#1729C2semidirect
6. #11346C2semidirect
7. #10964C2semidirect
8. #10922C2semidirect
9. #10921C2semidirect
108. C2×C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23922
C2×#18702
C2×#18692
C2×#17502
C2×#17292
#113462
#109642
#109222
#109212