Finite Group Databasebeta

← #383856

Subgroups — #383856

Order 512.

2382 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1924C212yesmaximal, complemented
2C2×#1779C212yesmaximal
3C4×#379C211yesmaximal, characteristic
4#11147C212yesmaximal, complemented
5#11042C212yesmaximal, complemented
6#11042C212yesmaximal, complemented
7#10922C211yesmaximal, characteristic, complemented
8#10922C211yesmaximal, characteristic, complemented
9#10922C212yesmaximal, complemented
10C25×C4C2211yescharacteristic
11C22×C4⋊C8C2212yescomplemented
12C22×C22⋊C8C2212yes—
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×#97C2211yescharacteristic, complemented
19C22×#97C2211yescharacteristic, complemented
20C2×#379C411yescharacteristic
21C2×#379C411yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1924—28no—
27#1779—28no—
28C2×#336C2212yescomplemented
29C2×#336C2212yescomplemented
30C2×#336C2212yescomplemented
31C2×#336C2211yescharacteristic, complemented
32C2×#336C2212yescomplemented
33C2×#336C2212yescomplemented
34C2×#336C2211yescharacteristic, complemented
35C22⋊#95C412yes—
36C22⋊#95C412yes—
37C22⋊#95C411yescharacteristic
38C22⋊#95C411yescharacteristic
39C26C2×C411yescharacteristic
40C24×C4—24no—
41C24×C4—24no—
42C24×C4—24no—
43C24×C4—41no—
44C24×C4—41no—
45C24×C4—42no—
46C24×C4—42no—
47C24×C4—42no—
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C412yes—
50C24×C4C2312yes—
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C22×C22⋊C4C2×C412yes—
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2312yes—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—42no—
69C2×C4⋊C8—42no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—42no—
80C2×C22⋊C8—42no—
81C2×C22⋊C8—42no—
82C2×C22⋊C8—42no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—42no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8D418yes—
88C2×#97—24no—
89C2×#97—41no—
90C2×#97—42no—
91C2×#97—41no—
92#379—24no—
93#379—24no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—42no—
97C4×C22⋊C4—41no—
98C4×C22⋊C4—41no—
99C4×C22⋊C4—42no—
100C4×C22⋊C4—42no—
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C412yes—
106C2×#95C2×C412yes—
107C2×#95C2×C412yes—
108#336—24no—
109#336—24no—
110#336—42no—
111#336—42no—
112#336—42no—
113#336—41no—
114#336—41no—
115C25—22no—
116C25—24no—
117C25—22no—
118C25—24no—
119C25—41no—
120C25—42no—
121C25—41no—
122C25—41no—
123C25—42no—
124C25—41no—
125C25M1614yes—
126C25C22×C411yescharacteristic
127C25C22×C412yes—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—41no—
157C23×C4—42no—
158C23×C4—41no—
159C23×C4—44no—
160C23×C4—48no—
161C23×C4—48no—
162C23×C4—48no—
163C23×C4—48no—
164C23×C4—48no—
165C23×C4—44no—
166C23×C4—48no—
167C23×C4—48no—
168C23×C4—48no—
169C23×C4—48no—
170C23×C4—48no—
171C23×C4—48no—
172C23×C4—48no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C412yes—
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C412yes—
182C23×C4C22×C412yes—
183C23×C4C2×D414yes—
184C23×C4#4114yes—
185C23×C4#4114yes—
186C23×C4C2411yescharacteristic
187C22×C8—22no—
188C22×C8—22no—
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×C8C2×D414yes—
214C22×C8C2×D414yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4M1614yes—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—22no—
253C4⋊C8—48no—
254C22⋊C8—48no—
255C22⋊C8—48no—
256C22⋊C8—48no—
257C22⋊C8—48no—
258C22⋊C8—48no—
259C22⋊C8—48no—
260#97—44no—
261#97—44no—
262#95—24no—
263#95—24no—
264#95—42no—
265#95—42no—
266#95—41no—
267#95—41no—
268#95—42no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—21no—
273C24—21no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—21no—
283C24—21no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—28no—
290C24—216no—
291C24—48no—
292C24—44no—
293C24—44no—
294C24—48no—
295C24—44no—
296C24—48no—
297C24—48no—
298C24—48no—
299C24—48no—
300C24—44no—
301C24—44no—
302C24—48no—
303C24—44no—
304C24—48no—
305C24—44no—
306C24—44no—
307C24—48no—
308C24—48no—
309C24—44no—
310C24—48no—
311C24—44no—
312C24#11712yes—
313C24C4×D414yes—
314C24C2×M1612yes—
315C24C2×M1612yes—
316C24C2×M1614yes—
317C24#13114yes—
318C24C23×C411yescharacteristic
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
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—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
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—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—48no—
381C22×C4—44no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—44no—
392C22×C4—432no—
393C22×C4—432no—
394C22×C4—432no—
395C22×C4—432no—
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—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—41no—
422C22×C4#11712yes—
423C22×C4C4×D414yes—
424C22×C4C2×M1614yes—
425C22×C4C22×D412yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C4H(5,𝔽2)12yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)11yescharacteristic
432C22×C4H(5,𝔽2)11yescharacteristic
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
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—
461C22⋊C4—44no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—48no—
469C42—42no—
470C42—42no—
471C42—42no—
472C42—42no—
473C42—41no—
474C42—42no—
475C42—41no—
476C42—42no—
477C42—41no—
478C42—41no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—432no—
510C23—416no—
511C23—432no—
512C23—416no—
513C23—416no—
514C23—416no—
515C23—416no—
516C23—432no—
517C23—432no—
518C23—416no—
519C23—416no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—42no—
526C23—41no—
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—41no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—42no—
546C23#43214yes—
547C23#43518yes—
548C23C2×#11711yescharacteristic
549C23C2×C4×D412yes—
550C23#51811yescharacteristic
551C23#51811yescharacteristic
552C23#51812yes—
553C23C22×M1611yescharacteristic
554C23C22×M1611yescharacteristic
555C23C2×#13112yes—
556C23#56811yescharacteristic
557C23#56811yescharacteristic
558C23#56812yes—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—432no—
568C2×C4—416no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4#53412yes—
620C2×C4#53512yes—
621C2×C4#53812yes—
622C2×C4#53812yes—
623C2×C4#53812yes—
624C2×C4#53812yes—
625C2×C4D4211yescharacteristic
626C2×C4#54612yes—
627C2×C4#54612yes—
628C2×C4#54612yes—
629C2×C4#55112yes—
630C2×C4#55112yes—
631C2×C4#55111yescharacteristic
632C2×C4#56012yes—
633C2×C4#56011yescharacteristic
634C2×C4H(3,𝔽4)11yescharacteristic
635C8—44no—
636C8—44no—
637C8—82no—
638C8—82no—
639C22—28no—
640C22—28no—
641C22—28no—
642C22—28no—
643C22—24no—
644C22—464no—
645C22—44no—
646C22—42no—
647C22—44no—
648C22—42no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—42no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—42no—
661C22—44no—
662C22—42no—
663C22—44no—
664C22—42no—
665C22—44no—
666C22#231211yescharacteristic
667C22#232112yes—
668C22#234212yes—
669C22#235811yescharacteristic
670C22#237511yescharacteristic
671C22C2×#43211yescharacteristic
672C22#292111yescharacteristic
673C22#292412yes—
674C22C2×#43512yes—
675C22#293012yes—
676C22#293312yes—
677C22D4×M1612yes—
678C22#297312yes—
679C22#297311yescharacteristic
680C22#297311yescharacteristic
681C22#297311yescharacteristic
682C22#297412yes—
683C22#297511yescharacteristic
684C22#297612yes—
685C22#297612yes—
686C22#297711yescharacteristic
687C22#297812yes—
688C22#297811yescharacteristic
689C4—44no—
690C4—42no—
691C4—44no—
692C4—81no—
693C4—81no—
694C4—82no—
695C4—81no—
696C4—81no—
697C4#267812yes—
698C4#268412yes—
699C4#268512yes—
700C4#273512yes—
701C2—48no—
702C2—44no—
703C2#1436511yescharacteristic
704C2#2190611yescharacteristic
705C2#2191011yescharacteristic
706C2#2203412yes—
707C2#2203612yes—
708C2#2255412yes—
709C2#2255912yes—
710C2#2284411yescharacteristic
711C2#2285211yescharacteristic
712C2#2330811yescharacteristic
713C2#2330911yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC26C2×C4
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
1. C2×#1924C2semidirect
4. #11147C2semidirect
5. #11042C2semidirect
6. #11042C2semidirect
7. #10922C2semidirect
8. #10922C2semidirect
9. #10922C2semidirect
11. C22×C4⋊C8C22semidirect
18. C22×#97C22semidirect
19. C22×#97C22semidirect
28. C2×#336C22semidirect
29. C2×#336C22semidirect
30. C2×#336C22semidirect
31. C2×#336C22semidirect
32. C2×#336C22semidirect
33. C2×#336C22semidirect
34. C2×#336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#19242
C2×#17792
C4×#3792
#111472
#110422
#109222