Finite Group Databasebeta

← C2×#10761 #353329

Subgroups — C2×#10761 #353329

Order 512.

6754 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D4C211yesmaximal, characteristic
2C23×#99C212yesmaximal
3C22×#342C212yesmaximal, complemented
4C2×C22⋊#95C212yesmaximal, complemented
5#10761C218yesmaximal, direct product factor
6C24×D4C2211yescharacteristic
7C25×C4C411yescharacteristic, complemented
8C25×C4C2211yescharacteristic
9C22×C4×D4—28no—
10C22×C4×D4—24no—
11C22×C4×D4—24no—
12C22×C4×D4—28no—
13C22×C4×D4C414yescomplemented
14C22×C4×D4C2214yes—
15C23×C4⋊C4C411yescharacteristic, complemented
16C23×C22⋊C4C411yescharacteristic, complemented
17C23×C22⋊C4C2214yes—
18C23×C22⋊C4C2211yescharacteristic
19C23×C42C411yescharacteristic, complemented
20C22×#99—28no—
21C22×#99—28no—
22C22×#99C4116yescomplemented
23C22×#99C2218yes—
24C22×#95—24no—
25C2×#342—28no—
26C2×#342C2218yescomplemented
27C22⋊#95—24no—
28C22⋊#95C2218yescomplemented
29C26C2×C411yescharacteristic
30C26C2311yescharacteristic
31C23×D4—22no—
32C23×D4—24no—
33C23×D4—22no—
34C23×D4—24no—
35C23×D4—24no—
36C23×D4C2×C418yes—
37C23×D4C2×C412yes—
38C23×D4D418yes—
39C23×D4D414yes—
40C23×D4Q814yes—
41C23×D4C2312yes—
42C24×C4—24no—
43C24×C4—24no—
44C24×C4—24no—
45C24×C4—24no—
46C24×C4—22no—
47C24×C4—22no—
48C24×C4—24no—
49C24×C4—48no—
50C24×C4—48no—
51C24×C4C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4C2×C411yescharacteristic
54C24×C4C2×C411yescharacteristic
55C24×C4C2×C412yes—
56C24×C4C2×C411yescharacteristic
57C24×C4C2×C414yescomplemented
58C24×C4D414yes—
59C24×C4D414yes—
60C24×C4C2312yes—
61C2×C4×D4—264no—
62C2×C4×D4—264no—
63C2×C4×D4—216no—
64C2×C4×D4—216no—
65C2×C4×D4—216no—
66C2×C4×D4—216no—
67C2×C4×D4—216no—
68C2×C4×D4—216no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4C2×C414yescomplemented
74C22×C22⋊C4—24no—
75C22×C22⋊C4—24no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—24no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—28no—
82C22×C22⋊C4—28no—
83C22×C22⋊C4—28no—
84C22×C22⋊C4—28no—
85C22×C22⋊C4—24no—
86C22×C22⋊C4—28no—
87C22×C22⋊C4C2×C412yes—
88C22×C22⋊C4C2×C414yescomplemented
89C22×C22⋊C4C2×C4116yescomplemented
90C22×C22⋊C4D418yes—
91C22×C22⋊C4C2318yes—
92C22×C22⋊C4C2312yes—
93C22×C42—24no—
94C22×C42—22no—
95C22×C42—24no—
96C22×C42—22no—
97C22×C42C2×C414yescomplemented
98C2×#99—264no—
99C2×#99—216no—
100C2×#99—216no—
101C2×#99C2×C4132yescomplemented
102C2×#95—216no—
103C2×#95—416no—
104#342—216no—
105C25—22no—
106C25—21no—
107C25—21no—
108C25—22no—
109C25—22no—
110C25—44no—
111C25—44no—
112C25—44no—
113C25—44no—
114C25C4214yes—
115C25C22⋊C414yes—
116C25C4⋊C414yes—
117C25C22×C411yescharacteristic
118C25C22×C412yes—
119C25C22×C411yescharacteristic
120C25C22×C414yes—
121C25C2×D412yes—
122C25C2×D414yes—
123C25C2×D412yes—
124C25C2×D412yes—
125C25C2×D414yes—
126C25C2×Q812yes—
127C25#4114yes—
128C25#4114yes—
129C25C2411yescharacteristic
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—24no—
134C22×D4—216no—
135C22×D4—216no—
136C22×D4—232no—
137C22×D4—216no—
138C22×D4—216no—
139C22×D4—28no—
140C22×D4—28no—
141C22×D4—216no—
142C22×D4—28no—
143C22×D4—28no—
144C22×D4—216no—
145C22×D4—232no—
146C22×D4—24no—
147C22×D4—24no—
148C22×D4—24no—
149C22×D4—24no—
150C22×D4C4218yes—
151C22×D4C22⋊C4132yes—
152C22×D4C22⋊C418yes—
153C22×D4C4⋊C4132yes—
154C22×D4C4⋊C418yes—
155C22×D4C22×C418yes—
156C22×D4C2×D418yes—
157C22×D4C2×D414yes—
158C22×D4C2×Q814yes—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—28no—
167C23×C4—216no—
168C23×C4—216no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—28no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—216no—
181C23×C4—216no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—216no—
185C23×C4—28no—
186C23×C4—28no—
187C23×C4—216no—
188C23×C4—28no—
189C23×C4—48no—
190C23×C4—416no—
191C23×C4—48no—
192C23×C4—44no—
193C23×C4—48no—
194C23×C4—432no—
195C23×C4—432no—
196C23×C4—432no—
197C23×C4—432no—
198C23×C4—48no—
199C23×C4—416no—
200C23×C4—416no—
201C23×C4—48no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—416no—
205C23×C4—416no—
206C23×C4—48no—
207C23×C4—48no—
208C23×C4C4214yes—
209C23×C4C22⋊C412yes—
210C23×C4C22⋊C412yes—
211C23×C4C22⋊C418yes—
212C23×C4C22⋊C414yes—
213C23×C4C22⋊C412yes—
214C23×C4C22⋊C414yes—
215C23×C4C22⋊C412yes—
216C23×C4C22⋊C414yes—
217C23×C4C4⋊C414yes—
218C23×C4C22×C412yes—
219C23×C4C22×C412yes—
220C23×C4C22×C412yes—
221C23×C4C22×C412yes—
222C23×C4C22×C412yes—
223C23×C4C2×D418yes—
224C4×D4—2128no—
225C4×D4—2128no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—216no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C22⋊C4—216no—
232C2×C22⋊C4—216no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—216no—
237C2×C22⋊C4—216no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—232no—
243C2×C22⋊C4—232no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—48no—
251C2×C22⋊C4—416no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—416no—
254C2×C22⋊C4C22⋊C418yes—
255C2×C22⋊C4C22×C4116yescomplemented
256C2×C22⋊C4C2×D418yes—
257C2×C42—216no—
258C2×C42—216no—
259C2×C42—28no—
260C2×C42—28no—
261C2×C42—28no—
262#99—264no—
263#95—432no—
264C24—28no—
265C24—24no—
266C24—28no—
267C24—24no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—24no—
273C24—28no—
274C24—24no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—28no—
279C24—24no—
280C24—28no—
281C24—42no—
282C24—44no—
283C24—48no—
284C24—48no—
285C24—42no—
286C24—42no—
287C24—416no—
288C24—48no—
289C24—44no—
290C24—48no—
291C24—48no—
292C24—48no—
293C24—416no—
294C24—416no—
295C24—42no—
296C24—48no—
297C24—44no—
298C24—416no—
299C24—48no—
300C24—416no—
301C24—44no—
302C24—44no—
303C24—48no—
304C24—416no—
305C24—44no—
306C24—416no—
307C24—48no—
308C24—44no—
309C24—44no—
310C24#95116yescomplemented
311C24C2×C4212yes—
312C24C2×C4214yes—
313C24C2×C4212yes—
314C24C2×C22⋊C412yes—
315C24C2×C22⋊C412yes—
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C412yes—
318C24C2×C22⋊C414yes—
319C24C2×C22⋊C412yes—
320C24C2×C22⋊C418yes—
321C24C2×C22⋊C412yes—
322C24C2×C22⋊C411yescharacteristic
323C24C2×C4⋊C412yes—
324C24C2×C4⋊C414yes—
325C24C2×C4⋊C412yes—
326C24C2×C4⋊C418yes—
327C24#11712yes—
328C24#11712yes—
329C24#11712yes—
330C24#11712yes—
331C24C4×D418yes—
332C24C4×D418yes—
333C24C4×D4116yes—
334C24#12018yes—
335C24#12018yes—
336C24#12114yes—
337C24#12114yes—
338C24#12118yes—
339C24C22⋊Q814yes—
340C24C22⋊Q814yes—
341C24C22⋊Q818yes—
342C24#123116yes—
343C24C23×C411yescharacteristic
344C24C23×C412yes—
345C24C22×D412yes—
346C24C22×D412yes—
347C24C22×D411yescharacteristic
348C24C22×D412yes—
349C24C22×Q811yescharacteristic
350C24C2×#4112yes—
351C24C2×#4112yes—
352C2×D4—216no—
353C2×D4—216no—
354C2×D4—216no—
355C2×D4—28no—
356C2×D4—28no—
357C2×D4—28no—
358C2×D4—28no—
359C2×D4—216no—
360C2×D4—216no—
361C2×D4—216no—
362C2×D4—216no—
363C2×D4—216no—
364C2×D4—216no—
365C2×D4—216no—
366C2×D4—264no—
367C2×D4—264no—
368C2×D4—264no—
369C2×D4—264no—
370C2×D4#95132yescomplemented
371C2×D4C2×C22⋊C4116yes—
372C2×D4C2×C4⋊C4116yes—
373C22×C4—216no—
374C22×C4—216no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—232no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—216no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—216no—
410C22×C4—28no—
411C22×C4—216no—
412C22×C4—216no—
413C22×C4—216no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—464no—
419C22×C4—416no—
420C22×C4—416no—
421C22×C4—464no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—432no—
425C22×C4—464no—
426C22×C4—432no—
427C22×C4—416no—
428C22×C4—464no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—416no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—464no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—432no—
448C22×C4—832no—
449C22×C4—816no—
450C22×C4—816no—
451C22×C4#95116yescomplemented
452C22×C4#9918yescomplemented
453C22×C4C2×C4214yes—
454C22×C4C2×C22⋊C414yes—
455C22×C4C2×C22⋊C414yes—
456C22×C4C2×C22⋊C414yes—
457C22×C4C2×C22⋊C414yes—
458C22×C4C2×C22⋊C414yes—
459C22×C4C2×C4⋊C414yes—
460C4⋊C4—232no—
461C22⋊C4—232no—
462C22⋊C4—216no—
463C22⋊C4—216no—
464C22⋊C4—216no—
465C22⋊C4—216no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—432no—
469C42—232no—
470C23—216no—
471C23—28no—
472C23—28no—
473C23—216no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—232no—
483C23—28no—
484C23—28no—
485C23—24no—
486C23—24no—
487C23—216no—
488C23—24no—
489C23—24no—
490C23—216no—
491C23—216no—
492C23—216no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—42no—
500C23—42no—
501C23—48no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—416no—
506C23—42no—
507C23—42no—
508C23—416no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—432no—
513C23—416no—
514C23—48no—
515C23—416no—
516C23—48no—
517C23—416no—
518C23—48no—
519C23—432no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—432no—
529C23—48no—
530C23—44no—
531C23—48no—
532C23—44no—
533C23—48no—
534C23—432no—
535C23—432no—
536C23—432no—
537C23—44no—
538C23—48no—
539C23—48no—
540C23—416no—
541C23—48no—
542C23—416no—
543C23—416no—
544C23—44no—
545C23—44no—
546C23—416no—
547C23—48no—
548C23—48no—
549C23C2×#9514yes—
550C23C2×#9514yes—
551C23C2×#9518yescomplemented
552C23C4×C22⋊C414yes—
553C23C4×C22⋊C412yes—
554C23C4×C22⋊C414yes—
555C23C4×C22⋊C414yes—
556C23C4×C22⋊C412yes—
557C23#37914yes—
558C23#37912yes—
559C23#37912yes—
560C23#38014yes—
561C23#38012yes—
562C23#38014yes—
563C23#38014yes—
564C23#38012yes—
565C23#38114yes—
566C23#38114yes—
567C23#385116yes—
568C23#385116yes—
569C23#386116yes—
570C23#386116yes—
571C23C2×#9914yes—
572C23#41014yes—
573C23C22×C4211yescharacteristic
574C23C22×C22⋊C411yescharacteristic
575C23C22×C22⋊C411yescharacteristic
576C23C22×C22⋊C412yes—
577C23C22×C22⋊C411yescharacteristic
578C23C22×C4⋊C411yescharacteristic
579C23C22×C4⋊C412yes—
580C23C2×#11711yescharacteristic
581C23C2×#11711yescharacteristic
582C23C2×C4×D414yes—
583C23C2×C4×D414yes—
584C23C2×#12014yes—
585C23C2×#12112yes—
586C23C2×#12112yes—
587C23C2×C22⋊Q812yes—
588C23C2×C22⋊Q812yes—
589C23C2×#12314yes—
590D4—264no—
591D4—232no—
592D4—232no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—216no—
596C2×C4—24no—
597C2×C4—216no—
598C2×C4—216no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—216no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—432no—
612C2×C4—432no—
613C2×C4—464no—
614C2×C4—44no—
615C2×C4—48no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—44no—
620C2×C4—48no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—432no—
629C2×C4—416no—
630C2×C4—864no—
631C2×C4—88no—
632C2×C4—88no—
633C2×C4—816no—
634C2×C4C2×#9518yescomplemented
635C2×C4C2×#9918yescomplemented
636C22—232no—
637C22—28no—
638C22—28no—
639C22—28no—
640C22—28no—
641C22—232no—
642C22—21no—
643C22—21no—
644C22—22no—
645C22—24no—
646C22—24no—
647C22—21no—
648C22—21no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—22no—
654C22—22no—
655C22—24no—
656C22—24no—
657C22—28no—
658C22—28no—
659C22—22no—
660C22—24no—
661C22—22no—
662C22—28no—
663C22—28no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22—42no—
668C22—42no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—416no—
675C22—416no—
676C22—48no—
677C22—432no—
678C22—48no—
679C22—48no—
680C22—48no—
681C22—432no—
682C22—44no—
683C22—44no—
684C22—48no—
685C22—48no—
686C22C22⋊#9514yes—
687C22C22⋊#9518yes—
688C22C22⋊#9514yes—
689C22C4×#9918yes—
690C22#178114yes—
691C22#178214yes—
692C22#179918yes—
693C22C22×#9511yescharacteristic
694C22C2×C4×C22⋊C411yescharacteristic
695C22C2×C4×C22⋊C411yescharacteristic
696C22C2×#37911yescharacteristic
697C22C2×#38011yescharacteristic
698C22C2×#38011yescharacteristic
699C22C2×#38111yescharacteristic
700C22C2×#38514yes—
701C22C2×#38614yes—
702C22C22×#9912yes—
703C22C2×#41012yes—
704C4—28no—
705C4—28no—
706C4—44no—
707C4—44no—
708C4—44no—
709C4—816no—
710C2—24no—
711C2—24no—
712C2—21no—
713C2—21no—
714C2—21no—
715C2—21no—
716C2—28no—
717C2—24no—
718C2—48no—
719C2#1076118yesdirect product factor
720C2C2×C22⋊#9511yescharacteristic
721C2C2×C4×#9912yes—
722C2C2×#178111yescharacteristic
723C2C2×#178211yescharacteristic
724C2C2×#179912yes—
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC24×D4C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#342C2semidirect
4. C2×C22⋊#95C2semidirect
5. #10761C2direct (N × H)
7. C25×C4C4semidirect
13. C22×C4×D4C4semidirect
15. C23×C4⋊C4C4semidirect
16. C23×C22⋊C4C4semidirect
19. C23×C42C4semidirect
22. C22×#99C4semidirect
26. C2×#342C22semidirect
28. C22⋊#95C22semidirect
57. C24×C4C2×C4semidirect
73. C22×C4⋊C4C2×C4semidirect
88. C22×C22⋊C4C2×C4semidirect
89. C22×C22⋊C4C2×C4semidirect
97. C22×C42C2×C4semidirect
101. C2×#99C2×C4semidirect
255. C2×C22⋊C4C22×C4semidirect
310. C24#95semidirect
370. C2×D4#95semidirect
451. C22×C4#95semidirect
452. C22×C4#99semidirect
551. C23C2×#95semidirect
634. C2×C4C2×#95semidirect
635. C2×C4C2×#99semidirect
719. C2#10761direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×D42
C23×#992
C22×#3422
C2×C22⋊#952
#107612