Finite Group Databasebeta

← C4×#2041 #359082

Subgroups — C4×#2041 #359082

Order 512.

2262 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#127C211yesmaximal, characteristic
2C2×#2041C211yesmaximal, characteristic
3C2×C4×#102C212yesmaximal, complemented
4C2×C4×#102C212yesmaximal, complemented
5C4×#389C211yesmaximal, characteristic, complemented
6#11267C211yesmaximal, characteristic, complemented
7#11207C211yesmaximal, characteristic, complemented
8#11189C212yesmaximal, complemented
9#11066C212yesmaximal, complemented
10#11008C211yesmaximal, characteristic
11C4×#336C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4—21no—
15C22×C4×D4C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C4×C8C2212yes—
21C4×#127—28no—
22C2×#390—21no—
23C2×#390—21no—
24C2×#390C2212yes—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#2041C418yesdirect product factor
33C4×#102—28no—
34C4×#102—28no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1436—21no—
43#1436—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C4×#95—21no—
47C23×D4—21no—
48C23×D4—21no—
49C23×D4C2312yes—
50C24×C4—22no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#127—24no—
54C2×#127—24no—
55C2×#127—22no—
56C2×#127C2×C414yes—
57C2×C4×D4—28no—
58C2×C4×D4—28no—
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—44no—
65C2×C4×D4—44no—
66C2×C4×D4—44no—
67C2×C4×D4—44no—
68C2×C4×D4D418yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102C2×C418yescomplemented
101C2×#102C2×C418yescomplemented
102C2×C4×C8—24no—
103C2×C4×C8—24no—
104C2×C4×C8—24no—
105#390—28no—
106#390—28no—
107#390—24no—
108#390—24no—
109#389—22no—
110#389—22no—
111#389—22no—
112#389C2×C414yescomplemented
113#384—24no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—44no—
117C4×C4⋊C4D414yescomplemented
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C43—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#336—24no—
132#336—22no—
133#336C2×C414yescomplemented
134C25—22no—
135C22×D4—24no—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4C22×C414yes—
161C22×D4C2×D414yes—
162C22×D4#4114yes—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—28no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C22×C8—22no—
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×C414yes—
195C22×C8#4114yes—
196#127—216no—
197#127—216no—
198C4×D4—216no—
199C4×D4—216no—
200C4×D4—432no—
201C4×D4—432no—
202C4×D4—48no—
203C4×D4—48no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4#4112yes—
239C2×C4⋊C4#4112yes—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—41no—
265C2×C42—41no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42D814yes—
272C2×C42SD1614yes—
273C2×C42C22×C412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D414yes—
276C2×C42#4112yes—
277C2×C42#4114yes—
278C2×C42#4112yes—
279#102—216no—
280#102—216no—
281#102—216no—
282#102—216no—
283C4×C8—48no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—42no—
288#95—42no—
289#95—42no—
290#95—42no—
291#95—42no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24C22×D411yescharacteristic
308C2×D4—216no—
309C2×D4—216no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4C4×D418yes—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—416no—
410C22×C4—416no—
411C22×C4—416no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4C4×D412yes—
417C22×C4C4×D412yes—
418C22×C4#12012yes—
419C22×C4#12012yes—
420C22×C4#12112yes—
421C22×C4#12112yes—
422C22×C4#12412yes—
423C22×C4#12412yes—
424C22×C4C2×D812yes—
425C22×C4C2×D812yes—
426C22×C4C2×D812yes—
427C22×C4C2×SD1612yes—
428C22×C4C2×SD1612yes—
429C22×C4C2×SD1612yes—
430C22×C4#13512yes—
431C22×C4#13512yes—
432C22×C4AGL(1,ℤ/8)12yes—
433C22×C4AGL(1,ℤ/8)12yes—
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4112yes—
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4112yes—
443C22×C4C2×#4111yescharacteristic
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—48no—
475C4⋊C4C4×D414yescomplemented
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C42—24no—
480C42—24no—
481C42—28no—
482C42—28no—
483C42—44no—
484C42—44no—
485C42—42no—
486C42—42no—
487C42—48no—
488C42—84no—
489C42#12118yes—
490C42#12318yes—
491C23—28no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—82no—
515C23—82no—
516C23—82no—
517C23—82no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23—82no—
526C23C2×C4×D411yescharacteristic
527C23C2×#12011yescharacteristic
528C23C2×#12111yescharacteristic
529C23#52511yescharacteristic
530C23#52511yescharacteristic
531C23C2×#12411yescharacteristic
532C23#53211yescharacteristic
533C23C22×D811yescharacteristic
534C23C22×SD1611yescharacteristic
535C23C2×#13511yescharacteristic
536C23C2×#13511yescharacteristic
537C23C2×AGL(1,ℤ/8)11yescharacteristic
538C23C2×AGL(1,ℤ/8)11yescharacteristic
539C23#57511yescharacteristic
540C23#57511yescharacteristic
541D4—28no—
542D4—28no—
543D4—432no—
544D4—432no—
545D4—44no—
546D4—44no—
547D4—44no—
548D4—44no—
549D4—44no—
550D4—44no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—832no—
619C2×C4#39414yes—
620C2×C4#39414yes—
621C2×C4C4×D814yes—
622C2×C4C4×SD1614yes—
623C2×C4#44714yes—
624C2×C4#45014yes—
625C2×C4#45914yes—
626C2×C4#46014yes—
627C2×C4#48614yes—
628C2×C4#49014yes—
629C2×C4C2×C4×D411yescharacteristic
630C2×C4C2×C4×D412yes—
631C2×C4C4×#4112yes—
632C2×C4C4×#4111yescharacteristic
633C2×C4C2×#12112yes—
634C2×C4C2×#12312yes—
635C2×C4#52512yes—
636C2×C4#52512yes—
637C2×C4#52512yes—
638C2×C4#52912yes—
639C2×C4#52912yes—
640C2×C4#52911yescharacteristic
641C2×C4#52912yes—
642C2×C4#52912yes—
643C2×C4#53211yescharacteristic
644C2×C4#53212yes—
645C8—44no—
646C8—44no—
647C22—48no—
648C22—48no—
649C22—48no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22C4×#12011yescharacteristic
663C22C4×#12111yescharacteristic
664C22C4×#12411yescharacteristic
665C22C2×#39411yescharacteristic
666C22#238811yescharacteristic
667C22#239211yescharacteristic
668C22#239811yescharacteristic
669C22C2×C4×D811yescharacteristic
670C22C2×C4×SD1611yescharacteristic
671C22C4×AGL(1,ℤ/8)11yescharacteristic
672C22C4×AGL(1,ℤ/8)11yescharacteristic
673C22C2×#44711yescharacteristic
674C22C2×#45011yescharacteristic
675C22#300211yescharacteristic
676C22#300211yescharacteristic
677C22C2×#45911yescharacteristic
678C22C2×#46011yescharacteristic
679C22#303511yescharacteristic
680C22#303511yescharacteristic
681C22#310011yescharacteristic
682C22#310011yescharacteristic
683C22#310111yescharacteristic
684C22#310311yescharacteristic
685C22#310511yescharacteristic
686C22#310511yescharacteristic
687C22#310611yescharacteristic
688C22#310611yescharacteristic
689C22C2×#48611yescharacteristic
690C22#313111yescharacteristic
691C22C2×#49011yescharacteristic
692C22#313511yescharacteristic
693C22#316711yescharacteristic
694C22#316911yescharacteristic
695C22#317111yescharacteristic
696C22#317111yescharacteristic
697C4—24no—
698C4—24no—
699C4—24no—
700C4—42no—
701C4—42no—
702C4—44no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—84no—
707C4#204118yesdirect product factor
708C4C4×#12112yes—
709C4C4×#12312yes—
710C4#239212yes—
711C4#239512yes—
712C2—88no—
713C2C4×#39411yescharacteristic
714C2C4×#44711yescharacteristic
715C2C4×#45011yescharacteristic
716C2C4×#45911yescharacteristic
717C2C4×#46011yescharacteristic
718C2C4×#48611yescharacteristic
719C2C4×#49011yescharacteristic
720C2C2×#204111yescharacteristic
721C2#2350411yescharacteristic
722C2#2356711yescharacteristic
723C2#2357011yescharacteristic
724C2#2367611yescharacteristic
725C2#2367811yescharacteristic
726C2#2381511yescharacteristic
727C2#2381911yescharacteristic
subgroupG/H
centerC23×C4C2×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC24C22×D4
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C4×#102C2semidirect
4. C2×C4×#102C2semidirect
5. C4×#389C2semidirect
6. #11267C2semidirect
7. #11207C2semidirect
8. #11189C2semidirect
9. #11066C2semidirect
11. C4×#336C2semidirect
32. #2041C4direct (N × H)
100. C2×#102C2×C4semidirect
101. C2×#102C2×C4semidirect
112. #389C2×C4semidirect
117. C4×C4⋊C4D4semidirect
133. #336C2×C4semidirect
475. C4⋊C4C4×D4semidirect
707. C4#2041direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#1272
C2×#20412
C2×C4×#1022
C4×#3892
#112672
#112072
#111892
#110662
#110082
C4×#3362