Finite Group Databasebeta

← C4×#1999 #358996

Subgroups — C4×#1999 #358996

Order 512.

2852 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D8C211yesmaximal, characteristic, complemented
2C2×C4×#121C212yesmaximal, complemented
3C2×#1999C211yesmaximal, characteristic
4C2×C4×#102C212yesmaximal, complemented
5C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
6#11328C211yesmaximal, characteristic, complemented
7#11096C211yesmaximal, characteristic, complemented
8#11066C212yesmaximal, complemented
9#11022C211yesmaximal, characteristic, complemented
10#10969C212yesmaximal, complemented
11C4×#340C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×#121C2212yes—
14C22×C4×D4—22no—
15C22×C4×D4C2212yes—
16C23×C42C2211yescharacteristic
17C2×C4×D8—24no—
18C2×C4×D8—28no—
19C22×#107C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C4×#121—28no—
28C4×#121—28no—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#388—22no—
32C2×#386—22no—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4—22no—
36#1999C418yesdirect product factor
37C4×#102—28no—
38C4×C22⋊C8—24no—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2211yescharacteristic
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C23×D4—22no—
47C23×D4C2312yes—
48C24×C4—22no—
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4C2311yescharacteristic
52C22×D8—24no—
53C22×D8—24no—
54C22×D8—22no—
55C22×D8—22no—
56C22×D8C2×C414yescomplemented
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×#121—24no—
62C2×#121—24no—
63C2×#121—24no—
64C2×#121—24no—
65C2×#121—24no—
66C2×#121C2×C418yescomplemented
67C2×C4×D4—216no—
68C2×C4×D4—24no—
69C2×C4×D4—24no—
70C2×C4×D4—24no—
71C2×C4×D4—24no—
72C2×C4×D4—24no—
73C2×C4×D4—48no—
74C2×C4×D4—48no—
75C2×C4×D4D418yescomplemented
76C22×C4⋊C4—22no—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C42—22no—
86C22×C42—22no—
87C22×C42—24no—
88C22×C42—24no—
89C22×C42—22no—
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C4×D8—232no—
95C2×#107—22no—
96C2×#107—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102C2×C418yescomplemented
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8C2×C414yescomplemented
112C2×C4×C8—24no—
113C2×C4×C8—22no—
114C2×C4×C8—24no—
115C2×C4×C8D414yescomplemented
116#390—24no—
117#390—24no—
118#389—24no—
119#389—24no—
120#388—216no—
121#386—216no—
122#380—24no—
123#380—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4D418yescomplemented
126C4×C22⋊C4—216no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130#340—22no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340C2×C414yescomplemented
135#336—22no—
136#336—22no—
137C25—22no—
138C25C2×D411yescharacteristic
139C22×D4—28no—
140C22×D4—28no—
141C22×D4—28no—
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—22no—
153C22×D4—44no—
154C22×D4—44no—
155C22×D4—44no—
156C22×D4—44no—
157C22×D4C22×C414yes—
158C22×D4C2×D414yes—
159C22×D4#4114yes—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—28no—
179C23×C4—24no—
180C23×C4—42no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4D814yescomplemented
188C23×C4C22×C412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4#4112yes—
195C23×C4C2411yescharacteristic
196C2×D8—28no—
197C2×D8—216no—
198C2×D8—28no—
199C2×D8—216no—
200C2×D8—28no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8C22×C412yes—
216C22×C8C22×C412yes—
217C22×C8C2×D412yes—
218C22×C8C2×D412yes—
219C22×C8#4112yes—
220C22×C8#4112yes—
221#121—216no—
222#121—216no—
223#121—216no—
224#121—216no—
225C4×D4—216no—
226C4×D4—216no—
227C4×D4—464no—
228C4×D4—48no—
229C4×D4—48no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C22×C414yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4#4114yes—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C42—28no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—21no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42—48no—
279C2×C42—44no—
280C2×C42C2×D414yes—
281C2×C42C2×D412yes—
282C2×C42C2×D414yes—
283C2×C42#4112yes—
284#107—28no—
285#102—216no—
286#102—216no—
287#102—216no—
288C22⋊C8—28no—
289C22⋊C8—28no—
290C4×C8—28no—
291C4×C8—44no—
292#95—28no—
293#95—28no—
294#95—28no—
295C24—22no—
296C24—21no—
297C24—22no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—22no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24C4×D412yes—
319C24C2×D812yes—
320C24#13512yes—
321C24C22×D411yescharacteristic
322C2×D4—232no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—416no—
339C2×D4—416no—
340C2×D4—416no—
341C2×D4—416no—
342C2×D4—416no—
343C2×D4—416no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4C4×D418yescomplemented
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—216no—
385C22×C4—28no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—44no—
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×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—416no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4C4×D412yes—
429C22×C4#12014yes—
430C22×C4#12014yes—
431C22×C4#12112yes—
432C22×C4#12112yes—
433C22×C4C2×D812yes—
434C22×C4C2×D812yes—
435C22×C4#13512yes—
436C22×C4#13512yes—
437C22×C4AGL(1,ℤ/8)14yes—
438C22×C4C23×C411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4112yes—
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448D8—232no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8C4×D414yescomplemented
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—416no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4C4×D418yescomplemented
482C22⋊C4—232no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—416no—
487C42—41no—
488C42—42no—
489C42—41no—
490C42—44no—
491C42—44no—
492C42—416no—
493C42#12014yes—
494C42#12118yes—
495C42#12714yes—
496C23—24no—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—28no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—82no—
544C23—82no—
545C23—82no—
546C23—82no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23—82no—
552C23—82no—
553C23C4×D814yescomplemented
554C23C2×C4×D411yescharacteristic
555C23C2×#12012yes—
556C23C2×#12111yescharacteristic
557C23#52512yes—
558C23#53211yescharacteristic
559C23C22×D811yescharacteristic
560C23C2×#13511yescharacteristic
561C23C2×#13511yescharacteristic
562C23C2×#13511yescharacteristic
563C23C2×AGL(1,ℤ/8)12yes—
564C23#57512yes—
565D4—28no—
566D4—28no—
567D4—464no—
568D4—44no—
569D4—44no—
570D4—44no—
571D4—44no—
572D4—44no—
573D4—44no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—416no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—416no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—832no—
623C2×C4#39214yes—
624C2×C4#39214yes—
625C2×C4#44718yes—
626C2×C4#44918yes—
627C2×C4#46614yes—
628C2×C4#46914yes—
629C2×C4C2×C4×D411yescharacteristic
630C2×C4C2×C4×D412yes—
631C2×C4C2×C4×D412yes—
632C2×C4C4×#4111yescharacteristic
633C2×C4C2×#12011yescharacteristic
634C2×C4C2×#12112yes—
635C2×C4#52512yes—
636C2×C4#52511yescharacteristic
637C2×C4#52512yes—
638C2×C4#52512yes—
639C2×C4#52511yescharacteristic
640C2×C4#52512yes—
641C2×C4#52911yescharacteristic
642C2×C4#52912yes—
643C2×C4C2×#12711yescharacteristic
644C2×C4#53211yescharacteristic
645C2×C4#53212yes—
646C2×C4#53212yes—
647C8—24no—
648C8—42no—
649C8—42no—
650C8—42no—
651C22—24no—
652C22—28no—
653C22—44no—
654C22—44no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22—48no—
663C22—48no—
664C22—48no—
665C22—84no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22C4×#12012yes—
678C22C4×#12111yescharacteristic
679C22C2×#39211yescharacteristic
680C22#238812yes—
681C22#239211yescharacteristic
682C22C2×C4×D811yescharacteristic
683C22C4×#13511yescharacteristic
684C22C4×AGL(1,ℤ/8)12yes—
685C22C2×#44712yes—
686C22C2×#44912yes—
687C22#300212yes—
688C22#300212yes—
689C22C2×#46611yescharacteristic
690C22#305011yescharacteristic
691C22C2×#46911yescharacteristic
692C22#305411yescharacteristic
693C22#310012yes—
694C22#310212yes—
695C22#310512yes—
696C22#310612yes—
697C22#316811yescharacteristic
698C22#316911yescharacteristic
699C22#317112yes—
700C4—24no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—48no—
705C4—84no—
706C4—84no—
707C4#199918yesdirect product factor
708C4C4×#12011yescharacteristic
709C4C4×#12112yes—
710C4C4×#12711yescharacteristic
711C4#238811yescharacteristic
712C4#239212yes—
713C4#239711yescharacteristic
714C2—44no—
715C2—88no—
716C2C4×#39211yescharacteristic
717C2C4×#44712yes—
718C2C4×#44912yes—
719C2C4×#46611yescharacteristic
720C2C4×#46911yescharacteristic
721C2C2×#199911yescharacteristic
722C2#2338611yescharacteristic
723C2#2356712yes—
724C2#2356912yes—
725C2#2369611yescharacteristic
726C2#2369711yescharacteristic
subgroupG/H
centerC23×C4C2×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC24C22×D4
prime radicalC2×#1999C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4×D8C2semidirect
2. C2×C4×#121C2semidirect
4. C2×C4×#102C2semidirect
5. C2×C4×C22⋊C8C2semidirect
6. #11328C2semidirect
7. #11096C2semidirect
8. #11066C2semidirect
9. #11022C2semidirect
10. #10969C2semidirect
11. C4×#340C2semidirect
36. #1999C4direct (N × H)
56. C22×D8C2×C4semidirect
66. C2×#121C2×C4semidirect
75. C2×C4×D4D4semidirect
105. C2×#102C2×C4semidirect
111. C2×C22⋊C8C2×C4semidirect
115. C2×C4×C8D4semidirect
125. C4×C4⋊C4D4semidirect
134. #340C2×C4semidirect
187. C23×C4D8semidirect
356. C2×D4C4×D4semidirect
468. C2×C8C4×D4semidirect
481. C4⋊C4C4×D4semidirect
553. C23C4×D8semidirect
707. C4#1999direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D82
C2×C4×#1212
C2×#19992
C2×C4×#1022
C2×C4×C22⋊C82
#113282
#110962
#110662
#110222
#109692
C4×#3402