Finite Group Databasebeta

← D4×#342 #356489

Subgroups — D4×#342 #356489

Order 512.

4168 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C22⋊C4C211yesmaximal, characteristic
2C2×D4×C22⋊C4C212yesmaximal
3C22×#342C212yesmaximal, complemented
4#10854C212yesmaximal, complemented
5#10846C211yesmaximal, characteristic, complemented
6#10735C214yesmaximal, complemented
7#10725C212yesmaximal, complemented
8C4×#342C211yesmaximal, characteristic, complemented
9C24×D4C2211yescharacteristic
10C22×C4×D4—22no—
11C22×C4×D4C412yescomplemented
12C23×C22⋊C4C2212yes—
13C23×C22⋊C4C2214yes—
14D4×C22⋊C4—22no—
15D4×C22⋊C4—24no—
16D4×C22⋊C4—22no—
17D4×C22⋊C4—28no—
18D4×C22⋊C4C4116yescomplemented
19C2×#390—22no—
20C2×#390C412yescomplemented
21C2×#386—24no—
22C2×#386C414yescomplemented
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25C2×#379C2214yes—
26C2×#379C2212yes—
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×#342—28no—
30C2×#342C2212yes—
31C2×#342C2214yes—
32C2×#342C2211yescharacteristic
33C2×#342C2212yes—
34C2×#342C2212yes—
35C2×#342C2214yes—
36C2×#342C2211yescharacteristic
37C26C2312yes—
38C23×D4—48no—
39C23×D4C2×C411yescharacteristic
40C23×D4C2×C418yes—
41C23×D4C2×C412yes—
42C23×D4C2×C411yescharacteristic
43C23×D4D412yes—
44C23×D4D414yes—
45C23×D4D414yes—
46C23×D4D412yes—
47C23×D4Q814yes—
48C24×C4—24no—
49C24×C4C2×C414yescomplemented
50C24×C4C2311yescharacteristic
51C2×C4×D4—24no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—216no—
55C2×C4×D4—416no—
56C2×C4×D4—48no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2×C412yescomplemented
59C22×C22⋊C4—28no—
60C22×C22⋊C4—28no—
61C22×C22⋊C4—24no—
62C22×C22⋊C4—24no—
63C22×C22⋊C4—28no—
64C22×C22⋊C4—24no—
65C22×C22⋊C4—24no—
66C22×C22⋊C4—24no—
67C22×C22⋊C4—24no—
68C22×C22⋊C4—24no—
69C22×C22⋊C4—44no—
70C22×C22⋊C4—44no—
71C22×C22⋊C4—44no—
72C22×C22⋊C4C2×C414yescomplemented
73C22×C22⋊C4C2×C4116yescomplemented
74C22×C22⋊C4C2×C414yescomplemented
75C22×C22⋊C4C2×C414yescomplemented
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2311yescharacteristic
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2311yescharacteristic
81C22×C22⋊C4C2312yes—
82C22×C42—22no—
83C22×C42C2×C412yescomplemented
84#390—24no—
85#390—24no—
86#390—216no—
87#386—28no—
88#386—28no—
89#386—232no—
90#380—24no—
91#380—22no—
92#380—22no—
93#380C2×C418yescomplemented
94#379—28no—
95#379—24no—
96#379—24no—
97#379C2×C4116yescomplemented
98C4×C22⋊C4—24no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4C2×C418yescomplemented
102C2×#95—22no—
103C2×#95—22no—
104C2×#95C2×C412yescomplemented
105C2×#95C2×C412yescomplemented
106#342—24no—
107#342—28no—
108#342—22no—
109#342—24no—
110#342—42no—
111#342—44no—
112#342D414yesdirect product factor
113C25—22no—
114C25—22no—
115C25—24no—
116C25—24no—
117C25—22no—
118C25—22no—
119C25—48no—
120C25—84no—
121C25C22×C412yes—
122C25C22×C412yes—
123C25C22×C418yes—
124C25C2×D414yes—
125C25C2×D414yes—
126C25C2×D414yes—
127C25C2×Q814yes—
128C25C2411yescharacteristic
129C22×D4—28no—
130C22×D4—216no—
131C22×D4—44no—
132C22×D4—48no—
133C22×D4—432no—
134C22×D4—416no—
135C22×D4—416no—
136C22×D4—48no—
137C22×D4—416no—
138C22×D4C4214yes—
139C22×D4C4214yes—
140C22×D4C4218yes—
141C22×D4C22⋊C418yes—
142C22×D4C22⋊C412yes—
143C22×D4C22⋊C411yescharacteristic
144C22×D4C22⋊C414yes—
145C22×D4C22⋊C412yes—
146C22×D4C22⋊C414yes—
147C22×D4C22⋊C4116yes—
148C22×D4C22⋊C418yes—
149C22×D4C22⋊C412yes—
150C22×D4C22⋊C411yescharacteristic
151C22×D4C4⋊C414yes—
152C22×D4C4⋊C418yes—
153C22×D4C4⋊C418yes—
154C22×D4C4⋊C414yes—
155C22×D4C4⋊C4116yes—
156C22×D4C4⋊C418yes—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—216no—
163C23×C4—28no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—48no—
167C23×C4—48no—
168C23×C4—416no—
169C23×C4—48no—
170C23×C4—88no—
171C23×C4—84no—
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C412yes—
174C23×C4C22×C414yes—
175C23×C4C22×C412yes—
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C414yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×Q812yes—
182C4×D4—464no—
183C4×D4—416no—
184C4×D4—48no—
185C4×D4—48no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4—44no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—232no—
205C2×C22⋊C4—216no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—216no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—216no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—416no—
224C2×C22⋊C4—416no—
225C2×C22⋊C4—48no—
226C2×C22⋊C4—48no—
227C2×C22⋊C4—416no—
228C2×C22⋊C4—48no—
229C2×C22⋊C4—48no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—48no—
232C2×C22⋊C4—48no—
233C2×C22⋊C4—416no—
234C2×C22⋊C4—48no—
235C2×C22⋊C4—88no—
236C2×C22⋊C4—84no—
237C2×C22⋊C4C22×C418yes—
238C2×C22⋊C4C22×C414yes—
239C2×C22⋊C4C22×C414yes—
240C2×C22⋊C4C2×D412yes—
241C2×C22⋊C4C2×D414yes—
242C2×C22⋊C4#4112yes—
243C2×C22⋊C4#4114yes—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—28no—
247C2×C42—48no—
248C2×C42—44no—
249#95—28no—
250#95—28no—
251#95—24no—
252#95—24no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—24no—
258C24—24no—
259C24—24no—
260C24—24no—
261C24—28no—
262C24—216no—
263C24—22no—
264C24—28no—
265C24—22no—
266C24—28no—
267C24—22no—
268C24—28no—
269C24—22no—
270C24—28no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—28no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—28no—
281C24—42no—
282C24—44no—
283C24—48no—
284C24—42no—
285C24—44no—
286C24—48no—
287C24—44no—
288C24—416no—
289C24—48no—
290C24—48no—
291C24—48no—
292C24—88no—
293C24—84no—
294C24—84no—
295C24—88no—
296C24—88no—
297C24—816no—
298C24—816no—
299C24—88no—
300C24—88no—
301C24C2×C4214yes—
302C24C2×C4214yes—
303C24C2×C22⋊C414yes—
304C24C2×C22⋊C412yes—
305C24C2×C22⋊C418yes—
306C24C2×C22⋊C412yes—
307C24C2×C22⋊C418yes—
308C24C2×C4⋊C414yes—
309C24C2×C4⋊C418yes—
310C24C2×C4⋊C414yes—
311C24C2×C4⋊C418yes—
312C24C23×C411yescharacteristic
313C24C23×C412yes—
314C24C22×D412yes—
315C24C22×D411yescharacteristic
316C24C22×D411yescharacteristic
317C24C22×Q811yescharacteristic
318C24C2×#4111yescharacteristic
319C24C2×#4112yes—
320C24H(5,𝔽2)12yes—
321C24H(5,𝔽2)11yescharacteristic
322C242−1+411yescharacteristic
323C2×D4—28no—
324C2×D4—216no—
325C2×D4—232no—
326C2×D4—216no—
327C2×D4—216no—
328C2×D4—232no—
329C2×D4—24no—
330C2×D4—28no—
331C2×D4—48no—
332C2×D4—416no—
333C2×D4—48no—
334C2×D4—416no—
335C2×D4—44no—
336C2×D4—48no—
337C2×D4—416no—
338C2×D4—432no—
339C2×D4—432no—
340C2×D4—416no—
341C2×D4#9518yes—
342C2×D4#95116yes—
343C2×D4#9514yes—
344C2×D4#9518yes—
345C2×D4#9518yes—
346C2×D4#95116yes—
347C2×D4#9514yes—
348C2×D4#9918yes—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—48no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—432no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—48no—
378C22×C4—44no—
379C22×C4—48no—
380C22×C4—432no—
381C22×C4—48no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—48no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4—816no—
394C22×C4—816no—
395C22×C4—832no—
396C22×C4—832no—
397C22×C4—816no—
398C22×C4C2×C4212yes—
399C22×C4C2×C4212yes—
400C22×C4C2×C22⋊C414yes—
401C22×C4C2×C22⋊C412yes—
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C414yes—
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C4⋊C412yes—
406C22×C4C2×C4⋊C414yes—
407C22×C4C2×C4⋊C412yes—
408C22×C4C2×C4⋊C414yes—
409C22×C4C4×D414yescomplemented
410C4⋊C4—44no—
411C4⋊C4—416no—
412C4⋊C4—44no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—28no—
419C22⋊C4—24no—
420C22⋊C4—28no—
421C22⋊C4—24no—
422C22⋊C4—416no—
423C22⋊C4—432no—
424C22⋊C4—432no—
425C22⋊C4—48no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—48no—
435C22⋊C4—88no—
436C22⋊C4—816no—
437C22⋊C4C4×D418yescomplemented
438C42—44no—
439C42—416no—
440C42—44no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—22no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—28no—
450C23—22no—
451C23—28no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—28no—
456C23—24no—
457C23—28no—
458C23—24no—
459C23—28no—
460C23—28no—
461C23—24no—
462C23—216no—
463C23—28no—
464C23—28no—
465C23—216no—
466C23—24no—
467C23—24no—
468C23—28no—
469C23—28no—
470C23—24no—
471C23—28no—
472C23—24no—
473C23—28no—
474C23—24no—
475C23—24no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—48no—
480C23—42no—
481C23—41no—
482C23—42no—
483C23—41no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—48no—
492C23—48no—
493C23—44no—
494C23—48no—
495C23—44no—
496C23—48no—
497C23—44no—
498C23—44no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—416no—
503C23—416no—
504C23—416no—
505C23—48no—
506C23—48no—
507C23—416no—
508C23—84no—
509C23—84no—
510C23—88no—
511C23—88no—
512C23—88no—
513C23—84no—
514C23—84no—
515C23—88no—
516C23—816no—
517C23—816no—
518C23—816no—
519C23—816no—
520C23—88no—
521C23—816no—
522C23—88no—
523C23—88no—
524C23—88no—
525C23—816no—
526C23C2×#9514yes—
527C23C2×#9518yes—
528C23C2×#9514yes—
529C23C2×#9918yes—
530C23C22×C4211yescharacteristic
531C23C22×C22⋊C412yes—
532C23C22×C22⋊C411yescharacteristic
533C23C22×C4⋊C411yescharacteristic
534C23C22×C4⋊C412yes—
535C23C2×C4×D411yescharacteristic
536C23C2×C4×D412yes—
537C23C4×#4111yescharacteristic
538C23C4×#4112yes—
539C23#51811yescharacteristic
540C23#51812yes—
541C23#52011yescharacteristic
542C23#52012yes—
543C23D4212yes—
544C23D4211yescharacteristic
545C23#54612yes—
546C23#54612yes—
547C23#54612yes—
548C23#54712yes—
549C23#54711yescharacteristic
550C23D4×Q811yescharacteristic
551C23#55411yescharacteristic
552C23#55412yes—
553D4—216no—
554D4—232no—
555D4—216no—
556D4—28no—
557D4—48no—
558D4—416no—
559D4#342116yesdirect product factor
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—816no—
593C2×C4—864no—
594C2×C4—816no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—88no—
598C2×C4—88no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4C2×#9512yes—
604C2×C4C2×#9514yes—
605C2×C4C2×#9512yes—
606C2×C4C2×#9914yes—
607C22—22no—
608C22—21no—
609C22—22no—
610C22—21no—
611C22—22no—
612C22—21no—
613C22—22no—
614C22—21no—
615C22—24no—
616C22—22no—
617C22—22no—
618C22—24no—
619C22—28no—
620C22—24no—
621C22—24no—
622C22—22no—
623C22—22no—
624C22—24no—
625C22—28no—
626C22—24no—
627C22—24no—
628C22—28no—
629C22—28no—
630C22—24no—
631C22—28no—
632C22—24no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22—42no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—42no—
643C22—416no—
644C22—416no—
645C22—48no—
646C22—416no—
647C22—48no—
648C22—416no—
649C22—48no—
650C22—48no—
651C22—44no—
652C22—48no—
653C22—44no—
654C22—48no—
655C22—88no—
656C22—88no—
657C22—84no—
658C22—88no—
659C22—84no—
660C22—88no—
661C22—84no—
662C22—84no—
663C22C2×#34218yescomplemented
664C22C22×#9511yescharacteristic
665C22C42×D411yescharacteristic
666C22#227511yescharacteristic
667C22D4×C22⋊C412yes—
668C22D4×C22⋊C411yescharacteristic
669C22#233912yes—
670C22#233911yescharacteristic
671C22D4×C4⋊C411yescharacteristic
672C22D4×C4⋊C412yes—
673C22#234911yescharacteristic
674C22#234912yes—
675C22C22×#9912yes—
676C22#288312yes—
677C4—22no—
678C4—24no—
679C4—48no—
680C4—44no—
681C4—84no—
682C4—88no—
683C4—84no—
684C4—82no—
685C4C2×#34214yescomplemented
686C2—22no—
687C2—22no—
688C2—21no—
689C2—22no—
690C2—21no—
691C2—22no—
692C2—21no—
693C2—21no—
694C2—28no—
695C2—48no—
696C2—44no—
697C2D4×#9511yescharacteristic
698C2C22×#34211yescharacteristic
699C2#2035411yescharacteristic
700C2D4×#9912yes—
701C2#2247612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC24×D4C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×D4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#342C2semidirect
4. #10854C2semidirect
5. #10846C2semidirect
6. #10735C2semidirect
7. #10725C2semidirect
8. C4×#342C2semidirect
11. C22×C4×D4C4semidirect
18. D4×C22⋊C4C4semidirect
20. C2×#390C4semidirect
22. C2×#386C4semidirect
49. C24×C4C2×C4semidirect
58. C22×C4⋊C4C2×C4semidirect
72. C22×C22⋊C4C2×C4semidirect
73. C22×C22⋊C4C2×C4semidirect
74. C22×C22⋊C4C2×C4semidirect
75. C22×C22⋊C4C2×C4semidirect
83. C22×C42C2×C4semidirect
93. #380C2×C4semidirect
97. #379C2×C4semidirect
101. C4×C22⋊C4C2×C4semidirect
104. C2×#95C2×C4semidirect
105. C2×#95C2×C4semidirect
112. #342D4direct (N × H)
409. C22×C4C4×D4semidirect
437. C22⋊C4C4×D4semidirect
559. D4#342direct (N × H)
663. C22C2×#342semidirect
685. C4C2×#342semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C22⋊C42
C22×#3422
#108542
#108462
#107352
#107252
C4×#3422