Finite Group Databasebeta

← D4×D4⋊C8 #219796

Subgroups — D4×D4⋊C8 #219796

Order 512.

1798 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D42C211yesmaximal, characteristic
2D4×C4⋊C8C211yesmaximal, characteristic, complemented
3C4×C8×D4C211yesmaximal, characteristic, complemented
4C22×D4⋊C8C212yesmaximal, complemented
5#9674C211yesmaximal, characteristic, complemented
6#9638C211yesmaximal, characteristic, complemented
7#9072C212yesmaximal, complemented
8#8939C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic, complemented
10#8310C212yesmaximal, complemented
11C4×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×D42C411yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4C2212yes—
15C2×C8×D4—21no—
16C2×C8×D4—21no—
17C22×C4⋊C8C2212yescomplemented
18C22×C4×C8C2212yescomplemented
19#2366C411yescharacteristic
20#2361—21no—
21#2361C412yes—
22#2358—22no—
23D4×C4⋊C4C411yescharacteristic
24D4×C4⋊C4C411yescharacteristic
25D4×C22⋊C4—21no—
26D4×C22⋊C4C412yes—
27C4×#127C2211yescharacteristic
28C4×#121—21no—
29C4×#121C2212yes—
30C4×#120—22no—
31C42×D4C2211yescharacteristic
32C42×D4C2211yescharacteristic
33#1939—21no—
34#1850—22no—
35#1842C2211yescharacteristic, complemented
36#1797C2212yescomplemented
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic, complemented
39C4×C4⋊C8C2211yescharacteristic, complemented
40C8×C22⋊C4—22no—
41C4×C22⋊C8C2212yescomplemented
42C42×C8C2211yescharacteristic, complemented
43C2×D4⋊C8—28no—
44C2×D4⋊C8C2212yes—
45C2×D4⋊C8C2211yescharacteristic
46C2×D4⋊C8C2211yescharacteristic
47C2×D4⋊C8C2212yescomplemented
48C2×D4⋊C8C2212yescomplemented
49C2×D4⋊C8C2212yescomplemented
50C2×D4⋊C8C2212yescomplemented
51#1279C2211yescharacteristic, complemented
52#1279C2212yescomplemented
53#1279C2211yescharacteristic, complemented
54C23×D4—21no—
55C23×D4C2×C412yes—
56C24×C4—22no—
57C23×C8—22no—
58C23×C8—22no—
59D42—24no—
60D42C818yescomplemented
61C2×#127C2×C411yescharacteristic
62C2×#121—21no—
63C2×#121C2×C412yes—
64C2×#120—22no—
65C2×C4×D4—21no—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4—22no—
69C2×C4×D4—24no—
70C2×C4×D4—24no—
71C2×C4×D4—21no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—21no—
75C2×C4×D4—42no—
76C2×C4×D4—42no—
77C2×C4×D4—44no—
78C2×C4×D4—44no—
79C2×C4×D4C2×C411yescharacteristic
80C2×C4×D4C2×C412yes—
81C2×C4×D4C2×C412yes—
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4C2×C411yescharacteristic
84C2×C4×D4D411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×C4×D4C2312yes—
87C2×C4×D4C2311yescharacteristic
88C22×C4⋊C4—21no—
89C22×C4⋊C4C2×C412yes—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C42—21no—
93C22×C42C2312yes—
94C8×D4—28no—
95C8×D4—44no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—42no—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2311yescharacteristic
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C4×C8—21no—
108C2×C4×C8—24no—
109C2×C4×C8—22no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—42no—
113C2×C4×C8C2312yes—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#390—21no—
117#390—21no—
118#390—21no—
119#390C2×C412yes—
120#390C2×C412yes—
121#389—21no—
122#389C2×C412yes—
123#388—22no—
124#388—22no—
125#386—22no—
126#386—22no—
127#386—22no—
128#386—22no—
129#386—22no—
130#385—22no—
131#385—22no—
132#385—22no—
133#384C2×C411yescharacteristic
134#380—21no—
135#380C2×C412yes—
136#379—22no—
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4D411yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4C2×C412yes—
148C4×C22⋊C4D412yes—
149C4×C22⋊C4D412yes—
150C43C2311yescharacteristic
151#336—22no—
152#336—21no—
153#336—22no—
154#336—21no—
155D4⋊C8—24no—
156D4⋊C8—22no—
157D4⋊C8—24no—
158D4⋊C8—24no—
159D4⋊C8—42no—
160D4⋊C8—42no—
161D4⋊C8D414yesdirect product factor
162C25—22no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—22no—
166C22×D4—28no—
167C22×D4—22no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—24no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4C22⋊C411yescharacteristic
176C22×D4C2×C818yescomplemented
177C22×D4C22×C412yes—
178C22×D4C22×C411yescharacteristic
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—82no—
191C23×C4—82no—
192C23×C4C22×C412yes—
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C22×C8—21no—
196C22×C8—28no—
197C22×C8—22no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—44no—
203C22×C8—44no—
204C22×C8—44no—
205C22×C8—44no—
206C22×C8—82no—
207#127—22no—
208#127C2×C814yescomplemented
209#121—24no—
210#121—28no—
211#121C2×C818yescomplemented
212#120—216no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—22no—
216C4×D4—24no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—44no—
225C4×D4—44no—
226C4×D4—42no—
227C4×D4—42no—
228C4×D4—41no—
229C4×D4—44no—
230C4×D4—44no—
231C4×D4—44no—
232C4×D4—44no—
233C4×D4—44no—
234C4×D4—44no—
235C4×D4—48no—
236C4×D4—42no—
237C4×D4—42no—
238C4×D4—81no—
239C4×D4—81no—
240C4×D4C2×C814yes—
241C4×D4C2×C814yescomplemented
242C4×D4M1614yes—
243C4×D4D814yes—
244C4×D4SD1614yes—
245C4×D4C2×D412yes—
246C4×D4#4112yes—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C22⋊C411yescharacteristic
255C2×C4⋊C4C22⋊C411yescharacteristic
256C2×C4⋊C4C22⋊C411yescharacteristic
257C2×C4⋊C4C22×C412yes—
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4C22⋊C412yes—
277C2×C22⋊C4C22⋊C412yes—
278C2×C22⋊C4C22⋊C412yes—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42C22×C412yes—
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C2×D412yes—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D412yes—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2411yescharacteristic
298C4⋊C8—24no—
299C4⋊C8—22no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—22no—
304C4⋊C8—44no—
305C4⋊C8—42no—
306C4⋊C8—81no—
307C4⋊C8C2×D412yescomplemented
308C22⋊C8—28no—
309C22⋊C8—44no—
310C4×C8—24no—
311C4×C8—22no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—22no—
316C4×C8—44no—
317C4×C8—42no—
318C4×C8—42no—
319C4×C8C2×D412yescomplemented
320#95—21no—
321#95—21no—
322#95—22no—
323#95—22no—
324#95—21no—
325#95C22⋊C412yes—
326#95C22⋊C412yes—
327#95C22⋊C412yes—
328C24—21no—
329C24—22no—
330C24—28no—
331C24—44no—
332C24—44no—
333C24—44no—
334C24—44no—
335C24—82no—
336C24—82no—
337C24C2×C22⋊C412yes—
338C2×D4—216no—
339C2×D4—28no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—22no—
343C2×D4—21no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4—22no—
348C2×D4—28no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—432no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—81no—
362C2×D4—81no—
363C2×D4C22⋊C814yes—
364C2×D4#10214yes—
365C2×D4#10414yes—
366C2×D4C4×D412yes—
367C2×D4C22×C814yes—
368C2×D4C22×C812yes—
369C2×D4#13112yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—81no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4—82no—
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C412yes—
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C412yes—
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C22×C814yes—
425C22×C4C2×M1614yes—
426C22×C4C2×D814yes—
427C22×C4C2×SD1614yes—
428C22×C4C23×C411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—416no—
438C2×C8—41no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—42no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—41no—
450C2×C8—88no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—81no—
462C4⋊C4C22⋊C814yes—
463C4⋊C4#10214yes—
464C4⋊C4C4×D412yes—
465C4⋊C4C22×C812yes—
466C22⋊C4—24no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—24no—
470C22⋊C4—28no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4C22⋊C818yes—
482C22⋊C4#10218yes—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—42no—
498C42—42no—
499C42C22×C812yes—
500C42C2×M1612yes—
501C42C2×D812yes—
502C42C2×SD1612yes—
503C42C22×D411yescharacteristic
504C42C2×#4111yescharacteristic
505C42H(5,𝔽2)11yescharacteristic
506C42H(5,𝔽2)11yescharacteristic
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—416no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—416no—
527C23—88no—
528C23—88no—
529C23—88no—
530C23—88no—
531C23—88no—
532C23C2×C22⋊C814yes—
533C23C2×#10214yes—
534C23C2×#10414yes—
535C23C22×C22⋊C411yescharacteristic
536D4—24no—
537D4—22no—
538D4—22no—
539D4—24no—
540D4—22no—
541D4—416no—
542D4—416no—
543D4—42no—
544D4—42no—
545D4—48no—
546D4—84no—
547D4D4⋊C8116yesdirect product factor
548D4C8×D414yescomplemented
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—88no—
593C2×C4—84no—
594C2×C4—88no—
595C2×C4—88no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4C2×C22⋊C812yes—
601C2×C4C2×C22⋊C814yes—
602C2×C4C2×C22⋊C812yes—
603C2×C4C2×#10212yes—
604C2×C4C2×#10214yes—
605C2×C4C2×#10212yes—
606C2×C4C2×#10412yes—
607C2×C4C2×C4×D411yescharacteristic
608C2×C4#51811yescharacteristic
609C2×C4D4211yescharacteristic
610C2×C4D4211yescharacteristic
611C2×C4#54611yescharacteristic
612C2×C4#54611yescharacteristic
613C2×C4C23×C811yescharacteristic
614C2×C4C22×M1611yescharacteristic
615C2×C4C2×#13111yescharacteristic
616C2×C4#56811yescharacteristic
617C2×C4C22×D811yescharacteristic
618C2×C4C22×SD1611yescharacteristic
619C2×C4#57611yescharacteristic
620C2×C4#57711yescharacteristic
621C8—24no—
622C8—42no—
623C8—44no—
624C8—42no—
625C8—82no—
626C8—81no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—832no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22C2×D4⋊C818yescomplemented
649C22D4×C22⋊C411yescharacteristic
650C22C22×C22⋊C811yescharacteristic
651C22#288011yescharacteristic
652C22C22×#10211yescharacteristic
653C22#289711yescharacteristic
654C22C22×#10411yescharacteristic
655C22#290111yescharacteristic
656C4—22no—
657C4—22no—
658C4—24no—
659C4—24no—
660C4—21no—
661C4—21no—
662C4—42no—
663C4—41no—
664C4—42no—
665C4—42no—
666C4—81no—
667C4—82no—
668C4C2×D4⋊C814yescomplemented
669C4C2×C8×D411yescharacteristic
670C4D4×M1611yescharacteristic
671C4#297211yescharacteristic
672C4#297411yescharacteristic
673C4D4×D811yescharacteristic
674C4D4×SD1611yescharacteristic
675C4#329411yescharacteristic
676C4#329511yescharacteristic
677C2—28no—
678C2—44no—
679C2—84no—
680C2C22×D4⋊C811yescharacteristic
681C2#1743811yescharacteristic
682C2#1750711yescharacteristic
683C2#1751711yescharacteristic
684C2D4×C22⋊C811yescharacteristic
685C2D4×#10211yescharacteristic
686C2D4×#10411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×D42C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×C4⋊C8C2semidirect
3. C4×C8×D4C2semidirect
4. C22×D4⋊C8C2semidirect
5. #9674C2semidirect
6. #9638C2semidirect
7. #9072C2semidirect
8. #8939C2semidirect
9. #8563C2semidirect
10. #8310C2semidirect
11. C4×D4⋊C8C2semidirect
17. C22×C4⋊C8C22semidirect
18. C22×C4×C8C22semidirect
35. #1842C22semidirect
36. #1797C22semidirect
38. C4×C4⋊C8C22semidirect
39. C4×C4⋊C8C22semidirect
41. C4×C22⋊C8C22semidirect
42. C42×C8C22semidirect
47. C2×D4⋊C8C22semidirect
48. C2×D4⋊C8C22semidirect
49. C2×D4⋊C8C22semidirect
50. C2×D4⋊C8C22semidirect
51. #1279C22semidirect
52. #1279C22semidirect
53. #1279C22semidirect
60. D42C8semidirect
161. D4⋊C8D4direct (N × H)
176. C22×D4C2×C8semidirect
208. #127C2×C8semidirect
211. #121C2×C8semidirect
241. C4×D4C2×C8semidirect
307. C4⋊C8C2×D4semidirect
319. C4×C8C2×D4semidirect
547. D4D4⋊C8direct (N × H)
548. D4C8×D4semidirect
648. C22C2×D4⋊C8semidirect
668. C4C2×D4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×D422
D4×C4⋊C82
C4×C8×D42
C22×D4⋊C82
#96742
#96382
#90722
#89392
#85632
#83102
C4×D4⋊C82