Finite Group Databasebeta

← D4×#400 #4306401

Subgroups — D4×#400 #4306401

Order 512.

3454 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#400C212yesmaximal, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3C2×D4×C4⋊C4C211yesmaximal, characteristic
4#15156C211yesmaximal, characteristic, complemented
5#15153C211yesmaximal, characteristic, complemented
6#15152C211yesmaximal, characteristic, complemented
7#15146C212yesmaximal, complemented
8#15139C212yesmaximal, complemented
9#14843C212yesmaximal, complemented
10#14826C212yesmaximal, complemented
11#14822C212yesmaximal, complemented
12#14767C214yesmaximal, complemented
13#14738C214yesmaximal, complemented
14C4×#400C211yesmaximal, characteristic, complemented
15D4×#95C212yesmaximal
16D4×#95C212yesmaximal
17D4×#95C211yesmaximal, characteristic
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24C23×C4⋊C4C2212yes—
25C2×#400—216no—
26C2×#400C2214yes—
27C2×#400C2214yes—
28C2×#400C2212yes—
29C2×#400C2212yes—
30C2×#400C2212yes—
31C2×#400C2212yes—
32C2×#400C2211yescharacteristic
33D4×C4⋊C4—28no—
34D4×C4⋊C4—28no—
35C2×#389C2211yescharacteristic
36C2×#389C2212yes—
37C2×#389C2211yescharacteristic
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#385C2212yes—
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C22×#95C2214yes—
48C22×#95C2214yes—
49C22×#95C2212yes—
50#1445C2211yescharacteristic
51#1445C2212yes—
52#1445C2214yes—
53#1443C2214yes—
54#1443C2212yes—
55#1443C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1442C2211yescharacteristic
60#1442C2214yes—
61#1442C2214yes—
62#1442C2212yes—
63#1441C2211yescharacteristic
64#1441C2214yes—
65#1441C2212yes—
66#1441C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C22⋊#95C2214yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C4×#95C2211yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2314yes—
89C24×C4C2314yes—
90C24×C4C2312yes—
91C24×C4C2312yes—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—28no—
95C2×C4×D4—24no—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—28no—
100C2×C4×D4—28no—
101C2×C4×D4—28no—
102C2×C4×D4—28no—
103C2×C4×D4D418yes—
104C2×C4×D4Q818yes—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—28no—
108C22×C4⋊C4—28no—
109C22×C4⋊C4—44no—
110C22×C4⋊C4—44no—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#400—28no—
138#400—28no—
139#400—216no—
140#400—216no—
141#400—28no—
142#400—44no—
143#400D418yesdirect product factor
144#389—24no—
145#389—28no—
146#389—24no—
147#385—28no—
148#385—28no—
149#385—216no—
150#384—24no—
151#384—24no—
152#380—28no—
153#380—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C2×#95—28no—
157C2×#95—216no—
158C2×#95—216no—
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C2×#95C2312yes—
192C25C2412yes—
193C22×D4C2×D414yes—
194C22×D4C2×Q814yes—
195C22×D4#4114yes—
196C22×D4#4118yes—
197C22×D4#4118yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4C2×D418yes—
228C23×C4C2×Q818yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C4×D4—232no—
246C4×D4—232no—
247C4×D4—416no—
248C4×D4—432no—
249C4×D4—432no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—432no—
283C2×C4⋊C4—432no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—41no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2×Q814yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4C2×D418yes—
307C2×C22⋊C4C2×Q818yes—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42C2×D414yes—
320C2×C42C2×Q814yes—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—22no—
336#95—24no—
337#95—42no—
338#95—44no—
339#95—42no—
340#95—44no—
341#95—42no—
342#95—42no—
343#95—44no—
344#95—42no—
345#95—42no—
346#95—42no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95C2×D414yes—
352#95C2×D418yes—
353#95C2×D418yes—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—24no—
360C24C22×D414yes—
361C24C22×Q814yes—
362C24C2×#4114yes—
363C24C2×#4118yes—
364C24C2×#4118yes—
365C24C2511yescharacteristic
366C2×D4C22⋊Q8116yes—
367C2×D4#123116yes—
368C2×D4#124116yes—
369C2×D4#125132yes—
370C2×D4#126132yes—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—416no—
460C22×C4—48no—
461C22×C4—416no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—416no—
468C22×C4—416no—
469C22×C4—48no—
470C22×C4—416no—
471C22×C4—416no—
472C22×C4—84no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4—88no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4—88no—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×Q812yes—
490C22×C4C22×Q814yes—
491C22×C4C22×Q812yes—
492C22×C4C22×Q812yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C42−1+412yes—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—416no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C22⋊C4—216no—
515C22⋊C4—216no—
516C22⋊C4—48no—
517C22⋊C4—416no—
518C22⋊C4—416no—
519C42—28no—
520C42—28no—
521C42—44no—
522C42—48no—
523C42—48no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—24no—
539C23—28no—
540C23—28no—
541C23—28no—
542C23C2×C22⋊Q818yes—
543C23C2×#12318yes—
544C23C2×#12418yes—
545C23C2×#125116yes—
546C23C2×#126116yes—
547C23C23×D411yescharacteristic
548C23C23×D411yescharacteristic
549C23C23×Q811yescharacteristic
550C23C22×#4111yescharacteristic
551C23C22×#4112yes—
552C23C22×#4112yes—
553C23C2×H(5,𝔽2)11yescharacteristic
554C23C2×2−1+411yescharacteristic
555C23#58512yes—
556C23#58511yescharacteristic
557C23#58512yes—
558D4#400164yesdirect product factor
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—432no—
589C2×C4—432no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—832no—
651C2×C4—832no—
652C2×C4—816no—
653C2×C4C2×C22⋊Q814yes—
654C2×C4C2×#12314yes—
655C2×C4C2×#12414yes—
656C2×C4C2×#12518yes—
657C2×C4C2×#12618yes—
658C2×C4D4214yes—
659C2×C4D4×Q814yes—
660C22—28no—
661C22—216no—
662C22—216no—
663C22—216no—
664C22—216no—
665C22—28no—
666C22—28no—
667C22—28no—
668C22—28no—
669C22—28no—
670C22C2×#400116yescomplemented
671C22C22×C22⋊Q811yescharacteristic
672C22C22×#12311yescharacteristic
673C22C22×#12411yescharacteristic
674C22C22×#12512yes—
675C22C22×#12612yes—
676C22C2×D4211yescharacteristic
677C22C2×D4×Q811yescharacteristic
678C22D4×#4112yes—
679C22D4×#4112yes—
680C22D4×#4111yescharacteristic
681C22#348511yescharacteristic
682C22#348911yescharacteristic
683C22#350011yescharacteristic
684C22#350111yescharacteristic
685C22#350412yes—
686C22#351312yes—
687C22#351412yes—
688C22#351512yes—
689C22#351512yes—
690C22#352011yescharacteristic
691C22#352112yes—
692C22#352112yes—
693C22#352112yes—
694C4—24no—
695C4—24no—
696C4—44no—
697C4—44no—
698C4—42no—
699C4—42no—
700C4—44no—
701C4—44no—
702C4—42no—
703C4—84no—
704C4—84no—
705C4—82no—
706C4—82no—
707C4—81no—
708C4—82no—
709C4C2×#40018yescomplemented
710C2—216no—
711C2C22×#40011yescharacteristic
712C2#3588812yes—
713C2#3596112yes—
714C2#3603111yescharacteristic
715C2#3604211yescharacteristic
716C2#3606711yescharacteristic
717C2D4×C22⋊Q811yescharacteristic
718C2D4×#12311yescharacteristic
719C2D4×#12411yescharacteristic
720C2D4×#12512yes—
721C2D4×#12612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#400C2semidirect
4. #15156C2semidirect
5. #15153C2semidirect
6. #15152C2semidirect
7. #15146C2semidirect
8. #15139C2semidirect
9. #14843C2semidirect
10. #14826C2semidirect
11. #14822C2semidirect
12. #14767C2semidirect
13. #14738C2semidirect
14. C4×#400C2semidirect
143. #400D4direct (N × H)
558. D4#400direct (N × H)
670. C22C2×#400semidirect
709. C4C2×#400semidirect

Maximal subgroups

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

subgroupindex
C22×#4002
C2×D4×C4⋊C42
#151562
#151532
#151522
#151462
#151392
#148432
#148262
#148222
#147672
#147382
C4×#4002
D4×#952