Finite Group Databasebeta

← D4×A4⋊Q8 #11714387

Subgroups — D4×A4⋊Q8 #11714387

Order 768.

1634 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1634 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×A4⋊Q8C212yesmaximal, complemented
2C4×D4×A4C211yesmaximal, characteristic
3D4×SL(2,ℤ/4)C212yesmaximal
4#85480C212yesmaximal, complemented
5#85303C212yesmaximal, complemented
6#85293C214yesmaximal, complemented
7#85283C211yesmaximal, characteristic, complemented
8C4×A4⋊Q8C211yesmaximal, characteristic, complemented
9D4×C22⋊Q8—31nomaximal, Sylow
10C2×D4×A4C2211yescharacteristic
11C22×C4×A4C2212yes—
12C22×SL(2,ℤ/4)C2214yes—
13C2×A4⋊Q8—28no—
14C2×A4⋊Q8C2212yes—
15C2×A4⋊Q8C2214yes—
16C2×A4⋊Q8C2211yescharacteristic
17D4×Dic6—41nomaximal
18A4×C4⋊C4C2211yescharacteristic
19A4×C22⋊C4C2212yes—
20C42×A4C2211yescharacteristic
21#5432C2214yes—
22C4⋊SL(2,ℤ/4)C2211yescharacteristic
23C4⋊SL(2,ℤ/4)C2212yes—
24C4⋊SL(2,ℤ/4)C2212yes—
25SL(2,ℤ/4)⋊C4C2212yes—
26SL(2,ℤ/4)⋊C4C2214yes—
27C4×SL(2,ℤ/4)C2212yes—
28C2×D4×Q8—31no—
29C22×C22⋊Q8—32no—
30C22×C4×D4D311yescharacteristic
31#2857—31no—
32#2848—32no—
33#2542—31no—
34#2538—31no—
35#2452—32no—
36#2449—34no—
37#2440—32no—
38#2437—34no—
39#2417—32no—
40#2409—32no—
41D4×C4⋊C4—31no—
42D4×C4⋊C4—32no—
43D4×C22⋊C4—32no—
44C4×C22⋊Q8—31no—
45C23×A4C2312yes—
46C22×Dic6—42no—
47D4×A4Q814yescomplemented
48C2×C4×A4—24no—
49C2×C4×A4C2312yes—
50C2×C4×A4C2311yescharacteristic
51C2×C4×A4C2311yescharacteristic
52C2×C4×A4C2311yescharacteristic
53C2×SL(2,ℤ/4)—28no—
54C2×SL(2,ℤ/4)C2314yes—
55C2×SL(2,ℤ/4)C2312yes—
56C2×SL(2,ℤ/4)C2312yes—
57A4⋊Q8—24no—
58A4⋊Q8—28no—
59A4⋊Q8D414yesdirect product factor
60C3×C4×D4—41no—
61D4×Dic3—42no—
62#924—42no—
63#888—42no—
64#878—44no—
65#869—41no—
66C4×Dic6—41no—
67C23×Q8—32no—
68C23×D4C2×D311yescharacteristic
69C24×C4C2×D312yes—
70D4×Q8—61no—
71D4×Q8—62no—
72D4×Q8—61no—
73D4×Q8—62no—
74D4×Q8—61no—
75D4×Q8—61no—
76C2×C4⋊Q8—31no—
77C2×C4⋊Q8—32no—
78C2×C22⋊Q8—31no—
79C2×C22⋊Q8—32no—
80C2×C22⋊Q8—32no—
81C2×C22⋊Q8—32no—
82C2×C22⋊Q8—34no—
83C2×C22⋊Q8—34no—
84C2×C22⋊Q8—32no—
85C2×C22⋊Q8—34no—
86C2×C22⋊Q8—32no—
87C2×C22⋊Q8—68no—
88C2×C22⋊Q8—68no—
89C2×C4×Q8—31no—
90C2×C4×D4—31no—
91C2×C4×D4—31no—
92C2×C4×D4—32no—
93C2×C4×D4—32no—
94C2×C4×D4—31no—
95C2×C4×D4—31no—
96C2×C4×D4—32no—
97C2×C4×D4—32no—
98C22×C4⋊C4—32no—
99C22×C4⋊C4—34no—
100C22×C4⋊C4C2×D311yescharacteristic
101C22×C22⋊C4—34no—
102C22×C22⋊C4C2×D312yes—
103C22×C42C2×D311yescharacteristic
104#399—34no—
105#398—34no—
106#396—34no—
107#396—34no—
108#395—34no—
109#393—34no—
110#391—32no—
111#391—31no—
112#390—32no—
113#390—32no—
114#389—32no—
115#389—32no—
116#389—32no—
117#389—32no—
118#389—31no—
119#387—32no—
120#386—34no—
121#386—34no—
122#385—34no—
123#385—34no—
124#385—34no—
125#385—34no—
126#385—34no—
127#385—32no—
128#384—31no—
129#384—32no—
130#380—32no—
131#380—32no—
132#380—34no—
133#380—31no—
134#380—32no—
135#380—32no—
136#379—34no—
137C4×C4⋊C4—32no—
138C4×C4⋊C4—31no—
139C4×C22⋊C4—32no—
140C22×A4—22no—
141C22×A4—22no—
142C22×A4C2×Q814yescomplemented
143C22×A4C2411yescharacteristic
144C2×C3×D4—41no—
145C22×C3×C4—42no—
146C22×Dic3—44no—
147C2×Dic6—41no—
148C2×Dic6—42no—
149C2×Dic6—44no—
150C2×Dic6—88no—
151C4×A4—22no—
152C4×A4—21no—
153C4×A4C2×D412yes—
154C4×A4C2×Q812yescomplemented
155SL(2,ℤ/4)—24no—
156SL(2,ℤ/4)—22no—
157SL(2,ℤ/4)C2×D414yes—
158C3×C4⋊C4—41no—
159C3×C22⋊C4—42no—
160C3×C42—41no—
161#217—44no—
162C4⋊Dic3—41no—
163C4⋊Dic3—42no—
164C4⋊Dic3—42no—
165Dic3⋊C4—42no—
166Dic3⋊C4—44no—
167C4×Dic3—42no—
168C25C22×D312yes—
169C22×Q8—31no—
170C22×Q8—32no—
171C22×Q8—34no—
172C22×Q8—62no—
173C22×Q8—62no—
174C22×Q8—62no—
175C22×Q8—68no—
176C22×Q8—62no—
177C22×Q8—124no—
178C22×D4—31no—
179C22×D4—32no—
180C22×D4—31no—
181C22×D4—34no—
182C22×D4Dic614yescomplemented
183C23×C4—24no—
184C23×C4—32no—
185C23×C4—32no—
186C23×C4—32no—
187C23×C4—32no—
188C23×C4—34no—
189C23×C4—34no—
190C23×C4—64no—
191C23×C4C22×D311yescharacteristic
192C23×C4C22×D312yes—
193C23×C4C22×D311yescharacteristic
194C23×C4C22×D311yescharacteristic
195C4⋊Q8—61no—
196C4⋊Q8—61no—
197C4⋊Q8—61no—
198C4⋊Q8—64no—
199C4⋊Q8—62no—
200C4⋊Q8—62no—
201C4⋊Q8—61no—
202C22⋊Q8—34no—
203C22⋊Q8—34no—
204C22⋊Q8—68no—
205C22⋊Q8—68no—
206C22⋊Q8—64no—
207C22⋊Q8—64no—
208C22⋊Q8—64no—
209C22⋊Q8—68no—
210C22⋊Q8—616no—
211C22⋊Q8—64no—
212C22⋊Q8—62no—
213C22⋊Q8—62no—
214C22⋊Q8—62no—
215C22⋊Q8—68no—
216C22⋊Q8—64no—
217C22⋊Q8—62no—
218C22⋊Q8—64no—
219C22⋊Q8—124no—
220C4×Q8—61no—
221C4×Q8—61no—
222C4×Q8—61no—
223C4×Q8—61no—
224C4×D4—31no—
225C4×D4—32no—
226C4×D4—31no—
227C4×D4—32no—
228C4×D4—31no—
229C4×D4—61no—
230C4×D4—61no—
231C4×D4—61no—
232C4×D4—62no—
233C4×D4—62no—
234C4×D4—68no—
235C4×D4—64no—
236C4×D4—62no—
237C4×D4—62no—
238C4×D4S411yescharacteristic
239C2×C4⋊C4—32no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—32no—
245C2×C4⋊C4—34no—
246C2×C4⋊C4—34no—
247C2×C4⋊C4—34no—
248C2×C4⋊C4—34no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—34no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—34no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—68no—
260C2×C4⋊C4—616no—
261C2×C22⋊C4—32no—
262C2×C22⋊C4—32no—
263C2×C22⋊C4—32no—
264C2×C22⋊C4—32no—
265C2×C22⋊C4—32no—
266C2×C22⋊C4—34no—
267C2×C22⋊C4—34no—
268C2×C22⋊C4—34no—
269C2×C22⋊C4—34no—
270C2×C22⋊C4—34no—
271C2×C22⋊C4—32no—
272C2×C22⋊C4—34no—
273C2×C22⋊C4—34no—
274C2×C22⋊C4—68no—
275C2×C22⋊C4—68no—
276C2×C42—31no—
277C2×C42—31no—
278C2×C42—31no—
279C2×C42—31no—
280C2×C42—32no—
281C2×C42—32no—
282#95—32no—
283#95—32no—
284#95—32no—
285#95—32no—
286#95—32no—
287#95—34no—
288#95—34no—
289#95—34no—
290#95—34no—
291C23×C3—42no—
292C2×A4—24no—
293C2×A4C22×D411yescharacteristic
294C2×A4C22×Q811yescharacteristic
295C2×A42−1+411yescharacteristic
296C3×D4—44no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—84no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—44no—
305C2×Dic3—88no—
306Dic6—44no—
307Dic6—88no—
308Dic6—84no—
309C24—22no—
310C24—22no—
311C24—32no—
312C24—32no—
313C24—34no—
314C24—62no—
315C24—62no—
316C24C2×Dic614yescomplemented
317C24C23×D311yescharacteristic
318C2×Q8—34no—
319C2×Q8—61no—
320C2×Q8—62no—
321C2×Q8—64no—
322C2×Q8—68no—
323C2×Q8—62no—
324C2×Q8—64no—
325C2×Q8—64no—
326C2×Q8—61no—
327C2×Q8—64no—
328C2×Q8—62no—
329C2×Q8—128no—
330C2×Q8—124no—
331C2×Q8—128no—
332C2×Q8—128no—
333C2×Q8—128no—
334C2×D4—32no—
335C2×D4—34no—
336C2×D4—34no—
337C2×D4—34no—
338C2×D4—38no—
339C2×D4—31no—
340C2×D4—62no—
341C2×D4—61no—
342C2×D4—64no—
343C2×D4C2×S411yescharacteristic
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—32no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—32no—
354C22×C4—32no—
355C22×C4—31no—
356C22×C4—32no—
357C22×C4—32no—
358C22×C4—32no—
359C22×C4—31no—
360C22×C4—32no—
361C22×C4—32no—
362C22×C4—32no—
363C22×C4—34no—
364C22×C4—34no—
365C22×C4—64no—
366C22×C4—64no—
367C22×C4—62no—
368C22×C4—64no—
369C22×C4—64no—
370C22×C4—64no—
371C22×C4—64no—
372C22×C4—68no—
373C22×C4—68no—
374C22×C4—62no—
375C22×C4—61no—
376C22×C4—64no—
377C22×C4—68no—
378C22×C4—64no—
379C22×C4—122no—
380C22×C4—122no—
381C22×C4—124no—
382C22×C4—124no—
383C22×C4C2×Dic612yescomplemented
384C22×C4D3×D412yes—
385C22×C4C2×S412yes—
386C4⋊C4—34no—
387C4⋊C4—38no—
388C4⋊C4—32no—
389C4⋊C4—31no—
390C4⋊C4—61no—
391C4⋊C4—64no—
392C4⋊C4—68no—
393C4⋊C4—68no—
394C4⋊C4—68no—
395C4⋊C4—68no—
396C4⋊C4—64no—
397C4⋊C4—62no—
398C4⋊C4—61no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C4⋊C4—64no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—61no—
406C4⋊C4—64no—
407C4⋊C4—62no—
408C4⋊C4—122no—
409C4⋊C4—122no—
410C4⋊C4—124no—
411C4⋊C4C2×S411yescharacteristic
412C22⋊C4—34no—
413C22⋊C4—34no—
414C22⋊C4—32no—
415C22⋊C4—32no—
416C22⋊C4—32no—
417C22⋊C4—62no—
418C22⋊C4—68no—
419C22⋊C4—68no—
420C22⋊C4—62no—
421C22⋊C4—64no—
422C22⋊C4—68no—
423C22⋊C4—64no—
424C22⋊C4—64no—
425C22⋊C4—62no—
426C22⋊C4—64no—
427C22⋊C4—124no—
428C22⋊C4—124no—
429C22⋊C4C2×S412yes—
430C42—31no—
431C42—31no—
432C42—31no—
433C42—61no—
434C42—64no—
435C42—62no—
436C42—62no—
437C42C2×S411yescharacteristic
438C22×C3—44no—
439C22×C3—41no—
440C22×C3—82no—
441C22×C3—82no—
442A4D4×Q811yescharacteristic, complemented
443C3×C4—42no—
444C3×C4—42no—
445C3×C4—82no—
446C3×C4—81no—
447Dic3—44no—
448Dic3—84no—
449Dic3—82no—
450C23—24no—
451C23—31no—
452C23—31no—
453C23—31no—
454C23—31no—
455C23—34no—
456C23—34no—
457C23—34no—
458C23—32no—
459C23—62no—
460C23—62no—
461C23—62no—
462C23—62no—
463C23—62no—
464C23—64no—
465C23—62no—
466C23—62no—
467C23—62no—
468C23—64no—
469C23—64no—
470C23—62no—
471C23—124no—
472C23C22×Dic611yescharacteristic
473C23C2×D3×D411yescharacteristic
474C23#101011yescharacteristic
475C23C22×S412yes—
476Q8—64no—
477Q8—64no—
478Q8—128no—
479Q8—124no—
480Q8—128no—
481Q8—124no—
482D4—38no—
483D4—34no—
484D4—64no—
485D4A4⋊Q814yesdirect product factor
486C2×C4—24no—
487C2×C4—31no—
488C2×C4—31no—
489C2×C4—32no—
490C2×C4—32no—
491C2×C4—32no—
492C2×C4—32no—
493C2×C4—32no—
494C2×C4—32no—
495C2×C4—34no—
496C2×C4—34no—
497C2×C4—31no—
498C2×C4—32no—
499C2×C4—64no—
500C2×C4—62no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—62no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—64no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—64no—
512C2×C4—62no—
513C2×C4—61no—
514C2×C4—62no—
515C2×C4—61no—
516C2×C4—62no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—64no—
525C2×C4—64no—
526C2×C4—62no—
527C2×C4—64no—
528C2×C4—124no—
529C2×C4—128no—
530C2×C4—1216no—
531C2×C4—128no—
532C2×C4C22×S411yescharacteristic
533C2×C4C22×S412yes—
534C2×C4C22×S411yescharacteristic
535C2×C4C22×S411yescharacteristic
536C2×C3—41no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3—84no—
540C22—22no—
541C22—22no—
542C22—31no—
543C22—31no—
544C22—31no—
545C22—31no—
546C22—31no—
547C22—31no—
548C22—31no—
549C22—31no—
550C22—31no—
551C22—34no—
552C22—64no—
553C22—64no—
554C22—64no—
555C22—64no—
556C22—61no—
557C22—64no—
558C22—64no—
559C22—62no—
560C22—62no—
561C22—122no—
562C22—122no—
563C22D4×Dic611yescharacteristic, complemented
564C22C2×A4⋊Q814yescomplemented
565C22C23×S411yescharacteristic
566C4—22no—
567C4—21no—
568C4—32no—
569C4—32no—
570C4—64no—
571C4—62no—
572C4—64no—
573C4—61no—
574C4—122no—
575C4—124no—
576C4—124no—
577C4—122no—
578C4C2×A4⋊Q812yescomplemented
579C4D4×S412yes—
580C3—41noSylow
581C2—24no—
582C2—31no—
583C2—31no—
584C2—31no—
585C2—31no—
586C2—64no—
587C2C22×A4⋊Q811yescharacteristic
588C2C2×D4×S411yescharacteristic
589C2#8749011yescharacteristic
subgroupG/H
center, FrattiniC22C23×S4
commutator, cosocleC22×A4C24
FittingC22×C4×D4D3
socleC24C23×D3
prime radicalC2×D4×A4C22
hypercenterC4×D4S4
hypocenterA4D4×Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×A4⋊Q8C2semidirect
4. #85480C2semidirect
5. #85303C2semidirect
6. #85293C2semidirect
7. #85283C2semidirect
8. C4×A4⋊Q8C2semidirect
47. D4×A4Q8semidirect
59. A4⋊Q8D4direct (N × H)
142. C22×A4C2×Q8semidirect
154. C4×A4C2×Q8semidirect
182. C22×D4Dic6semidirect
316. C24C2×Dic6semidirect
383. C22×C4C2×Dic6semidirect
442. A4D4×Q8semidirect
485. D4A4⋊Q8direct (N × H)
563. C22D4×Dic6semidirect
564. C22C2×A4⋊Q8semidirect
578. C4C2×A4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×A4⋊Q82
C4×D4×A42
D4×SL(2,ℤ/4)2
#854802
#853032
#852932
#852832
C4×A4⋊Q82
D4×C22⋊Q83
D4×Dic64