Finite Group Databasebeta

← D3×Q8×SD16 #11675603

Subgroups — D3×Q8×SD16 #11675603

Order 768.

2058 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×Q82C211yesmaximal, characteristic, complemented
2D3×D4×Q8C211yesmaximal, characteristic
3C3×Q8×SD16C211yesmaximal, characteristic, complemented
4Q8×#215C211yesmaximal, characteristic, complemented
5Q8×#214C211yesmaximal, characteristic, complemented
6#81903C213yesmaximal, complemented
7D3×#498C213yesmaximal, complemented
8D3×#476C213yesmaximal
9#81053C213yesmaximal, complemented
10#81039C213yesmaximal, complemented
11D3×#475C213yesmaximal, complemented
12Q8×#204C211yesmaximal, characteristic, complemented
13D3×C8×Q8C211yesmaximal, characteristic, complemented
14C4×D3×SD16C213yesmaximal
15SD16×Dic6C213yesmaximal, complemented
16C2×Q8×SD16—31nomaximal, Sylow
17C22×D3×Q8—21no—
18C3×Q82C2211yescharacteristic, complemented
19C3×D4×Q8C2211yescharacteristic
20Q8×#67—21no—
21C2×D3×SD16C2211yescharacteristic
22#5727C2213yes—
23D3×C4⋊Q8—23no—
24D3×C4⋊Q8C2213yes—
25D3×C4⋊Q8C2213yes—
26#5722—23no—
27#5722C2213yescomplemented
28#5632—23no—
29D3×C22⋊Q8—23no—
30Q8×D12C2211yescharacteristic
31C4×D3×Q8—21no—
32C4×D3×Q8C2213yes—
33C4×D3×Q8C2211yescharacteristic
34Q8×Dic6—21no—
35Q8×Dic6C2213yescomplemented
36Q8×Dic6C2211yescharacteristic, complemented
37C4×D3×D4C2213yes—
38D4×Dic6C2213yes—
39C3×#498C2213yescomplemented
40C3×#476C2213yes—
41C3×#475C2213yescomplemented
42C3×C8×Q8C2211yescharacteristic, complemented
43C3×C4×SD16C2213yes—
44Dic3×SD16C2213yes—
45#5094C2213yescomplemented
46#5084C2213yes—
47#5080C2213yescomplemented
48#5080C2213yescomplemented
49C4×#215C2213yes—
50Q8×C3⋊C8C2211yescharacteristic, complemented
51#5020C2213yescomplemented
52C4×#214C2213yes—
53#5008C2213yes—
54#4870C2216yes—
55D3×#106C2213yes—
56D3×#106C2216yes—
57#4855C2216yescomplemented
58#4855C2216yescomplemented
59#4854C2216yescomplemented
60#4852C2213yes—
61#4851C2213yescomplemented
62#4842C2213yes—
63D3×C4⋊C8C2213yes—
64#4830C2213yescomplemented
65D3×#103C2213yes—
66#4791C2216yescomplemented
67#4788C2213yes—
68D3×#102C2213yes—
69#4761C2216yes—
70#4758C2213yes—
71C4×#204C2213yes—
72C4×D3×C8C2213yes—
73#4680C2213yescomplemented
74C8×Dic6C2213yescomplemented
75C2×Q82—31no—
76C2×D4×Q8—31no—
77Q8×SD16—33no—
78Q8×SD16—33no—
79Q8×SD16—33no—
80Q8×SD16—31no—
81Q8×SD16—33no—
82Q8×SD16—31no—
83Q8×SD16—31no—
84Q8×SD16D311yescharacteristic, direct product factor
85C2×#498—33no—
86C2×#475—33no—
87C2×#476—33no—
88C2×C8×Q8—31no—
89C2×C4×SD16—33no—
90C22×C3×Q8—21no—
91C2×D3×Q8—21no—
92C2×D3×Q8—23no—
93C2×D3×Q8—23no—
94C2×D3×Q8—23no—
95C2×D3×Q8—48no—
96C2×D3×Q8D411yescharacteristic
97C2×D3×Q8C2311yescharacteristic
98C2×D3×D4C2311yescharacteristic
99C22×C4×D3—23no—
100C22×Dic6—23no—
101C2×C3×SD16C2311yescharacteristic
102C3×C4⋊Q8—23no—
103C3×C4⋊Q8C2313yes—
104C3×C4⋊Q8C2313yes—
105C3×C22⋊Q8—23no—
106C3×C4×Q8—21no—
107C3×C4×Q8C2313yes—
108C3×C4×Q8C2311yescharacteristic
109C3×C4×D4C2313yes—
110#946—23no—
111Q8×Dic3—21no—
112Q8×Dic3D411yescharacteristic
113Q8×Dic3C2313yes—
114#944—26no—
115#944—23no—
116#944—23no—
117C2×#215C2311yescharacteristic
118D4×Dic3C2313yes—
119C2×#214C2311yescharacteristic
120C4×#67—23no—
121#924—23no—
122D3×SD16Q818yesdirect product factor
123C2×#204C2311yescharacteristic
124C2×D3×C8C2313yes—
125C2×D3×C8C2311yescharacteristic
126#897—23no—
127#896—26no—
128#893C2313yes—
129D3×C4⋊C4—23no—
130D3×C4⋊C4—23no—
131D3×C4⋊C4—23no—
132D3×C4⋊C4D413yes—
133D3×C4⋊C4C2313yes—
134D3×C4⋊C4C2313yes—
135#888—23no—
136#888—23no—
137#888—26no—
138#888—23no—
139#888—23no—
140#888C2316yes—
141#888C2316yes—
142#887—23no—
143#887—23no—
144#887D413yes—
145#887C2313yes—
146#67⋊C4—23no—
147D3×C22⋊C4—23no—
148#878—26no—
149C4×D12C2313yes—
150C42×D3—23no—
151C42×D3C2313yes—
152#869—23no—
153#869C2313yes—
154#869C2313yes—
155C4×Dic6—23no—
156C4×Dic6—23no—
157C4×Dic6C2313yes—
158C4×Dic6C2313yes—
159C3×#106C2316yes—
160C3×#106C2313yes—
161C3×C4⋊C8C2313yes—
162C3×#103C2313yes—
163C3×#102C2313yes—
164C3×C4×C8C2313yes—
165#835C2313yes—
166#832C2313yes—
167#821C2313yes—
168#817C2313yes—
169#817C2316yes—
170#816C2313yes—
171Dic3⋊C8C2316yes—
172C8×Dic3C2313yes—
173C8×Dic3C2313yes—
174#810C2313yes—
175#809C2313yes—
176#808C2316yes—
177#808C2313yes—
178#808C2316yes—
179#808C2313yes—
180C4⋊(C3⋊C8)C2313yes—
181C4×C3⋊C8C2313yes—
182C23×Q8—61no—
183C22×SD16—31no—
184Q82—33no—
185Q82—33no—
186Q82—31no—
187Q82—61no—
188Q82—63no—
189Q82C2×D311yescharacteristic, complemented
190D4×Q8—33no—
191D4×Q8—31no—
192D4×Q8—33no—
193D4×Q8—63no—
194D4×Q8—61no—
195D4×Q8C2×D311yescharacteristic
196C2×C4⋊Q8—33no—
197C2×C4⋊Q8—33no—
198C2×C4⋊Q8—63no—
199C2×C22⋊Q8—63no—
200C2×C4×Q8—31no—
201C2×C4×Q8—33no—
202C2×C4×Q8—61no—
203C2×C4×D4—33no—
204#498—33no—
205#498—33no—
206#498—36no—
207#498—36no—
208#498—33no—
209#498C2×D313yescomplemented
210#476—33no—
211#476—33no—
212#476—36no—
213#476—33no—
214#476—36no—
215#476C2×D313yes—
216#475—33no—
217#475—33no—
218#475—36no—
219#475—33no—
220#475—36no—
221#475C2×D313yescomplemented
222C8×Q8—31no—
223C8×Q8—33no—
224C8×Q8—33no—
225C8×Q8C2×D311yescharacteristic, complemented
226C4×SD16—33no—
227C4×SD16—33no—
228C4×SD16—33no—
229C4×SD16—33no—
230C4×SD16—33no—
231C4×SD16—33no—
232C4×SD16—33no—
233C4×SD16C2×D313yes—
234C2×#106—36no—
235C2×#106—33no—
236C2×C4⋊C8—33no—
237C2×#103—33no—
238C2×#102—33no—
239C2×C4×C8—33no—
240C23×D3—21no—
241C2×C3×Q8—23no—
242C2×C3×Q8—23no—
243C2×C3×Q8—44no—
244C2×C3×Q8C2×D411yescharacteristic
245C2×C3×Q8C2411yescharacteristic
246C2×C3×D4C2411yescharacteristic
247C22×C3×C4—23no—
248C2×#67—21no—
249C22×Dic3—23no—
250D3×Q8—22no—
251D3×Q8—44no—
252D3×Q8—412no—
253D3×Q8—412no—
254D3×Q8—86no—
255D3×Q8SD1618yesdirect product factor
256D3×Q8C2×Q814yescomplemented
257D3×D4—22no—
258D3×D4C2×Q814yes—
259C2×D12C2411yescharacteristic
260C2×C4×D3—21no—
261C2×C4×D3—23no—
262C2×C4×D3—23no—
263C2×C4×D3—23no—
264C2×C4×D3—23no—
265C2×C4×D3—412no—
266C2×C4×D3C2×D413yes—
267C2×C4×D3C2×D413yes—
268C2×C4×D3C2411yescharacteristic
269C2×Dic6—21no—
270C2×Dic6—23no—
271C2×Dic6—26no—
272C2×Dic6—23no—
273C2×Dic6—26no—
274C2×Dic6—412no—
275C2×Dic6C2×D413yes—
276C2×Dic6C2411yescharacteristic
277C3×SD16C2×Q814yescomplemented
278C2×C3×C8C2413yes—
279C2×C3×C8C2411yescharacteristic
280C3×C4⋊C4—23no—
281C3×C4⋊C4—23no—
282C3×C4⋊C4—23no—
283C3×C4⋊C4C2×D413yes—
284C3×C4⋊C4C2413yes—
285C3×C4⋊C4C2413yes—
286C3×C22⋊C4—23no—
287C3×C42—23no—
288C3×C42C2413yes—
289#217—23no—
290#215C2×Q814yescomplemented
291#214C2×Q814yescomplemented
292(C2×D3)⋊C4—23no—
293(C2×D3)⋊C4—23no—
294C4⋊Dic3—23no—
295C4⋊Dic3—23no—
296C4⋊Dic3—23no—
297C4⋊Dic3C2×D413yes—
298C4⋊Dic3C2413yes—
299C4⋊Dic3C2413yes—
300Dic3⋊C4—23no—
301Dic3⋊C4—23no—
302Dic3⋊C4—26no—
303Dic3⋊C4—23no—
304Dic3⋊C4—26no—
305Dic3⋊C4—23no—
306Dic3⋊C4C2×D416yes—
307C4×Dic3—23no—
308C4×Dic3—23no—
309C4×Dic3—23no—
310C4×Dic3C2×D413yes—
311C4×Dic3C2×D413yes—
312C4×Dic3C2413yes—
313C2×C3⋊C8C2413yes—
314C2×C3⋊C8C2411yescharacteristic
315#204C2×Q814yescomplemented
316D3×C8—26no—
317D3×C8C2×Q814yescomplemented
318C22×Q8—21no—
319C22×Q8—31no—
320C22×Q8—31no—
321C22×Q8—63no—
322C22×Q8—63no—
323C22×Q8—63no—
324C22×Q8—63no—
325C22×Q8—61no—
326C22×Q8—128no—
327C22×D4—31no—
328C23×C4—63no—
329C2×SD16—31no—
330C2×SD16—31no—
331C2×SD16—38no—
332C2×SD16—31no—
333C2×SD16C22×D311yescharacteristic
334C22×C8—31no—
335C22×C8—33no—
336C4⋊Q8—23no—
337C4⋊Q8—36no—
338C4⋊Q8—36no—
339C4⋊Q8—33no—
340C4⋊Q8—33no—
341C4⋊Q8—63no—
342C4⋊Q8—63no—
343C4⋊Q8—63no—
344C4⋊Q8—63no—
345C4⋊Q8—63no—
346C4⋊Q8—66no—
347C4⋊Q8—63no—
348C4⋊Q8—66no—
349C4⋊Q8—63no—
350C4⋊Q8C22×D313yes—
351C4⋊Q8C22×D313yes—
352C22⋊Q8—23no—
353C22⋊Q8—66no—
354C22⋊Q8—66no—
355C22⋊Q8—63no—
356C22⋊Q8—63no—
357C22⋊Q8—63no—
358C4×Q8—21no—
359C4×Q8—31no—
360C4×Q8—33no—
361C4×Q8—33no—
362C4×Q8—33no—
363C4×Q8—33no—
364C4×Q8—33no—
365C4×Q8—63no—
366C4×Q8—63no—
367C4×Q8—63no—
368C4×Q8—63no—
369C4×Q8—61no—
370C4×Q8C22×D313yes—
371C4×Q8C22×D311yescharacteristic
372C4×D4—33no—
373C4×D4—33no—
374C4×D4—33no—
375C4×D4—63no—
376C4×D4—63no—
377C4×D4C22×D313yes—
378C2×C4⋊C4—33no—
379C2×C4⋊C4—33no—
380C2×C4⋊C4—33no—
381C2×C4⋊C4—63no—
382C2×C4⋊C4—63no—
383C2×C4⋊C4—63no—
384C2×C22⋊C4—63no—
385C2×C42—33no—
386C2×C42—63no—
387#106—33no—
388#106—36no—
389#106—33no—
390#106—33no—
391#106—36no—
392#106—36no—
393#106C22×D313yes—
394#106C22×D316yes—
395C4⋊C8—36no—
396C4⋊C8—33no—
397C4⋊C8C22×D313yes—
398#103—33no—
399#103—33no—
400#103—33no—
401#103C22×D313yes—
402#102—33no—
403#102—33no—
404#102—33no—
405#102C22×D313yes—
406C4×C8—33no—
407C4×C8—33no—
408C4×C8—33no—
409C4×C8C22×D313yes—
410C23×C3—21no—
411C22×D3—21no—
412C22×D3—24no—
413C22×D3—42no—
414C22×D3—42no—
415C22×D3C22×D411yescharacteristic
416C3×Q8—21no—
417C3×Q8—46no—
418C3×Q8—83no—
419C3×Q8C2×SD1614yescomplemented
420C3×Q8C22×Q812yescomplemented
421C3×D4—21no—
422C3×D4C22×Q812yes—
423C2×C3×C4—21no—
424C2×C3×C4—23no—
425C2×C3×C4—23no—
426C2×C3×C4—46no—
427C2×C3×C4C22×D413yes—
428C2×C3×C4C22×D413yes—
429C2×C3×C4C2511yescharacteristic
430#67—24no—
431C2×Dic3—21no—
432C2×Dic3—23no—
433C2×Dic3—23no—
434C2×Dic3—46no—
435C2×Dic3C22×D413yes—
436C2×Dic3C22×D411yescharacteristic
437C2×Dic3C22×D413yes—
438D12—21no—
439D12C22×Q812yes—
440C4×D3—24no—
441C4×D3—26no—
442C4×D3—46no—
443C4×D3—46no—
444C4×D3—46no—
445C4×D3—46no—
446C4×D3C2×SD16112yes—
447C4×D3C22×Q812yes—
448C4×D32−1+412yes—
449Dic6—21no—
450Dic6—24no—
451Dic6—46no—
452Dic6—412no—
453Dic6—86no—
454Dic6—83no—
455Dic6C2×SD16112yescomplemented
456Dic6C22×Q812yescomplemented
457C3×C8—23no—
458C3×C8C22×Q812yescomplemented
459C3⋊C8—23no—
460C3⋊C8C22×Q812yescomplemented
461C24—61no—
462C2×Q8—23no—
463C2×Q8—23no—
464C2×Q8—38no—
465C2×Q8—33no—
466C2×Q8—34no—
467C2×Q8—31no—
468C2×Q8—44no—
469C2×Q8—61no—
470C2×Q8—66no—
471C2×Q8—66no—
472C2×Q8—62no—
473C2×Q8—63no—
474C2×Q8—63no—
475C2×Q8—1212no—
476C2×Q8—1212no—
477C2×Q8—1212no—
478C2×Q8—124no—
479C2×Q8—246no—
480C2×Q8D3×D411yescharacteristic
481C2×Q8C23×D311yescharacteristic
482C2×D4—34no—
483C2×D4—31no—
484C2×D4—62no—
485C2×D4—61no—
486C2×D4C23×D311yescharacteristic
487C22×C4—23no—
488C22×C4—33no—
489C22×C4—31no—
490C22×C4—33no—
491C22×C4—63no—
492C22×C4—61no—
493C22×C4—63no—
494C22×C4—63no—
495C22×C4—63no—
496C22×C4—63no—
497C22×C4—1212no—
498SD16—34no—
499SD16—34no—
500SD16—34no—
501SD16D3×Q814yesdirect product factor
502C2×C8—31no—
503C2×C8—34no—
504C2×C8—33no—
505C2×C8—66no—
506C2×C8C23×D313yes—
507C2×C8C23×D311yescharacteristic
508C4⋊C4—23no—
509C4⋊C4—23no—
510C4⋊C4—23no—
511C4⋊C4—36no—
512C4⋊C4—33no—
513C4⋊C4—33no—
514C4⋊C4—33no—
515C4⋊C4—63no—
516C4⋊C4—63no—
517C4⋊C4—63no—
518C4⋊C4—63no—
519C4⋊C4—66no—
520C4⋊C4—66no—
521C4⋊C4—63no—
522C4⋊C4—63no—
523C4⋊C4—63no—
524C4⋊C4D3×D413yes—
525C4⋊C4C23×D313yes—
526C4⋊C4C23×D313yes—
527C22⋊C4—23no—
528C22⋊C4—63no—
529C22⋊C4—63no—
530C22⋊C4—63no—
531C42—23no—
532C42—33no—
533C42—33no—
534C42—33no—
535C42—63no—
536C42—63no—
537C42—63no—
538C42C23×D313yes—
539C22×C3—22no—
540C22×C3—41no—
541C22×C3—41no—
542C22×C3C23×D411yescharacteristic
543C2×D3—22no—
544C2×D3—41no—
545C2×D3—41no—
546C2×D3—48no—
547C2×D3D4×Q812yes—
548C2×D3C22×SD1612yes—
549C2×D3#57712yes—
550C3×C4—22no—
551C3×C4—23no—
552C3×C4—43no—
553C3×C4—43no—
554C3×C4C22×SD1616yes—
555C3×C4C23×Q811yescharacteristic
556C3×C4C2×2−1+411yescharacteristic
557Dic3—22no—
558Dic3—23no—
559Dic3—43no—
560Dic3—43no—
561Dic3D4×Q812yes—
562Dic3C22×SD1616yes—
563C23—21no—
564C23—31no—
565C23—64no—
566C23—61no—
567C23—122no—
568C23—122no—
569Q8—21no—
570Q8—312no—
571Q8—32no—
572Q8—46no—
573Q8—64no—
574Q8—61no—
575Q8—83no—
576Q8—1212no—
577Q8—126no—
578Q8—243no—
579Q8—246no—
580Q8D3×SD1614yesdirect product factor
581Q8C2×D3×Q812yescomplemented
582D4—21no—
583D4—32no—
584D4—64no—
585D4—61no—
586D4C2×D3×Q812yes—
587C2×C4—21no—
588C2×C4—23no—
589C2×C4—23no—
590C2×C4—31no—
591C2×C4—33no—
592C2×C4—32no—
593C2×C4—32no—
594C2×C4—312no—
595C2×C4—33no—
596C2×C4—46no—
597C2×C4—66no—
598C2×C4—64no—
599C2×C4—61no—
600C2×C4—63no—
601C2×C4—63no—
602C2×C4—126no—
603C2×C4—126no—
604C2×C4—126no—
605C2×C4—126no—
606C2×C4—126no—
607C2×C4C2×D3×D413yes—
608C2×C4C2×D3×D413yes—
609C2×C4C24×D311yescharacteristic
610C8—23no—
611C8—32no—
612C8—63no—
613C8C2×D3×Q812yescomplemented
614C2×C3—42no—
615C2×C3C2×D4×Q811yescharacteristic
616C2×C3C23×SD1611yescharacteristic
617C2×C3C2×#57711yescharacteristic
618D3—42no—
619D3Q8×SD1614yesdirect product factor
620C22—22no—
621C22—32no—
622C22—32no—
623C22—32no—
624C22—41no—
625C22—41no—
626C22—62no—
627C22—128no—
628C22—121no—
629C22—121no—
630C22C22×D3×D411yescharacteristic
631C4—23no—
632C4—22no—
633C4—36no—
634C4—32no—
635C4—43no—
636C4—43no—
637C4—63no—
638C4—62no—
639C4—123no—
640C4—123no—
641C4C2×D3×SD1616yes—
642C4C22×D3×Q811yescharacteristic
643C4D3×2−1+411yescharacteristic
644C3C2×Q8×SD1611yescharacteristic, complemented, Sylow
645C2—34no—
646C2—42no—
647C2—122no—
648C2D3×D4×Q811yescharacteristic
649C2C22×D3×SD1611yescharacteristic
650C2D3×#57711yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×Q8×SD16C2
socleC22×C3C23×D4
prime radicalC2×D3×D4C23
hypercenterQ8×SD16D3
hypocenterC3C2×Q8×SD16
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×Q8 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
1. D3×Q82C2semidirect
3. C3×Q8×SD16C2semidirect
4. Q8×#215C2semidirect
5. Q8×#214C2semidirect
6. #81903C2semidirect
7. D3×#498C2semidirect
9. #81053C2semidirect
10. #81039C2semidirect
11. D3×#475C2semidirect
12. Q8×#204C2semidirect
13. D3×C8×Q8C2semidirect
15. SD16×Dic6C2semidirect
18. C3×Q82C22semidirect
27. #5722C22semidirect
35. Q8×Dic6C22semidirect
36. Q8×Dic6C22semidirect
39. C3×#498C22semidirect
41. C3×#475C22semidirect
42. C3×C8×Q8C22semidirect
45. #5094C22semidirect
47. #5080C22semidirect
48. #5080C22semidirect
50. Q8×C3⋊C8C22semidirect
51. #5020C22semidirect
57. #4855C22semidirect
58. #4855C22semidirect
59. #4854C22semidirect
61. #4851C22semidirect
64. #4830C22semidirect
66. #4791C22semidirect
73. #4680C22semidirect
74. C8×Dic6C22semidirect
84. Q8×SD16D3direct (N × H)
122. D3×SD16Q8direct (N × H)
189. Q82C2×D3semidirect
209. #498C2×D3semidirect
221. #475C2×D3semidirect
225. C8×Q8C2×D3semidirect
255. D3×Q8SD16direct (N × H)
256. D3×Q8C2×Q8semidirect
277. C3×SD16C2×Q8semidirect
290. #215C2×Q8semidirect
291. #214C2×Q8semidirect
315. #204C2×Q8semidirect
317. D3×C8C2×Q8semidirect
419. C3×Q8C2×SD16semidirect
420. C3×Q8C22×Q8semidirect
455. Dic6C2×SD16semidirect
456. Dic6C22×Q8semidirect
458. C3×C8C22×Q8semidirect
460. C3⋊C8C22×Q8semidirect
501. SD16D3×Q8direct (N × H)
580. Q8D3×SD16direct (N × H)
581. Q8C2×D3×Q8semidirect
613. C8C2×D3×Q8semidirect
619. D3Q8×SD16direct (N × H)
644. C3C2×Q8×SD16semidirect

Maximal subgroups

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

subgroupindex
D3×Q822
D3×D4×Q82
C3×Q8×SD162
Q8×#2152
Q8×#2142
#819032
D3×#4982
D3×#4762
#810532
#810392
D3×#4752
Q8×#2042
D3×C8×Q82
C4×D3×SD162
SD16×Dic62
C2×Q8×SD163