Finite Group Databasebeta

← D3×#2853 #11001068

Subgroups — D3×#2853 #11001068

Order 768.

3354 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#123C211yesmaximal, characteristic
2C22×C42×D3C211yesmaximal, characteristic, complemented
3C3×#2853C211yesmaximal, characteristic, complemented
4#82890C211yesmaximal, characteristic, complemented
5#82079C211yesmaximal, characteristic, complemented
6#75959C214yesmaximal, complemented
7D3×#388C214yesmaximal, complemented
8#75447C214yesmaximal, complemented
9D3×#387C214yesmaximal, complemented
10D3×#386C212yesmaximal
11#74871C212yesmaximal, complemented
12D3×#385C212yesmaximal, complemented
13#74588C212yesmaximal, complemented
14D3×#381C211yesmaximal, characteristic, complemented
15#74116C211yesmaximal, characteristic, complemented
16C2×#2853—31nomaximal, Sylow
17C22×D3×D4C2211yescharacteristic
18C23×C4×D3C2212yes—
19C23×C4×D3C2211yescharacteristic
20C2×C3×#123C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic, complemented
22C2×#935C2211yescharacteristic
23C2×#929C2211yescharacteristic
24C22×C4×Dic3C2212yescomplemented
25C22×C4×Dic3C2211yescharacteristic, complemented
26D3×#123—28no—
27C2×#894C2212yes—
28C2×D3×C4⋊C4C2212yes—
29C2×D3×C4⋊C4C2214yes—
30C2×#886C2211yescharacteristic
31C2×#883C2212yes—
32C2×D3×C22⋊C4C2214yes—
33C2×D3×C22⋊C4C2212yes—
34C2×D3×C22⋊C4C2211yescharacteristic
35C2×D3×C22⋊C4C2212yes—
36C2×C42×D3—28no—
37C2×C42×D3C2214yes—
38C2×C42×D3C2214yescomplemented
39C3×#388C2214yescomplemented
40C3×#387C2214yescomplemented
41C3×#386C2212yes—
42C3×#385C2212yescomplemented
43C3×#381C2211yescharacteristic, complemented
44#5220C2212yes—
45#5213C2212yes—
46#5211C2211yescharacteristic, complemented
47#5210C2212yescomplemented
48#4990C2214yescomplemented
49#4989C2214yescomplemented
50#4980C2214yescomplemented
51#4978C2214yescomplemented
52#4958C2212yes—
53#4956C2214yes—
54#4951C2214yescomplemented
55#4950C2212yescomplemented
56#4946C2212yescomplemented
57#4945C2214yescomplemented
58#4944C2214yescomplemented
59#4943C2212yescomplemented
60#4940C2214yescomplemented
61#4933C2214yescomplemented
62#4670C2214yescomplemented
63#4669C2214yescomplemented
64#4668C2214yescomplemented
65#4665C2214yes—
66#4664C2214yescomplemented
67D3×#95C2214yes—
68D3×#95C2214yes—
69D3×#95C2214yes—
70#4655C2214yescomplemented
71#4654C2214yescomplemented
72#4653C2214yescomplemented
73#4652C2214yescomplemented
74C22×#123—31no—
75C23×C42—31no—
76#2853—31no—
77#2853—34no—
78#2853—34no—
79#2853—32no—
80#2853—32no—
81#2853—31no—
82#2853—31no—
83#2853D311yescharacteristic, direct product factor
84C2×#388—34no—
85C2×#387—34no—
86C2×#386—32no—
87C2×#385—32no—
88C2×#381—31no—
89C24×D3C2311yescharacteristic
90C24×D3C2311yescharacteristic
91C22×C3×D4C2311yescharacteristic
92C23×C3×C4C2312yes—
93C23×C3×C4C2311yescharacteristic
94C22×#67C2311yescharacteristic
95C22×#67C2311yescharacteristic
96C23×Dic3C2312yes—
97C23×Dic3C2311yescharacteristic
98C2×D3×D4—24no—
99C2×D3×D4—28no—
100C22×D12C2311yescharacteristic
101C22×C4×D3—28no—
102C22×C4×D3—28no—
103C22×C4×D3—216no—
104C22×C4×D3D418yes—
105C22×C4×D3C2311yescharacteristic
106C22×C4×D3C2314yes—
107C22×C4×D3C2312yes—
108C22×C4×D3C2314yes—
109C22×C4×D3C2312yes—
110C22×C4×D3C2312yes—
111C22×C4×D3C2314yes—
112C3×#123—24no—
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C4⋊C4C2314yes—
115C2×C3×C22⋊C4C2312yes—
116C2×C3×C22⋊C4C2312yes—
117C2×C3×C22⋊C4C2314yes—
118C2×C3×C22⋊C4C2311yescharacteristic
119C2×C3×C42—24no—
120C2×C3×C42C2314yes—
121C2×#217C2312yes—
122C2×#217C2314yes—
123C2×#217C2312yes—
124C2×#217C2311yescharacteristic
125#935—24no—
126#929—24no—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2312yes—
131C2×(C2×D3)⋊C4C2312yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×C4⋊Dic3C2314yes—
134C2×C4⋊Dic3C2312yes—
135C2×Dic3⋊C4C2312yes—
136C2×Dic3⋊C4C2314yes—
137C2×Dic3⋊C4C2312yes—
138C2×Dic3⋊C4C2314yes—
139C2×C4×Dic3—24no—
140C2×C4×Dic3—28no—
141C2×C4×Dic3C2314yes—
142C2×C4×Dic3C2314yes—
143C2×C4×Dic3C2314yes—
144#894—28no—
145D3×C4⋊C4—28no—
146D3×C4⋊C4—216no—
147#886—24no—
148#883—28no—
149D3×C22⋊C4—24no—
150D3×C22⋊C4—28no—
151D3×C22⋊C4—28no—
152D3×C22⋊C4—216no—
153C42×D3—216no—
154C42×D3—216no—
155C3×#95C2314yes—
156C3×#95C2314yes—
157C3×#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C3⋊#95C2314yes—
163C3⋊#95C2314yes—
164C3⋊#95C2314yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167C23×D4—31no—
168C24×C4—31no—
169C24×C4—32no—
170C2×#123—31no—
171C2×#123—31no—
172C2×#123—32no—
173C2×#123—32no—
174C2×#123—31no—
175C2×#123—68no—
176C2×#123C2×D311yescharacteristic
177C22×C4⋊C4—32no—
178C22×C4⋊C4—34no—
179C22×C22⋊C4—31no—
180C22×C22⋊C4—32no—
181C22×C22⋊C4—32no—
182C22×C22⋊C4—34no—
183C22×C42—32no—
184C22×C42—31no—
185C22×C42—34no—
186C22×C42—34no—
187C22×C42—68no—
188C22×C42C2×D311yescharacteristic, complemented
189#388—34no—
190#388—34no—
191#388—34no—
192#388—34no—
193#388—34no—
194#388—34no—
195#388—34no—
196#388C2×D314yescomplemented
197#387—34no—
198#387—34no—
199#387—34no—
200#387—34no—
201#387—34no—
202#387—34no—
203#387—34no—
204#387C2×D314yescomplemented
205#386—32no—
206#386—32no—
207#386—34no—
208#386—32no—
209#386—34no—
210#386C2×D312yes—
211#385—32no—
212#385—34no—
213#385—32no—
214#385—32no—
215#385—34no—
216#385C2×D312yescomplemented
217#381—31no—
218#381—32no—
219#381—34no—
220#381C2×D311yescharacteristic, complemented
221C2×#95—34no—
222C2×#95—34no—
223C2×#95—34no—
224C24×C3C2411yescharacteristic
225C24×C3C2411yescharacteristic
226C23×D3—28no—
227C23×D3—22no—
228C23×D3—28no—
229C23×D3—24no—
230C23×D3C2×D414yes—
231C23×D3#4118yes—
232C23×D3C2411yescharacteristic
233C23×D3C2411yescharacteristic
234C23×D3C2411yescharacteristic
235C2×C3×D4—22no—
236C2×C3×D4—24no—
237C22×C3×C4—24no—
238C22×C3×C4—24no—
239C22×C3×C4—28no—
240C22×C3×C4C2×D414yes—
241C22×C3×C4C2411yescharacteristic
242C22×C3×C4C2414yes—
243C22×C3×C4C2412yes—
244C22×C3×C4C2412yes—
245C22×C3×C4C2414yes—
246C2×#67—22no—
247C2×#67—24no—
248C2×#67—22no—
249C2×#67—24no—
250C22×Dic3—24no—
251C22×Dic3—24no—
252C22×Dic3—28no—
253C22×Dic3C2×D414yes—
254C22×Dic3C2411yescharacteristic
255C22×Dic3C2414yes—
256C22×Dic3C2412yes—
257C22×Dic3C2412yes—
258C22×Dic3C2414yes—
259D3×D4—416no—
260C2×D12—22no—
261C2×D12—24no—
262C2×C4×D3—264no—
263C2×C4×D3—232no—
264C2×C4×D3—28no—
265C2×C4×D3—28no—
266C2×C4×D3—28no—
267C2×C4×D3—28no—
268C2×C4×D3—28no—
269C2×C4×D3—28no—
270C2×C4×D3—28no—
271C2×C4×D3—24no—
272C2×C4×D3—22no—
273C2×C4×D3—28no—
274C2×C4×D3—28no—
275C2×C4×D3—28no—
276C2×C4×D3—28no—
277C2×C4×D3—24no—
278C2×C4×D3C2×D418yes—
279C2×C4×D3C2×D418yes—
280C2×C4×D3#41116yes—
281C3×C4⋊C4—24no—
282C3×C4⋊C4—28no—
283C3×C22⋊C4—22no—
284C3×C22⋊C4—24no—
285C3×C22⋊C4—24no—
286C3×C22⋊C4—28no—
287C3×C42—28no—
288#217—22no—
289#217—24no—
290#217—24no—
291#217—28no—
292(C2×D3)⋊C4—24no—
293(C2×D3)⋊C4—28no—
294(C2×D3)⋊C4—24no—
295(C2×D3)⋊C4—28no—
296(C2×D3)⋊C4—24no—
297(C2×D3)⋊C4—28no—
298C4⋊Dic3—24no—
299C4⋊Dic3—28no—
300Dic3⋊C4—24no—
301Dic3⋊C4—28no—
302Dic3⋊C4—24no—
303Dic3⋊C4—28no—
304C4×Dic3—28no—
305C4×Dic3—28no—
306C4×Dic3—28no—
307C25—31no—
308C25—31no—
309C22×D4—31no—
310C22×D4—31no—
311C22×D4—31no—
312C22×D4—64no—
313C22×D4—68no—
314C22×D4C22×D311yescharacteristic
315C23×C4—31no—
316C23×C4—38no—
317C23×C4—32no—
318C23×C4—34no—
319C23×C4—32no—
320C23×C4—32no—
321C23×C4—32no—
322C23×C4—34no—
323C23×C4—31no—
324C23×C4—34no—
325C23×C4—616no—
326C23×C4—68no—
327C23×C4—68no—
328C23×C4C22×D312yes—
329C23×C4C22×D311yescharacteristic
330#123—24no—
331#123—68no—
332#123—64no—
333#123—68no—
334#123—64no—
335#123—64no—
336C2×C4⋊C4—32no—
337C2×C4⋊C4—32no—
338C2×C4⋊C4—32no—
339C2×C4⋊C4—34no—
340C2×C4⋊C4—34no—
341C2×C4⋊C4—34no—
342C2×C4⋊C4—68no—
343C2×C4⋊C4—616no—
344C2×C4⋊C4C22×D312yes—
345C2×C4⋊C4C22×D314yes—
346C2×C22⋊C4—31no—
347C2×C22⋊C4—32no—
348C2×C22⋊C4—32no—
349C2×C22⋊C4—34no—
350C2×C22⋊C4—32no—
351C2×C22⋊C4—32no—
352C2×C22⋊C4—32no—
353C2×C22⋊C4—34no—
354C2×C22⋊C4—34no—
355C2×C22⋊C4—34no—
356C2×C22⋊C4—68no—
357C2×C22⋊C4—616no—
358C2×C22⋊C4—64no—
359C2×C22⋊C4—68no—
360C2×C22⋊C4C22×D311yescharacteristic
361C2×C22⋊C4C22×D312yes—
362C2×C22⋊C4C22×D312yes—
363C2×C22⋊C4C22×D314yes—
364C2×C42—24no—
365C2×C42—34no—
366C2×C42—34no—
367C2×C42—34no—
368C2×C42—616no—
369C2×C42—616no—
370C2×C42—68no—
371C2×C42—64no—
372C2×C42C22×D314yes—
373#95—34no—
374#95—34no—
375#95—34no—
376#95—34no—
377#95—34no—
378#95—34no—
379#95—34no—
380#95—34no—
381#95—34no—
382#95C22×D314yes—
383#95C22×D314yes—
384#95C22×D314yes—
385C23×C3—24no—
386C23×C3—21no—
387C23×C3—24no—
388C23×C3—22no—
389C23×C3C22×D412yes—
390C23×C3C2×#4114yes—
391C23×C3C2511yescharacteristic
392C22×D3—24no—
393C22×D3—216no—
394C22×D3—232no—
395C22×D3—21no—
396C22×D3—24no—
397C22×D3—22no—
398C22×D3—44no—
399C22×D3—48no—
400C22×D3—416no—
401C22×D3#123116yes—
402C22×D3C22×D412yes—
403C22×D3C22×D412yes—
404C22×D3C2×#4114yes—
405C22×D3C2×#4114yes—
406C22×D3C2×#4118yes—
407C3×D4—44no—
408C2×C3×C4—216no—
409C2×C3×C4—28no—
410C2×C3×C4—24no—
411C2×C3×C4—24no—
412C2×C3×C4—24no—
413C2×C3×C4—21no—
414C2×C3×C4—22no—
415C2×C3×C4—24no—
416C2×C3×C4—24no—
417C2×C3×C4—24no—
418C2×C3×C4—24no—
419C2×C3×C4—22no—
420C2×C3×C4C22×D414yes—
421C2×C3×C4C2×#4118yes—
422#67—44no—
423#67—44no—
424C2×Dic3—216no—
425C2×Dic3—28no—
426C2×Dic3—24no—
427C2×Dic3—24no—
428C2×Dic3—24no—
429C2×Dic3—21no—
430C2×Dic3—22no—
431C2×Dic3—24no—
432C2×Dic3—24no—
433C2×Dic3—24no—
434C2×Dic3—24no—
435C2×Dic3—22no—
436C2×Dic3C22×D414yes—
437C2×Dic3C2×#4118yes—
438D12—44no—
439C4×D3—232no—
440C4×D3—216no—
441C4×D3—216no—
442C4×D3—232no—
443C4×D3—44no—
444C4×D3—48no—
445C4×D3—416no—
446C24—31no—
447C24—34no—
448C24—38no—
449C24—31no—
450C24—31no—
451C24—68no—
452C24—62no—
453C24—68no—
454C24—64no—
455C24C23×D311yescharacteristic
456C24C23×D311yescharacteristic
457C2×D4—22no—
458C2×D4—24no—
459C2×D4—62no—
460C2×D4—64no—
461C2×D4—62no—
462C2×D4—64no—
463C2×D4—62no—
464C2×D4—64no—
465C2×D4—1216no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—28no—
469C22×C4—34no—
470C22×C4—32no—
471C22×C4—34no—
472C22×C4—38no—
473C22×C4—38no—
474C22×C4—316no—
475C22×C4—34no—
476C22×C4—31no—
477C22×C4—32no—
478C22×C4—68no—
479C22×C4—68no—
480C22×C4—68no—
481C22×C4—632no—
482C22×C4—68no—
483C22×C4—68no—
484C22×C4—68no—
485C22×C4—664no—
486C22×C4—64no—
487C22×C4—68no—
488C22×C4—68no—
489C22×C4—62no—
490C22×C4—68no—
491C22×C4—68no—
492C22×C4—64no—
493C22×C4—68no—
494C22×C4—64no—
495C22×C4—64no—
496C22×C4—68no—
497C22×C4D3×D414yes—
498C22×C4C23×D314yes—
499C22×C4C23×D311yescharacteristic
500C22×C4C23×D312yes—
501C22×C4C23×D312yes—
502C22×C4C23×D314yes—
503C4⋊C4—24no—
504C4⋊C4—28no—
505C4⋊C4—64no—
506C4⋊C4—64no—
507C4⋊C4—68no—
508C4⋊C4—68no—
509C4⋊C4—64no—
510C4⋊C4—68no—
511C22⋊C4—22no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—28no—
515C22⋊C4—64no—
516C22⋊C4—68no—
517C22⋊C4—64no—
518C22⋊C4—68no—
519C22⋊C4—64no—
520C22⋊C4—68no—
521C22⋊C4—62no—
522C22⋊C4—64no—
523C22⋊C4—64no—
524C22⋊C4—68no—
525C42—28no—
526C42—68no—
527C42—68no—
528C42—68no—
529C22×C3—24no—
530C22×C3—28no—
531C22×C3—41no—
532C22×C3—42no—
533C22×C3—44no—
534C22×C3C2×#12314yes—
535C22×C3C23×D411yescharacteristic
536C22×C3C22×#4112yes—
537C22×C3C22×#4114yes—
538C2×D3—24no—
539C2×D3—28no—
540C2×D3—232no—
541C2×D3—41no—
542C2×D3—42no—
543C2×D3—44no—
544C2×D3—88no—
545C2×D3C2×#12314yes—
546C2×D3C2×#12314yes—
547C2×D3#52518yes—
548C2×D3#529116yes—
549C3×C4—24no—
550C3×C4—28no—
551C3×C4—41no—
552C3×C4—42no—
553C3×C4—44no—
554Dic3—24no—
555Dic3—28no—
556Dic3—41no—
557Dic3—42no—
558Dic3—44no—
559C23—24no—
560C23—21no—
561C23—24no—
562C23—22no—
563C23—32no—
564C23—38no—
565C23—34no—
566C23—316no—
567C23—32no—
568C23—34no—
569C23—616no—
570C23—632no—
571C23—64no—
572C23—61no—
573C23—64no—
574C23—62no—
575C23—124no—
576C23—128no—
577C23—1216no—
578C23C2×D3×D412yes—
579C23D3×#4114yes—
580C23C24×D311yescharacteristic
581D4—44no—
582D4—124no—
583D4—124no—
584D4—124no—
585C2×C4—216no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—34no—
598C2×C4—38no—
599C2×C4—632no—
600C2×C4—616no—
601C2×C4—616no—
602C2×C4—632no—
603C2×C4—616no—
604C2×C4—68no—
605C2×C4—64no—
606C2×C4—64no—
607C2×C4—64no—
608C2×C4—62no—
609C2×C4—61no—
610C2×C4—64no—
611C2×C4—64no—
612C2×C4—64no—
613C2×C4—64no—
614C2×C4—62no—
615C2×C4—124no—
616C2×C4—128no—
617C2×C4—1216no—
618C2×C4C2×D3×D414yes—
619C2×C4D3×#4118yes—
620C2×C3—24no—
621C2×C3—81no—
622C2×C3C22×#12311yescharacteristic
623C2×C3C2×#52512yes—
624C2×C3C2×#52914yes—
625D3—24no—
626D3—81no—
627D3#285318yesdirect product factor
628C22—24no—
629C22—28no—
630C22—38no—
631C22—316no—
632C22—34no—
633C22—34no—
634C22—42no—
635C22—44no—
636C22—41no—
637C22—632no—
638C22—64no—
639C22—68no—
640C22—122no—
641C22—124no—
642C22—121no—
643C22—248no—
644C22D3×#12314yes—
645C22C22×D3×D411yescharacteristic
646C22C2×D3×#4114yes—
647C22C2×D3×#4112yes—
648C4—28no—
649C4—24no—
650C4—42no—
651C4—44no—
652C4—41no—
653C4—68no—
654C4—64no—
655C4—122no—
656C4—124no—
657C4—121no—
658C3C2×#285311yescharacteristic, complemented, Sylow
659C2—24no—
660C2—38no—
661C2—64no—
662C2—81no—
663C2—241no—
664C2C2×D3×#12311yescharacteristic
665C2D3×#52512yes—
666C2D3×#52914yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2853C2
prime radicalC22×D3×D4C22
hypercenter#2853D3
hypocenterC3C2×#2853
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2853

Complements

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

NH ≅ G/Ntype
2. C22×C42×D3C2semidirect
3. C3×#2853C2semidirect
4. #82890C2semidirect
5. #82079C2semidirect
6. #75959C2semidirect
7. D3×#388C2semidirect
8. #75447C2semidirect
9. D3×#387C2semidirect
11. #74871C2semidirect
12. D3×#385C2semidirect
13. #74588C2semidirect
14. D3×#381C2semidirect
15. #74116C2semidirect
21. C22×C3×C42C22semidirect
24. C22×C4×Dic3C22semidirect
25. C22×C4×Dic3C22semidirect
38. C2×C42×D3C22semidirect
39. C3×#388C22semidirect
40. C3×#387C22semidirect
42. C3×#385C22semidirect
43. C3×#381C22semidirect
46. #5211C22semidirect
47. #5210C22semidirect
48. #4990C22semidirect
49. #4989C22semidirect
50. #4980C22semidirect
51. #4978C22semidirect
54. #4951C22semidirect
55. #4950C22semidirect
56. #4946C22semidirect
57. #4945C22semidirect
58. #4944C22semidirect
59. #4943C22semidirect
60. #4940C22semidirect
61. #4933C22semidirect
62. #4670C22semidirect
63. #4669C22semidirect
64. #4668C22semidirect
66. #4664C22semidirect
70. #4655C22semidirect
71. #4654C22semidirect
72. #4653C22semidirect
73. #4652C22semidirect
83. #2853D3direct (N × H)
188. C22×C42C2×D3semidirect
196. #388C2×D3semidirect
204. #387C2×D3semidirect
216. #385C2×D3semidirect
220. #381C2×D3semidirect
627. D3#2853direct (N × H)
658. C3C2×#2853semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#1232
C22×C42×D32
C3×#28532
#828902
#820792
#759592
D3×#3882
#754472
D3×#3872
D3×#3862
#748712
D3×#3852
#745882
D3×#3812
#741162
C2×#28533