Finite Group Databasebeta

← Dic3×#464 #10801423

Subgroups — Dic3×#464 #10801423

Order 768.

1016 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#464C211yesmaximal, characteristic
2C2×Dic3×Q16C211yesmaximal, characteristic, complemented
3C2×Dic3×SD16C211yesmaximal, characteristic
4Dic3×#124C211yesmaximal, characteristic
5C4×Q8×Dic3C211yesmaximal, characteristic, complemented
6#72065C211yesmaximal, characteristic, complemented
7#72048C211yesmaximal, characteristic
8#71721C211yesmaximal, characteristic
9#71627C211yesmaximal, characteristic, complemented
10#70455C211yesmaximal, characteristic, complemented
11Dic3×C4⋊C8C211yesmaximal, characteristic, complemented
12#70223C211yesmaximal, characteristic, complemented
13Dic3×#103C211yesmaximal, characteristic, complemented
14#70155C211yesmaximal, characteristic
15Dic3×#102C211yesmaximal, characteristic
16C4×#464—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C22×C3×SD16C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×Q8×Dic3C2211yescharacteristic
23C2×Q8×Dic3C2211yescharacteristic
24C2×D4×Dic3C2211yescharacteristic
25C3×#464C418yescomplemented
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5221C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37Dic3×Q16—24no—
38Dic3×SD16—24no—
39C2×#818C2211yescharacteristic
40C2×#817C2211yescharacteristic
41C2×C8×Dic3C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43Dic3×C4⋊C4—21no—
44Dic3×C4⋊C4—21no—
45Dic3×C4⋊C4C2211yescharacteristic
46Dic3×C4⋊C4C2211yescharacteristic
47#4951—21no—
48#4951—21no—
49Dic3×C22⋊C4—21no—
50Dic3×C22⋊C4—21no—
51#4936C2211yescharacteristic
52C4×C4⋊Dic3—21no—
53C4×C4⋊Dic3C2211yescharacteristic
54C42×Dic3—21no—
55C42×Dic3C2211yescharacteristic
56C2×C4⋊(C3⋊C8)C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#464D311yescharacteristic
64C2×C4×Q16—31no—
65C2×C4×SD16—31no—
66C4×#124—31no—
67C42×Q8—31no—
68#1983—31no—
69#1949—31no—
70#1927—31no—
71#1925—31no—
72#1880—31no—
73#1878—31no—
74#1805—31no—
75C4×C4⋊C8—31no—
76C4×#103—31no—
77C4×#102—31no—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×Q8C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×Dic3—21no—
82C2×C3×Q16—22no—
83C2×C3×Q16—22no—
84C2×C3×Q16C2×C414yescomplemented
85C2×C3×SD16—22no—
86C2×C3×SD16—22no—
87C2×C3×SD16C2×C414yes—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×#124—22no—
91C3×#124C2×C414yes—
92C3×C4×Q8—22no—
93C3×C4×Q8C2×C414yescomplemented
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103Q8×Dic3—22no—
104Q8×Dic3—22no—
105Q8×Dic3—22no—
106Q8×Dic3—22no—
107Q8×Dic3—22no—
108Q8×Dic3D414yes—
109D4×Dic3—22no—
110D4×Dic3—22no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3—21no—
124C2×C4×Dic3D411yescharacteristic
125C2×C4×Dic3D411yescharacteristic
126C2×C4×Dic3D411yescharacteristic
127C2×C4×Dic3D411yescharacteristic
128C2×C4×Dic3C2311yescharacteristic
129C2×C4×Dic3C2311yescharacteristic
130C22×C3⋊C8C2311yescharacteristic
131C22×C3⋊C8C2311yescharacteristic
132C3×C4⋊C8C2×C414yescomplemented
133C3×#103C2×C414yescomplemented
134C3×#102C2×C414yes—
135#835—22no—
136#835—22no—
137#835—22no—
138#835—22no—
139#832—22no—
140#832—22no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148#818—22no—
149#817—22no—
150C8×Dic3—22no—
151C8×Dic3—22no—
152C4⋊(C3⋊C8)—22no—
153C22×Q16C2×D311yescharacteristic
154C22×SD16C2×D311yescharacteristic
155C2×#124C2×D311yescharacteristic
156C2×C4×Q8—31no—
157C2×C4×Q8—31no—
158C2×C4×Q8—31no—
159C2×C4×Q8C2×D311yescharacteristic
160C2×C4×D4—31no—
161#464Dic318yesdirect product factor
162C4×Q16—64no—
163C4×SD16—64no—
164C2×#107—31no—
165C2×#106—31no—
166C2×C4⋊C8—31no—
167C2×C4⋊C8C2×D311yescharacteristic
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103C2×D311yescharacteristic
173C2×#102—31no—
174C2×#102—31no—
175C2×#102C2×D311yescharacteristic
176C2×C4×C8—31no—
177C2×C4×C8—31no—
178#391—31no—
179#390—31no—
180#388—61no—
181#388—61no—
182#382—31no—
183C4×C4⋊C4—31no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—61no—
187C4×C4⋊C4—61no—
188C4×C4⋊C4—61no—
189C4×C22⋊C4—61no—
190C4×C22⋊C4—61no—
191C43—31no—
192C43—61no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#336—31no—
198#336—31no—
199C24×C3—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8C22×C412yes—
209C2×C3×Q8C22×C412yes—
210C2×C3×Q8C2×D412yes—
211C2×C3×Q8#4112yes—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4C22×C412yes—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—42no—
229C22×Dic3—42no—
230C22×Dic3—42no—
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234C22×Dic3C2×D411yescharacteristic
235C3×Q16—28no—
236C3×SD16—28no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8C22×C412yes—
242C2×C3×C8C22×C412yes—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4C22×C412yes—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—24no—
248C3×C42—24no—
249C3×C42C22×C412yes—
250C3×C42#4112yes—
251#217—42no—
252#217—42no—
253#217—42no—
254C4⋊Dic3—24no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—24no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—42no—
263C4⋊Dic3C2×D412yes—
264C4⋊Dic3#4112yes—
265C4×Dic3—24no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—24no—
269C4×Dic3—22no—
270C4×Dic3—22no—
271C4×Dic3—22no—
272C4×Dic3—41no—
273C4×Dic3—42no—
274C4×Dic3—42no—
275C4×Dic3—42no—
276C4×Dic3—42no—
277C4×Dic3C2×D412yes—
278C4×Dic3#4112yes—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C22×Q8C22×D311yescharacteristic
286C22×Q8C22×D311yescharacteristic
287C22×D4C22×D311yescharacteristic
288C23×C4—61no—
289C2×Q16—22no—
290C2×Q16—22no—
291C2×Q16C2×Dic314yescomplemented
292C2×SD16—22no—
293C2×SD16—22no—
294C2×SD16C2×Dic314yes—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8C22×D311yescharacteristic
298C22×C8C22×D311yescharacteristic
299#124—22no—
300#124C2×Dic314yes—
301C4×Q8—22no—
302C4×Q8—34no—
303C4×Q8—62no—
304C4×Q8—62no—
305C4×Q8—62no—
306C4×Q8—62no—
307C4×Q8—62no—
308C4×Q8C2×Dic314yescomplemented
309C4×D4—62no—
310C4×D4—62no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C42—21no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—31no—
332C2×C42—61no—
333C2×C42—61no—
334C2×C42—61no—
335C2×C42—61no—
336C2×C42—61no—
337C2×C42—61no—
338C2×C42C22×D311yescharacteristic
339#107—62no—
340#106—62no—
341C4⋊C8—62no—
342C4⋊C8C2×Dic314yescomplemented
343#103—62no—
344#103—62no—
345#103—62no—
346#103—62no—
347#103C2×Dic314yescomplemented
348#102—62no—
349#102—62no—
350#102C2×Dic314yes—
351C4×C8—62no—
352C4×C8—62no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3—22no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3C22×D411yescharacteristic
368C3×Q8—22no—
369C3×Q8—22no—
370C3×Q8—22no—
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—41no—
374C3×Q8—41no—
375C3×Q8C4×D414yes—
376C3×D4—22no—
377C3×D4—22no—
378C3×D4—41no—
379C3×D4—41no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4C4×D412yes—
394C2×C3×C4C4×D412yes—
395C2×C3×C4C23×C411yescharacteristic
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—22no—
406C2×Dic3—21no—
407C2×Dic3—22no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—84no—
421C2×Dic3#12112yes—
422C2×Dic3#12112yes—
423C2×Dic3#13512yes—
424C2×Dic3#13512yes—
425C2×Dic3#13712yes—
426C3×C8—22no—
427C3×C8—22no—
428C3⋊C8—41no—
429C3⋊C8—41no—
430C24—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8D3×D412yes—
440C2×Q8#23712yes—
441C2×Q8C22×Dic312yes—
442C2×Q8C22×Dic312yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C22×Dic312yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—31no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—122no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4D3×D411yescharacteristic
464C22×C4D3×D411yescharacteristic
465C22×C4C23×D311yescharacteristic
466Q16—28no—
467SD16—28no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8C22×Dic312yes—
479C2×C8C22×Dic312yes—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—32no—
483C4⋊C4—32no—
484C4⋊C4—64no—
485C4⋊C4—64no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—122no—
493C4⋊C4C22×Dic312yes—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C22⋊C4—122no—
499C42—24no—
500C42—32no—
501C42—32no—
502C42—62no—
503C42—62no—
504C42—62no—
505C42—64no—
506C42—64no—
507C42—61no—
508C42—61no—
509C42—121no—
510C42—122no—
511C42—122no—
512C42—122no—
513C42—122no—
514C42#23712yes—
515C42C22×Dic312yes—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3C2×C4×D411yescharacteristic
524C22×C3C2×#12111yescharacteristic
525C22×C3#52511yescharacteristic
526C22×C3C2×#13511yescharacteristic
527C22×C3C2×#13511yescharacteristic
528C22×C3C2×#13711yescharacteristic
529C22×C3#57511yescharacteristic
530C3×C4—22no—
531C3×C4—22no—
532C3×C4—24no—
533C3×C4—42no—
534C3×C4—42no—
535C3×C4C2×C4×D411yescharacteristic
536C3×C4C4×#4111yescharacteristic
537C3×C4#52911yescharacteristic
538C3×C4#53211yescharacteristic
539Dic3—22no—
540Dic3—22no—
541Dic3—41no—
542Dic3—42no—
543Dic3—42no—
544Dic3—81no—
545Dic3—81no—
546Dic3#46414yesdirect product factor
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C2×D3×D411yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—22no—
559Q8—22no—
560Q8—41no—
561Q8—41no—
562Q8D4×Dic314yes—
563D4—22no—
564D4—22no—
565D4—41no—
566D4—41no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—32no—
577C2×C4—32no—
578C2×C4—32no—
579C2×C4—32no—
580C2×C4—32no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—61no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—244no—
605C2×C4D4×Dic312yes—
606C2×C4D4×Dic312yes—
607C2×C4C2×D3×D411yescharacteristic
608C2×C4C2×#23711yescharacteristic
609C2×C4C2×#23711yescharacteristic
610C2×C4D3×#4111yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4C23×Dic311yescharacteristic
613C8—22no—
614C8—22no—
615C8—121no—
616C8—121no—
617C2×C3—82no—
618C2×C3C4×#12111yescharacteristic
619C2×C3C4×#13511yescharacteristic
620C2×C3C4×#13711yescharacteristic
621C2×C3C2×#46411yescharacteristic
622C2×C3#303511yescharacteristic
623C2×C3#310211yescharacteristic
624C2×C3#310811yescharacteristic
625C22—42no—
626C22—42no—
627C22—42no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22D3×#12111yescharacteristic
633C22#560611yescharacteristic
634C22D3×#13511yescharacteristic
635C22D3×#13511yescharacteristic
636C22D3×#13711yescharacteristic
637C22#577711yescharacteristic
638C22C2×D4×Dic311yescharacteristic
639C4—22no—
640C4—24no—
641C4—22no—
642C4—34no—
643C4—42no—
644C4—42no—
645C4—62no—
646C4—62no—
647C4—121no—
648C4—122no—
649C4—122no—
650C4—241no—
651C4—241no—
652C4#559611yescharacteristic
653C4#560011yescharacteristic
654C4C2×D4×Dic311yescharacteristic
655C4Dic3×#4111yescharacteristic
656C3C4×#46411yescharacteristic, complemented, Sylow
657C2—82no—
658C2#8056811yescharacteristic
659C2#8056911yescharacteristic
660C2D3×#46411yescharacteristic
661C2#8059711yescharacteristic
662C2Dic3×#12111yescharacteristic
663C2Dic3×#13511yescharacteristic
664C2Dic3×#13711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#464C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenterC2×#464D3
hypocenterC3C4×#464
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×Q16C2semidirect
5. C4×Q8×Dic3C2semidirect
6. #72065C2semidirect
9. #71627C2semidirect
10. #70455C2semidirect
11. Dic3×C4⋊C8C2semidirect
12. #70223C2semidirect
13. Dic3×#103C2semidirect
25. C3×#464C4semidirect
84. C2×C3×Q16C2×C4semidirect
93. C3×C4×Q8C2×C4semidirect
132. C3×C4⋊C8C2×C4semidirect
133. C3×#103C2×C4semidirect
161. #464Dic3direct (N × H)
291. C2×Q16C2×Dic3semidirect
308. C4×Q8C2×Dic3semidirect
342. C4⋊C8C2×Dic3semidirect
347. #103C2×Dic3semidirect
546. Dic3#464direct (N × H)
656. C3C4×#464semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4642
C2×Dic3×Q162
C2×Dic3×SD162
Dic3×#1242
C4×Q8×Dic32
#720652
#720482
#717212
#716272
#704552
Dic3×C4⋊C82
#702232
Dic3×#1032
#701552
Dic3×#1022
C4×#4643