Finite Group Databasebeta

← Dic3×#951 #11860421

Subgroups — Dic3×#951 #11860421

Order 1152.

1049 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Dic3×#41C211yesmaximal, characteristic
2C2×C3×#951C211yesmaximal, characteristic
3C2×Dic3×Dic6C211yesmaximal, characteristic, complemented
4#10592931C211yesmaximal, characteristic
5#10591576C212yesmaximal, complemented
6Dic3×#216C212yesmaximal, complemented
7#10591440C212yesmaximal, complemented
8Dic3×#214C212yesmaximal, complemented
9#10591131C211yesmaximal, characteristic, complemented
10#10591099C211yesmaximal, characteristic, complemented
11Dic3×#208C211yesmaximal, characteristic, complemented
12Dic3×#137—31nomaximal
13C4×#951—31nomaximal
14C2×C32×#41C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×#3748C2211yescharacteristic
18C3×#951C418yescomplemented
19C3×Q8×Dic3C2211yescharacteristic
20C3×Q8×Dic3C2211yescharacteristic
21C2×C3×#216C2212yes—
22C3×D4×Dic3—21no—
23C3×D4×Dic3—21no—
24C3×D4×Dic3C2211yescharacteristic
25C3×D4×Dic3C2211yescharacteristic
26C2×C3×#214C2212yes—
27C3×#926—21no—
28C3×#926C2211yescharacteristic
29C2×C3×C4×Dic3—21no—
30C2×C3×C4×Dic3C2211yescharacteristic
31C2×C3×#208C2211yescharacteristic
32C2×Dic3⋊Dic3—21no—
33C2×Dic32—21no—
34Dic3×Dic6—21no—
35Dic3×Dic6—21no—
36Dic3×Dic6C2212yes—
37Dic3×Dic6C2212yes—
38#63662C2212yes—
39#63659C2212yes—
40#63575C2212yes—
41#63573C2212yes—
42#63565C2212yes—
43#63565C2212yes—
44#63554C2212yes—
45Dic3×C3⋊C8C2212yes—
46C2×C3×#137—31no—
47C3×C4×#41—31no—
48Dic3×#41D311yescharacteristic
49C2×#951D311yescharacteristic
50C2×Q8×Dic3—31no—
51C2×C4×Dic6—31no—
52#5232—31no—
53#5232—31no—
54#5232—61no—
55#5225—31no—
56#5184—32no—
57Dic3×Q16—32no—
58#5164—32no—
59Dic3×SD16—32no—
60#5121—31no—
61Dic3×M16—31no—
62#5030—32no—
63C4×#216—32no—
64#5018—32no—
65C4×#214—32no—
66#5004—31no—
67#4995—31no—
68C4×#208—31no—
69C32×#41—22no—
70C32×#41C2×C414yes—
71C2×C32×Q8C2311yescharacteristic
72C2×C32×D4—21no—
73C2×C32×D4C2311yescharacteristic
74C22×C32×C4—21no—
75C22×C32×C4C2311yescharacteristic
76C22×C3⋊Dic3—21no—
77C22×C3×Dic3—21no—
78C22×C3×Dic3—21no—
79C22×C3×Dic3—21no—
80C22×C3×Dic3D411yescharacteristic
81C22×C3×Dic3D411yescharacteristic
82C2×C3×Dic6—21no—
83C2×C3×Dic6—22no—
84C2×C3×Dic6C2×C414yescomplemented
85C2×C3×Dic6C2312yes—
86C4⋊(C3⋊Dic3)—21no—
87C4⋊(C3⋊Dic3)C2312yes—
88#3748—22no—
89C2×C3⋊(C3⋊C8)C2312yes—
90C3×#217—21no—
91C3×#217—21no—
92C3×#217—21no—
93C3×#216C2×C418yescomplemented
94C3×#214C2×C418yescomplemented
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3—21no—
97C3×C4⋊Dic3C2311yescharacteristic
98C3×C4⋊Dic3C2311yescharacteristic
99C3×C4×Dic3—21no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3—21no—
102C3×C4×Dic3D411yescharacteristic
103C3×C4×Dic3D411yescharacteristic
104C3×C4×Dic3C2311yescharacteristic
105C3×C4×Dic3C2311yescharacteristic
106C3×#208C2×C414yescomplemented
107C2×C3×C3⋊C8C2312yes—
108Dic3⋊Dic3—22no—
109Dic3⋊Dic3—22no—
110Dic32—22no—
111Dic32—22no—
112C4×#137—91noSylow
113C2×C3×#41—21no—
114C2×C3×#41C2×D311yescharacteristic
115C2×C3×#41C2×D311yescharacteristic
116C22×C3×Q8—31no—
117C22×Dic6C2×D311yescharacteristic
118C3×#137—38no—
119C2×C3×Q16—32no—
120C2×C3×SD16—32no—
121C2×C3×M16—31no—
122C3×C4×Q8—31no—
123C3×C4×Q8—31no—
124C3×C4×D4—31no—
125C3×C4×D4—31no—
126C3×C4×D4—61no—
127C3×C4×D4—61no—
128C3×#117—31no—
129C3×#117—61no—
130C2×C3×C42—31no—
131C2×C3×C42—61no—
132#951Dic318yesdirect product factor
133Q8×Dic3—32no—
134Q8×Dic3—32no—
135Q8×Dic3—61no—
136Q8×Dic3—61no—
137Q8×Dic3C2×D311yescharacteristic
138Q8×Dic3C2×D311yescharacteristic
139C2×#216C2×D312yes—
140D4×Dic3—21no—
141D4×Dic3—21no—
142D4×Dic3C2×D311yescharacteristic
143D4×Dic3C2×D311yescharacteristic
144C2×#214C2×D312yes—
145#926—21no—
146#926C2×D311yescharacteristic
147C2×C4⋊Dic3—31no—
148C2×C4⋊Dic3—31no—
149C2×C4⋊Dic3—61no—
150C2×C4⋊Dic3—61no—
151C2×Dic3⋊C4—61no—
152C2×C4×Dic3—21no—
153C2×C4×Dic3—61no—
154C2×C4×Dic3—61no—
155C2×C4×Dic3C2×D311yescharacteristic
156C2×#208—31no—
157C2×#208—31no—
158C2×#208—61no—
159C2×#208C2×D311yescharacteristic
160C4×Dic6—32no—
161C4×Dic6—32no—
162C4×Dic6—61no—
163C4×Dic6—61no—
164#835—32no—
165#835—32no—
166#835—32no—
167#835—32no—
168#835—62no—
169#832—32no—
170#832—32no—
171#832—62no—
172#818—32no—
173#817—32no—
174#815—32no—
175C8×Dic3—32no—
176#810—32no—
177#810—32no—
178#808—32no—
179#807—32no—
180#803—32no—
181C4×C3⋊C8—32no—
182C23×C32—21no—
183C23×C32C2×D411yescharacteristic
184C32×Q8—21no—
185C32×Q8C22×C412yes—
186C32×D4—24no—
187C32×D4—21no—
188C32×D4C22×C412yes—
189C2×C32×C4—21no—
190C2×C32×C4—24no—
191C2×C32×C4C22×C412yes—
192C2×C32×C4C2×D411yescharacteristic
193C2×C32×C4#4112yes—
194C2×C32×C4C2411yescharacteristic
195C2×C3⋊Dic3—22no—
196C2×C3⋊Dic3—42no—
197C2×C3×Dic3—24no—
198C2×C3×Dic3—22no—
199C2×C3×Dic3—22no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—21no—
202C2×C3×Dic3—41no—
203C2×C3×Dic3—42no—
204C2×C3×Dic3—42no—
205C2×C3×Dic3C2×D411yescharacteristic
206C2×C3×Dic3C2×D411yescharacteristic
207C2×C3×Dic3C2×D411yescharacteristic
208C2×C3×Dic3C2×D411yescharacteristic
209C3×Dic6—24no—
210C3×Dic6—22no—
211C3×Dic6C22×C414yes—
212C3⋊(C3⋊C8)—22no—
213C3×C3⋊C8C22×C414yes—
214C2×#137—31no—
215C4×#41—31no—
216C2×C4×Q8—91no—
217#441—92no—
218#440—92no—
219C4×Q16—92no—
220C4×SD16—92no—
221#428—91no—
222#419—91no—
223#417—91no—
224C4×M16—91no—
225C3×#41—22no—
226C3×#41—22no—
227C3×#41—24no—
228C3×#41—42no—
229C3×#41C4×D314yes—
230C3×#41C2×Dic314yes—
231C2×C3×Q8—21no—
232C2×C3×Q8—34no—
233C2×C3×Q8—32no—
234C2×C3×Q8—62no—
235C2×C3×Q8—61no—
236C2×C3×Q8C22×D311yescharacteristic
237C2×C3×Q8C22×D311yescharacteristic
238C2×C3×D4—21no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—41no—
242C2×C3×D4C22×D311yescharacteristic
243C2×C3×D4C22×D311yescharacteristic
244C22×C3×C4—21no—
245C22×C3×C4—21no—
246C22×C3×C4—21no—
247C22×C3×C4—31no—
248C22×C3×C4—31no—
249C22×C3×C4—41no—
250C22×C3×C4—61no—
251C22×C3×C4—61no—
252C22×C3×C4—61no—
253C22×C3×C4C22×D311yescharacteristic
254C22×C3×C4C22×D311yescharacteristic
255C22×Dic3—21no—
256C22×Dic3—21no—
257C22×Dic3—21no—
258C22×Dic3—61no—
259C22×Dic3—61no—
260C22×Dic3—121no—
261C22×Dic3#6711yescharacteristic
262C22×Dic3#6711yescharacteristic
263C2×Dic6—21no—
264C2×Dic6—22no—
265C2×Dic6C2×Dic314yescomplemented
266C2×Dic6C22×D312yes—
267C3×Q16—38no—
268C3×SD16—38no—
269C3×M16—34no—
270C2×C3×C8—32no—
271C3×C4⋊C4—31no—
272C3×C4⋊C4—31no—
273C3×C4⋊C4—61no—
274C3×C4⋊C4—61no—
275C3×C22⋊C4—61no—
276C3×C22⋊C4—61no—
277C3×C22⋊C4—61no—
278C3×C42—31no—
279C3×C42—31no—
280C3×C42—31no—
281C3×C42—31no—
282C3×C42—61no—
283C3×C42—61no—
284C3×C42—61no—
285#217—21no—
286#217—21no—
287#217—21no—
288#216C2×Dic318yescomplemented
289#214C2×Dic318yescomplemented
290C4⋊Dic3—21no—
291C4⋊Dic3—21no—
292C4⋊Dic3—32no—
293C4⋊Dic3—32no—
294C4⋊Dic3—62no—
295C4⋊Dic3—61no—
296C4⋊Dic3—61no—
297C4⋊Dic3—62no—
298C4⋊Dic3—62no—
299C4⋊Dic3—121no—
300C4⋊Dic3C22×D311yescharacteristic
301C4⋊Dic3C22×D311yescharacteristic
302Dic3⋊C4—62no—
303Dic3⋊C4—62no—
304C4×Dic3—21no—
305C4×Dic3—21no—
306C4×Dic3—21no—
307C4×Dic3—62no—
308C4×Dic3—62no—
309C4×Dic3—62no—
310C4×Dic3—62no—
311C4×Dic3#6711yescharacteristic
312C4×Dic3#6711yescharacteristic
313C4×Dic3C22×D311yescharacteristic
314C4×Dic3C22×D311yescharacteristic
315#208—62no—
316#208—62no—
317#208—122no—
318#208C2×Dic314yescomplemented
319C2×C3⋊C8—32no—
320C2×C3⋊C8—32no—
321C2×C3⋊C8—62no—
322C2×C3⋊C8C22×D312yes—
323C22×C32—21no—
324C22×C32—21no—
325C22×C32—22no—
326C22×C32—41no—
327C22×C32—41no—
328C22×C32C4×D412yes—
329C22×C32C22×D411yescharacteristic
330C32×C4—22no—
331C32×C4C4×D412yes—
332C32×C4C23×C411yescharacteristic
333C32×C4C2×#4111yescharacteristic
334C3⋊Dic3—42no—
335C3×Dic3—24no—
336C3×Dic3—21no—
337C3×Dic3—21no—
338C3×Dic3—21no—
339C3×Dic3—41no—
340C3×Dic3—41no—
341C3×Dic3#13712yescomplemented
342C2×#41D3211yescharacteristic
343C22×Q8—31no—
344#137—38no—
345C2×Q16—32no—
346C2×SD16—32no—
347C2×M16—31no—
348C2×M16—91no—
349C4×Q8—31no—
350C4×Q8—31no—
351C4×Q8—92no—
352C4×Q8—92no—
353C4×Q8—181no—
354C4×Q8—181no—
355C4×D4—31no—
356C4×D4—31no—
357C4×D4—61no—
358C4×D4—61no—
359#117—31no—
360#117—61no—
361C2×C4⋊C4—91no—
362C2×C4⋊C4—181no—
363C2×C42—31no—
364C2×C42—61no—
365C2×C42—181no—
366#107—92no—
367#106—92no—
368#103—92no—
369#103—92no—
370#103—92no—
371#102—92no—
372#97—92no—
373C4×C8—92no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—41no—
378C23×C3D3×D411yescharacteristic
379C23×C3C2×#6711yescharacteristic
380C3×Q8—21no—
381C3×Q8—21no—
382C3×Q8—22no—
383C3×Q8—34no—
384C3×Q8—41no—
385C3×Q8—64no—
386C3×Q8—62no—
387C3×Q8C2×C4×D312yes—
388C3×Q8C22×Dic312yes—
389C3×D4—24no—
390C3×D4—24no—
391C3×D4—21no—
392C3×D4—21no—
393C3×D4—22no—
394C3×D4—41no—
395C3×D4—44no—
396C3×D4C2×C4×D312yes—
397C3×D4C22×Dic312yes—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—24no—
401C2×C3×C4—24no—
402C2×C3×C4—21no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—21no—
406C2×C3×C4—31no—
407C2×C3×C4—31no—
408C2×C3×C4—31no—
409C2×C3×C4—31no—
410C2×C3×C4—44no—
411C2×C3×C4—41no—
412C2×C3×C4—62no—
413C2×C3×C4—64no—
414C2×C3×C4—62no—
415C2×C3×C4—61no—
416C2×C3×C4—61no—
417C2×C3×C4—122no—
418C2×C3×C4—121no—
419C2×C3×C4—122no—
420C2×C3×C4C2×C4×D312yes—
421C2×C3×C4#23512yes—
422C2×C3×C4D3×D411yescharacteristic
423C2×C3×C4#23712yes—
424C2×C3×C4C22×Dic312yes—
425C2×C3×C4C2×#6711yescharacteristic
426C2×C3×C4C23×D311yescharacteristic
427C2×C3×C4C23×D311yescharacteristic
428C2×Dic3—24no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—41no—
434C2×Dic3—42no—
435C2×Dic3—42no—
436C2×Dic3—62no—
437C2×Dic3—62no—
438C2×Dic3—122no—
439C2×Dic3—122no—
440C2×Dic3—122no—
441C2×Dic3—242no—
442C2×Dic3C2×#6711yescharacteristic
443C2×Dic3C2×#6711yescharacteristic
444C2×Dic3C2×#6711yescharacteristic
445C2×Dic3C2×#6711yescharacteristic
446Dic6—24no—
447Dic6—22no—
448Dic6C22×Dic314yes—
449C3×C8—34no—
450C3⋊C8—62no—
451C3⋊C8—62no—
452C3⋊C8—122no—
453C3⋊C8C22×Dic314yes—
454C2×C32—22no—
455C2×C32—42no—
456C2×C32C2×C4×D411yescharacteristic
457C2×C32C2×#13711yescharacteristic
458C2×C32#57511yescharacteristic
459#41—22no—
460#41D3×Dic314yes—
461C2×Q8—32no—
462C2×Q8—34no—
463C2×Q8—62no—
464C2×Q8—61no—
465C2×Q8C2×D3211yescharacteristic
466C2×D4—21no—
467C2×D4C2×D3211yescharacteristic
468C22×C4—21no—
469C22×C4—31no—
470C22×C4—31no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—181no—
475C22×C4C2×D3211yescharacteristic
476Q16—38no—
477SD16—38no—
478M16—34no—
479M16—182no—
480C2×C8—32no—
481C2×C8—92no—
482C4⋊C4—31no—
483C4⋊C4—31no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—92no—
487C4⋊C4—182no—
488C4⋊C4—182no—
489C4⋊C4—181no—
490C22⋊C4—61no—
491C22⋊C4—61no—
492C22⋊C4—61no—
493C42—31no—
494C42—31no—
495C42—31no—
496C42—31no—
497C42—61no—
498C42—61no—
499C42—61no—
500C42—182no—
501C42—182no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—21no—
509C22×C3—22no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—42no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3C4×#6712yes—
520C22×C3D4×Dic312yes—
521C22×C3C2×D3×D411yescharacteristic
522C22×C3C22×#6711yescharacteristic
523C3×C4—22no—
524C3×C4—22no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—22no—
528C3×C4—32no—
529C3×C4—42no—
530C3×C4—61no—
531C3×C4—61no—
532C3×C4—61no—
533C3×C4—64no—
534C3×C4—121no—
535C3×C4—121no—
536C3×C4C4×#6712yes—
537C3×C4D4×Dic312yes—
538C3×C4C22×C4×D311yescharacteristic
539C3×C4C2×#23511yescharacteristic
540C3×C4C2×#23711yescharacteristic
541C3×C4C23×Dic311yescharacteristic
542Dic3—24no—
543Dic3—21no—
544Dic3—21no—
545Dic3—21no—
546Dic3—41no—
547Dic3—41no—
548Dic3—122no—
549Dic3—122no—
550Dic3—242no—
551Dic3#95112yesdirect product factor
552C32C4×#13711yescharacteristic, complemented, Sylow
553C23—21no—
554C23D3×#6711yescharacteristic
555Q8—21no—
556Q8—34no—
557Q8—62no—
558Q8—64no—
559Q8C2×D3×Dic312yes—
560D4—24no—
561D4—21no—
562D4C2×D3×Dic312yes—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—31no—
566C2×C4—31no—
567C2×C4—31no—
568C2×C4—31no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—64no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—122no—
575C2×C4—121no—
576C2×C4—122no—
577C2×C4—182no—
578C2×C4—362no—
579C2×C4C2×D3×Dic312yes—
580C2×C4#380412yes—
581C2×C4D3×#6711yescharacteristic
582C2×C4C22×D3211yescharacteristic
583C8—34no—
584C8—182no—
585C2×C3—22no—
586C2×C3—22no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—42no—
591C2×C3—42no—
592C2×C3—42no—
593C2×C3—82no—
594C2×C3D3×#13711yescharacteristic
595C2×C3#577711yescharacteristic
596C2×C3C2×C4×#6711yescharacteristic
597C2×C3C2×D4×Dic311yescharacteristic
598C2×C3#582211yescharacteristic
599C2×C3C2×#95111yescharacteristic
600C22—21no—
601C22—21no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22Dic3×#6712yes—
606C22C2×D3×#6711yescharacteristic
607C4—22no—
608C4—32no—
609C4—61no—
610C4—61no—
611C4—61no—
612C4—64no—
613C4—121no—
614C4—121no—
615C4—362no—
616C4Dic3×#6712yes—
617C4C22×D3×Dic311yescharacteristic
618C4C2×#380411yescharacteristic
619C3—21no—
620C3C4×#95111yescharacteristic, complemented
621C3Dic3×#13711yescharacteristic, complemented
622C2—22no—
623C2—42no—
624C2D3×#95111yescharacteristic
625C2#1059547011yescharacteristic
626C2C2×Dic3×#6711yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutatorC32×C4C23×C4
FrattiniC2×C4C22×D32
Fitting, prime radicalC2×C32×#41C22
socleC22×C32C22×D4
cosocleC2×C32×C4C24
hypercenterC2×#41D32
hypocenterC32C4×#137
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
3. C2×Dic3×Dic6C2semidirect
5. #10591576C2semidirect
6. Dic3×#216C2semidirect
7. #10591440C2semidirect
8. Dic3×#214C2semidirect
9. #10591131C2semidirect
10. #10591099C2semidirect
11. Dic3×#208C2semidirect
18. C3×#951C4semidirect
84. C2×C3×Dic6C2×C4semidirect
93. C3×#216C2×C4semidirect
94. C3×#214C2×C4semidirect
106. C3×#208C2×C4semidirect
132. #951Dic3direct (N × H)
265. C2×Dic6C2×Dic3semidirect
288. #216C2×Dic3semidirect
289. #214C2×Dic3semidirect
318. #208C2×Dic3semidirect
341. C3×Dic3#137semidirect
551. Dic3#951direct (N × H)
552. C32C4×#137semidirect
620. C3C4×#951semidirect
621. C3Dic3×#137semidirect

Maximal subgroups

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

subgroupindex
C3×Dic3×#412
C2×C3×#9512
C2×Dic3×Dic62
#105929312
#105915762
Dic3×#2162
#105914402
Dic3×#2142
#105911312
#105910992
Dic3×#2082
Dic3×#1373
C4×#9513