Finite Group Databasebeta

← Dic3×#886 #11802168

Subgroups — Dic3×#886 #11802168

Order 1152.

1493 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#886C211yesmaximal, characteristic
2C2×Dic3×#67C211yesmaximal, characteristic
3C22×Dic32C211yesmaximal, characteristic, complemented
4#10592649C211yesmaximal, characteristic, complemented
5C3×Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
6#10591713C211yesmaximal, characteristic
7#10591696C211yesmaximal, characteristic, complemented
8Dic3×(C2×D3)⋊C4C212yesmaximal, complemented
9#10591316C212yesmaximal, complemented
10#10591269C212yesmaximal, complemented
11Dic3×C4⋊Dic3C212yesmaximal, complemented
12Dic3×#123—31nomaximal
13C4×#886—31nomaximal
14C23×C3⋊Dic3C2211yescharacteristic
15C22×C3×#67C2211yescharacteristic
16C23×C3×Dic3C2211yescharacteristic
17C23×C3×Dic3C2211yescharacteristic
18C22×D3×Dic3C2211yescharacteristic
19C2×C32×C22⋊C4C2211yescharacteristic
20C2×#3757C2211yescharacteristic
21C2×C4⋊(C3⋊Dic3)C2212yes—
22C2×C3×#217C2211yescharacteristic
23C2×C3×(C2×D3)⋊C4C2212yes—
24C2×C3×C4⋊Dic3C2212yes—
25C2×C3×C4×Dic3C2212yes—
26C3×#886C418yescomplemented
27Dic3×#67—24no—
28C2×Dic3⋊Dic3C2211yescharacteristic
29C2×(C2×D3)⋊Dic3C2211yescharacteristic
30C2×(C2×D3)⋊Dic3C2212yes—
31C2×Dic32—24no—
32C2×Dic32C2211yescharacteristic
33C2×Dic32C2212yes—
34C2×Dic32C2212yes—
35#63658C2212yes—
36C3×C3⋊#95C2212yes—
37#63579C2212yes—
38#63579C2212yes—
39#63579C2212yes—
40C2×C3×#123—31no—
41C2×D4×Dic3—31no—
42C2×C4×#67—31no—
43C22×C4×Dic3—31no—
44C22×C4×Dic3—31no—
45C2×#886D311yescharacteristic
46C3×C4×C22⋊C4—31no—
47#5220—31no—
48#5211—31no—
49#4980—32no—
50Dic3×C4⋊C4—32no—
51#4958—31no—
52#4951—32no—
53#4950—31no—
54#4950—31no—
55#4950—61no—
56#4943—31no—
57Dic3×C22⋊C4—32no—
58Dic3×C22⋊C4D311yescharacteristic, complemented
59C4×(C2×D3)⋊C4—32no—
60C4×C4⋊Dic3—32no—
61#4670—32no—
62#4653—32no—
63C24×C32C2311yescharacteristic
64C23×C3×D3C2311yescharacteristic
65C22×C32×C4C2312yes—
66C22×C3⋊Dic3—22no—
67C22×C3⋊Dic3—24no—
68C22×C3⋊Dic3C2311yescharacteristic
69C22×C3⋊Dic3C2311yescharacteristic
70C22×C3⋊Dic3C2312yes—
71C2×C3×#67—22no—
72C2×C3×#67—22no—
73C2×C3×#67C2×C414yes—
74C22×C3×Dic3—22no—
75C22×C3×Dic3—22no—
76C22×C3×Dic3—24no—
77C22×C3×Dic3C2×C414yescomplemented
78C22×C3×Dic3D414yes—
79C22×C3×Dic3C2312yes—
80C22×C3×Dic3C2311yescharacteristic
81C22×C3×Dic3C2312yes—
82C22×C3×Dic3C2311yescharacteristic
83C22×C3×Dic3C2311yescharacteristic
84C2×D3×Dic3—22no—
85C2×D3×Dic3—24no—
86C32×C22⋊C4C2×C414yescomplemented
87#3757—22no—
88C4⋊(C3⋊Dic3)—24no—
89C3×#217—22no—
90C3×(C2×D3)⋊C4C2×C418yescomplemented
91C3×C4⋊Dic3C2×C418yescomplemented
92C3×C4×Dic3—24no—
93Dic3⋊Dic3—22no—
94(C2×D3)⋊Dic3—24no—
95(C2×D3)⋊Dic3—22no—
96Dic32—24no—
97Dic32—24no—
98Dic32—22no—
99C4×#123—91noSylow
100C22×C3×D4—31no—
101C23×C3×C4—31no—
102C23×C3×C4—31no—
103C22×#67C2×D311yescharacteristic
104C23×Dic3—31no—
105C23×Dic3—31no—
106C23×Dic3—31no—
107C23×Dic3—61no—
108C23×Dic3C2×D311yescharacteristic
109C23×Dic3C2×D311yescharacteristic
110C22×C4×D3—31no—
111C3×#123—38no—
112C2×C3×C4⋊C4—32no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—32no—
115C2×C3×C22⋊C4—31no—
116C2×C3×C22⋊C4C2×D311yescharacteristic
117C2×C3×C22⋊C4C2×D311yescharacteristic
118C2×C3×C42—32no—
119C2×#217—31no—
120C2×#217—31no—
121C2×#217—32no—
122C2×#217—31no—
123C2×#217—61no—
124C2×#217C2×D311yescharacteristic
125D4×Dic3—64no—
126C4×#67—64no—
127C2×(C2×D3)⋊C4—31no—
128C2×(C2×D3)⋊C4—32no—
129C2×(C2×D3)⋊C4C2×D312yes—
130C2×C4⋊Dic3—32no—
131C2×C4⋊Dic3—32no—
132C2×C4⋊Dic3—31no—
133C2×C4⋊Dic3—62no—
134C2×C4⋊Dic3C2×D312yes—
135C2×Dic3⋊C4—31no—
136C2×C4×Dic3—32no—
137C2×C4×Dic3—31no—
138C2×C4×Dic3—32no—
139C2×C4×Dic3—32no—
140C2×C4×Dic3—32no—
141C2×C4×Dic3—31no—
142C2×C4×Dic3—64no—
143C2×C4×Dic3—64no—
144C2×C4×Dic3C2×D312yes—
145#886Dic318yesdirect product factor
146C3×#95—32no—
147C3⋊#95—32no—
148C3⋊#95—32no—
149C3⋊#95—32no—
150C3⋊#95—32no—
151C3⋊#95—32no—
152C3⋊#95—32no—
153C3⋊#95—32no—
154C3⋊#95—32no—
155C3⋊#95—62no—
156C3⋊#95C2×D312yes—
157C23×C32—22no—
158C23×C32—22no—
159C23×C32C22×C412yes—
160C23×C32C2×D412yes—
161C23×C32#4112yes—
162C23×C32C2411yescharacteristic
163C22×C3×D3—21no—
164C22×C3×D3—22no—
165C22×C3×D3—22no—
166C22×C3×D3—21no—
167C22×C3×D3C22×C412yes—
168C2×C32×C4—22no—
169C2×C32×C4—22no—
170C2×C32×C4C22×C414yes—
171C2×C3⋊Dic3—28no—
172C2×C3⋊Dic3—22no—
173C2×C3⋊Dic3—22no—
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3—22no—
177C2×C3⋊Dic3—22no—
178C2×C3⋊Dic3#4114yes—
179C3×#67—28no—
180C2×C3×Dic3—28no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—28no—
187C2×C3×Dic3—22no—
188C2×C3×Dic3—22no—
189C2×C3×Dic3—22no—
190C2×C3×Dic3C22×C412yes—
191C2×C3×Dic3C22×C414yes—
192C2×C3×Dic3C2×D412yes—
193C2×C3×Dic3C2×D412yes—
194C2×C3×Dic3#4114yes—
195D3×Dic3—44no—
196C2×#123—31no—
197C2×C4×D4—91no—
198C22×C42—91no—
199#388—92no—
200#387—92no—
201#386—91no—
202#385—91no—
203#381—91no—
204C4×C4⋊C4—92no—
205C4×C22⋊C4—31no—
206C4×C22⋊C4—92no—
207C24×C3—21no—
208C24×C3—31no—
209C24×C3C22×D311yescharacteristic
210C24×C3C22×D311yescharacteristic
211C23×D3C22×D311yescharacteristic
212C2×C3×D4—34no—
213C2×C3×D4—62no—
214C2×C3×D4—62no—
215C22×C3×C4—22no—
216C22×C3×C4—31no—
217C22×C3×C4—34no—
218C22×C3×C4—34no—
219C22×C3×C4—32no—
220C22×C3×C4—31no—
221C22×C3×C4—31no—
222C22×C3×C4—32no—
223C22×C3×C4—64no—
224C22×C3×C4—62no—
225C22×C3×C4—62no—
226C22×C3×C4C22×D312yes—
227C22×C3×C4C22×D312yes—
228C2×#67—22no—
229C2×#67—22no—
230C2×#67C2×Dic314yes—
231C22×Dic3—22no—
232C22×Dic3—22no—
233C22×Dic3—24no—
234C22×Dic3—32no—
235C22×Dic3—32no—
236C22×Dic3—31no—
237C22×Dic3—31no—
238C22×Dic3—31no—
239C22×Dic3—31no—
240C22×Dic3—62no—
241C22×Dic3—61no—
242C22×Dic3—61no—
243C22×Dic3—64no—
244C22×Dic3—64no—
245C22×Dic3—62no—
246C22×Dic3—62no—
247C22×Dic3—62no—
248C22×Dic3—64no—
249C22×Dic3—124no—
250C22×Dic3—122no—
251C22×Dic3D1214yes—
252C22×Dic3C2×Dic314yescomplemented
253C22×Dic3C22×D312yes—
254C22×Dic3C22×D311yescharacteristic
255C22×Dic3C22×D312yes—
256C22×Dic3C22×D311yescharacteristic
257C22×Dic3C22×D311yescharacteristic
258C2×C4×D3—62no—
259C2×C4×D3—64no—
260C3×C4⋊C4—38no—
261C3×C22⋊C4—24no—
262C3×C22⋊C4—38no—
263C3×C22⋊C4—62no—
264C3×C22⋊C4C4×D314yescomplemented
265C3×C22⋊C4C2×Dic314yescomplemented
266C3×C42—64no—
267#217—22no—
268#217—64no—
269#217—62no—
270#217—62no—
271#217—62no—
272#217—122no—
273(C2×D3)⋊C4—64no—
274(C2×D3)⋊C4—62no—
275(C2×D3)⋊C4C2×Dic318yescomplemented
276C4⋊Dic3—64no—
277C4⋊Dic3—64no—
278C4⋊Dic3—62no—
279C4⋊Dic3—124no—
280C4⋊Dic3C2×Dic318yescomplemented
281Dic3⋊C4—62no—
282C4×Dic3—24no—
283C4×Dic3—64no—
284C4×Dic3—64no—
285C4×Dic3—62no—
286C4×Dic3—64no—
287C4×Dic3—64no—
288C4×Dic3—62no—
289C22×C32—24no—
290C22×C32—22no—
291C22×C32—22no—
292C22×C32—24no—
293C22×C32C4×D414yes—
294C22×C32C23×C411yescharacteristic
295C22×C32C22×D411yescharacteristic
296C22×C32C2×#4111yescharacteristic
297C22×C32C2×#4112yes—
298C22×C32C2×#4112yes—
299C2×C3×D3—22no—
300C2×C3×D3—24no—
301C2×C3×D3—41no—
302C2×C3×D3—41no—
303C2×C3×D3—42no—
304C32×C4—24no—
305C3⋊Dic3—24no—
306C3⋊Dic3—41no—
307C3⋊Dic3—41no—
308C3×Dic3—24no—
309C3×Dic3—22no—
310C3×Dic3—22no—
311C3×Dic3—42no—
312C3×Dic3#12314yescomplemented
313C22×D4—31no—
314C23×C4—31no—
315C23×C4—31no—
316C23×C4—91no—
317C23×C4—91no—
318#123—38no—
319C4×D4—184no—
320C2×C4⋊C4—32no—
321C2×C4⋊C4—91no—
322C2×C4⋊C4—92no—
323C2×C22⋊C4—32no—
324C2×C22⋊C4—31no—
325C2×C22⋊C4—91no—
326C2×C22⋊C4—92no—
327C2×C22⋊C4—91no—
328C2×C22⋊C4D3211yescharacteristic
329C2×C42—32no—
330C2×C42—91no—
331C2×C42—92no—
332C2×C42—92no—
333C2×C42—184no—
334#95—32no—
335#95—92no—
336#95—92no—
337#95—92no—
338#95—92no—
339C23×C3—22no—
340C23×C3—22no—
341C23×C3—22no—
342C23×C3—22no—
343C23×C3—21no—
344C23×C3—22no—
345C23×C3—22no—
346C23×C3—22no—
347C23×C3—32no—
348C23×C3—42no—
349C23×C3—42no—
350C23×C3—61no—
351C23×C3—61no—
352C23×C3—62no—
353C23×C3—62no—
354C23×C3C2×C4×D312yes—
355C23×C3C2×D1212yes—
356C23×C3#23512yes—
357C23×C3D3×D412yes—
358C23×C3#23712yes—
359C23×C3C22×Dic312yes—
360C23×C3C23×D311yescharacteristic
361C23×C3C23×D311yescharacteristic
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—22no—
365C22×D3—22no—
366C22×D3C22×Dic312yes—
367C3×D4—68no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—24no—
373C2×C3×C4—32no—
374C2×C3×C4—34no—
375C2×C3×C4—34no—
376C2×C3×C4—32no—
377C2×C3×C4—32no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4—68no—
381C2×C3×C4—68no—
382C2×C3×C4—62no—
383C2×C3×C4—61no—
384C2×C3×C4—62no—
385C2×C3×C4—61no—
386C2×C3×C4—62no—
387C2×C3×C4—62no—
388C2×C3×C4—62no—
389C2×C3×C4—62no—
390C2×C3×C4C2×C4×D314yes—
391C2×C3×C4C22×Dic314yes—
392#67—28no—
393C2×Dic3—28no—
394C2×Dic3—22no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—21no—
399C2×Dic3—28no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—34no—
404C2×Dic3—34no—
405C2×Dic3—64no—
406C2×Dic3—68no—
407C2×Dic3—68no—
408C2×Dic3—62no—
409C2×Dic3—62no—
410C2×Dic3—62no—
411C2×Dic3—62no—
412C2×Dic3—61no—
413C2×Dic3—61no—
414C2×Dic3—61no—
415C2×Dic3—61no—
416C2×Dic3—62no—
417C2×Dic3—62no—
418C2×Dic3—62no—
419C2×Dic3—62no—
420C2×Dic3—122no—
421C2×Dic3—122no—
422C2×Dic3—128no—
423C2×Dic3—122no—
424C2×Dic3—121no—
425C2×Dic3—121no—
426C2×Dic3—122no—
427C2×Dic3—124no—
428C2×Dic3C2×D1212yes—
429C2×Dic3C2×D1212yes—
430C2×Dic3#23714yes—
431C2×Dic3C22×Dic312yes—
432C2×Dic3C22×Dic314yes—
433C4×D3—124no—
434C2×C32—24no—
435C2×C32C2×C4×D411yescharacteristic
436C2×C32C4×#4112yes—
437C2×C32C2×#12311yescharacteristic
438C2×C32#52511yescharacteristic
439C2×C32#52912yes—
440C3×D3—42no—
441C24—31no—
442C24C2×D3211yescharacteristic
443C2×D4—34no—
444C2×D4—62no—
445C2×D4—62no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—32no—
449C22×C4—31no—
450C22×C4—34no—
451C22×C4—34no—
452C22×C4—32no—
453C22×C4—64no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—92no—
457C22×C4—91no—
458C22×C4—91no—
459C22×C4—182no—
460C22×C4—184no—
461C22×C4—184no—
462C22×C4—182no—
463C22×C4C2×D3212yes—
464C4⋊C4—38no—
465C4⋊C4—184no—
466C4⋊C4—182no—
467C22⋊C4—38no—
468C22⋊C4—62no—
469C22⋊C4—182no—
470C22⋊C4—184no—
471C22⋊C4—182no—
472C22⋊C4D3×Dic314yescomplemented
473C42—64no—
474C42—184no—
475C42—184no—
476C42—182no—
477C22×C3—24no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—24no—
483C22×C3—24no—
484C22×C3—24no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—21no—
490C22×C3—24no—
491C22×C3—42no—
492C22×C3—44no—
493C22×C3—44no—
494C22×C3—42no—
495C22×C3—62no—
496C22×C3—64no—
497C22×C3—121no—
498C22×C3—121no—
499C22×C3—122no—
500C22×C3C4×D1214yes—
501C22×C3D4×Dic314yes—
502C22×C3C22×C4×D311yescharacteristic
503C22×C3C22×D1211yescharacteristic
504C22×C3C2×#23511yescharacteristic
505C22×C3C2×D3×D411yescharacteristic
506C22×C3C2×#23711yescharacteristic
507C22×C3C2×#23712yes—
508C22×C3D3×#4112yes—
509C22×C3D3×#4112yes—
510C22×C3D3×#4112yes—
511C22×C3C23×Dic311yescharacteristic
512C2×D3—22no—
513C2×D3—24no—
514C2×D3—42no—
515C2×D3—41no—
516C2×D3—41no—
517C3×C4—24no—
518C3×C4—24no—
519C3×C4—34no—
520C3×C4—44no—
521C3×C4—62no—
522C3×C4—64no—
523C3×C4—62no—
524C3×C4—122no—
525Dic3—22no—
526Dic3—24no—
527Dic3—22no—
528Dic3—42no—
529Dic3—64no—
530Dic3—64no—
531Dic3—124no—
532Dic3—121no—
533Dic3—121no—
534Dic3—121no—
535Dic3—121no—
536Dic3—241no—
537Dic3—241no—
538Dic3#88614yesdirect product factor
539C32C4×#12311yescharacteristic, complemented, Sylow
540C23—22no—
541C23—22no—
542C23—32no—
543C23—61no—
544C23—61no—
545C23—62no—
546C23—62no—
547C23#379512yes—
548C23D3×D1212yes—
549C23C2×D3×Dic312yes—
550C23C22×D3211yescharacteristic
551D4—68no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—32no—
555C2×C4—34no—
556C2×C4—32no—
557C2×C4—34no—
558C2×C4—32no—
559C2×C4—68no—
560C2×C4—62no—
561C2×C4—61no—
562C2×C4—62no—
563C2×C4—61no—
564C2×C4—62no—
565C2×C4—68no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—94no—
570C2×C4—188no—
571C2×C4—182no—
572C2×C4—182no—
573C2×C4—181no—
574C2×C4—181no—
575C2×C4—182no—
576C2×C4—182no—
577C2×C4—364no—
578C2×C4C2×D3×Dic314yes—
579C2×C3—24no—
580C2×C3—24no—
581C2×C3—21no—
582C2×C3—22no—
583C2×C3—21no—
584C2×C3—22no—
585C2×C3—21no—
586C2×C3—44no—
587C2×C3—122no—
588C2×C3C2×C4×D1211yescharacteristic
589C2×C3C2×#88611yescharacteristic
590C2×C3#552411yescharacteristic
591C2×C3C4×#23712yes—
592C2×C3#553812yes—
593C2×C3#564912yes—
594C2×C3D3×#12311yescharacteristic
595C2×C3#565611yescharacteristic
596C2×C3C2×D4×Dic311yescharacteristic
597C2×C3Dic3×#4112yes—
598D3—42no—
599C22—22no—
600C22—24no—
601C22—24no—
602C22—22no—
603C22—62no—
604C22—64no—
605C22—121no—
606C22—122no—
607C22—121no—
608C22Dic3×D1214yes—
609C22C2×#379511yescharacteristic
610C22C2×D3×D1211yescharacteristic
611C22D3×#23712yes—
612C22#6431212yes—
613C22C22×D3×Dic311yescharacteristic
614C4—24no—
615C4—34no—
616C4—64no—
617C4—62no—
618C4—62no—
619C4—122no—
620C4—184no—
621C4—361no—
622C4—361no—
623C3—21no—
624C3C4×#88611yescharacteristic, complemented
625C3Dic3×#12311yescharacteristic, complemented
626C2—24no—
627C2—122no—
628C2#1059407911yescharacteristic
629C2#1059413112yes—
630C2D3×#88611yescharacteristic
631C2C2×Dic3×D1211yescharacteristic
632C2Dic3×#23712yes—
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C22⋊C4C22
socle, cosocleC23×C32C24
prime radicalC22×C3×#67C22
hypercenterC2×C22⋊C4D32
hypocenterC32C4×#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C22×Dic32C2semidirect
4. #10592649C2semidirect
5. C3×Dic3×C22⋊C4C2semidirect
7. #10591696C2semidirect
8. Dic3×(C2×D3)⋊C4C2semidirect
9. #10591316C2semidirect
10. #10591269C2semidirect
11. Dic3×C4⋊Dic3C2semidirect
26. C3×#886C4semidirect
58. Dic3×C22⋊C4D3semidirect
77. C22×C3×Dic3C2×C4semidirect
86. C32×C22⋊C4C2×C4semidirect
90. C3×(C2×D3)⋊C4C2×C4semidirect
91. C3×C4⋊Dic3C2×C4semidirect
145. #886Dic3direct (N × H)
252. C22×Dic3C2×Dic3semidirect
264. C3×C22⋊C4C4×D3semidirect
265. C3×C22⋊C4C2×Dic3semidirect
275. (C2×D3)⋊C4C2×Dic3semidirect
280. C4⋊Dic3C2×Dic3semidirect
312. C3×Dic3#123semidirect
472. C22⋊C4D3×Dic3semidirect
538. Dic3#886direct (N × H)
539. C32C4×#123semidirect
624. C3C4×#886semidirect
625. C3Dic3×#123semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8862
C2×Dic3×#672
C22×Dic322
#105926492
C3×Dic3×C22⋊C42
#105917132
#105916962
Dic3×(C2×D3)⋊C42
#105913162
#105912692
Dic3×C4⋊Dic32
Dic3×#1233
C4×#8863