Finite Group Databasebeta

← C3⋊D3×#886 #13175929

Subgroups — C3⋊D3×#886 #13175929

Order 1728.

2364 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3⋊D3×#67C211yesmaximal, characteristic
2C22×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
3#11723927C211yesmaximal, characteristic, complemented
4C3×C22⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
5C32×#886C211yesmaximal, characteristic, complemented
6#11723612C211yesmaximal, characteristic, complemented
7#11723603C211yesmaximal, characteristic, complemented
8C3⋊D3×(C2×D3)⋊C4C212yesmaximal, complemented
9#11723523C212yesmaximal, complemented
10#11723493C212yesmaximal, complemented
11C3⋊D3×C4⋊Dic3C212yesmaximal, complemented
12C3⋊D3×#123—31nomaximal
13D3×#886—34nomaximal
14C23×C3×C3⋊D3C2211yescharacteristic
15C22×D3×C3⋊D3C2211yescharacteristic
16C2×#6312C2211yescharacteristic
17C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
18C2×C3×C4×C3⋊D3C2212yes—
19C2×C32×#67C2211yescharacteristic
20C22×C32×Dic3C2211yescharacteristic, complemented
21C3⋊D3×#67—24no—
22C2×#6292C2211yescharacteristic
23C2×#6290C2211yescharacteristic
24C2×Dic3×C3⋊D3—24no—
25C2×Dic3×C3⋊D3C2212yes—
26C2×Dic3×C3⋊D3C2211yescharacteristic
27C2×Dic3×C3⋊D3C2212yescomplemented
28C33×C22⋊C4C2211yescharacteristic, complemented
29#88859C2212yescomplemented
30C4⋊(C3⋊(C3⋊Dic3))C2212yescomplemented
31C3×#3757C2211yescharacteristic, complemented
32C3×#3752C2212yescomplemented
33C32×(C2×D3)⋊C4C2212yescomplemented
34C32×C4⋊Dic3C2212yescomplemented
35#88807C2212yescomplemented
36#88807C2212yescomplemented
37#88804C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C2×D4×C3⋊D3—31no—
40C22×C4×C3⋊D3—31no—
41C2×D3×#67—34no—
42C22×D3×Dic3—34no—
43C32×#123—31no—
44#64144—31no—
45#64138—31no—
46#64103—32no—
47C4⋊C4×C3⋊D3—32no—
48#64095—31no—
49#64095—34no—
50#64095—64no—
51#64092—32no—
52C22⋊C4×C3⋊D3—32no—
53C22⋊C4×C3⋊D3D311yescharacteristic, complemented
54C3×#886D314yescomplemented
55C3×D3×C22⋊C4—34no—
56#63966—34no—
57#63962—34no—
58D3×(C2×D3)⋊C4—38no—
59#63891—38no—
60#63890—38no—
61D3×C4⋊Dic3—38no—
62C23×C33C2311yescharacteristic
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C3×C3⋊D3—24no—
65C22×C3×C3⋊D3D414yes—
66C22×C3×C3⋊D3C2311yescharacteristic
67C22×C3×C3⋊D3C2311yescharacteristic
68C22×C32×D3C2311yescharacteristic
69C2×D3×C3⋊D3—22no—
70C2×D3×C3⋊D3—24no—
71C2×C33×C4C2312yes—
72#6312—22no—
73C2×C3⋊(C3⋊Dic3)—22no—
74C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
75C2×C3⋊(C3⋊Dic3)C2312yes—
76C2×C3×C3⋊Dic3C2312yes—
77C3×C4×C3⋊D3—24no—
78C32×#67—22no—
79C2×C32×Dic3—22no—
80C2×C32×Dic3C2312yes—
81C2×C32×Dic3C2311yescharacteristic
82#6292—22no—
83#6290—22no—
84Dic3×C3⋊D3—24no—
85Dic3×C3⋊D3—24no—
86Dic3×C3⋊D3—22no—
87D3×#123—94no—
88C2×#886—91no—
89C23×C3⋊D3—31no—
90C23×C3⋊D3C2×D311yescharacteristic
91C23×C3×D3—34no—
92C22×D32—34no—
93C2×C32×D4—31no—
94C22×C32×C4—31no—
95C2×#638—31no—
96C2×#638—34no—
97C2×#638—31no—
98C2×#638—64no—
99C22×C3⋊Dic3—31no—
100C22×C3⋊Dic3—34no—
101C22×C3⋊Dic3—64no—
102D4×C3⋊D3—64no—
103C2×C4⋊(C3⋊D3)—31no—
104C2×C4×C3⋊D3—32no—
105C2×C4×C3⋊D3—32no—
106C2×C4×C3⋊D3—31no—
107C2×C4×C3⋊D3—64no—
108C2×C4×C3⋊D3C2×D312yes—
109C2×C3×#67C2×D314yes—
110C22×C3×Dic3C2×D314yescomplemented
111C2×C3×C4×D3—38no—
112D3×#67—616no—
113C2×#626—34no—
114C2×#625—34no—
115C2×D3×Dic3—38no—
116C2×D3×Dic3—38no—
117C2×D3×Dic3—34no—
118C2×D3×Dic3—616no—
119C32×C4⋊C4—32no—
120C32×C22⋊C4—24no—
121C32×C22⋊C4—32no—
122C32×C22⋊C4C2×D314yescomplemented
123C32×C22⋊C4C2×D311yescharacteristic, complemented
124#3757—32no—
125#3757C2×D311yescharacteristic, complemented
126#3752—32no—
127#3752—38no—
128#3752—32no—
129#3752—68no—
130#3752C2×D312yescomplemented
131C4⋊(C3⋊Dic3)—32no—
132C4⋊(C3⋊Dic3)—38no—
133C4⋊(C3⋊Dic3)—68no—
134#3750—32no—
135#3750—32no—
136C3×#217—34no—
137C3×(C2×D3)⋊C4—38no—
138C3×(C2×D3)⋊C4C2×D318yescomplemented
139C3×C4⋊Dic3C2×D318yescomplemented
140Dic3⋊Dic3—38no—
141Dic3⋊Dic3—38no—
142#3722—38no—
143(C2×D3)⋊Dic3—38no—
144C22×C33—22no—
145C22×C33C2×D412yes—
146C22×C33C2411yescharacteristic
147C2×C3⋊(C3⋊D3)—21no—
148C2×C3⋊(C3⋊D3)—22no—
149C2×C3×C3⋊D3—22no—
150C2×C3×C3⋊D3—28no—
151C2×C3×C3⋊D3C2×D412yes—
152C2×C3×C3⋊D3C2×D412yes—
153C2×C3×C3⋊D3#4114yes—
154C2×C32×D3—21no—
155C2×C32×D3—22no—
156D3×C3⋊D3—44no—
157C33×C4—22no—
158C3⋊(C3⋊Dic3)—22no—
159C3⋊(C3⋊Dic3)—21no—
160C3×C3⋊Dic3—22no—
161C32×Dic3—22no—
162C32×Dic3—21no—
163C22×#67—91no—
164C23×Dic3—91no—
165C2×D3×D4—94no—
166C22×C4×D3—94no—
167C3×#123—34no—
168C2×C3×C22⋊C4—91no—
169#935—94no—
170#929—94no—
171C2×(C2×D3)⋊C4—92no—
172C2×C4⋊Dic3—92no—
173#894—98no—
174D3×C4⋊C4—98no—
175#886—91no—
176#886—91no—
177#886—94no—
178#886—91no—
179#886—92no—
180#886—92no—
181#886—184no—
182#886C3⋊D311yescharacteristic, direct product factor
183#883—98no—
184D3×C22⋊C4—34no—
185D3×C22⋊C4—98no—
186C23×C32—24no—
187C23×C32—31no—
188C23×C32C22×D314yes—
189C23×C32C22×D311yescharacteristic
190C22×C3⋊D3—24no—
191C22×C3⋊D3—31no—
192C22×C3⋊D3—34no—
193C22×C3⋊D3—64no—
194C22×C3⋊D3—62no—
195C22×C3⋊D3—64no—
196C22×C3⋊D3D1214yes—
197C22×C3⋊D3C22×D311yescharacteristic
198C22×C3⋊D3C22×D311yescharacteristic
199C22×C3×D3—34no—
200C22×C3×D3—316no—
201C22×C3×D3—34no—
202C22×C3×D3—616no—
203C22×C3×D3C22×D314yes—
204C2×D32—616no—
205C2×D32—68no—
206C32×D4—62no—
207C2×C32×C4—28no—
208C2×C32×C4—31no—
209C2×C32×C4—32no—
210C2×C32×C4—62no—
211C2×C32×C4C22×D318yes—
212C2×C32×C4C22×D312yes—
213#638—62no—
214#638—62no—
215#638—68no—
216#638—128no—
217C2×C3⋊Dic3—31no—
218C2×C3⋊Dic3—32no—
219C2×C3⋊Dic3—38no—
220C2×C3⋊Dic3—34no—
221C2×C3⋊Dic3—64no—
222C2×C3⋊Dic3—68no—
223C2×C3⋊Dic3—62no—
224C2×C3⋊Dic3—68no—
225C2×C3⋊Dic3—128no—
226C2×C3⋊Dic3C22×D312yes—
227C4⋊(C3⋊D3)—62no—
228C4×C3⋊D3—24no—
229C4×C3⋊D3—64no—
230C4×C3⋊D3—64no—
231C4×C3⋊D3—62no—
232C3×#67—28no—
233C2×C3×Dic3—28no—
234C2×C3×Dic3—38no—
235C2×C3×Dic3C22×D318yes—
236C2×C3×Dic3C22×D314yes—
237C3×C4×D3—616no—
238#626—68no—
239#625—68no—
240D3×Dic3—616no—
241D3×Dic3—616no—
242D3×Dic3—68no—
243C2×#123—271noSylow
244C2×C33—22no—
245C2×C33C22×D411yescharacteristic
246C2×C33C2×#4112yes—
247C3⋊(C3⋊D3)—41no—
248C3×C3⋊D3—22no—
249C3×C3⋊D3#12314yescomplemented
250C32×D3—41no—
251C24×C3—91no—
252C23×D3—34no—
253C23×D3—94no—
254C23×D3—91no—
255C2×C3×D4—34no—
256C22×C3×C4—34no—
257C22×C3×C4—92no—
258C2×#67—94no—
259C2×#67—94no—
260C2×#67—91no—
261C2×#67—91no—
262C2×#67—91no—
263C2×#67—184no—
264C2×#67—184no—
265C2×#67C2×C3⋊D311yescharacteristic
266C22×Dic3—94no—
267C22×Dic3—91no—
268C22×Dic3—92no—
269C22×Dic3—91no—
270C22×Dic3—92no—
271C22×Dic3—184no—
272C22×Dic3—184no—
273C22×Dic3C2×C3⋊D311yescharacteristic, complemented
274D3×D4—1816no—
275C2×D12—94no—
276C2×C4×D3—38no—
277C2×C4×D3—98no—
278C2×C4×D3—98no—
279C2×C4×D3—94no—
280C2×C4×D3—1816no—
281C3×C4⋊C4—38no—
282C3×C22⋊C4—24no—
283C3×C22⋊C4—38no—
284C3×C22⋊C4—91no—
285C3×C22⋊C4—92no—
286C3×C22⋊C4D3214yescomplemented
287C3×C22⋊C4C2×C3⋊D311yescharacteristic, complemented
288#217—34no—
289#217—98no—
290(C2×D3)⋊C4—38no—
291(C2×D3)⋊C4—98no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—98no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—188no—
297(C2×D3)⋊C4C2×C3⋊D312yescomplemented
298C4⋊Dic3—98no—
299C4⋊Dic3—92no—
300C4⋊Dic3—92no—
301C4⋊Dic3—92no—
302C4⋊Dic3—188no—
303C4⋊Dic3C2×C3⋊D312yescomplemented
304Dic3⋊C4—98no—
305Dic3⋊C4—98no—
306C22×C32—22no—
307C22×C32—24no—
308C22×C32—28no—
309C22×C32—28no—
310C22×C32—48no—
311C22×C32—61no—
312C22×C32—62no—
313C22×C32C2×D1212yes—
314C22×C32D3×D418yes—
315C22×C32C23×D314yes—
316C22×C32C23×D311yescharacteristic
317C2×C3⋊D3—22no—
318C2×C3⋊D3—28no—
319C2×C3⋊D3—61no—
320C2×C3⋊D3—62no—
321C2×C3⋊D3—64no—
322C2×C3⋊D3—68no—
323C2×C3⋊D3—124no—
324C2×C3⋊D3—128no—
325C2×C3⋊D3—124no—
326C2×C3⋊D3C2×D1212yes—
327C2×C3⋊D3C2×D1212yes—
328C2×C3⋊D3#23714yes—
329C2×C3×D3—24no—
330C2×C3×D3—28no—
331C2×C3×D3—38no—
332C2×C3×D3—316no—
333C2×C3×D3—38no—
334C2×C3×D3—632no—
335C2×C3×D3—68no—
336D32—1216no—
337C32×C4—22no—
338C32×C4—28no—
339C32×C4—48no—
340C32×C4—62no—
341C32×C4—61no—
342C3⋊Dic3—22no—
343C3⋊Dic3—62no—
344C3⋊Dic3—61no—
345C3⋊Dic3—68no—
346C3⋊Dic3—64no—
347C3⋊Dic3—128no—
348C3⋊Dic3—124no—
349C3×Dic3—28no—
350C3×Dic3—24no—
351C3×Dic3—68no—
352C22×D4—271no—
353C23×C4—271no—
354#123—31no—
355#123—272no—
356#123—272no—
357#123—271no—
358#123—271no—
359#123—271no—
360C2×C4⋊C4—272no—
361C2×C22⋊C4—91no—
362C2×C22⋊C4—272no—
363C33C2×#12311yescharacteristic, complemented, Sylow
364C23×C3—24no—
365C23×C3—34no—
366C23×C3—91no—
367C23×C3—91no—
368C23×C3—94no—
369C23×C3—184no—
370C23×C3C2×D3214yes—
371C23×C3C22×C3⋊D311yescharacteristic
372C22×D3—34no—
373C22×D3—34no—
374C22×D3—316no—
375C22×D3—616no—
376C22×D3—94no—
377C22×D3—91no—
378C22×D3—184no—
379C22×D3—182no—
380C22×D3—184no—
381C22×D3—188no—
382C22×D3—1816no—
383C22×D3C22×C3⋊D311yescharacteristic
384C3×D4—68no—
385C2×C3×C4—28no—
386C2×C3×C4—34no—
387C2×C3×C4—38no—
388C2×C3×C4—68no—
389C2×C3×C4—92no—
390C2×C3×C4—184no—
391C2×C3×C4C2×D3218yes—
392C2×C3×C4C22×C3⋊D312yes—
393#67—22no—
394#67—182no—
395#67—182no—
396#67—188no—
397#67—188no—
398#67—182no—
399#67—368no—
400C2×Dic3—22no—
401C2×Dic3—38no—
402C2×Dic3—94no—
403C2×Dic3—98no—
404C2×Dic3—91no—
405C2×Dic3—92no—
406C2×Dic3—184no—
407C2×Dic3—188no—
408C2×Dic3—184no—
409C2×Dic3—184no—
410C2×Dic3—182no—
411C2×Dic3—188no—
412C2×Dic3—182no—
413C2×Dic3—368no—
414C2×Dic3C22×C3⋊D311yescharacteristic
415C2×Dic3C22×C3⋊D312yes—
416D12—188no—
417C4×D3—616no—
418C4×D3—1816no—
419C4×D3—1816no—
420C4×D3—188no—
421C2×C32—22no—
422C2×C32—24no—
423C2×C32—28no—
424C2×C32—28no—
425C2×C32—48no—
426C2×C32—121no—
427C2×C32C22×D1211yescharacteristic
428C2×C32C2×D3×D414yes—
429C2×C32C2×#23712yes—
430C2×C32D3×#4118yes—
431C3⋊D3—22no—
432C3⋊D3—121no—
433C3⋊D3—124no—
434C3⋊D3—244no—
435C3⋊D3#88614yesdirect product factor
436C3×D3—316no—
437C3×D3—44no—
438C3×D3—68no—
439C24—91no—
440C24—271no—
441C2×D4—31no—
442C2×D4—271no—
443C2×D4—271no—
444C2×D4—271no—
445C2×D4—544no—
446C22×C4—31no—
447C22×C4—92no—
448C22×C4—272no—
449C22×C4—271no—
450C22×C4—272no—
451C22×C4—271no—
452C22×C4—544no—
453C4⋊C4—32no—
454C4⋊C4—272no—
455C4⋊C4—272no—
456C4⋊C4—272no—
457C22⋊C4—32no—
458C22⋊C4—91no—
459C22⋊C4—92no—
460C22⋊C4—272no—
461C22⋊C4—272no—
462C22⋊C4—272no—
463C22⋊C4D3×C3⋊D311yescharacteristic, complemented
464C22×C3—28no—
465C22×C3—22no—
466C22×C3—24no—
467C22×C3—28no—
468C22×C3—48no—
469C22×C3—64no—
470C22×C3—68no—
471C22×C3—92no—
472C22×C3—94no—
473C22×C3—92no—
474C22×C3—188no—
475C22×C3—182no—
476C22×C3D3×D1218yes—
477C22×C3D4×C3⋊D312yes—
478C22×C3C22×D3214yes—
479C22×C3C23×C3⋊D311yescharacteristic
480C2×D3—21no—
481C2×D3—22no—
482C2×D3—38no—
483C2×D3—316no—
484C2×D3—38no—
485C2×D3—632no—
486C2×D3—68no—
487C2×D3—184no—
488C2×D3—188no—
489C2×D3—181no—
490C2×D3—182no—
491C2×D3—364no—
492C2×D3—368no—
493C2×D3—3616no—
494C2×D3—364no—
495C3×C4—28no—
496C3×C4—22no—
497C3×C4—48no—
498C3×C4—68no—
499C3×C4—64no—
500C3×C4—182no—
501Dic3—22no—
502Dic3—21no—
503Dic3—68no—
504Dic3—188no—
505Dic3—184no—
506Dic3—182no—
507Dic3—181no—
508Dic3—368no—
509Dic3—364no—
510C32—24no—
511C32C2×#88611yescharacteristic, complemented
512C32D3×#12314yescomplemented
513C23—31no—
514C23—91no—
515C23—91no—
516C23—94no—
517C23—184no—
518C23—271no—
519C23—542no—
520C23—544no—
521C23C2×D3×C3⋊D311yescharacteristic
522D4—62no—
523D4—542no—
524D4—542no—
525D4—542no—
526C2×C4—32no—
527C2×C4—31no—
528C2×C4—62no—
529C2×C4—92no—
530C2×C4—184no—
531C2×C4—272no—
532C2×C4—271no—
533C2×C4—544no—
534C2×C4—544no—
535C2×C4—542no—
536C2×C4—542no—
537C2×C4C2×D3×C3⋊D312yes—
538C2×C3—22no—
539C2×C3—28no—
540C2×C3—24no—
541C2×C3—28no—
542C2×C3—48no—
543C2×C3—94no—
544C2×C3—124no—
545C2×C3—182no—
546C2×C3C2×D3×D1214yes—
547C2×C3D3×#23718yes—
548C2×C3C2×D4×C3⋊D311yescharacteristic
549C2×C3#41×C3⋊D312yes—
550D3—316no—
551D3—41no—
552D3—68no—
553D3—361no—
554D3—364no—
555D3—724no—
556C22—22no—
557C22—61no—
558C22—62no—
559C22—92no—
560C22—94no—
561C22—92no—
562C22—188no—
563C22—182no—
564C22—541no—
565C22—542no—
566C22—1084no—
567C22C3⋊D3×D1212yes—
568C22C22×D3×C3⋊D311yescharacteristic
569C4—22no—
570C4—62no—
571C4—61no—
572C4—182no—
573C4—542no—
574C4—541no—
575C3—24no—
576C3D3×#88614yescomplemented
577C3C3⋊D3×#12311yescharacteristic, complemented
578C2—22no—
579C2—94no—
580C2—121no—
581C2—182no—
582C2—1081no—
583C2C2×C3⋊D3×D1211yescharacteristic
584C2C3⋊D3×#23712yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC2×C3⋊D3×#67C2
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33C2×#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×Dic3×C3⋊D3C2semidirect
3. #11723927C2semidirect
4. C3×C22⋊C4×C3⋊D3C2semidirect
5. C32×#886C2semidirect
6. #11723612C2semidirect
7. #11723603C2semidirect
8. C3⋊D3×(C2×D3)⋊C4C2semidirect
9. #11723523C2semidirect
10. #11723493C2semidirect
11. C3⋊D3×C4⋊Dic3C2semidirect
17. C22×C3⋊(C3⋊Dic3)C22semidirect
20. C22×C32×Dic3C22semidirect
27. C2×Dic3×C3⋊D3C22semidirect
28. C33×C22⋊C4C22semidirect
29. #88859C22semidirect
30. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
31. C3×#3757C22semidirect
32. C3×#3752C22semidirect
33. C32×(C2×D3)⋊C4C22semidirect
34. C32×C4⋊Dic3C22semidirect
35. #88807C22semidirect
36. #88807C22semidirect
37. #88804C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
53. C22⋊C4×C3⋊D3D3semidirect
54. C3×#886D3semidirect
110. C22×C3×Dic3C2×D3semidirect
122. C32×C22⋊C4C2×D3semidirect
123. C32×C22⋊C4C2×D3semidirect
125. #3757C2×D3semidirect
130. #3752C2×D3semidirect
138. C3×(C2×D3)⋊C4C2×D3semidirect
139. C3×C4⋊Dic3C2×D3semidirect
182. #886C3⋊D3direct (N × H)
249. C3×C3⋊D3#123semidirect
273. C22×Dic3C2×C3⋊D3semidirect
286. C3×C22⋊C4D32semidirect
287. C3×C22⋊C4C2×C3⋊D3semidirect
297. (C2×D3)⋊C4C2×C3⋊D3semidirect
303. C4⋊Dic3C2×C3⋊D3semidirect
363. C33C2×#123semidirect
435. C3⋊D3#886direct (N × H)
463. C22⋊C4D3×C3⋊D3semidirect
511. C32C2×#886semidirect
512. C32D3×#123semidirect
576. C3D3×#886semidirect
577. C3C3⋊D3×#123semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×#672
C22×Dic3×C3⋊D32
#117239272
C3×C22⋊C4×C3⋊D32
C32×#8862
#117236122
#117236032
C3⋊D3×(C2×D3)⋊C42
#117235232
#117234932
C3⋊D3×C4⋊Dic32
C3⋊D3×#1233
D3×#8863