Finite Group Databasebeta

← C2×#86290 #11702640

Subgroups — C2×#86290 #11702640

Order 768.

6554 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×C3×#533C211yesmaximal, characteristic, complemented
2#86290C2132yesmaximal, direct product factor
3C2×#5735C214yesmaximal, complemented
4C2×#5693C214yesmaximal, complemented
5C2×#5638C214yesmaximal, complemented
6C2×#5640C214yesmaximal, complemented
7C2×#5579C212yesmaximal, complemented
8C2×#5585C212yesmaximal, complemented
9C2×#5567C212yesmaximal, complemented
10C2×#5584C212yesmaximal, complemented
11C2×#5534C212yesmaximal, complemented
12C2×#5535C211yesmaximal, characteristic
13C2×#5518C211yesmaximal, characteristic, complemented
14C2×#5479C212yesmaximal, complemented
15C2×#3557—31nomaximal, Sylow
16C3×#533C22116yescomplemented
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×#125C2212yes—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2212yes—
23C2×C3×#117C2211yescharacteristic
24C2×C3×#117C2211yescharacteristic
25C22×C3×C42C2211yescharacteristic
26C2×#947C2214yes—
27C2×#946C2214yes—
28C2×#946C2214yes—
29C2×#944C2214yes—
30C2×#930C2212yes—
31C2×#929C2214yes—
32C2×C4×#67C2214yes—
33C2×#924C2212yes—
34#5735C22164yescomplemented
35#5693C22164yescomplemented
36#5640C22164yescomplemented
37#5638C22164yescomplemented
38#5585C22132yescomplemented
39#5584C22132yescomplemented
40#5579C22132yescomplemented
41#5567C22132yescomplemented
42#5535C22116yes—
43#5534C22132yescomplemented
44#5518C22116yescomplemented
45C2×#898C2214yes—
46C2×#898C2214yes—
47C2×#898C2214yes—
48C2×#896C2214yes—
49C2×#896C2214yes—
50C2×#896C2214yes—
51C2×#894C2214yes—
52C2×#894C2214yes—
53C2×#894C2214yes—
54C2×#893C2214yes—
55C2×#892C2214yes—
56C2×D3×C4⋊C4C2214yes—
57C2×#889C2214yes—
58C2×#889C2214yes—
59C2×#889C2214yes—
60C2×#887C2214yes—
61C2×#885C2214yes—
62C2×#884C2214yes—
63C2×#883C2214yes—
64C2×#879C2214yes—
65#5479C22132yescomplemented
66C2×#876C2214yes—
67C2×#875C2212yes—
68C2×#875C2212yes—
69C2×#874C2211yescharacteristic
70C2×C4×D12C2212yes—
71C2×C4×D12C2212yes—
72C2×C4×D12C2212yes—
73C2×#872C2214yes—
74C2×#870C2211yescharacteristic
75C2×#870C2212yes—
76C2×#870C2211yescharacteristic
77C2×#869C2211yescharacteristic
78C2×#869C2212yes—
79C2×C4×Dic6C2212yes—
80C2×C4×Dic6C2212yes—
81C2×C4×Dic6C2212yes—
82#3557—332no—
83C2×#563—34no—
84C2×#559—32no—
85C2×#557—32no—
86C2×#556—32no—
87C2×#554—32no—
88C2×#552—34no—
89C2×#542—34no—
90C2×#542—32no—
91C2×#541—31no—
92C2×#540—31no—
93C2×#539—34no—
94C2×#533D311yescharacteristic, complemented
95C2×#529—32no—
96C22×C3×Q8C2312yes—
97C22×C3×Q8C2312yes—
98C23×C3×C4C2311yescharacteristic
99C23×C3×C4C2312yes—
100C22×#67C2314yes—
101C22×D12C2312yes—
102C22×D12C2314yes—
103C22×C4×D3C2314yes—
104C22×C4×D3C2314yes—
105C22×C4×D3C2314yes—
106C22×Dic6C2312yes—
107C22×Dic6C2314yes—
108C3×C4⋊Q8C23116yes—
109C3×#125C23116yes—
110C3×C22⋊Q8C23116yes—
111C3×C22⋊Q8C23116yes—
112C3×C4×Q8C23116yes—
113C3×C4×Q8C23116yes—
114C3×#117C2318yes—
115C3×#117C2318yes—
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C4⋊C4C2312yes—
118C2×C3×C4⋊C4C2312yes—
119C2×C3×C4⋊C4C2312yes—
120C2×C3×C4⋊C4C2312yes—
121C2×C3×C4⋊C4C2312yes—
122C2×C3×C4⋊C4C2312yes—
123C2×C3×C4⋊C4C2312yes—
124C2×C3×C22⋊C4C2312yes—
125C2×C3×C22⋊C4C2311yescharacteristic
126C2×C3×C22⋊C4C2311yescharacteristic
127C2×C3×C42C2311yescharacteristic
128C2×C3×C42C2312yes—
129C2×C3×C42C2312yes—
130C2×C3×C42C2311yescharacteristic
131C2×C3×C42C2311yescharacteristic
132C2×C3×C42C2311yescharacteristic
133C2×C3×C42C2318yes—
134C2×#217C2314yes—
135#947C23132yes—
136#946C23132yes—
137#946C23132yes—
138#944C23132yes—
139#930C23116yes—
140#929C23132yes—
141C4×#67C23132yes—
142C2×(C2×D3)⋊C4C2314yes—
143C2×(C2×D3)⋊C4C2314yes—
144C2×(C2×D3)⋊C4C2314yes—
145C2×(C2×D3)⋊C4C2314yes—
146C2×(C2×D3)⋊C4C2314yes—
147C2×(C2×D3)⋊C4C2314yes—
148C2×(C2×D3)⋊C4C2314yes—
149C2×(C2×D3)⋊C4C2314yes—
150C2×C4⋊Dic3C2312yes—
151C2×C4⋊Dic3C2312yes—
152C2×C4⋊Dic3C2314yes—
153C2×C4⋊Dic3C2314yes—
154#924C23116yes—
155C2×Dic3⋊C4C2314yes—
156C2×Dic3⋊C4C2314yes—
157C2×Dic3⋊C4C2314yes—
158C2×Dic3⋊C4C2314yes—
159C2×Dic3⋊C4C2314yes—
160C2×Dic3⋊C4C2314yes—
161C2×Dic3⋊C4C2314yes—
162C2×Dic3⋊C4C2314yes—
163C2×C4×Dic3C2314yes—
164C2×C4×Dic3C2314yes—
165C2×C4×Dic3C2314yes—
166#898C23132yes—
167#898C23132yes—
168#898C23132yes—
169#896C23132yes—
170#896C23132yes—
171#896C23132yes—
172#894C23132yes—
173#894C23132yes—
174#894C23132yes—
175#893C23132yes—
176#892C23132yes—
177D3×C4⋊C4C23132yes—
178#889C23132yes—
179#889C23132yes—
180#889C23132yes—
181#887C23132yes—
182#885C23132yes—
183#884C23132yes—
184#883C23132yes—
185#879C23132yes—
186#876C23132yes—
187#875C23116yes—
188#875C23116yes—
189#874C2318yes—
190C4×D12C23116yes—
191C4×D12C23116yes—
192C4×D12C23116yes—
193#872C23132yes—
194#870C23116yes—
195#870C2318yes—
196#870C2318yes—
197#869C2318yes—
198#869C23116yes—
199C4×Dic6C23116yes—
200C4×Dic6C23116yes—
201C4×Dic6C23116yes—
202#563—364no—
203#559—332no—
204#557—332no—
205#556—332no—
206#554—332no—
207#552—364no—
208#542—364no—
209#542—332no—
210#541—316no—
211#540—316no—
212#539—364no—
213#533C2×D3116yescomplemented
214C2×C4⋊Q8—34no—
215C2×C4⋊Q8—31no—
216C2×C4⋊Q8—32no—
217C2×C4⋊Q8C2×D312yes—
218C2×#127—31no—
219#529—332no—
220C2×#126—34no—
221C2×#126—34no—
222C2×#126—34no—
223C2×#126—34no—
224C2×#126—34no—
225C2×#125—34no—
226C2×#125—34no—
227C2×#125—34no—
228C2×#125—32no—
229C2×#125—31no—
230C2×#125—31no—
231C2×#125C2×D312yes—
232C2×#124—34no—
233C2×#124—34no—
234C2×#124—32no—
235C2×#124—32no—
236C2×#123—34no—
237C2×#123—34no—
238C2×#123—34no—
239C2×#123—34no—
240C2×#123—34no—
241C2×C22⋊Q8—34no—
242C2×C22⋊Q8—34no—
243C2×C22⋊Q8—32no—
244C2×C22⋊Q8—34no—
245C2×C22⋊Q8—34no—
246C2×C22⋊Q8—34no—
247C2×C22⋊Q8C2×D312yes—
248C2×C22⋊Q8C2×D312yes—
249C2×#121—32no—
250C2×#121—34no—
251C2×C4×Q8—34no—
252C2×C4×Q8—32no—
253C2×C4×Q8—32no—
254C2×C4×Q8—32no—
255C2×C4×Q8C2×D312yes—
256C2×C4×Q8C2×D312yes—
257C2×C4×D4—34no—
258C2×C4×D4—34no—
259C2×C4×D4—32no—
260C2×C4×D4—32no—
261C2×C4×D4—32no—
262C2×#117—34no—
263C2×#117—34no—
264C2×#117C2×D311yescharacteristic
265C2×#117C2×D311yescharacteristic
266C22×C4⋊C4—34no—
267C22×C42C2×D311yescharacteristic
268C24×C3C2411yescharacteristic
269C23×D3C2414yes—
270C2×C3×Q8—44no—
271C2×C3×Q8—44no—
272C2×C3×Q8C2418yes—
273C2×C3×Q8C2418yes—
274C22×C3×C4—24no—
275C22×C3×C4—44no—
276C22×C3×C4C2412yes—
277C22×C3×C4C2412yes—
278C22×C3×C4C2412yes—
279C22×C3×C4C2411yescharacteristic
280C22×C3×C4C2412yes—
281C22×C3×C4C2412yes—
282C22×C3×C4C2412yes—
283C22×C3×C4C2414yes—
284C22×C3×C4C2418yes—
285C22×C3×C4C2411yescharacteristic
286C2×#67—48no—
287C2×#67C24116yes—
288C22×Dic3C2414yes—
289C22×Dic3C2414yes—
290C22×Dic3C2414yes—
291C2×D12—28no—
292C2×D12—48no—
293C2×D12C2418yes—
294C2×D12C24116yes—
295C2×C4×D3—48no—
296C2×C4×D3—48no—
297C2×C4×D3—48no—
298C2×C4×D3C24116yes—
299C2×C4×D3C24116yes—
300C2×C4×D3C24116yes—
301C2×Dic6—28no—
302C2×Dic6—48no—
303C2×Dic6C2418yes—
304C2×Dic6C24116yes—
305C3×C4⋊C4C2418yes—
306C3×C4⋊C4C2418yes—
307C3×C4⋊C4C2418yes—
308C3×C4⋊C4C2418yes—
309C3×C4⋊C4C2418yes—
310C3×C4⋊C4C2418yes—
311C3×C4⋊C4C2418yes—
312C3×C4⋊C4C2418yes—
313C3×C22⋊C4C2418yes—
314C3×C22⋊C4C2414yes—
315C3×C22⋊C4C2414yes—
316C3×C42C2414yes—
317C3×C42C2418yes—
318C3×C42C2418yes—
319C3×C42C2414yes—
320C3×C42C2414yes—
321C3×C42C2414yes—
322#217C24116yes—
323(C2×D3)⋊C4C24116yes—
324(C2×D3)⋊C4C24116yes—
325(C2×D3)⋊C4C24116yes—
326(C2×D3)⋊C4C24116yes—
327(C2×D3)⋊C4C24116yes—
328(C2×D3)⋊C4C24116yes—
329(C2×D3)⋊C4C24116yes—
330(C2×D3)⋊C4C24116yes—
331C4⋊Dic3C2418yes—
332C4⋊Dic3C2418yes—
333C4⋊Dic3C24116yes—
334C4⋊Dic3C24116yes—
335Dic3⋊C4C24116yes—
336Dic3⋊C4C24116yes—
337Dic3⋊C4C24116yes—
338Dic3⋊C4C24116yes—
339Dic3⋊C4C24116yes—
340Dic3⋊C4C24116yes—
341Dic3⋊C4C24116yes—
342Dic3⋊C4C24116yes—
343C4×Dic3C24116yes—
344C4×Dic3C24116yes—
345C4×Dic3C24116yes—
346C22×Q8—32no—
347C22×Q8—34no—
348C22×Q8C22×D312yes—
349C22×Q8C22×D312yes—
350C22×D4—34no—
351C22×D4—32no—
352C22×D4—34no—
353C23×C4—34no—
354C23×C4—34no—
355C23×C4—34no—
356C23×C4C22×D311yescharacteristic
357C23×C4C22×D312yes—
358C4⋊Q8—332no—
359C4⋊Q8—38no—
360C4⋊Q8—316no—
361C4⋊Q8C22×D3116yes—
362#127—38no—
363#126—332no—
364#126—332no—
365#126—332no—
366#126—332no—
367#126—332no—
368#125—332no—
369#125—332no—
370#125—332no—
371#125—316no—
372#125—38no—
373#125—38no—
374#125C22×D3116yes—
375#124—332no—
376#124—332no—
377#124—316no—
378#124—316no—
379#123—332no—
380#123—332no—
381#123—332no—
382#123—332no—
383#123—332no—
384C22⋊Q8—332no—
385C22⋊Q8—332no—
386C22⋊Q8—316no—
387C22⋊Q8—332no—
388C22⋊Q8—332no—
389C22⋊Q8—332no—
390C22⋊Q8C22×D3116yes—
391C22⋊Q8C22×D3116yes—
392#121—316no—
393#121—332no—
394C4×Q8—332no—
395C4×Q8—316no—
396C4×Q8—316no—
397C4×Q8—316no—
398C4×Q8C22×D3116yes—
399C4×Q8C22×D3116yes—
400C4×D4—332no—
401C4×D4—332no—
402C4×D4—316no—
403C4×D4—316no—
404C4×D4—316no—
405#117—332no—
406#117—332no—
407#117C22×D318yes—
408#117C22×D318yes—
409C2×C4⋊C4—34no—
410C2×C4⋊C4—34no—
411C2×C4⋊C4—34no—
412C2×C4⋊C4—34no—
413C2×C4⋊C4—34no—
414C2×C4⋊C4—32no—
415C2×C4⋊C4—32no—
416C2×C4⋊C4—34no—
417C2×C4⋊C4—34no—
418C2×C4⋊C4—34no—
419C2×C4⋊C4—34no—
420C2×C4⋊C4—34no—
421C2×C4⋊C4—332no—
422C2×C4⋊C4C22×D312yes—
423C2×C4⋊C4C22×D312yes—
424C2×C4⋊C4C22×D312yes—
425C2×C4⋊C4C22×D312yes—
426C2×C4⋊C4C22×D312yes—
427C2×C4⋊C4C22×D312yes—
428C2×C4⋊C4C22×D312yes—
429C2×C4⋊C4C22×D312yes—
430C2×C22⋊C4—34no—
431C2×C22⋊C4—34no—
432C2×C22⋊C4—34no—
433C2×C22⋊C4—34no—
434C2×C22⋊C4—34no—
435C2×C22⋊C4—34no—
436C2×C22⋊C4—34no—
437C2×C22⋊C4—34no—
438C2×C22⋊C4—34no—
439C2×C22⋊C4C22×D312yes—
440C2×C22⋊C4C22×D311yescharacteristic
441C2×C22⋊C4C22×D311yescharacteristic
442C2×C42—34no—
443C2×C42—34no—
444C2×C42—34no—
445C2×C42C22×D311yescharacteristic
446C2×C42C22×D311yescharacteristic
447C2×C42C22×D312yes—
448C2×C42C22×D311yescharacteristic
449C2×C42C22×D312yes—
450C2×C42C22×D311yescharacteristic
451C2×C42C22×D318yes—
452C23×C3—22no—
453C23×C3—24no—
454C23×C3C2512yes—
455C23×C3C2511yescharacteristic
456C22×D3—28no—
457C22×D3—28no—
458C22×D3—28no—
459C22×D3C2518yes—
460C3×Q8—48no—
461C3×Q8—48no—
462C2×C3×C4—22no—
463C2×C3×C4—24no—
464C2×C3×C4—24no—
465C2×C3×C4—24no—
466C2×C3×C4—24no—
467C2×C3×C4—24no—
468C2×C3×C4—24no—
469C2×C3×C4—28no—
470C2×C3×C4—48no—
471C2×C3×C42−1+414yes—
472C2×C3×C4C2514yes—
473C2×C3×C4C2514yes—
474C2×C3×C4C2514yes—
475C2×C3×C4C2512yes—
476C2×C3×C4C2514yes—
477C2×C3×C4C2514yes—
478C2×C3×C4C2514yes—
479C2×C3×C4C2512yes—
480#67—416no—
481C2×Dic3—28no—
482C2×Dic3—28no—
483C2×Dic3—28no—
484C2×Dic3C2518yes—
485C2×Dic3C2518yes—
486C2×Dic3C2518yes—
487D12—216no—
488D12—416no—
489C4×D3—416no—
490C4×D3—416no—
491C4×D3—416no—
492Dic6—216no—
493Dic6—416no—
494C24—34no—
495C24C23×D311yescharacteristic
496C2×Q8—38no—
497C2×Q8—316no—
498C2×Q8—44no—
499C2×Q8—44no—
500C2×Q8—68no—
501C2×Q8—128no—
502C2×Q8C23×D318yes—
503C2×Q8C23×D318yes—
504C2×D4—316no—
505C2×D4—38no—
506C2×D4—316no—
507C2×D4—68no—
508C2×D4—128no—
509C2×D4—128no—
510C22×C4—24no—
511C22×C4—34no—
512C22×C4—34no—
513C22×C4—34no—
514C22×C4—316no—
515C22×C4—316no—
516C22×C4—316no—
517C22×C4—44no—
518C22×C4—128no—
519C22×C4—128no—
520C22×C4—128no—
521C22×C4C23×D314yes—
522C22×C4C23×D312yes—
523C22×C4C23×D312yes—
524C22×C4C23×D312yes—
525C22×C4C23×D311yescharacteristic
526C22×C4C23×D312yes—
527C22×C4C23×D312yes—
528C22×C4C23×D312yes—
529C22×C4C23×D318yes—
530C22×C4C23×D311yescharacteristic
531C4⋊C4—316no—
532C4⋊C4—316no—
533C4⋊C4—316no—
534C4⋊C4—316no—
535C4⋊C4—316no—
536C4⋊C4—38no—
537C4⋊C4—38no—
538C4⋊C4—316no—
539C4⋊C4—316no—
540C4⋊C4—316no—
541C4⋊C4—316no—
542C4⋊C4—316no—
543C4⋊C4C23×D318yes—
544C4⋊C4C23×D318yes—
545C4⋊C4C23×D318yes—
546C4⋊C4C23×D318yes—
547C4⋊C4C23×D318yes—
548C4⋊C4C23×D318yes—
549C4⋊C4C23×D318yes—
550C4⋊C4C23×D318yes—
551C22⋊C4—316no—
552C22⋊C4—316no—
553C22⋊C4—316no—
554C22⋊C4—316no—
555C22⋊C4—316no—
556C22⋊C4—316no—
557C22⋊C4—316no—
558C22⋊C4—316no—
559C22⋊C4—316no—
560C22⋊C4C23×D318yes—
561C22⋊C4C23×D314yes—
562C22⋊C4C23×D314yes—
563C42—316no—
564C42—316no—
565C42—316no—
566C42C23×D314yes—
567C42C23×D318yes—
568C42C23×D314yes—
569C42C23×D314yes—
570C42C23×D318yes—
571C42C23×D314yes—
572C22×C3—22no—
573C22×C3—24no—
574C22×C3—44no—
575C22×C3C2×2−1+412yes—
576C22×C3#58514yes—
577C22×C3C2611yescharacteristic
578C2×D3—28no—
579C2×D3—28no—
580C2×D3—28no—
581C2×D3—416no—
582C3×C4—22no—
583C3×C4—24no—
584C3×C4—24no—
585C3×C4—24no—
586C3×C4—24no—
587C3×C4—24no—
588C3×C4—24no—
589C3×C4C2×2−1+414yes—
590Dic3—28no—
591Dic3—28no—
592Dic3—28no—
593C23—22no—
594C23—24no—
595C23—38no—
596C23—68no—
597C23—68no—
598C23—68no—
599C23C24×D312yes—
600C23C24×D311yescharacteristic
601Q8—48no—
602Q8—48no—
603Q8—616no—
604Q8—1216no—
605D4—616no—
606D4—1216no—
607D4—1216no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—28no—
616C2×C4—38no—
617C2×C4—38no—
618C2×C4—38no—
619C2×C4—48no—
620C2×C4—68no—
621C2×C4—68no—
622C2×C4—68no—
623C2×C4—1216no—
624C2×C4—1216no—
625C2×C4—1216no—
626C2×C4#100714yes—
627C2×C4C24×D314yes—
628C2×C4C24×D314yes—
629C2×C4C24×D314yes—
630C2×C4C24×D312yes—
631C2×C4C24×D314yes—
632C2×C4C24×D314yes—
633C2×C4C24×D314yes—
634C2×C4C24×D312yes—
635C2×C3—42no—
636C2×C3#355714yescomplemented
637C2×C3C22×2−1+411yescharacteristic
638C2×C3C2×#58512yes—
639D3—48no—
640C22—22no—
641C22—24no—
642C22—44no—
643C22—68no—
644C22—68no—
645C22—68no—
646C22—1216no—
647C22C2×#100712yes—
648C22#596414yes—
649C22C25×D311yescharacteristic
650C4—22no—
651C4—24no—
652C4—24no—
653C4—24no—
654C4—24no—
655C4—24no—
656C4—24no—
657C4—68no—
658C4—68no—
659C4—68no—
660C4C2×#100714yes—
661C3C2×#355711yescharacteristic, complemented, Sylow
662C2—42no—
663C2—128no—
664C2#8629014yesdirect product factor
665C2C22×#100711yescharacteristic
666C2C2×#596412yes—
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×#533C2
socleC23×C3C25
prime radicalC2×#5535C2
hypercenterC2×#533D3
hypocenterC3C2×#3557
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×C3×#533C2semidirect
2. #86290C2direct (N × H)
3. C2×#5735C2semidirect
4. C2×#5693C2semidirect
5. C2×#5638C2semidirect
6. C2×#5640C2semidirect
7. C2×#5579C2semidirect
8. C2×#5585C2semidirect
9. C2×#5567C2semidirect
10. C2×#5584C2semidirect
11. C2×#5534C2semidirect
13. C2×#5518C2semidirect
14. C2×#5479C2semidirect
16. C3×#533C22semidirect
34. #5735C22semidirect
35. #5693C22semidirect
36. #5640C22semidirect
37. #5638C22semidirect
38. #5585C22semidirect
39. #5584C22semidirect
40. #5579C22semidirect
41. #5567C22semidirect
43. #5534C22semidirect
44. #5518C22semidirect
65. #5479C22semidirect
94. C2×#533D3semidirect
213. #533C2×D3semidirect
636. C2×C3#3557semidirect
661. C3C2×#3557semidirect
664. C2#86290direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
#862902
C2×#57352
C2×#56932
C2×#56382
C2×#56402
C2×#55792
C2×#55852
C2×#55672
C2×#55842
C2×#55342
C2×#55352
C2×#55182
C2×#54792
C2×#35573