Finite Group Databasebeta

← C2×C3×#5141 #11867853

Subgroups — C2×C3×#5141 #11867853

Order 1152.

1069 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#131C211yesmaximal, characteristic, complemented
2C2×C3×#951C211yesmaximal, characteristic, complemented
3C2×C3×#949C211yesmaximal, characteristic
4C22×C3×#204C211yesmaximal, characteristic, complemented
5C3×#5141C218yesmaximal, direct product factor
6C2×C3×#824C211yesmaximal, characteristic, complemented
7C2×C3×#823C211yesmaximal, characteristic, complemented
8C2×C3×#819C211yesmaximal, characteristic, complemented
9C2×C3×#471—31nomaximal
10C2×#5141C311yesmaximal, characteristic, direct product factor
11C2×C32×#41C2211yescharacteristic
12C22×C3×D12C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C32×#131—22no—
15C32×#131C2214yescomplemented
16C2×C32×M16—21no—
17C2×C32×M16C2211yescharacteristic
18C22×C32×C8—21no—
19C22×C32×C8C2211yescharacteristic
20C3×#951—22no—
21C3×#951C2214yescomplemented
22C3×#949—22no—
23C3×#949C2214yes—
24C2×C3×#216—21no—
25C2×C3×#215—21no—
26C2×C3×#214—21no—
27C2×C3×#213—21no—
28C2×C3×#208C2211yescharacteristic
29C2×C3×#208C2211yescharacteristic
30C2×C3×#204—21no—
31C2×C3×#204—21no—
32C2×C3×#204—22no—
33C2×C3×#204C2214yescomplemented
34C3×#824C2214yescomplemented
35C3×#823C2214yescomplemented
36C3×#819C2214yescomplemented
37C2×C3×#137—31no—
38C2×C3×AGL(1,ℤ/8)—31no—
39C22×C3×SD16—31no—
40C2×C3×#131—21no—
41C2×C3×#131D311yescharacteristic, complemented
42C2×C3×#131C2×C311yescharacteristic, complemented
43C2×#951C2×C311yescharacteristic, complemented
44C2×#949C2×C311yescharacteristic
45C22×#204C2×C311yescharacteristic, complemented
46C3×#471—38no—
47C2×C3×#108—31no—
48C2×C3×#101—31no—
49C2×C3×#100—31no—
50#5141C2×C318yesdirect product factor
51C2×#824C2×C311yescharacteristic, complemented
52C2×#823C2×C311yescharacteristic, complemented
53C2×#819C2×C311yescharacteristic, complemented
54C23×C3×D3—21no—
55C32×#41—21no—
56C32×#41—21no—
57C32×#41D412yes—
58C32×#41C2312yes—
59C2×C32×Q8D411yescharacteristic
60C2×C32×D4—21no—
61C2×C32×D4D411yescharacteristic
62C22×C32×C4—21no—
63C22×C32×C4C2311yescharacteristic
64C22×C3×Dic3—21no—
65C2×C3×D12—21no—
66C2×C3×D12—21no—
67C2×C3×D12—21no—
68C2×C3×D12—21no—
69C2×C3×D12—21no—
70C2×C3×D12C2312yes—
71C2×C3×Dic6—21no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6—21no—
76C2×C3×Dic6C2312yes—
77C32×M16—21no—
78C32×M16—24no—
79C32×M16C2312yes—
80C2×C32×C8—21no—
81C2×C32×C8—24no—
82C2×C32×C8D411yescharacteristic
83C2×C32×C8D411yescharacteristic
84C2×C32×C8D412yes—
85C2×C32×C8C2312yes—
86C3×#216—24no—
87C3×#215—24no—
88C3×#214—24no—
89C3×#213—24no—
90C3×#208—21no—
91C3×#208—21no—
92C3×#208C2312yes—
93C3×#208C2312yes—
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8—21no—
96C3×#204—24no—
97C3×#204—24no—
98C2×#471—31noSylow
99C2×C3×#41—21no—
100C2×C3×#41C2×D311yescharacteristic
101C2×C3×#41C22×C311yescharacteristic
102C22×C3×Q8—31no—
103C22×C3×D4—31no—
104C22×D12C22×C311yescharacteristic
105C22×Dic6C22×C311yescharacteristic
106C3×#137—34no—
107C3×#137—62no—
108C3×AGL(1,ℤ/8)—34no—
109C3×AGL(1,ℤ/8)—62no—
110C2×C3×Q16—61no—
111C2×C3×SD16—34no—
112C2×C3×SD16—61no—
113C2×C3×SD16—61no—
114C2×C3×SD16—61no—
115C2×C3×SD16—61no—
116C2×C3×SD16—62no—
117C2×C3×D8—61no—
118C3×#131—24no—
119C3×#131—22no—
120C3×#131—22no—
121C3×#131—24no—
122C3×#131C2×D314yescomplemented
123C3×#131C22×C314yescomplemented
124C2×C3×M16—22no—
125C2×C3×M16—21no—
126C2×C3×M16—21no—
127C2×C3×M16—21no—
128C2×C3×M16—31no—
129C2×C3×M16—31no—
130C2×C3×M16C2×D311yescharacteristic
131C2×C3×M16C22×C311yescharacteristic
132C22×C3×C8—22no—
133C22×C3×C8—21no—
134C22×C3×C8—21no—
135C22×C3×C8—21no—
136C22×C3×C8C2×D311yescharacteristic
137C22×C3×C8C22×C311yescharacteristic
138#951—22no—
139#951C22×C314yescomplemented
140#949—22no—
141#949C22×C314yes—
142C2×#216—21no—
143C2×#215—21no—
144C2×#214—21no—
145C2×#213—21no—
146C2×#208C22×C311yescharacteristic
147C2×#208C22×C311yescharacteristic
148C2×#204—21no—
149C2×#204—21no—
150C2×#204—22no—
151C2×#204C22×C314yescomplemented
152C3×#108—34no—
153C3×#101—34no—
154C3×#100—34no—
155#824C22×C314yescomplemented
156#823C22×C314yescomplemented
157#819C22×C314yescomplemented
158C23×C32—21no—
159C23×C32#4111yescharacteristic
160C22×C3×D3—22no—
161C22×C3×D3—41no—
162C22×C3×D3—41no—
163C22×C3×D3—82no—
164C32×Q8—21no—
165C32×Q8C2×D412yes—
166C32×D4—22no—
167C32×D4—22no—
168C32×D4—21no—
169C32×D4C2×D412yes—
170C2×C32×C4—21no—
171C2×C32×C4—22no—
172C2×C32×C4—22no—
173C2×C32×C4C2×D411yescharacteristic
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4#4111yescharacteristic
178C2×C32×C4C2411yescharacteristic
179C2×C3×Dic3—22no—
180C2×C3×Dic3—41no—
181C2×C3×Dic3—41no—
182C3×D12—22no—
183C3×D12—22no—
184C3×D12—41no—
185C3×D12—41no—
186C3×Dic6—22no—
187C3×Dic6—22no—
188C3×Dic6—41no—
189C3×Dic6—41no—
190C32×C8—22no—
191C32×C8C2×D412yes—
192C32×C8C2×D412yes—
193C3×C3⋊C8—22no—
194C3×C3⋊C8—22no—
195C2×#137—31no—
196C2×AGL(1,ℤ/8)—31no—
197C22×SD16—31no—
198C2×#131C3×D311yescharacteristic, complemented
199#471—38no—
200C2×#108—31no—
201C2×#101—31no—
202C2×#100—31no—
203C24×C3—61no—
204C23×D3—21no—
205C3×#41—22no—
206C3×#41—22no—
207C3×#41—21no—
208C3×#41—21no—
209C3×#41—21no—
210C3×#41—21no—
211C3×#41—22no—
212C3×#41—22no—
213C3×#41D1212yes—
214C3×#41C3×D412yes—
215C3×#41C22×D312yes—
216C3×#41C23×C312yes—
217C2×C3×Q8—21no—
218C2×C3×Q8—32no—
219C2×C3×Q8—61no—
220C2×C3×Q8—61no—
221C2×C3×Q8—61no—
222C2×C3×Q8—61no—
223C2×C3×Q8—61no—
224C2×C3×Q8D1211yescharacteristic
225C2×C3×Q8C3×D411yescharacteristic
226C2×C3×D4—22no—
227C2×C3×D4—21no—
228C2×C3×D4—21no—
229C2×C3×D4—21no—
230C2×C3×D4—32no—
231C2×C3×D4—61no—
232C2×C3×D4—61no—
233C2×C3×D4—61no—
234C2×C3×D4—61no—
235C2×C3×D4—61no—
236C2×C3×D4D1211yescharacteristic
237C2×C3×D4C3×D411yescharacteristic
238C22×C3×C4—22no—
239C22×C3×C4—21no—
240C22×C3×C4—21no—
241C22×C3×C4—21no—
242C22×C3×C4—61no—
243C22×C3×C4C22×D311yescharacteristic
244C22×C3×C4C23×C311yescharacteristic
245C22×Dic3—21no—
246C2×D12—21no—
247C2×D12—21no—
248C2×D12—21no—
249C2×D12—21no—
250C2×D12—21no—
251C2×D12C23×C312yes—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6—21no—
255C2×Dic6—21no—
256C2×Dic6—21no—
257C2×Dic6C23×C312yes—
258C3×Q16—64no—
259C3×SD16—64no—
260C3×SD16—64no—
261C3×SD16—64no—
262C3×SD16—64no—
263C3×D8—64no—
264C3×M16—22no—
265C3×M16—28no—
266C3×M16—21no—
267C3×M16—21no—
268C3×M16—24no—
269C3×M16—24no—
270C3×M16—22no—
271C3×M16—32no—
272C3×M16—32no—
273C3×M16—61no—
274C3×M16—61no—
275C3×M16C22×D312yes—
276C3×M16C23×C312yes—
277C2×C3×C8—22no—
278C2×C3×C8—28no—
279C2×C3×C8—21no—
280C2×C3×C8—21no—
281C2×C3×C8—24no—
282C2×C3×C8—24no—
283C2×C3×C8—21no—
284C2×C3×C8—21no—
285C2×C3×C8—22no—
286C2×C3×C8—22no—
287C2×C3×C8—61no—
288C2×C3×C8—61no—
289C2×C3×C8#6711yescharacteristic
290C2×C3×C8#6711yescharacteristic
291C2×C3×C8#6712yes—
292C2×C3×C8C3×D411yescharacteristic
293C2×C3×C8C3×D411yescharacteristic
294C2×C3×C8C3×D412yes—
295C2×C3×C8C22×D312yes—
296C2×C3×C8C23×C312yes—
297#216—24no—
298#215—24no—
299#214—24no—
300#213—24no—
301#208—21no—
302#208—21no—
303#208C23×C312yes—
304#208C23×C312yes—
305C2×C3⋊C8—21no—
306C2×C3⋊C8—21no—
307#204—24no—
308#204—24no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32—42no—
312C22×C32#12111yescharacteristic
313C22×C32#12111yescharacteristic
314C22×C32C2×#4111yescharacteristic
315C2×C3×D3—42no—
316C2×C3×D3—82no—
317C2×C3×D3—82no—
318C2×C3×D3—82no—
319C32×C4—22no—
320C32×C4#12111yescharacteristic
321C32×C4#12111yescharacteristic
322C32×C4C22×D411yescharacteristic
323C32×C4C22×D411yescharacteristic
324C3×Dic3—42no—
325C2×#41C2×C3×D311yescharacteristic
326C22×Q8—31no—
327C22×D4—31no—
328#137—34no—
329#137—62no—
330AGL(1,ℤ/8)—34no—
331AGL(1,ℤ/8)—62no—
332C2×Q16—61no—
333C2×SD16—34no—
334C2×SD16—61no—
335C2×SD16—61no—
336C2×SD16—61no—
337C2×SD16—61no—
338C2×SD16—62no—
339C2×D8—61no—
340#131—22no—
341#131C2×C3×D314yescomplemented
342C2×M16—21no—
343C2×M16—31no—
344C2×M16—31no—
345C2×M16C2×C3×D311yescharacteristic
346C22×C8—21no—
347C22×C8C2×C3×D311yescharacteristic
348#108—34no—
349#101—34no—
350#100—34no—
351C23×C3—22no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—62no—
356C23×C3—121no—
357C23×C3—121no—
358C23×C3—242no—
359C23×C3#23511yescharacteristic
360C23×C3C3×#4111yescharacteristic
361C22×D3—22no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—82no—
365C3×Q8—22no—
366C3×Q8—21no—
367C3×Q8—21no—
368C3×Q8—22no—
369C3×Q8—62no—
370C3×Q8—62no—
371C3×Q8—121no—
372C3×Q8—121no—
373C3×Q8C2×D1212yes—
374C3×Q8C2×C3×D412yes—
375C3×D4—24no—
376C3×D4—24no—
377C3×D4—22no—
378C3×D4—22no—
379C3×D4—22no—
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—21no—
383C3×D4—21no—
384C3×D4—22no—
385C3×D4—62no—
386C3×D4—62no—
387C3×D4—121no—
388C3×D4—121no—
389C3×D4C2×D1212yes—
390C3×D4C2×C3×D412yes—
391C2×C3×C4—24no—
392C2×C3×C4—22no—
393C2×C3×C4—24no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—62no—
407C2×C3×C4—121no—
408C2×C3×C4—121no—
409C2×C3×C4C2×D1211yescharacteristic
410C2×C3×C4C2×D1211yescharacteristic
411C2×C3×C4#23511yescharacteristic
412C2×C3×C4C2×#6711yescharacteristic
413C2×C3×C4C2×#6711yescharacteristic
414C2×C3×C4C2×C3×D411yescharacteristic
415C2×C3×C4C2×C3×D411yescharacteristic
416C2×C3×C4C2×C3×D411yescharacteristic
417C2×C3×C4C2×C3×D411yescharacteristic
418C2×C3×C4C3×#4111yescharacteristic
419C2×C3×C4C23×D311yescharacteristic
420C2×C3×C4C24×C311yescharacteristic
421C2×Dic3—22no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424D12—22no—
425D12—22no—
426D12—41no—
427D12—41no—
428Dic6—22no—
429Dic6—22no—
430Dic6—41no—
431Dic6—41no—
432C3×C8—24no—
433C3×C8—22no—
434C3×C8—22no—
435C3×C8—22no—
436C3×C8—22no—
437C3×C8—62no—
438C3×C8—62no—
439C3×C8C2×#6712yes—
440C3×C8C2×#6712yes—
441C3×C8C2×C3×D412yes—
442C3×C8C2×C3×D412yes—
443C3⋊C8—22no—
444C3⋊C8—22no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—42no—
448C2×C32#47112yescomplemented
449C2×C32C2×#12111yescharacteristic
450C3×D3—82no—
451C24—61no—
452#41—21no—
453#41—21no—
454#41C3×D1212yes—
455#41C22×C3×D312yes—
456C2×Q8—32no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×Q8C3×D1211yescharacteristic
463C2×D4—21no—
464C2×D4—32no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—61no—
468C2×D4—61no—
469C2×D4—61no—
470C2×D4C3×D1211yescharacteristic
471C22×C4—21no—
472C22×C4—61no—
473C22×C4C22×C3×D311yescharacteristic
474Q16—64no—
475SD16—64no—
476SD16—64no—
477SD16—64no—
478SD16—64no—
479D8—64no—
480M16—21no—
481M16—24no—
482M16—32no—
483M16—32no—
484M16—61no—
485M16—61no—
486M16C22×C3×D312yes—
487C2×C8—21no—
488C2×C8—24no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8C3×#6711yescharacteristic
492C2×C8C3×#6711yescharacteristic
493C2×C8C3×#6712yes—
494C2×C8C22×C3×D312yes—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—24no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—42no—
504C22×C3—44no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—122no—
508C22×C3—242no—
509C22×C3—242no—
510C22×C3—242no—
511C22×C3#93011yescharacteristic
512C22×C3#93011yescharacteristic
513C22×C3C3×#12111yescharacteristic
514C22×C3C3×#12111yescharacteristic
515C22×C3C2×#23511yescharacteristic
516C22×C3C2×C3×#4111yescharacteristic
517C2×D3—42no—
518C2×D3—82no—
519C2×D3—82no—
520C2×D3—82no—
521C3×C4—24no—
522C3×C4—22no—
523C3×C4—22no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—21no—
528C3×C4—122no—
529C3×C4#93011yescharacteristic
530C3×C4#93011yescharacteristic
531C3×C4C3×#12111yescharacteristic
532C3×C4C3×#12111yescharacteristic
533C3×C4C22×D1211yescharacteristic
534C3×C4C22×#6711yescharacteristic
535C3×C4C22×C3×D411yescharacteristic
536C3×C4C22×C3×D411yescharacteristic
537Dic3—42no—
538C32C2×#47111yescharacteristic, complemented, Sylow
539C23—21no—
540C23—62no—
541C23—121no—
542C23—121no—
543C23—242no—
544C23C3×#23511yescharacteristic
545Q8—21no—
546Q8—62no—
547Q8—62no—
548Q8—121no—
549Q8—121no—
550Q8C2×C3×D1212yes—
551D4—22no—
552D4—22no—
553D4—21no—
554D4—62no—
555D4—62no—
556D4—121no—
557D4—121no—
558D4C2×C3×D1212yes—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—62no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4C2×C3×D1211yescharacteristic
566C2×C4C2×C3×D1211yescharacteristic
567C2×C4C3×#23511yescharacteristic
568C2×C4C2×C3×#6711yescharacteristic
569C2×C4C2×C3×#6711yescharacteristic
570C2×C4C23×C3×D311yescharacteristic
571C8—22no—
572C8—62no—
573C8—62no—
574C8C2×C3×#6712yes—
575C8C2×C3×#6712yes—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—22no—
582C2×C3—41no—
583C2×C3—41no—
584C2×C3—44no—
585C2×C3—42no—
586C2×C3—42no—
587C2×C3—242no—
588C2×C3#514112yesdirect product factor
589C2×C3C3×#47112yescomplemented
590C2×C3C2×#93011yescharacteristic
591C2×C3C2×C3×#12111yescharacteristic
592D3—82no—
593C22—22no—
594C22—22no—
595C22—42no—
596C22—122no—
597C22—242no—
598C22—242no—
599C22—242no—
600C22C3×#93011yescharacteristic
601C22C3×#93011yescharacteristic
602C22C2×C3×#23511yescharacteristic
603C4—22no—
604C4—122no—
605C4C3×#93011yescharacteristic
606C4C3×#93011yescharacteristic
607C4C22×C3×D1211yescharacteristic
608C4C22×C3×#6711yescharacteristic
609C3—21no—
610C3C2×#514111yescharacteristic, direct product factor
611C3C2×C3×#47111yescharacteristic, complemented
612C2—21no—
613C2—21no—
614C2—42no—
615C2—242no—
616C2C3×#514112yesdirect product factor
617C2C2×C3×#93011yescharacteristic
subgroupG/H
centerC22×C3#930
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#131C2
socleC22×C32#121
prime radicalC2×C3×#949C2
hypercenterC2×C3×#131D3
hypocenterC3C2×C3×#471
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131

Complements

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

NH ≅ G/Ntype
1. C2×C32×#131C2semidirect
2. C2×C3×#951C2semidirect
4. C22×C3×#204C2semidirect
5. C3×#5141C2direct (N × H)
6. C2×C3×#824C2semidirect
7. C2×C3×#823C2semidirect
8. C2×C3×#819C2semidirect
10. C2×#5141C3direct (N × H)
15. C32×#131C22semidirect
21. C3×#951C22semidirect
33. C2×C3×#204C22semidirect
34. C3×#824C22semidirect
35. C3×#823C22semidirect
36. C3×#819C22semidirect
41. C2×C3×#131D3semidirect
42. C2×C3×#131C2×C3semidirect
43. C2×#951C2×C3semidirect
45. C22×#204C2×C3semidirect
50. #5141C2×C3direct (N × H)
51. C2×#824C2×C3semidirect
52. C2×#823C2×C3semidirect
53. C2×#819C2×C3semidirect
122. C3×#131C2×D3semidirect
123. C3×#131C22×C3semidirect
139. #951C22×C3semidirect
151. C2×#204C22×C3semidirect
155. #824C22×C3semidirect
156. #823C22×C3semidirect
157. #819C22×C3semidirect
198. C2×#131C3×D3semidirect
341. #131C2×C3×D3semidirect
448. C2×C32#471semidirect
538. C32C2×#471semidirect
588. C2×C3#5141direct (N × H)
589. C2×C3C3×#471semidirect
610. C3C2×#5141direct (N × H)
611. C3C2×C3×#471semidirect
616. C2C3×#5141direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1312
C2×C3×#9512
C2×C3×#9492
C22×C3×#2042
C3×#51412
C2×C3×#8242
C2×C3×#8232
C2×C3×#8192
C2×C3×#4713
C2×#51413