Finite Group Databasebeta

← C2×C3×#4749 #11867915

Subgroups — C2×C3×#4749 #11867915

Order 1152.

1069 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#932C211yesmaximal, characteristic, complemented
2C22×C3×#213C211yesmaximal, characteristic, complemented
3C2×C3×#908C211yesmaximal, characteristic, complemented
4C2×C3×#907C211yesmaximal, characteristic, complemented
5C2×C32×#100C211yesmaximal, characteristic, complemented
6C2×C3×#823C211yesmaximal, characteristic, complemented
7C2×C3×#822C211yesmaximal, characteristic, complemented
8C3×#4749C218yesmaximal, direct product factor
9C2×C3×#472—31nomaximal
10C2×#4749C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C2×C3×#235C2211yescharacteristic
13C22×C3×D12C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16C2×C3×#214—21no—
17C3×#932—22no—
18C3×#932C2214yescomplemented
19C2×C3×#213—21no—
20C2×C3×#213—21no—
21C2×C3×#213—21no—
22C2×C3×#213—22no—
23C2×C3×#213C2214yescomplemented
24C2×C3×#208C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×#908—22no—
27C3×#908C2214yescomplemented
28C3×#907—22no—
29C3×#907C2214yescomplemented
30C2×C3×D24—21no—
31C2×C3×#204—21no—
32C2×C3×ΓL(1,𝔽25)—21no—
33C2×C3×D3×C8—21no—
34C32×#100C2214yescomplemented
35C3×#823C2214yescomplemented
36C3×#822C2214yescomplemented
37C2×C3×AGL(1,ℤ/8)—31no—
38C2×C3×AGL(1,ℤ/8)—31no—
39C22×C3×D8—31no—
40C2×C3×#131—31no—
41C2×#932C2×C311yescharacteristic, complemented
42C22×#213C2×C311yescharacteristic, complemented
43C2×#908C2×C311yescharacteristic, complemented
44C2×#907C2×C311yescharacteristic, complemented
45C3×#472—38no—
46C2×C3×#108—31no—
47C2×C3×#100—21no—
48C2×C3×#100—31no—
49C2×C3×#100D311yescharacteristic, complemented
50C2×C3×#100C2×C311yescharacteristic, complemented
51C2×#823C2×C311yescharacteristic, complemented
52C2×#822C2×C311yescharacteristic, complemented
53#4749C2×C318yesdirect product factor
54C24×C32—21no—
55C23×C3×D3—21no—
56C2×C32×D4—21no—
57C2×C32×D4—21no—
58C2×C32×D4—21no—
59C2×C32×D4—21no—
60C2×C32×D4—21no—
61C2×C32×D4C2312yes—
62C22×C32×C4C2311yescharacteristic
63C2×C3×#67—21no—
64C3×#235—21no—
65C3×#235—21no—
66C3×#235D412yes—
67C3×#235C2312yes—
68C2×C3×D12—21no—
69C2×C3×D12—21no—
70C2×C3×D12—21no—
71C2×C3×D12—21no—
72C2×C3×D12—21no—
73C2×C3×D12D411yescharacteristic
74C2×C3×D12C2312yes—
75C2×C3×C4×D3—21no—
76C2×C3×Dic6D411yescharacteristic
77C32×M16—21no—
78C32×M16—21no—
79C32×M16C2312yes—
80C32×M16C2312yes—
81C2×C32×C8—21no—
82C2×C32×C8—21no—
83C3×#214—24no—
84C3×#213—24no—
85C3×#213—24no—
86C3×#213—24no—
87C3×#208—21no—
88C3×#208C2312yes—
89C2×C3×C3⋊C8—21no—
90C2×C3×C3⋊C8D411yescharacteristic
91C2×C3×C3⋊C8D411yescharacteristic
92C2×C3×C3⋊C8D412yes—
93C2×C3×C3⋊C8C2312yes—
94C3×D24—24no—
95C3×#204—24no—
96C3×ΓL(1,𝔽25)—24no—
97C3×D3×C8—24no—
98C2×#472—31noSylow
99C2×C3×#41—31no—
100C22×C3×D4—21no—
101C22×C3×D4—31no—
102C22×C3×D4C2×D311yescharacteristic
103C22×C3×D4C22×C311yescharacteristic
104C2×#235C22×C311yescharacteristic
105C22×D12C22×C311yescharacteristic
106C3×AGL(1,ℤ/8)—34no—
107C3×AGL(1,ℤ/8)—34no—
108C3×AGL(1,ℤ/8)—62no—
109C3×AGL(1,ℤ/8)—62no—
110C2×C3×SD16—61no—
111C2×C3×SD16—61no—
112C2×C3×D8—34no—
113C2×C3×D8—61no—
114C2×C3×D8—61no—
115C2×C3×D8—61no—
116C2×C3×D8—62no—
117C2×C3×D8—61no—
118C3×#131—34no—
119C3×#131—62no—
120C2×C3×M16—21no—
121C2×C3×M16—21no—
122C2×C3×M16—31no—
123C2×C3×M16—61no—
124C2×C3×M16C2×D311yescharacteristic
125C2×C3×M16C2×D311yescharacteristic
126C2×C3×M16C22×C311yescharacteristic
127C2×C3×M16C22×C311yescharacteristic
128C22×C3×C8—31no—
129C22×C3×C8—61no—
130C2×#214—21no—
131#932—22no—
132#932C22×C314yescomplemented
133C2×#213—21no—
134C2×#213—21no—
135C2×#213—21no—
136C2×#213—22no—
137C2×#213C22×C314yescomplemented
138C2×#208C22×C311yescharacteristic
139C22×C3⋊C8C22×C311yescharacteristic
140#908—22no—
141#908C22×C314yescomplemented
142#907—22no—
143#907C22×C314yescomplemented
144C2×D24—21no—
145C2×#204—21no—
146C2×ΓL(1,𝔽25)—21no—
147C2×D3×C8—21no—
148C3×#108—34no—
149C3×#100—24no—
150C3×#100—34no—
151C3×#100C2×D314yescomplemented
152C3×#100C22×C314yescomplemented
153#823C22×C314yescomplemented
154#822C22×C314yescomplemented
155C23×C32—22no—
156C23×C32—41no—
157C23×C32—41no—
158C23×C32—82no—
159C23×C32#4111yescharacteristic
160C22×C3×D3—21no—
161C22×C3×D3—22no—
162C22×C3×D3—41no—
163C22×C3×D3—41no—
164C22×C3×D3—82no—
165C32×D4—22no—
166C32×D4—22no—
167C32×D4—41no—
168C32×D4—41no—
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2×D411yescharacteristic
173C2×C32×C4#4111yescharacteristic
174C2×C32×C4C2411yescharacteristic
175C3×#67—22no—
176C3×#67—22no—
177C2×C3×Dic3—21no—
178C3×D12—21no—
179C3×D12—22no—
180C3×D12—22no—
181C3×D12—41no—
182C3×D12—41no—
183C3×D12C2×D412yes—
184C3×C4×D3—22no—
185C3×C4×D3—22no—
186C3×Dic6—21no—
187C3×Dic6C2×D412yes—
188C32×C8—22no—
189C32×C8—22no—
190C3×C3⋊C8—22no—
191C3×C3⋊C8C2×D412yes—
192C3×C3⋊C8C2×D412yes—
193C2×AGL(1,ℤ/8)—31no—
194C2×AGL(1,ℤ/8)—31no—
195C22×D8—31no—
196C2×#131—31no—
197#472—38no—
198C2×#108—31no—
199C2×#100—31no—
200C2×#100C3×D311yescharacteristic, complemented
201C24×C3—22no—
202C24×C3—21no—
203C24×C3—21no—
204C24×C3—61no—
205C23×D3—21no—
206C3×#41—32no—
207C3×#41—32no—
208C3×#41—61no—
209C3×#41—61no—
210C2×C3×Q8—31no—
211C2×C3×D4—22no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—22no—
223C2×C3×D4—31no—
224C2×C3×D4—32no—
225C2×C3×D4—41no—
226C2×C3×D4—41no—
227C2×C3×D4—41no—
228C2×C3×D4—41no—
229C2×C3×D4—61no—
230C2×C3×D4—61no—
231C2×C3×D4—61no—
232C2×C3×D4—61no—
233C2×C3×D4—61no—
234C2×C3×D4—61no—
235C2×C3×D4C22×D312yes—
236C2×C3×D4C23×C312yes—
237C22×C3×C4—21no—
238C22×C3×C4—61no—
239C22×C3×C4C22×D311yescharacteristic
240C22×C3×C4C23×C311yescharacteristic
241C2×#67—21no—
242#235—21no—
243#235—21no—
244#235C3×D412yes—
245#235C23×C312yes—
246C2×D12—21no—
247C2×D12—21no—
248C2×D12—21no—
249C2×D12—21no—
250C2×D12—21no—
251C2×D12C3×D411yescharacteristic
252C2×D12C23×C312yes—
253C2×C4×D3—21no—
254C2×Dic6C3×D411yescharacteristic
255C3×SD16—64no—
256C3×SD16—64no—
257C3×D8—64no—
258C3×D8—64no—
259C3×D8—64no—
260C3×D8—64no—
261C3×M16—21no—
262C3×M16—21no—
263C3×M16—21no—
264C3×M16—21no—
265C3×M16—22no—
266C3×M16—22no—
267C3×M16—32no—
268C3×M16—41no—
269C3×M16—41no—
270C3×M16—64no—
271C3×M16—61no—
272C3×M16C22×D312yes—
273C3×M16C22×D312yes—
274C3×M16C23×C312yes—
275C3×M16C23×C312yes—
276C2×C3×C8—21no—
277C2×C3×C8—21no—
278C2×C3×C8—21no—
279C2×C3×C8—21no—
280C2×C3×C8—31no—
281C2×C3×C8—31no—
282C2×C3×C8—32no—
283C2×C3×C8—32no—
284C2×C3×C8—41no—
285C2×C3×C8—41no—
286C2×C3×C8—64no—
287C2×C3×C8—61no—
288#214—24no—
289#213—24no—
290#213—24no—
291#213—24no—
292#208—21no—
293#208C23×C312yes—
294C2×C3⋊C8—21no—
295C2×C3⋊C8C3×D411yescharacteristic
296C2×C3⋊C8C3×D411yescharacteristic
297C2×C3⋊C8C3×D412yes—
298C2×C3⋊C8C23×C312yes—
299D24—24no—
300#204—24no—
301ΓL(1,𝔽25)—24no—
302D3×C8—24no—
303C22×C32—22no—
304C22×C32—42no—
305C22×C32—82no—
306C22×C32—82no—
307C22×C32—82no—
308C22×C32#12111yescharacteristic
309C22×C32#12111yescharacteristic
310C22×C32C2×#4111yescharacteristic
311C2×C3×D3—22no—
312C2×C3×D3—42no—
313C2×C3×D3—42no—
314C2×C3×D3—82no—
315C2×C3×D3—82no—
316C2×C3×D3—82no—
317C32×C4#12111yescharacteristic
318C32×C4#12111yescharacteristic
319C32×C4C22×D411yescharacteristic
320C32×C4C22×D411yescharacteristic
321C3×Dic3—22no—
322C2×#41—31no—
323C22×D4—31no—
324C22×D4C2×C3×D311yescharacteristic
325AGL(1,ℤ/8)—34no—
326AGL(1,ℤ/8)—34no—
327AGL(1,ℤ/8)—62no—
328AGL(1,ℤ/8)—62no—
329C2×SD16—61no—
330C2×SD16—61no—
331C2×D8—34no—
332C2×D8—61no—
333C2×D8—61no—
334C2×D8—61no—
335C2×D8—62no—
336C2×D8—61no—
337#131—34no—
338#131—62no—
339C2×M16—31no—
340C2×M16—61no—
341C2×M16C2×C3×D311yescharacteristic
342C2×M16C2×C3×D311yescharacteristic
343C22×C8—31no—
344C22×C8—61no—
345#108—34no—
346#100—34no—
347#100C2×C3×D314yescomplemented
348C23×C3—24no—
349C23×C3—22no—
350C23×C3—22no—
351C23×C3—21no—
352C23×C3—42no—
353C23×C3—42no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—61no—
359C23×C3—62no—
360C23×C3—82no—
361C23×C3—82no—
362C23×C3—84no—
363C23×C3—121no—
364C23×C3—121no—
365C23×C3—242no—
366C23×C3#23511yescharacteristic
367C23×C3C3×#4111yescharacteristic
368C22×D3—21no—
369C22×D3—22no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—82no—
373C3×Q8—32no—
374C3×Q8—61no—
375C3×D4—22no—
376C3×D4—22no—
377C3×D4—22no—
378C3×D4—22no—
379C3×D4—32no—
380C3×D4—42no—
381C3×D4—42no—
382C3×D4—42no—
383C3×D4—42no—
384C3×D4—41no—
385C3×D4—41no—
386C3×D4—41no—
387C3×D4—41no—
388C3×D4—62no—
389C3×D4—62no—
390C3×D4—61no—
391C3×D4—62no—
392C3×D4—62no—
393C3×D4—121no—
394C3×D4—121no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—61no—
402C2×C3×C4—62no—
403C2×C3×C4—62no—
404C2×C3×C4#23511yescharacteristic
405C2×C3×C4D3×D411yescharacteristic
406C2×C3×C4D3×D411yescharacteristic
407C2×C3×C4D3×D411yescharacteristic
408C2×C3×C4D3×D411yescharacteristic
409C2×C3×C4C2×C3×D411yescharacteristic
410C2×C3×C4C2×C3×D411yescharacteristic
411C2×C3×C4C2×C3×D411yescharacteristic
412C2×C3×C4C2×C3×D411yescharacteristic
413C2×C3×C4C3×#4111yescharacteristic
414C2×C3×C4C23×D311yescharacteristic
415C2×C3×C4C24×C311yescharacteristic
416#67—22no—
417#67—22no—
418C2×Dic3—21no—
419D12—21no—
420D12—22no—
421D12—22no—
422D12—41no—
423D12—41no—
424D12C2×C3×D412yes—
425C4×D3—22no—
426C4×D3—22no—
427Dic6—21no—
428Dic6C2×C3×D412yes—
429C3×C8—22no—
430C3×C8—22no—
431C3×C8—22no—
432C3×C8—22no—
433C3×C8—32no—
434C3×C8—32no—
435C3×C8—42no—
436C3×C8—42no—
437C3×C8—62no—
438C3⋊C8—22no—
439C3⋊C8C2×C3×D412yes—
440C3⋊C8C2×C3×D412yes—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—82no—
444C2×C32#47212yescomplemented
445C2×C32C2×#12111yescharacteristic
446C3×D3—42no—
447C3×D3—82no—
448C24—21no—
449C24—61no—
450#41—32no—
451#41—32no—
452#41—61no—
453#41—61no—
454C2×Q8—31no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—32no—
461C2×D4—31no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—61no—
468C2×D4C22×C3×D312yes—
469C22×C4—61no—
470C22×C4C22×C3×D311yescharacteristic
471SD16—64no—
472SD16—64no—
473D8—64no—
474D8—64no—
475D8—64no—
476D8—64no—
477M16—21no—
478M16—21no—
479M16—32no—
480M16—64no—
481M16—61no—
482M16C22×C3×D312yes—
483M16C22×C3×D312yes—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—32no—
487C2×C8—31no—
488C2×C8—31no—
489C2×C8—32no—
490C2×C8—64no—
491C2×C8—61no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—42no—
498C22×C3—44no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—62no—
502C22×C3—82no—
503C22×C3—82no—
504C22×C3—82no—
505C22×C3—82no—
506C22×C3—84no—
507C22×C3—84no—
508C22×C3—84no—
509C22×C3—82no—
510C22×C3—82no—
511C22×C3—122no—
512C22×C3—122no—
513C22×C3—242no—
514C22×C3—242no—
515C22×C3—242no—
516C22×C3#88411yescharacteristic
517C22×C3#88411yescharacteristic
518C22×C3C3×#12111yescharacteristic
519C22×C3C3×#12111yescharacteristic
520C22×C3C2×#23511yescharacteristic
521C22×C3C2×C3×#4111yescharacteristic
522C2×D3—22no—
523C2×D3—42no—
524C2×D3—42no—
525C2×D3—82no—
526C2×D3—82no—
527C2×D3—82no—
528C3×C4—21no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—62no—
533C3×C4#88411yescharacteristic
534C3×C4#88411yescharacteristic
535C3×C4C3×#12111yescharacteristic
536C3×C4C3×#12111yescharacteristic
537C3×C4C2×D3×D411yescharacteristic
538C3×C4C2×D3×D411yescharacteristic
539C3×C4C22×C3×D411yescharacteristic
540C3×C4C22×C3×D411yescharacteristic
541Dic3—22no—
542C32C2×#47211yescharacteristic, complemented, Sylow
543C23—22no—
544C23—41no—
545C23—41no—
546C23—61no—
547C23—62no—
548C23—82no—
549C23—121no—
550C23—121no—
551C23—242no—
552C23C3×#23511yescharacteristic
553Q8—32no—
554Q8—61no—
555D4—22no—
556D4—22no—
557D4—32no—
558D4—41no—
559D4—41no—
560D4—62no—
561D4—62no—
562D4—61no—
563D4—62no—
564D4—62no—
565D4—121no—
566D4—121no—
567C2×C4—61no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4C3×#23511yescharacteristic
571C2×C4C3×D3×D411yescharacteristic
572C2×C4C3×D3×D411yescharacteristic
573C2×C4C3×D3×D411yescharacteristic
574C2×C4C3×D3×D411yescharacteristic
575C2×C4C23×C3×D311yescharacteristic
576C8—22no—
577C8—22no—
578C8—32no—
579C8—32no—
580C8—62no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—22no—
587C2×C3—41no—
588C2×C3—41no—
589C2×C3—84no—
590C2×C3—82no—
591C2×C3—82no—
592C2×C3—122no—
593C2×C3—242no—
594C2×C3#474912yesdirect product factor
595C2×C3C3×#47212yescomplemented
596C2×C3C2×#88411yescharacteristic
597C2×C3C2×C3×#12111yescharacteristic
598D3—42no—
599D3—82no—
600C22—22no—
601C22—42no—
602C22—62no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—122no—
607C22—122no—
608C22—242no—
609C22—242no—
610C22—242no—
611C22C3×#88411yescharacteristic
612C22C3×#88411yescharacteristic
613C22C2×C3×#23511yescharacteristic
614C4—62no—
615C4C3×#88411yescharacteristic
616C4C3×#88411yescharacteristic
617C4C2×C3×D3×D411yescharacteristic
618C4C2×C3×D3×D411yescharacteristic
619C3—21no—
620C3C2×#474911yescharacteristic, direct product factor
621C3C2×C3×#47211yescharacteristic, complemented
622C2—21no—
623C2—21no—
624C2—82no—
625C2—122no—
626C2—242no—
627C2C3×#474912yesdirect product factor
628C2C2×C3×#88411yescharacteristic
subgroupG/H
centerC22×C3#884
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#100C2
socleC22×C32#121
hypercenterC2×C3×#100D3
hypocenterC3C2×C3×#472
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#100

Complements

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

NH ≅ G/Ntype
1. C2×C3×#932C2semidirect
2. C22×C3×#213C2semidirect
3. C2×C3×#908C2semidirect
4. C2×C3×#907C2semidirect
5. C2×C32×#100C2semidirect
6. C2×C3×#823C2semidirect
7. C2×C3×#822C2semidirect
8. C3×#4749C2direct (N × H)
10. C2×#4749C3direct (N × H)
18. C3×#932C22semidirect
23. C2×C3×#213C22semidirect
27. C3×#908C22semidirect
29. C3×#907C22semidirect
34. C32×#100C22semidirect
35. C3×#823C22semidirect
36. C3×#822C22semidirect
41. C2×#932C2×C3semidirect
42. C22×#213C2×C3semidirect
43. C2×#908C2×C3semidirect
44. C2×#907C2×C3semidirect
49. C2×C3×#100D3semidirect
50. C2×C3×#100C2×C3semidirect
51. C2×#823C2×C3semidirect
52. C2×#822C2×C3semidirect
53. #4749C2×C3direct (N × H)
132. #932C22×C3semidirect
137. C2×#213C22×C3semidirect
141. #908C22×C3semidirect
143. #907C22×C3semidirect
151. C3×#100C2×D3semidirect
152. C3×#100C22×C3semidirect
153. #823C22×C3semidirect
154. #822C22×C3semidirect
200. C2×#100C3×D3semidirect
347. #100C2×C3×D3semidirect
444. C2×C32#472semidirect
542. C32C2×#472semidirect
594. C2×C3#4749direct (N × H)
595. C2×C3C3×#472semidirect
620. C3C2×#4749direct (N × H)
621. C3C2×C3×#472semidirect
627. C2C3×#4749direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9322
C22×C3×#2132
C2×C3×#9082
C2×C3×#9072
C2×C32×#1002
C2×C3×#8232
C2×C3×#8222
C3×#47492
C2×C3×#4723
C2×#47493