Finite Group Databasebeta

← C2×C3×#4748 #11867850

Subgroups — C2×C3×#4748 #11867850

Order 1152.

1029 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#214C211yesmaximal, characteristic, complemented
2C2×C3×#932C211yesmaximal, characteristic
3C2×C3×#909C211yesmaximal, characteristic, complemented
4C2×C3×#907C211yesmaximal, characteristic, complemented
5C2×C32×#100C211yesmaximal, characteristic, complemented
6C2×C3×#824C211yesmaximal, characteristic, complemented
7C2×C3×#822C211yesmaximal, characteristic, complemented
8C3×#4748C218yesmaximal, direct product factor
9C2×C3×#471—31nomaximal
10C2×#4748C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C2×C3×#235C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16C2×C3×#214—21no—
17C2×C3×#214—21no—
18C2×C3×#214—21no—
19C2×C3×#214—22no—
20C2×C3×#214C2214yescomplemented
21C3×#932—22no—
22C3×#932C2214yes—
23C2×C3×#213—21no—
24C2×C3×#208C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×#909—22no—
27C3×#909C2214yescomplemented
28C3×#907—22no—
29C3×#907C2214yescomplemented
30C2×C3×Dic12—21no—
31C2×C3×#204—21no—
32C2×C3×ΓL(1,𝔽25)—21no—
33C2×C3×D3×C8—21no—
34C32×#100C2214yescomplemented
35C3×#824C2214yescomplemented
36C3×#822C2214yescomplemented
37C2×C3×#137—31no—
38C2×C3×AGL(1,ℤ/8)—31no—
39C22×C3×SD16—31no—
40C2×C3×#131—31no—
41C22×#214C2×C311yescharacteristic, complemented
42C2×#932C2×C311yescharacteristic
43C2×#909C2×C311yescharacteristic, complemented
44C2×#907C2×C311yescharacteristic, complemented
45C3×#471—38no—
46C2×C3×#108—31no—
47C2×C3×#101—31no—
48C2×C3×#100—21no—
49C2×C3×#100D311yescharacteristic, complemented
50C2×C3×#100C2×C311yescharacteristic, complemented
51C2×#824C2×C311yescharacteristic, complemented
52C2×#822C2×C311yescharacteristic, complemented
53#4748C2×C318yesdirect product factor
54C24×C32—21no—
55C2×C32×D4—21no—
56C2×C32×D4—21no—
57C2×C32×D4—21no—
58C2×C32×D4—21no—
59C2×C32×D4—21no—
60C2×C32×D4C2312yes—
61C22×C32×C4C2311yescharacteristic
62C2×C3×#67—21no—
63C22×C3×Dic3—21no—
64C3×#235—21no—
65C3×#235—21no—
66C3×#235D412yes—
67C3×#235C2312yes—
68C2×C3×D12D411yescharacteristic
69C2×C3×C4×D3—21no—
70C2×C3×Dic6—21no—
71C2×C3×Dic6—21no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6D411yescharacteristic
76C2×C3×Dic6C2312yes—
77C32×M16—21no—
78C32×M16—21no—
79C32×M16C2312yes—
80C32×M16C2312yes—
81C2×C32×C8—21no—
82C2×C32×C8—21no—
83C3×#214—24no—
84C3×#214—24no—
85C3×#214—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×Dic12—24no—
95C3×#204—24no—
96C3×ΓL(1,𝔽25)—24no—
97C3×D3×C8—24no—
98C2×#471—31noSylow
99C2×C3×#41—31no—
100C22×C3×Q8—31no—
101C22×C3×D4—21no—
102C22×C3×D4C2×D311yescharacteristic
103C22×C3×D4C22×C311yescharacteristic
104C2×#235C22×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—62no—
116C2×C3×SD16—61no—
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—
131C2×#214—21no—
132C2×#214—21no—
133C2×#214—22no—
134C2×#214C22×C314yescomplemented
135#932—22no—
136#932C22×C314yes—
137C2×#213—21no—
138C2×#208C22×C311yescharacteristic
139C22×C3⋊C8C22×C311yescharacteristic
140#909—22no—
141#909C22×C314yescomplemented
142#907—22no—
143#907C22×C314yescomplemented
144C2×Dic12—21no—
145C2×#204—21no—
146C2×ΓL(1,𝔽25)—21no—
147C2×D3×C8—21no—
148C3×#108—34no—
149C3×#101—34no—
150C3×#100—24no—
151C3×#100C2×D314yescomplemented
152C3×#100C22×C314yescomplemented
153#824C22×C314yescomplemented
154#822C22×C314yescomplemented
155C23×C32—22no—
156C23×C32—41no—
157C23×C32—41no—
158C23×C32—82no—
159C23×C32#4111yescharacteristic
160C22×C3×D3—21no—
161C32×D4—22no—
162C32×D4—22no—
163C32×D4—41no—
164C32×D4—41no—
165C2×C32×C4C2×D411yescharacteristic
166C2×C32×C4C2×D411yescharacteristic
167C2×C32×C4C2×D411yescharacteristic
168C2×C32×C4C2×D411yescharacteristic
169C2×C32×C4#4111yescharacteristic
170C2×C32×C4C2411yescharacteristic
171C3×#67—22no—
172C3×#67—22no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—41no—
176C2×C3×Dic3—41no—
177C3×D12—21no—
178C3×D12C2×D412yes—
179C3×C4×D3—22no—
180C3×C4×D3—22no—
181C3×Dic6—21no—
182C3×Dic6—22no—
183C3×Dic6—22no—
184C3×Dic6—41no—
185C3×Dic6—41no—
186C3×Dic6C2×D412yes—
187C32×C8—22no—
188C32×C8—22no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8C2×D412yes—
191C3×C3⋊C8C2×D412yes—
192C2×#137—31no—
193C2×AGL(1,ℤ/8)—31no—
194C22×SD16—31no—
195C2×#131—31no—
196#471—38no—
197C2×#108—31no—
198C2×#101—31no—
199C2×#100C3×D311yescharacteristic, complemented
200C24×C3—22no—
201C24×C3—21no—
202C24×C3—21no—
203C3×#41—32no—
204C3×#41—32no—
205C3×#41—61no—
206C3×#41—61no—
207C2×C3×Q8—31no—
208C2×C3×Q8—32no—
209C2×C3×Q8—61no—
210C2×C3×Q8—61no—
211C2×C3×Q8—61no—
212C2×C3×Q8—61no—
213C2×C3×Q8—61no—
214C2×C3×D4—22no—
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—21no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—22no—
226C2×C3×D4—31no—
227C2×C3×D4—41no—
228C2×C3×D4—41no—
229C2×C3×D4—41no—
230C2×C3×D4—41no—
231C2×C3×D4—61no—
232C2×C3×D4C22×D312yes—
233C2×C3×D4C23×C312yes—
234C22×C3×C4—21no—
235C22×C3×C4—61no—
236C22×C3×C4—61no—
237C22×C3×C4C22×D311yescharacteristic
238C22×C3×C4C23×C311yescharacteristic
239C2×#67—21no—
240C22×Dic3—21no—
241#235—21no—
242#235—21no—
243#235C3×D412yes—
244#235C23×C312yes—
245C2×D12C3×D411yescharacteristic
246C2×C4×D3—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6C3×D411yescharacteristic
253C2×Dic6C23×C312yes—
254C3×Q16—64no—
255C3×SD16—64no—
256C3×SD16—64no—
257C3×SD16—64no—
258C3×SD16—64no—
259C3×D8—64no—
260C3×M16—21no—
261C3×M16—21no—
262C3×M16—21no—
263C3×M16—21no—
264C3×M16—22no—
265C3×M16—22no—
266C3×M16—32no—
267C3×M16—41no—
268C3×M16—41no—
269C3×M16—64no—
270C3×M16—61no—
271C3×M16C22×D312yes—
272C3×M16C22×D312yes—
273C3×M16C23×C312yes—
274C3×M16C23×C312yes—
275C2×C3×C8—21no—
276C2×C3×C8—21no—
277C2×C3×C8—21no—
278C2×C3×C8—21no—
279C2×C3×C8—31no—
280C2×C3×C8—31no—
281C2×C3×C8—32no—
282C2×C3×C8—32no—
283C2×C3×C8—41no—
284C2×C3×C8—41no—
285C2×C3×C8—64no—
286C2×C3×C8—61no—
287#214—24no—
288#214—24no—
289#214—24no—
290#213—24no—
291#208—21no—
292#208C23×C312yes—
293C2×C3⋊C8—21no—
294C2×C3⋊C8C3×D411yescharacteristic
295C2×C3⋊C8C3×D411yescharacteristic
296C2×C3⋊C8C3×D412yes—
297C2×C3⋊C8C23×C312yes—
298Dic12—24no—
299#204—24no—
300ΓL(1,𝔽25)—24no—
301D3×C8—24no—
302C22×C32—22no—
303C22×C32—42no—
304C22×C32—82no—
305C22×C32—82no—
306C22×C32—82no—
307C22×C32#12111yescharacteristic
308C22×C32#12111yescharacteristic
309C22×C32C2×#4111yescharacteristic
310C2×C3×D3—22no—
311C2×C3×D3—42no—
312C32×C4#12111yescharacteristic
313C32×C4#12111yescharacteristic
314C32×C4C22×D411yescharacteristic
315C32×C4C22×D411yescharacteristic
316C3×Dic3—22no—
317C3×Dic3—42no—
318C2×#41—31no—
319C22×Q8—31no—
320C22×D4C2×C3×D311yescharacteristic
321#137—34no—
322#137—62no—
323AGL(1,ℤ/8)—34no—
324AGL(1,ℤ/8)—62no—
325C2×Q16—61no—
326C2×SD16—34no—
327C2×SD16—61no—
328C2×SD16—61no—
329C2×SD16—61no—
330C2×SD16—62no—
331C2×SD16—61no—
332C2×D8—61no—
333#131—34no—
334#131—62no—
335C2×M16—31no—
336C2×M16—61no—
337C2×M16C2×C3×D311yescharacteristic
338C2×M16C2×C3×D311yescharacteristic
339C22×C8—31no—
340C22×C8—61no—
341#108—34no—
342#101—34no—
343#100C2×C3×D314yescomplemented
344C23×C3—24no—
345C23×C3—22no—
346C23×C3—22no—
347C23×C3—21no—
348C23×C3—42no—
349C23×C3—42no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—61no—
355C23×C3—82no—
356C23×C3—82no—
357C23×C3—84no—
358C23×C3#23511yescharacteristic
359C23×C3C3×#4111yescharacteristic
360C22×D3—21no—
361C3×Q8—32no—
362C3×Q8—61no—
363C3×Q8—62no—
364C3×Q8—62no—
365C3×Q8—121no—
366C3×Q8—121no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—22no—
371C3×D4—32no—
372C3×D4—42no—
373C3×D4—42no—
374C3×D4—42no—
375C3×D4—42no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—62no—
381C3×D4—62no—
382C3×D4—61no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—61no—
390C2×C3×C4—62no—
391C2×C3×C4—62no—
392C2×C3×C4—62no—
393C2×C3×C4—121no—
394C2×C3×C4—121no—
395C2×C3×C4#23511yescharacteristic
396C2×C3×C4D3×D411yescharacteristic
397C2×C3×C4D3×D411yescharacteristic
398C2×C3×C4D3×D411yescharacteristic
399C2×C3×C4D3×D411yescharacteristic
400C2×C3×C4C2×C3×D411yescharacteristic
401C2×C3×C4C2×C3×D411yescharacteristic
402C2×C3×C4C2×C3×D411yescharacteristic
403C2×C3×C4C2×C3×D411yescharacteristic
404C2×C3×C4C3×#4111yescharacteristic
405C2×C3×C4C23×D311yescharacteristic
406C2×C3×C4C24×C311yescharacteristic
407#67—22no—
408#67—22no—
409C2×Dic3—21no—
410C2×Dic3—22no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413D12—21no—
414D12C2×C3×D412yes—
415C4×D3—22no—
416C4×D3—22no—
417Dic6—21no—
418Dic6—22no—
419Dic6—22no—
420Dic6—41no—
421Dic6—41no—
422Dic6C2×C3×D412yes—
423C3×C8—22no—
424C3×C8—22no—
425C3×C8—22no—
426C3×C8—22no—
427C3×C8—32no—
428C3×C8—32no—
429C3×C8—42no—
430C3×C8—42no—
431C3×C8—62no—
432C3⋊C8—22no—
433C3⋊C8C2×C3×D412yes—
434C3⋊C8C2×C3×D412yes—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—82no—
438C2×C32#47112yescomplemented
439C2×C32C2×#12111yescharacteristic
440C3×D3—42no—
441C24—21no—
442#41—32no—
443#41—32no—
444#41—61no—
445#41—61no—
446C2×Q8—32no—
447C2×Q8—31no—
448C2×Q8—61no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×Q8—61no—
452C2×Q8—61no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—31no—
459C2×D4—61no—
460C2×D4C22×C3×D312yes—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4C22×C3×D311yescharacteristic
464Q16—64no—
465SD16—64no—
466SD16—64no—
467SD16—64no—
468SD16—64no—
469D8—64no—
470M16—21no—
471M16—21no—
472M16—32no—
473M16—64no—
474M16—61no—
475M16C22×C3×D312yes—
476M16C22×C3×D312yes—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—32no—
480C2×C8—31no—
481C2×C8—31no—
482C2×C8—32no—
483C2×C8—64no—
484C2×C8—61no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—42no—
491C22×C3—44no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—62no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3—82no—
499C22×C3—84no—
500C22×C3—84no—
501C22×C3—84no—
502C22×C3—82no—
503C22×C3—82no—
504C22×C3—122no—
505C22×C3#88411yescharacteristic
506C22×C3#88411yescharacteristic
507C22×C3C3×#12111yescharacteristic
508C22×C3C3×#12111yescharacteristic
509C22×C3C2×#23511yescharacteristic
510C22×C3C2×C3×#4111yescharacteristic
511C2×D3—22no—
512C2×D3—42no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—62no—
518C3×C4—122no—
519C3×C4#88411yescharacteristic
520C3×C4#88411yescharacteristic
521C3×C4C3×#12111yescharacteristic
522C3×C4C3×#12111yescharacteristic
523C3×C4C2×D3×D411yescharacteristic
524C3×C4C2×D3×D411yescharacteristic
525C3×C4C22×C3×D411yescharacteristic
526C3×C4C22×C3×D411yescharacteristic
527Dic3—22no—
528Dic3—42no—
529C32C2×#47111yescharacteristic, complemented, Sylow
530C23—22no—
531C23—41no—
532C23—41no—
533C23—61no—
534C23—82no—
535C23C3×#23511yescharacteristic
536Q8—32no—
537Q8—61no—
538Q8—62no—
539Q8—62no—
540Q8—121no—
541Q8—121no—
542D4—22no—
543D4—22no—
544D4—32no—
545D4—41no—
546D4—41no—
547D4—62no—
548D4—62no—
549D4—61no—
550C2×C4—61no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4C3×#23511yescharacteristic
557C2×C4C3×D3×D411yescharacteristic
558C2×C4C3×D3×D411yescharacteristic
559C2×C4C3×D3×D411yescharacteristic
560C2×C4C3×D3×D411yescharacteristic
561C2×C4C23×C3×D311yescharacteristic
562C8—22no—
563C8—22no—
564C8—32no—
565C8—32no—
566C8—62no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—22no—
573C2×C3—41no—
574C2×C3—41no—
575C2×C3—84no—
576C2×C3—82no—
577C2×C3—82no—
578C2×C3—122no—
579C2×C3#474812yesdirect product factor
580C2×C3C3×#47112yescomplemented
581C2×C3C2×#88411yescharacteristic
582C2×C3C2×C3×#12111yescharacteristic
583D3—42no—
584C22—22no—
585C22—42no—
586C22—62no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—122no—
591C22C3×#88411yescharacteristic
592C22C3×#88411yescharacteristic
593C22C2×C3×#23511yescharacteristic
594C4—62no—
595C4—122no—
596C4C3×#88411yescharacteristic
597C4C3×#88411yescharacteristic
598C4C2×C3×D3×D411yescharacteristic
599C4C2×C3×D3×D411yescharacteristic
600C3—21no—
601C3C2×#474811yescharacteristic, direct product factor
602C3C2×C3×#47111yescharacteristic, complemented
603C2—21no—
604C2—21no—
605C2—82no—
606C2—122no—
607C2C3×#474812yesdirect product factor
608C2C2×C3×#88411yescharacteristic
subgroupG/H
centerC22×C3#884
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#100C2
socleC22×C32#121
prime radicalC2×C3×#932C2
hypercenterC2×C3×#100D3
hypocenterC3C2×C3×#471
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. C22×C3×#214C2semidirect
3. C2×C3×#909C2semidirect
4. C2×C3×#907C2semidirect
5. C2×C32×#100C2semidirect
6. C2×C3×#824C2semidirect
7. C2×C3×#822C2semidirect
8. C3×#4748C2direct (N × H)
10. C2×#4748C3direct (N × H)
20. C2×C3×#214C22semidirect
27. C3×#909C22semidirect
29. C3×#907C22semidirect
34. C32×#100C22semidirect
35. C3×#824C22semidirect
36. C3×#822C22semidirect
41. C22×#214C2×C3semidirect
43. C2×#909C2×C3semidirect
44. C2×#907C2×C3semidirect
49. C2×C3×#100D3semidirect
50. C2×C3×#100C2×C3semidirect
51. C2×#824C2×C3semidirect
52. C2×#822C2×C3semidirect
53. #4748C2×C3direct (N × H)
134. C2×#214C22×C3semidirect
141. #909C22×C3semidirect
143. #907C22×C3semidirect
151. C3×#100C2×D3semidirect
152. C3×#100C22×C3semidirect
153. #824C22×C3semidirect
154. #822C22×C3semidirect
199. C2×#100C3×D3semidirect
343. #100C2×C3×D3semidirect
438. C2×C32#471semidirect
529. C32C2×#471semidirect
579. C2×C3#4748direct (N × H)
580. C2×C3C3×#471semidirect
601. C3C2×#4748direct (N × H)
602. C3C2×C3×#471semidirect
607. C2C3×#4748direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2142
C2×C3×#9322
C2×C3×#9092
C2×C3×#9072
C2×C32×#1002
C2×C3×#8242
C2×C3×#8222
C3×#47482
C2×C3×#4713
C2×#47483