Finite Group Databasebeta

← C2×C3×#5204 #11867959

Subgroups — C2×C3×#5204 #11867959

Order 1152.

1025 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#137C211yesmaximal, characteristic
2C2×C3×#951C211yesmaximal, characteristic
3C2×C3×#948C211yesmaximal, characteristic
4C22×C3×#216C211yesmaximal, characteristic, complemented
5C2×C3×#836C211yesmaximal, characteristic, complemented
6C3×#5204C218yesmaximal, direct product factor
7C2×C3×#824C211yesmaximal, characteristic, complemented
8C2×C3×#822C211yesmaximal, characteristic, complemented
9C2×C3×#473—31nomaximal
10C2×#5204C311yesmaximal, characteristic, direct product factor
11C2×C32×#41C2211yescharacteristic
12C22×C32×Q8C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C32×#137—22no—
15C32×#137C2214yes—
16C2×C32×Q16—21no—
17C2×C32×SD16—21no—
18C2×C32×M16C2211yescharacteristic
19C3×#951—22no—
20C3×#951C2214yes—
21C3×#948—22no—
22C3×#948C2214yes—
23C2×C3×#216—21no—
24C2×C3×#216—21no—
25C2×C3×#216—21no—
26C2×C3×#216—22no—
27C2×C3×#216C2214yescomplemented
28C2×C3×#214—21no—
29C2×C3×#208—21no—
30C2×C3×#208C2211yescharacteristic
31C2×C3×#208C2211yescharacteristic
32C22×C3×C3⋊C8—21no—
33C22×C3×C3⋊C8C2211yescharacteristic
34C3×#836C2214yescomplemented
35C3×#824C2214yescomplemented
36C3×#822C2214yescomplemented
37C2×C3×#137—21no—
38C2×C3×#137—31no—
39C2×C3×#137D311yescharacteristic
40C2×C3×#137C2×C311yescharacteristic
41C22×C3×Q16—31no—
42C2×C3×#131—31no—
43C2×#951C2×C311yescharacteristic
44C2×#948C2×C311yescharacteristic
45C22×#216C2×C311yescharacteristic, complemented
46C3×#473—38no—
47C2×C3×#108—31no—
48C2×C3×#101—31no—
49C2×C3×#101—31no—
50C2×#836C2×C311yescharacteristic, complemented
51#5204C2×C318yesdirect product factor
52C2×#824C2×C311yescharacteristic, complemented
53C2×#822C2×C311yescharacteristic, complemented
54C32×#41—21no—
55C32×#41—21no—
56C32×#41D412yes—
57C32×#41C2312yes—
58C2×C32×Q8—21no—
59C2×C32×Q8—21no—
60C2×C32×Q8—21no—
61C2×C32×Q8—21no—
62C2×C32×Q8—21no—
63C2×C32×Q8D411yescharacteristic
64C2×C32×Q8C2312yes—
65C2×C32×D4—21no—
66C2×C32×D4D411yescharacteristic
67C22×C32×C4—21no—
68C22×C32×C4—21no—
69C22×C32×C4C2311yescharacteristic
70C22×C3×Dic3—21no—
71C2×C3×Dic6—21no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6—21no—
76C2×C3×Dic6C2312yes—
77C32×Q16—24no—
78C32×SD16—24no—
79C32×M16—21no—
80C32×M16C2312yes—
81C2×C32×C8—21no—
82C3×#216—24no—
83C3×#216—24no—
84C3×#216—24no—
85C3×#214—24no—
86C3×#208—21no—
87C3×#208—21no—
88C3×#208—24no—
89C3×#208C2312yes—
90C3×#208C2312yes—
91C2×C3×C3⋊C8—21no—
92C2×C3×C3⋊C8—21no—
93C2×C3×C3⋊C8—24no—
94C2×C3×C3⋊C8D411yescharacteristic
95C2×C3×C3⋊C8D411yescharacteristic
96C2×C3×C3⋊C8D412yes—
97C2×C3×C3⋊C8C2312yes—
98C2×#473—31noSylow
99C2×C3×#41—21no—
100C2×C3×#41C2×D311yescharacteristic
101C2×C3×#41C22×C311yescharacteristic
102C22×C3×Q8—21no—
103C22×C3×Q8—31no—
104C22×C3×Q8C2×D311yescharacteristic
105C22×C3×Q8C22×C311yescharacteristic
106C22×Dic6C22×C311yescharacteristic
107C3×#137—24no—
108C3×#137—22no—
109C3×#137—22no—
110C3×#137—24no—
111C3×#137—34no—
112C3×#137—62no—
113C3×#137C2×D314yes—
114C3×#137C22×C314yes—
115C2×C3×Q16—22no—
116C2×C3×Q16—21no—
117C2×C3×Q16—21no—
118C2×C3×Q16—34no—
119C2×C3×Q16—61no—
120C2×C3×Q16—61no—
121C2×C3×Q16—62no—
122C2×C3×Q16—61no—
123C2×C3×SD16—22no—
124C2×C3×SD16—21no—
125C2×C3×SD16—21no—
126C2×C3×SD16—61no—
127C3×#131—34no—
128C3×#131—62no—
129C2×C3×M16—21no—
130C2×C3×M16—31no—
131C2×C3×M16—31no—
132C2×C3×M16—61no—
133C2×C3×M16C2×D311yescharacteristic
134C2×C3×M16C22×C311yescharacteristic
135C22×C3×C8—31no—
136C22×C3×C8—61no—
137#951—22no—
138#951C22×C314yes—
139#948—22no—
140#948C22×C314yes—
141C2×#216—21no—
142C2×#216—21no—
143C2×#216—21no—
144C2×#216—22no—
145C2×#216C22×C314yescomplemented
146C2×#214—21no—
147C2×#208—21no—
148C2×#208C22×C311yescharacteristic
149C2×#208C22×C311yescharacteristic
150C22×C3⋊C8—21no—
151C22×C3⋊C8C22×C311yescharacteristic
152C3×#108—34no—
153C3×#101—34no—
154C3×#101—34no—
155#836C22×C314yescomplemented
156#824C22×C314yescomplemented
157#822C22×C314yescomplemented
158C23×C32—21no—
159C23×C32#4111yescharacteristic
160C32×Q8—22no—
161C32×Q8—21no—
162C32×Q8—22no—
163C32×Q8—41no—
164C32×Q8—41no—
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×C4—22no—
174C2×C32×C4—41no—
175C2×C32×C4—41no—
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4C2×D411yescharacteristic
178C2×C32×C4C2×D411yescharacteristic
179C2×C32×C4C2×D411yescharacteristic
180C2×C32×C4#4111yescharacteristic
181C2×C32×C4C2411yescharacteristic
182C2×C3×Dic3—22no—
183C2×C3×Dic3—41no—
184C2×C3×Dic3—41no—
185C3×Dic6—22no—
186C3×Dic6—22no—
187C3×Dic6—41no—
188C3×Dic6—41no—
189C32×C8—22no—
190C3×C3⋊C8—22no—
191C3×C3⋊C8—22no—
192C3×C3⋊C8C2×D412yes—
193C3×C3⋊C8C2×D412yes—
194C2×#137—31no—
195C2×#137C3×D311yescharacteristic
196C22×Q16—31no—
197C2×#131—31no—
198#473—38no—
199C2×#108—31no—
200C2×#101—31no—
201C2×#101—31no—
202C3×#41—21no—
203C3×#41—21no—
204C3×#41—21no—
205C3×#41—21no—
206C3×#41—22no—
207C3×#41—22no—
208C3×#41—41no—
209C3×#41—41no—
210C3×#41#6712yes—
211C3×#41C3×D412yes—
212C3×#41C22×D312yes—
213C3×#41C23×C312yes—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—22no—
217C2×C3×Q8—22no—
218C2×C3×Q8—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8—21no—
224C2×C3×Q8—21no—
225C2×C3×Q8—21no—
226C2×C3×Q8—21no—
227C2×C3×Q8—22no—
228C2×C3×Q8—32no—
229C2×C3×Q8—41no—
230C2×C3×Q8—41no—
231C2×C3×Q8—41no—
232C2×C3×Q8—61no—
233C2×C3×Q8—61no—
234C2×C3×Q8—61no—
235C2×C3×Q8—61no—
236C2×C3×Q8—61no—
237C2×C3×Q8#6711yescharacteristic
238C2×C3×Q8C3×D411yescharacteristic
239C2×C3×Q8C22×D312yes—
240C2×C3×Q8C23×C312yes—
241C2×C3×D4—21no—
242C2×C3×D4—21no—
243C2×C3×D4—21no—
244C2×C3×D4—41no—
245C2×C3×D4#6711yescharacteristic
246C2×C3×D4C3×D411yescharacteristic
247C22×C3×C4—21no—
248C22×C3×C4—21no—
249C22×C3×C4—21no—
250C22×C3×C4—21no—
251C22×C3×C4—21no—
252C22×C3×C4—41no—
253C22×C3×C4—41no—
254C22×C3×C4—61no—
255C22×C3×C4C22×D311yescharacteristic
256C22×C3×C4C23×C311yescharacteristic
257C22×Dic3—21no—
258C2×Dic6—21no—
259C2×Dic6—21no—
260C2×Dic6—21no—
261C2×Dic6—21no—
262C2×Dic6—21no—
263C2×Dic6C23×C312yes—
264C3×Q16—28no—
265C3×Q16—24no—
266C3×Q16—24no—
267C3×Q16—64no—
268C3×Q16—64no—
269C3×Q16—64no—
270C3×SD16—28no—
271C3×SD16—24no—
272C3×SD16—24no—
273C3×SD16—64no—
274C3×M16—22no—
275C3×M16—21no—
276C3×M16—21no—
277C3×M16—22no—
278C3×M16—32no—
279C3×M16—32no—
280C3×M16—64no—
281C3×M16—61no—
282C3×M16—61no—
283C3×M16C22×D312yes—
284C3×M16C23×C312yes—
285C2×C3×C8—22no—
286C2×C3×C8—21no—
287C2×C3×C8—21no—
288C2×C3×C8—31no—
289C2×C3×C8—31no—
290C2×C3×C8—32no—
291C2×C3×C8—32no—
292C2×C3×C8—64no—
293C2×C3×C8—61no—
294C2×C3×C8—61no—
295#216—24no—
296#216—24no—
297#216—24no—
298#214—24no—
299#208—21no—
300#208—21no—
301#208—24no—
302#208C23×C312yes—
303#208C23×C312yes—
304C2×C3⋊C8—21no—
305C2×C3⋊C8—21no—
306C2×C3⋊C8—24no—
307C2×C3⋊C8C3×D411yescharacteristic
308C2×C3⋊C8C3×D411yescharacteristic
309C2×C3⋊C8C3×D412yes—
310C2×C3⋊C8C23×C312yes—
311C22×C32—22no—
312C22×C32—22no—
313C22×C32—42no—
314C22×C32#12111yescharacteristic
315C22×C32#12111yescharacteristic
316C22×C32C2×#4111yescharacteristic
317C32×C4—22no—
318C32×C4—42no—
319C32×C4#12111yescharacteristic
320C32×C4#12111yescharacteristic
321C32×C4C22×D411yescharacteristic
322C32×C4C22×D411yescharacteristic
323C3×Dic3—42no—
324C2×#41C2×C3×D311yescharacteristic
325C22×Q8—31no—
326C22×Q8C2×C3×D311yescharacteristic
327#137—22no—
328#137—34no—
329#137—62no—
330#137C2×C3×D314yes—
331C2×Q16—21no—
332C2×Q16—34no—
333C2×Q16—61no—
334C2×Q16—61no—
335C2×Q16—62no—
336C2×Q16—61no—
337C2×SD16—21no—
338C2×SD16—61no—
339#131—34no—
340#131—62no—
341C2×M16—31no—
342C2×M16—31no—
343C2×M16—61no—
344C2×M16C2×C3×D311yescharacteristic
345C22×C8—31no—
346C22×C8—61no—
347#108—34no—
348#101—34no—
349#101—34no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—41no—
354C23×C3#23711yescharacteristic
355C23×C3C3×#4111yescharacteristic
356C3×Q8—21no—
357C3×Q8—21no—
358C3×Q8—24no—
359C3×Q8—22no—
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—22no—
363C3×Q8—22no—
364C3×Q8—42no—
365C3×Q8—41no—
366C3×Q8—42no—
367C3×Q8—42no—
368C3×Q8—41no—
369C3×Q8—41no—
370C3×Q8—41no—
371C3×Q8—41no—
372C3×Q8—62no—
373C3×Q8—62no—
374C3×Q8—121no—
375C3×Q8—121no—
376C3×Q8C2×#6712yes—
377C3×Q8C2×C3×D412yes—
378C3×D4—22no—
379C3×D4—22no—
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—21no—
383C3×D4—21no—
384C3×D4—22no—
385C3×D4—41no—
386C3×D4—42no—
387C3×D4—42no—
388C3×D4C2×#6712yes—
389C3×D4C2×C3×D412yes—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—42no—
405C2×C3×C4—42no—
406C2×C3×C4—41no—
407C2×C3×C4—42no—
408C2×C3×C4—42no—
409C2×C3×C4—42no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—62no—
415C2×C3×C4—121no—
416C2×C3×C4—121no—
417C2×C3×C4D3×D411yescharacteristic
418C2×C3×C4D3×D411yescharacteristic
419C2×C3×C4#23711yescharacteristic
420C2×C3×C4C2×#6711yescharacteristic
421C2×C3×C4C2×#6711yescharacteristic
422C2×C3×C4C2×C3×D411yescharacteristic
423C2×C3×C4C2×C3×D411yescharacteristic
424C2×C3×C4C2×C3×D411yescharacteristic
425C2×C3×C4C2×C3×D411yescharacteristic
426C2×C3×C4C3×#4111yescharacteristic
427C2×C3×C4C23×D311yescharacteristic
428C2×C3×C4C24×C311yescharacteristic
429C2×Dic3—22no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432Dic6—22no—
433Dic6—22no—
434Dic6—41no—
435Dic6—41no—
436C3×C8—24no—
437C3×C8—22no—
438C3×C8—22no—
439C3×C8—32no—
440C3×C8—32no—
441C3×C8—62no—
442C3×C8—62no—
443C3⋊C8—22no—
444C3⋊C8—22no—
445C3⋊C8C2×C3×D412yes—
446C3⋊C8C2×C3×D412yes—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—42no—
450C2×C32#47312yescomplemented
451C2×C32C2×#12111yescharacteristic
452#41—21no—
453#41—21no—
454#41C3×#6712yes—
455#41C22×C3×D312yes—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—32no—
462C2×Q8—61no—
463C2×Q8—61no—
464C2×Q8—61no—
465C2×Q8—61no—
466C2×Q8—61no—
467C2×Q8C3×#6711yescharacteristic
468C2×Q8C22×C3×D312yes—
469C2×D4—21no—
470C2×D4C3×#6711yescharacteristic
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—61no—
474C22×C4C22×C3×D311yescharacteristic
475Q16—24no—
476Q16—64no—
477Q16—64no—
478Q16—64no—
479SD16—24no—
480SD16—64no—
481M16—21no—
482M16—32no—
483M16—32no—
484M16—64no—
485M16—61no—
486M16—61no—
487M16C22×C3×D312yes—
488C2×C8—21no—
489C2×C8—32no—
490C2×C8—31no—
491C2×C8—31no—
492C2×C8—32no—
493C2×C8—64no—
494C2×C8—61no—
495C2×C8—61no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—82no—
508C22×C3#93911yescharacteristic
509C22×C3#93911yescharacteristic
510C22×C3C3×#12111yescharacteristic
511C22×C3C3×#12111yescharacteristic
512C22×C3C2×#23711yescharacteristic
513C22×C3C2×C3×#4111yescharacteristic
514C3×C4—22no—
515C3×C4—22no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—42no—
521C3×C4—44no—
522C3×C4—42no—
523C3×C4—42no—
524C3×C4—122no—
525C3×C4#93911yescharacteristic
526C3×C4#93911yescharacteristic
527C3×C4C3×#12111yescharacteristic
528C3×C4C3×#12111yescharacteristic
529C3×C4C2×D3×D411yescharacteristic
530C3×C4C22×#6711yescharacteristic
531C3×C4C22×C3×D411yescharacteristic
532C3×C4C22×C3×D411yescharacteristic
533Dic3—42no—
534C32C2×#47311yescharacteristic, complemented, Sylow
535C23—21no—
536C23C3×#23711yescharacteristic
537Q8—21no—
538Q8—22no—
539Q8—22no—
540Q8—41no—
541Q8—41no—
542Q8—62no—
543Q8—62no—
544Q8—121no—
545Q8—121no—
546Q8C2×C3×#6712yes—
547D4—22no—
548D4—22no—
549D4—21no—
550D4C2×C3×#6712yes—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—62no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4C3×D3×D411yescharacteristic
561C2×C4C3×D3×D411yescharacteristic
562C2×C4C3×#23711yescharacteristic
563C2×C4C2×C3×#6711yescharacteristic
564C2×C4C2×C3×#6711yescharacteristic
565C2×C4C23×C3×D311yescharacteristic
566C8—22no—
567C8—32no—
568C8—32no—
569C8—62no—
570C8—62no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—22no—
576C2×C3—21no—
577C2×C3—41no—
578C2×C3—41no—
579C2×C3—42no—
580C2×C3—42no—
581C2×C3—82no—
582C2×C3#520412yesdirect product factor
583C2×C3C3×#47312yescomplemented
584C2×C3C2×#93911yescharacteristic
585C2×C3C2×C3×#12111yescharacteristic
586C22—22no—
587C22—22no—
588C22—42no—
589C22C3×#93911yescharacteristic
590C22C3×#93911yescharacteristic
591C22C2×C3×#23711yescharacteristic
592C4—22no—
593C4—42no—
594C4—122no—
595C4C3×#93911yescharacteristic
596C4C3×#93911yescharacteristic
597C4C2×C3×D3×D411yescharacteristic
598C4C22×C3×#6711yescharacteristic
599C3—21no—
600C3C2×#520411yescharacteristic, direct product factor
601C3C2×C3×#47311yescharacteristic, complemented
602C2—21no—
603C2—21no—
604C2—42no—
605C2C3×#520412yesdirect product factor
606C2C2×C3×#93911yescharacteristic
subgroupG/H
centerC22×C3#939
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#137C2
socleC22×C32#121
prime radicalC2×C32×#41C22
hypercenterC2×C3×#137D3
hypocenterC3C2×C3×#473
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#137

Complements

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

NH ≅ G/Ntype
4. C22×C3×#216C2semidirect
5. C2×C3×#836C2semidirect
6. C3×#5204C2direct (N × H)
7. C2×C3×#824C2semidirect
8. C2×C3×#822C2semidirect
10. C2×#5204C3direct (N × H)
27. C2×C3×#216C22semidirect
34. C3×#836C22semidirect
35. C3×#824C22semidirect
36. C3×#822C22semidirect
45. C22×#216C2×C3semidirect
50. C2×#836C2×C3semidirect
51. #5204C2×C3direct (N × H)
52. C2×#824C2×C3semidirect
53. C2×#822C2×C3semidirect
145. C2×#216C22×C3semidirect
155. #836C22×C3semidirect
156. #824C22×C3semidirect
157. #822C22×C3semidirect
450. C2×C32#473semidirect
534. C32C2×#473semidirect
582. C2×C3#5204direct (N × H)
583. C2×C3C3×#473semidirect
600. C3C2×#5204direct (N × H)
601. C3C2×C3×#473semidirect
605. C2C3×#5204direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1372
C2×C3×#9512
C2×C3×#9482
C22×C3×#2162
C2×C3×#8362
C3×#52042
C2×C3×#8242
C2×C3×#8222
C2×C3×#4733
C2×#52043