Finite Group Databasebeta

← C3×#71612 #11787427

Subgroups — C3×#71612 #11787427

Order 1152.

1017 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic
2C22×C3×#214C211yesmaximal, characteristic
3C2×C3×#832C211yesmaximal, characteristic
4C2×C3×#810C211yesmaximal, characteristic, complemented
5C3×#4929C211yesmaximal, characteristic, complemented
6C3×#4529C211yesmaximal, characteristic, complemented
7C3×#4524C211yesmaximal, characteristic, complemented
8C3×#1877—31nomaximal
9#71612C311yesmaximal, characteristic, direct product factor
10C22×C32×D4C2211yescharacteristic
11C23×C32×C4—21no—
12C22×C3×Dic6C2211yescharacteristic
13C32×C4×D4—22no—
14C32×C4×D4—22no—
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C32×C22⋊C4—21no—
17C2×C32×C42C2211yescharacteristic
18C2×C3×#214—22no—
19C2×C3×#214—22no—
20C2×C3×#214C414yescomplemented
21C2×C3×C4⋊Dic3C2211yescharacteristic
22C22×C3×C3⋊C8C2211yescharacteristic
23C22×C3×C3⋊C8C2211yescharacteristic
24C3×#832—22no—
25C3×C3⋊#95—21no—
26C3×C3⋊#95—21no—
27C3×#810—22no—
28C22×C3×SD16—31no—
29C2×C3×C4×D4—21no—
30C2×C3×C4×D4D311yescharacteristic
31C2×C3×C4×D4C2×C311yescharacteristic
32C22×#214C2×C311yescharacteristic
33C2×C3×#103—31no—
34C2×C3×#102—31no—
35C3×#391—31no—
36C2×#832C2×C311yescharacteristic
37C2×#810C2×C311yescharacteristic, complemented
38#4929C2×C311yescharacteristic, complemented
39C3×#340—31no—
40C3×#336—31no—
41#4529C2×C311yescharacteristic, complemented
42#4524C2×C311yescharacteristic, complemented
43C24×C32—21no—
44C2×C32×D4—21no—
45C2×C32×D4—21no—
46C2×C32×D4—21no—
47C2×C32×D4C2×C412yes—
48C2×C32×D4D412yes—
49C2×C32×D4D412yes—
50C22×C32×C4—24no—
51C22×C32×C4—21no—
52C22×C32×C4—42no—
53C22×C32×C4—42no—
54C22×C32×C4D411yescharacteristic
55C22×C32×C4D411yescharacteristic
56C22×C32×C4C2311yescharacteristic
57C22×C3×Dic3—21no—
58C22×C3×Dic3—21no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6C2×C412yescomplemented
65C32×C4⋊C4—21no—
66C32×C4⋊C4D412yes—
67C32×C22⋊C4—24no—
68C32×C42—21no—
69C32×C42—21no—
70C3×#214—28no—
71C3×C4⋊Dic3—21no—
72C3×C4⋊Dic3—21no—
73C2×C3×C3⋊C8—21no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8C2×C412yescomplemented
79#1877—31noSylow
80C22×C3×Q8—31no—
81C22×C3×D4—21no—
82C22×C3×D4C2×D311yescharacteristic
83C22×C3×D4C22×C311yescharacteristic
84C23×C3×C4—22no—
85C23×C3×C4—21no—
86C23×C3×C4—21no—
87C22×Dic6C22×C311yescharacteristic
88C2×C3×SD16—34no—
89C2×C3×SD16—62no—
90C2×C3×SD16—62no—
91C22×C3×C8—31no—
92C22×C3×C8—31no—
93C3×C4×D4—24no—
94C3×C4×D4—24no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—31no—
101C2×C3×C4⋊C4C2×D311yescharacteristic
102C2×C3×C4⋊C4C22×C311yescharacteristic
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C42—21no—
107C2×C3×C42C2×D311yescharacteristic
108C2×C3×C42C22×C311yescharacteristic
109C2×#214—22no—
110C2×#214—22no—
111C2×#214C3×C414yescomplemented
112C2×C4⋊Dic3C22×C311yescharacteristic
113C22×C3⋊C8C22×C311yescharacteristic
114C22×C3⋊C8C22×C311yescharacteristic
115C3×#103—62no—
116C3×#102—62no—
117C3×#95—61no—
118C3×#95—61no—
119#832—22no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122#810—22no—
123C23×C32—22no—
124C23×C32—22no—
125C23×C32—22no—
126C23×C32—41no—
127C23×C32—41no—
128C23×C32—41no—
129C23×C32—41no—
130C23×C32C2×D411yescharacteristic
131C32×D4—22no—
132C32×D4—22no—
133C32×D4—22no—
134C32×D4C22⋊C414yes—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—22no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—48no—
142C2×C32×C4—41no—
143C2×C32×C4—41no—
144C2×C32×C4SD1612yescomplemented
145C2×C32×C4C22×C411yescharacteristic
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4C2×D411yescharacteristic
149C2×C32×C4#4111yescharacteristic
150C2×C32×C4#4111yescharacteristic
151C2×C3×Dic3—22no—
152C2×C3×Dic3—41no—
153C2×C3×Dic3—41no—
154C2×C3×Dic3—41no—
155C2×C3×Dic3—41no—
156C2×C3×Dic3—41no—
157C3×Dic6—22no—
158C3×Dic6—22no—
159C3×Dic6—41no—
160C3×Dic6—41no—
161C3×C3⋊C8—22no—
162C3×C3⋊C8—41no—
163C22×SD16—31no—
164C2×C4×D4C3×D311yescharacteristic
165C2×#103—31no—
166C2×#102—31no—
167#391—31no—
168#340—31no—
169#336—31no—
170C24×C3—22no—
171C24×C3—21no—
172C24×C3—21no—
173C2×C3×Q8—32no—
174C2×C3×Q8—61no—
175C2×C3×Q8—61no—
176C2×C3×Q8—61no—
177C2×C3×Q8—61no—
178C2×C3×Q8—61no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4C4×D312yes—
192C2×C3×D4D1212yes—
193C2×C3×D4#6712yes—
194C2×C3×D4C2×C3×C412yes—
195C2×C3×D4C3×D412yes—
196C2×C3×D4C3×D412yes—
197C22×C3×C4—22no—
198C22×C3×C4—28no—
199C22×C3×C4—24no—
200C22×C3×C4—24no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—44no—
207C22×C3×C4—42no—
208C22×C3×C4—42no—
209C22×C3×C4—44no—
210C22×C3×C4—42no—
211C22×C3×C4—42no—
212C22×C3×C4—61no—
213C22×C3×C4—61no—
214C22×C3×C4#6711yescharacteristic
215C22×C3×C4#6711yescharacteristic
216C22×C3×C4C3×D411yescharacteristic
217C22×C3×C4C3×D411yescharacteristic
218C22×C3×C4C22×D311yescharacteristic
219C22×C3×C4C23×C311yescharacteristic
220C22×Dic3—21no—
221C22×Dic3—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6C2×C3×C412yescomplemented
228C3×SD16—68no—
229C2×C3×C8—32no—
230C2×C3×C8—61no—
231C2×C3×C8—61no—
232C2×C3×C8—61no—
233C2×C3×C8—61no—
234C2×C3×C8—61no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—61no—
240C3×C4⋊C4—61no—
241C3×C4⋊C4#6712yes—
242C3×C4⋊C4C3×D412yes—
243C3×C22⋊C4—28no—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—24no—
246C3×C42—22no—
247C3×C42—22no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252#214—28no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8C2×C3×C412yescomplemented
261C22×C32—24no—
262C22×C32—42no—
263C22×C32—42no—
264C22×C32—42no—
265C22×C32—42no—
266C22×C32—42no—
267C22×C32—42no—
268C22×C32C4×D411yescharacteristic
269C22×C32#12011yescharacteristic
270C22×C32#12111yescharacteristic
271C22×C32C2×SD1611yescharacteristic
272C22×C32#13511yescharacteristic
273C22×C32AGL(1,ℤ/8)11yescharacteristic
274C22×C32#13711yescharacteristic
275C32×C4—22no—
276C32×C4—42no—
277C32×C4—41no—
278C32×C4C2×C22⋊C411yescharacteristic
279C32×C4C4×D411yescharacteristic
280C32×C4#12111yescharacteristic
281C32×C4#12311yescharacteristic
282C3×Dic3—42no—
283C3×Dic3—81no—
284C22×Q8—31no—
285C22×D4C2×C3×D311yescharacteristic
286C23×C4—21no—
287C2×SD16—34no—
288C2×SD16—62no—
289C2×SD16—62no—
290C22×C8—31no—
291C22×C8—31no—
292C4×D4—22no—
293C4×D4—22no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4C2×C3×D311yescharacteristic
296C2×C22⋊C4—21no—
297C2×C42C2×C3×D311yescharacteristic
298#103—62no—
299#102—62no—
300#95—61no—
301#95—61no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—21no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3C2×#6711yescharacteristic
325C23×C3C2×C3×D411yescharacteristic
326C3×Q8—62no—
327C3×Q8—62no—
328C3×Q8—121no—
329C3×Q8—121no—
330C3×D4—24no—
331C3×D4—24no—
332C3×D4—24no—
333C3×D4—22no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—24no—
340C3×D4(C2×D3)⋊C414yes—
341C3×D4C3×C22⋊C414yes—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—416no—
368C2×C3×C4—48no—
369C2×C3×C4—48no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—62no—
377C2×C3×C4—121no—
378C2×C3×C4—121no—
379C2×C3×C4—121no—
380C2×C3×C4—121no—
381C2×C3×C4—121no—
382C2×C3×C4#21412yescomplemented
383C2×C3×C4C3×SD1612yescomplemented
384C2×C3×C4C2×C4×D311yescharacteristic
385C2×C3×C4C2×D1211yescharacteristic
386C2×C3×C4#23511yescharacteristic
387C2×C3×C4#23511yescharacteristic
388C2×C3×C4C2×#6711yescharacteristic
389C2×C3×C4C2×#6711yescharacteristic
390C2×C3×C4C22×C3×C411yescharacteristic
391C2×C3×C4C2×C3×D411yescharacteristic
392C2×C3×C4C2×C3×D411yescharacteristic
393C2×C3×C4C2×C3×D411yescharacteristic
394C2×C3×C4C3×#4111yescharacteristic
395C2×C3×C4C3×#4111yescharacteristic
396C2×Dic3—22no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402Dic6—22no—
403Dic6—22no—
404Dic6—41no—
405Dic6—41no—
406C3×C8—62no—
407C3×C8—121no—
408C3⋊C8—22no—
409C3⋊C8—41no—
410C2×C32—44no—
411C2×C32#38611yescharacteristic
412C2×C32C4×SD1611yescharacteristic
413C2×C32#44011yescharacteristic
414C2×C32#45011yescharacteristic
415C2×C32#45211yescharacteristic
416C2×C32#46111yescharacteristic
417C2×C32#46311yescharacteristic
418C24—21no—
419C2×Q8—32no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4C3×C4×D312yes—
429C2×D4C3×D1212yes—
430C2×D4C3×#6712yes—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4C3×#6711yescharacteristic
438C22×C4C3×#6711yescharacteristic
439C22×C4C22×C3×D311yescharacteristic
440SD16—68no—
441C2×C8—32no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C4⋊C4—21no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4C3×#6712yes—
451C22⋊C4—24no—
452C42—21no—
453C42—21no—
454C22×C3—28no—
455C22×C3—24no—
456C22×C3—24no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—44no—
465C22×C3—44no—
466C22×C3—44no—
467C22×C3—44no—
468C22×C3—44no—
469C22×C3—44no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3C4×#6711yescharacteristic
483C22×C3#93011yescharacteristic
484C22×C3#93211yescharacteristic
485C22×C3C2×#21411yescharacteristic
486C22×C3#95011yescharacteristic
487C22×C3#95111yescharacteristic
488C22×C3#95311yescharacteristic
489C22×C3C3×C4×D411yescharacteristic
490C22×C3C3×#12011yescharacteristic
491C22×C3C3×#12111yescharacteristic
492C22×C3C2×C3×SD1611yescharacteristic
493C22×C3C3×#13511yescharacteristic
494C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
495C22×C3C3×#13711yescharacteristic
496C3×C4—24no—
497C3×C4—22no—
498C3×C4—22no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—44no—
504C3×C4—42no—
505C3×C4—42no—
506C3×C4—41no—
507C3×C4—42no—
508C3×C4—41no—
509C3×C4—122no—
510C3×C4—241no—
511C3×C4C2×(C2×D3)⋊C411yescharacteristic
512C3×C4C4×#6711yescharacteristic
513C3×C4#92911yescharacteristic
514C3×C4#93011yescharacteristic
515C3×C4C2×C3×C22⋊C411yescharacteristic
516C3×C4C3×C4×D411yescharacteristic
517C3×C4C3×#12111yescharacteristic
518C3×C4C3×#12311yescharacteristic
519Dic3—42no—
520Dic3—81no—
521C32#187711yescharacteristic, complemented, Sylow
522C23—22no—
523C23—22no—
524C23—22no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C2×C3×#6711yescharacteristic
530Q8—62no—
531Q8—62no—
532Q8—121no—
533Q8—121no—
534D4—22no—
535D4—22no—
536D4—22no—
537D4C3×(C2×D3)⋊C414yes—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—48no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—62no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4C3×#21412yescomplemented
554C2×C4C2×C3×C4×D311yescharacteristic
555C2×C4C2×C3×D1211yescharacteristic
556C2×C4C3×#23511yescharacteristic
557C2×C4C3×#23511yescharacteristic
558C2×C4C2×C3×#6711yescharacteristic
559C2×C4C2×C3×#6711yescharacteristic
560C8—62no—
561C8—121no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—48no—
570C2×C3—44no—
571C2×C3—44no—
572C2×C3#501611yescharacteristic
573C2×C3C4×#21411yescharacteristic
574C2×C3#501811yescharacteristic
575C2×C3#501911yescharacteristic
576C2×C3#521311yescharacteristic
577C2×C3#521811yescharacteristic
578C2×C3#523911yescharacteristic
579C2×C3C3×#38611yescharacteristic
580C2×C3C3×C4×SD1611yescharacteristic
581C2×C3C3×#44011yescharacteristic
582C2×C3C3×#45011yescharacteristic
583C2×C3C3×#45211yescharacteristic
584C2×C3C3×#46111yescharacteristic
585C2×C3C3×#46311yescharacteristic
586C22—24no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22C3×C4×#6711yescharacteristic
594C22C3×#93011yescharacteristic
595C22C3×#93211yescharacteristic
596C22C2×C3×#21411yescharacteristic
597C22C3×#95011yescharacteristic
598C22C3×#95111yescharacteristic
599C22C3×#95311yescharacteristic
600C4—22no—
601C4—42no—
602C4—41no—
603C4—122no—
604C4—241no—
605C4C2×C3×(C2×D3)⋊C411yescharacteristic
606C4C3×C4×#6711yescharacteristic
607C4C3×#92911yescharacteristic
608C4C3×#93011yescharacteristic
609C3—21no—
610C3#7161211yescharacteristic, direct product factor
611C3C3×#187711yescharacteristic, complemented
612C2—44no—
613C2C3×#501611yescharacteristic
614C2C3×C4×#21411yescharacteristic
615C2C3×#501811yescharacteristic
616C2C3×#501911yescharacteristic
617C2C3×#521311yescharacteristic
618C2C3×#521811yescharacteristic
619C2C3×#523911yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×C4×D4C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×C4×D4D3
hypocenterC3C3×#1877
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
4. C2×C3×#810C2semidirect
5. C3×#4929C2semidirect
6. C3×#4529C2semidirect
7. C3×#4524C2semidirect
9. #71612C3direct (N × H)
20. C2×C3×#214C4semidirect
37. C2×#810C2×C3semidirect
38. #4929C2×C3semidirect
41. #4529C2×C3semidirect
42. #4524C2×C3semidirect
64. C2×C3×Dic6C2×C4semidirect
78. C2×C3×C3⋊C8C2×C4semidirect
111. C2×#214C3×C4semidirect
144. C2×C32×C4SD16semidirect
227. C2×Dic6C2×C3×C4semidirect
260. C2×C3⋊C8C2×C3×C4semidirect
382. C2×C3×C4#214semidirect
383. C2×C3×C4C3×SD16semidirect
521. C32#1877semidirect
553. C2×C4C3×#214semidirect
610. C3#71612direct (N × H)
611. C3C3×#1877semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×D42
C22×C3×#2142
C2×C3×#8322
C2×C3×#8102
C3×#49292
C3×#45292
C3×#45242
C3×#18773
#716123