Finite Group Databasebeta

← C3×#70171 #11787421

Subgroups — C3×#70171 #11787421

Order 1152.

905 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Dic3C211yesmaximal, characteristic
2C22×C3×#214C211yesmaximal, characteristic
3C2×C32×#102C211yesmaximal, characteristic
4C2×C3×#816C211yesmaximal, characteristic, complemented
5C3×#4973C211yesmaximal, characteristic, complemented
6C3×#4550C211yesmaximal, characteristic, complemented
7C3×#4529C211yesmaximal, characteristic, complemented
8C3×#1877—31nomaximal
9#70171C311yesmaximal, characteristic, direct product factor
10C22×C32×D4C2211yescharacteristic
11C23×C3×Dic3—21no—
12C22×C3×Dic6C2211yescharacteristic
13C22×C32×C8C2211yescharacteristic
14C2×C32×C4⋊C4C2211yescharacteristic
15C2×C3×#217—21no—
16C3×D4×Dic3—22no—
17C3×D4×Dic3—22no—
18C2×C3×#214—22no—
19C2×C3×#214—22no—
20C2×C3×#214C414yescomplemented
21C2×C3×C4⋊Dic3C2211yescharacteristic
22C2×C3×C4×Dic3C2211yescharacteristic
23C22×C3×C3⋊C8C2211yescharacteristic
24C32×#102—22no—
25C3×C3⋊#95—21no—
26C3×C3⋊#95—21no—
27C3×#816—22no—
28C22×C3×SD16—31no—
29C2×C3×C4×D4—31no—
30C2×D4×Dic3C2×C311yescharacteristic
31C22×#214C2×C311yescharacteristic
32C2×C3×#103—31no—
33C2×C3×#102—21no—
34C2×C3×#102D311yescharacteristic
35C2×C3×#102C2×C311yescharacteristic
36C3×#391—31no—
37C2×#816C2×C311yescharacteristic, complemented
38#4973C2×C311yescharacteristic, complemented
39C3×#340—31no—
40C3×#336—31no—
41#4550C2×C311yescharacteristic, complemented
42#4529C2×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—21no—
51C22×C32×C4C2311yescharacteristic
52C22×C3×Dic3—24no—
53C22×C3×Dic3—21no—
54C22×C3×Dic3—21no—
55C22×C3×Dic3—42no—
56C22×C3×Dic3—42no—
57C22×C3×Dic3D411yescharacteristic
58C22×C3×Dic3D411yescharacteristic
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6C2×C412yescomplemented
65C2×C32×C8—21no—
66C2×C32×C8—21no—
67C2×C32×C8—21no—
68C32×C4⋊C4—21no—
69C32×C4⋊C4—21no—
70C3×#217—24no—
71C3×#214—28no—
72C3×C4⋊Dic3—21no—
73C3×C4⋊Dic3D412yes—
74C3×C4×Dic3—21no—
75C3×C4×Dic3—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—61no—
85C23×Dic3—21no—
86C22×Dic6C22×C311yescharacteristic
87C2×C3×SD16—34no—
88C2×C3×SD16—62no—
89C2×C3×SD16—62no—
90C22×C3×C8—21no—
91C22×C3×C8—31no—
92C22×C3×C8C2×D311yescharacteristic
93C22×C3×C8C22×C311yescharacteristic
94C3×C4×D4—62no—
95C3×C4×D4—62no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—31no—
98C2×C3×C4⋊C4C2×D311yescharacteristic
99C2×C3×C4⋊C4C22×C311yescharacteristic
100C2×C3×C22⋊C4—61no—
101C2×C3×C42—31no—
102C2×#217—21no—
103D4×Dic3—22no—
104D4×Dic3—22no—
105C2×#214—22no—
106C2×#214—22no—
107C2×#214C3×C414yescomplemented
108C2×C4⋊Dic3C22×C311yescharacteristic
109C2×C4×Dic3C22×C311yescharacteristic
110C22×C3⋊C8C22×C311yescharacteristic
111C3×#103—62no—
112C3×#102—24no—
113C3×#102—22no—
114C3×#102—22no—
115C3×#95—61no—
116C3×#95—61no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119#816—22no—
120C23×C32—22no—
121C23×C32—22no—
122C23×C32—22no—
123C23×C32—41no—
124C23×C32—41no—
125C23×C32—41no—
126C23×C32—41no—
127C23×C32C2×D411yescharacteristic
128C32×D4—22no—
129C32×D4—22no—
130C32×D4—22no—
131C32×D4C22⋊C414yes—
132C2×C32×C4—41no—
133C2×C32×C4—41no—
134C2×C32×C4—41no—
135C2×C32×C4C22×C411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4#4111yescharacteristic
140C2×C32×C4#4111yescharacteristic
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—21no—
148C2×C3×Dic3—48no—
149C2×C3×Dic3—41no—
150C2×C3×Dic3—41no—
151C2×C3×Dic3—41no—
152C2×C3×Dic3—41no—
153C2×C3×Dic3SD1612yescomplemented
154C3×Dic6—22no—
155C3×Dic6—22no—
156C3×Dic6—41no—
157C3×Dic6—41no—
158C32×C8—41no—
159C3×C3⋊C8—22no—
160C22×SD16—31no—
161C2×C4×D4—31no—
162C2×#103—31no—
163C2×#102C3×D311yescharacteristic
164#391—31no—
165#340—31no—
166#336—31no—
167C24×C3—21no—
168C24×C3—21no—
169C24×C3—41no—
170C2×C3×Q8—32no—
171C2×C3×Q8—61no—
172C2×C3×Q8—61no—
173C2×C3×Q8—61no—
174C2×C3×Q8—61no—
175C2×C3×Q8—61no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—41no—
186C2×C3×D4—41no—
187C2×C3×D4—41no—
188C2×C3×D4C4×D312yes—
189C2×C3×D4D1212yes—
190C2×C3×D4#6712yes—
191C2×C3×D4C2×C3×C412yes—
192C2×C3×D4C3×D412yes—
193C2×C3×D4C3×D412yes—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—31no—
198C22×C3×C4—31no—
199C22×C3×C4—41no—
200C22×C3×C4—64no—
201C22×C3×C4—61no—
202C22×C3×C4—61no—
203C22×C3×C4—122no—
204C22×C3×C4—122no—
205C22×C3×C4C22×D311yescharacteristic
206C22×C3×C4C23×C311yescharacteristic
207C22×Dic3—24no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—42no—
211C22×Dic3—42no—
212C22×Dic3C3×D411yescharacteristic
213C22×Dic3C3×D411yescharacteristic
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6C2×C3×C412yescomplemented
220C3×SD16—68no—
221C2×C3×C8—22no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—32no—
229C2×C3×C8—41no—
230C2×C3×C8—41no—
231C2×C3×C8—61no—
232C2×C3×C8—61no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—32no—
239C3×C4⋊C4—41no—
240C3×C4⋊C4—61no—
241C3×C22⋊C4—64no—
242C3×C42—61no—
243C3×C42—61no—
244#217—24no—
245#214—28no—
246C4⋊Dic3—21no—
247C4⋊Dic3C3×D412yes—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8C2×C3×C412yescomplemented
253C22×C32—24no—
254C22×C32—42no—
255C22×C32—42no—
256C22×C32—42no—
257C22×C32—42no—
258C22×C32—42no—
259C22×C32—42no—
260C22×C32C4×D411yescharacteristic
261C22×C32#12011yescharacteristic
262C22×C32#12111yescharacteristic
263C22×C32C2×SD1611yescharacteristic
264C22×C32#13511yescharacteristic
265C22×C32AGL(1,ℤ/8)11yescharacteristic
266C22×C32#13711yescharacteristic
267C32×C4—81no—
268C32×C4C2×C22⋊C411yescharacteristic
269C32×C4C4×D411yescharacteristic
270C32×C4#12111yescharacteristic
271C32×C4#12311yescharacteristic
272C3×Dic3—22no—
273C3×Dic3—42no—
274C3×Dic3—42no—
275C3×Dic3—41no—
276C22×Q8—31no—
277C22×D4C2×C3×D311yescharacteristic
278C23×C4—61no—
279C2×SD16—34no—
280C2×SD16—62no—
281C2×SD16—62no—
282C22×C8—31no—
283C22×C8C2×C3×D311yescharacteristic
284C4×D4—62no—
285C4×D4—62no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4C2×C3×D311yescharacteristic
288C2×C22⋊C4—61no—
289C2×C42—31no—
290#103—62no—
291#102—22no—
292#95—61no—
293#95—61no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3—81no—
313C23×C3—81no—
314C23×C3—81no—
315C23×C3—81no—
316C23×C3D3×D411yescharacteristic
317C23×C3C2×C3×D411yescharacteristic
318C3×Q8—62no—
319C3×Q8—62no—
320C3×Q8—121no—
321C3×Q8—121no—
322C3×D4—22no—
323C3×D4—22no—
324C3×D4—22no—
325C3×D4—22no—
326C3×D4—22no—
327C3×D4—22no—
328C3×D4—24no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—42no—
332C3×D4(C2×D3)⋊C414yes—
333C3×D4C3×C22⋊C414yes—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—32no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—62no—
351C2×C3×C4—61no—
352C2×C3×C4—62no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—81no—
356C2×C3×C4—81no—
357C2×C3×C4—128no—
358C2×C3×C4—121no—
359C2×C3×C4—121no—
360C2×C3×C4—121no—
361C2×C3×C4—121no—
362C2×C3×C4C2×C4×D311yescharacteristic
363C2×C3×C4C2×D1211yescharacteristic
364C2×C3×C4D3×D411yescharacteristic
365C2×C3×C4#23711yescharacteristic
366C2×C3×C4#23711yescharacteristic
367C2×C3×C4C2×#6711yescharacteristic
368C2×C3×C4C22×C3×C411yescharacteristic
369C2×C3×C4C2×C3×D411yescharacteristic
370C2×C3×C4C2×C3×D411yescharacteristic
371C2×C3×C4C2×C3×D411yescharacteristic
372C2×C3×C4C3×#4111yescharacteristic
373C2×C3×C4C3×#4111yescharacteristic
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3—22no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—48no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3C3×SD1612yescomplemented
387Dic6—22no—
388Dic6—22no—
389Dic6—41no—
390Dic6—41no—
391C3×C8—42no—
392C3×C8—41no—
393C3×C8—41no—
394C3×C8—62no—
395C3⋊C8—22no—
396C2×C32—44no—
397C2×C32#38611yescharacteristic
398C2×C32C4×SD1611yescharacteristic
399C2×C32#44011yescharacteristic
400C2×C32#45011yescharacteristic
401C2×C32#45211yescharacteristic
402C2×C32#46111yescharacteristic
403C2×C32#46311yescharacteristic
404C24—21no—
405C2×Q8—32no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4C3×C4×D312yes—
415C2×D4C3×D1212yes—
416C2×D4C3×#6712yes—
417C22×C4—21no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—64no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—122no—
424C22×C4—122no—
425C22×C4C22×C3×D311yescharacteristic
426SD16—68no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—32no—
431C2×C8—61no—
432C2×C8—61no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—32no—
436C4⋊C4—61no—
437C22⋊C4—64no—
438C42—61no—
439C42—61no—
440C22×C3—24no—
441C22×C3—24no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—44no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—82no—
463C22×C3—82no—
464C22×C3—82no—
465C22×C3—82no—
466C22×C3—82no—
467C22×C3—82no—
468C22×C3#67⋊C411yescharacteristic
469C22×C3#88211yescharacteristic
470C22×C3#91111yescharacteristic
471C22×C3#91211yescharacteristic
472C22×C3D3×SD1611yescharacteristic
473C22×C3#91511yescharacteristic
474C22×C3#93911yescharacteristic
475C22×C3C3×C4×D411yescharacteristic
476C22×C3C3×#12011yescharacteristic
477C22×C3C3×#12111yescharacteristic
478C22×C3C2×C3×SD1611yescharacteristic
479C22×C3C3×#13511yescharacteristic
480C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
481C22×C3C3×#13711yescharacteristic
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—62no—
487C3×C4—82no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4—122no—
491C3×C4—122no—
492C3×C4—121no—
493C3×C4#67⋊C411yescharacteristic
494C3×C4#88611yescharacteristic
495C3×C4C2×(C2×D3)⋊C411yescharacteristic
496C3×C4#93911yescharacteristic
497C3×C4C2×C3×C22⋊C411yescharacteristic
498C3×C4C3×C4×D411yescharacteristic
499C3×C4C3×#12111yescharacteristic
500C3×C4C3×#12311yescharacteristic
501Dic3—22no—
502Dic3—42no—
503Dic3—42no—
504Dic3—41no—
505C32#187711yescharacteristic, complemented, Sylow
506C23—22no—
507C23—22no—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C3×D3×D411yescharacteristic
514Q8—62no—
515Q8—62no—
516Q8—121no—
517Q8—121no—
518D4—22no—
519D4—22no—
520D4—22no—
521D4C3×(C2×D3)⋊C414yes—
522C2×C4—32no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—62no—
529C2×C4—61no—
530C2×C4—62no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—128no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4C2×C3×C4×D311yescharacteristic
539C2×C4C2×C3×D1211yescharacteristic
540C2×C4C3×D3×D411yescharacteristic
541C2×C4C3×#23711yescharacteristic
542C2×C4C3×#23711yescharacteristic
543C2×C4C2×C3×#6711yescharacteristic
544C8—41no—
545C8—62no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—44no—
554C2×C3—44no—
555C2×C3—84no—
556C2×C3#475711yescharacteristic
557C2×C3#475811yescharacteristic
558C2×C3#477611yescharacteristic
559C2×C3#478311yescharacteristic
560C2×C3#495811yescharacteristic
561C2×C3#515311yescharacteristic
562C2×C3#516211yescharacteristic
563C2×C3C3×#38611yescharacteristic
564C2×C3C3×C4×SD1611yescharacteristic
565C2×C3C3×#44011yescharacteristic
566C2×C3C3×#45011yescharacteristic
567C2×C3C3×#45211yescharacteristic
568C2×C3C3×#46111yescharacteristic
569C2×C3C3×#46311yescharacteristic
570C22—24no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22C3×#67⋊C411yescharacteristic
578C22C3×#88211yescharacteristic
579C22C3×#91111yescharacteristic
580C22C3×#91211yescharacteristic
581C22C3×D3×SD1611yescharacteristic
582C22C3×#91511yescharacteristic
583C22C3×#93911yescharacteristic
584C4—62no—
585C4—81no—
586C4—122no—
587C4—122no—
588C4—121no—
589C4C3×#67⋊C411yescharacteristic
590C4C3×#88611yescharacteristic
591C4C2×C3×(C2×D3)⋊C411yescharacteristic
592C4C3×#93911yescharacteristic
593C3—21no—
594C3#7017111yescharacteristic, direct product factor
595C3C3×#187711yescharacteristic, complemented
596C2—44no—
597C2C3×#475711yescharacteristic
598C2C3×#475811yescharacteristic
599C2C3×#477611yescharacteristic
600C2C3×#478311yescharacteristic
601C2C3×#495811yescharacteristic
602C2C3×#515311yescharacteristic
603C2C3×#516211yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#102C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#102D3
hypocenterC3C3×#1877
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#102

Complements

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

NH ≅ G/Ntype
4. C2×C3×#816C2semidirect
5. C3×#4973C2semidirect
6. C3×#4550C2semidirect
7. C3×#4529C2semidirect
9. #70171C3direct (N × H)
20. C2×C3×#214C4semidirect
37. C2×#816C2×C3semidirect
38. #4973C2×C3semidirect
41. #4550C2×C3semidirect
42. #4529C2×C3semidirect
64. C2×C3×Dic6C2×C4semidirect
78. C2×C3×C3⋊C8C2×C4semidirect
107. C2×#214C3×C4semidirect
153. C2×C3×Dic3SD16semidirect
219. C2×Dic6C2×C3×C4semidirect
252. C2×C3⋊C8C2×C3×C4semidirect
386. C2×Dic3C3×SD16semidirect
505. C32#1877semidirect
594. C3#70171direct (N × H)
595. C3C3×#1877semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×Dic32
C22×C3×#2142
C2×C32×#1022
C2×C3×#8162
C3×#49732
C3×#45502
C3×#45292
C3×#18773
#701713