Finite Group Databasebeta

← C3×#70269 #11787267

Subgroups — C3×#70269 #11787267

Order 1152.

801 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#946C211yesmaximal, characteristic
2C2×C3×#892C211yesmaximal, characteristic
3C2×C32×#103C211yesmaximal, characteristic, complemented
4C2×C3×#835C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D3)⋊C8C211yesmaximal, characteristic
6C3×#4551C211yesmaximal, characteristic, complemented
7C3×#4529C211yesmaximal, characteristic, complemented
8C3×#1873—31nomaximal
9#70269C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8C2211yescharacteristic
11C22×C3×C4×D3C2211yescharacteristic
12C22×C32×C8C2211yescharacteristic
13C2×C32×C4⋊C4C2211yescharacteristic
14C3×#946—22no—
15C3×#946C414yescomplemented
16C2×C3×(C2×D3)⋊C4—21no—
17C2×C3×(C2×D3)⋊C4—21no—
18C2×C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×Dic3⋊C4—21no—
21C2×C3×C4×Dic3—21no—
22C22×C3×C3⋊C8C2211yescharacteristic
23C3×#892—22no—
24C3×#892—22no—
25C32×#103—22no—
26C3×#835—22no—
27C3×(C2×D3)⋊C8—22no—
28C2×C3×C22⋊Q8—31no—
29C2×C3×#117—31no—
30C2×#946C2×C311yescharacteristic
31C2×#892C2×C311yescharacteristic
32C2×C3×#103—21no—
33C2×C3×#103—31no—
34C2×C3×#103D311yescharacteristic, complemented
35C2×C3×#103C2×C311yescharacteristic, complemented
36C2×C3×C22⋊C8—31no—
37C2×#835C2×C311yescharacteristic, complemented
38C2×(C2×D3)⋊C8C2×C311yescharacteristic
39C3×#340—31no—
40C3×#340—31no—
41#4551C2×C311yescharacteristic, complemented
42#4529C2×C311yescharacteristic, complemented
43C23×C3×D3D411yescharacteristic
44C2×C32×Q8—21no—
45C2×C32×Q8—21no—
46C2×C32×Q8—21no—
47C2×C32×Q8—21no—
48C2×C32×Q8—21no—
49C2×C32×Q8C2×C412yes—
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4C2311yescharacteristic
53C22×C3×Dic3—21no—
54C22×C3×Dic3—21no—
55C22×C3×Dic3D411yescharacteristic
56C2×C3×C4×D3—21no—
57C2×C3×C4×D3—21no—
58C2×C3×C4×D3—21no—
59C2×C3×C4×D3—21no—
60C2×C3×C4×D3C2×C412yes—
61C2×C3×C4×D3D412yes—
62C2×C32×C8—21no—
63C2×C32×C8—21no—
64C2×C32×C8—21no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4D412yes—
67C3×(C2×D3)⋊C4—24no—
68C3×(C2×D3)⋊C4—24no—
69C3×C4⋊Dic3—21no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3C2×C412yes—
72C3×C4⋊Dic3D412yes—
73C3×Dic3⋊C4—24no—
74C3×C4×Dic3—24no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8—21no—
78#1873—31noSylow
79C22×C3×Q8—21no—
80C22×C3×Q8C2×D311yescharacteristic
81C22×C3×Q8C22×C311yescharacteristic
82C23×C3×C4—31no—
83C22×C4×D3C22×C311yescharacteristic
84C22×C3×C8—21no—
85C22×C3×C8—31no—
86C22×C3×C8C2×D311yescharacteristic
87C22×C3×C8C22×C311yescharacteristic
88C3×C22⋊Q8—34no—
89C3×C22⋊Q8—62no—
90C3×#117—62no—
91C3×#117—62no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—31no—
94C2×C3×C4⋊C4—31no—
95C2×C3×C4⋊C4—61no—
96C2×C3×C4⋊C4C2×D311yescharacteristic
97C2×C3×C4⋊C4C22×C311yescharacteristic
98C2×C3×C22⋊C4—61no—
99C2×C3×C22⋊C4—61no—
100C2×C3×C42—61no—
101#946—22no—
102#946C3×C414yescomplemented
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3C22×C311yescharacteristic
106C2×C4⋊Dic3C22×C311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C22×C3⋊C8C22×C311yescharacteristic
110#892—22no—
111#892—22no—
112C3×#103—24no—
113C3×#103—22no—
114C3×#103—22no—
115C3×#103—62no—
116C3×C22⋊C8—62no—
117#835—22no—
118(C2×D3)⋊C8—22no—
119C23×C32C2×D411yescharacteristic
120C22×C3×D3—21no—
121C22×C3×D3—21no—
122C22×C3×D3—21no—
123C22×C3×D3—21no—
124C22×C3×D3—21no—
125C22×C3×D3—21no—
126C22×C3×D3C22⋊C412yes—
127C32×Q8—22no—
128C32×Q8—22no—
129C32×Q8—41no—
130C32×Q8—41no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—41no—
134C2×C32×C4—41no—
135C2×C32×C4—41no—
136C2×C32×C4—41no—
137C2×C32×C4C22×C411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4#4111yescharacteristic
142C2×C32×C4#4111yescharacteristic
143C2×C3×Dic3—21no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—41no—
148C2×C3×Dic3—41no—
149C2×C3×Dic3—41no—
150C2×C3×Dic3—41no—
151C2×C3×Dic3C22⋊C412yes—
152C3×C4×D3—22no—
153C3×C4×D3—22no—
154C3×C4×D3—22no—
155C3×C4×D3—22no—
156C32×C8—41no—
157C3×C3⋊C8—41no—
158C2×C22⋊Q8—31no—
159C2×#117—31no—
160C2×#103—31no—
161C2×#103C3×D311yescharacteristic, complemented
162C2×C22⋊C8—31no—
163#340—31no—
164#340—31no—
165C24×C3—31no—
166C23×D3C3×D411yescharacteristic
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8—22no—
180C2×C3×Q8—41no—
181C2×C3×Q8—41no—
182C2×C3×Q8—41no—
183C2×C3×Q8C4×D312yes—
184C2×C3×Q8C2×C3×C412yes—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—31no—
191C22×C3×C4—32no—
192C22×C3×C4—32no—
193C22×C3×C4—41no—
194C22×C3×C4—41no—
195C22×C3×C4—61no—
196C22×C3×C4—61no—
197C22×C3×C4—61no—
198C22×C3×C4—61no—
199C22×C3×C4—61no—
200C22×C3×C4—61no—
201C22×C3×C4C22×D311yescharacteristic
202C22×C3×C4C23×C311yescharacteristic
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3C3×D411yescharacteristic
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3C2×C3×C412yes—
211C2×C4×D3C3×D412yes—
212C2×C3×C8—22no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—41no—
220C2×C3×C8—41no—
221C2×C3×C8—61no—
222C2×C3×C8—61no—
223C2×C3×C8—61no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—32no—
228C3×C4⋊C4—32no—
229C3×C4⋊C4—41no—
230C3×C4⋊C4—64no—
231C3×C4⋊C4—61no—
232C3×C4⋊C4—61no—
233C3×C4⋊C4#6712yes—
234C3×C4⋊C4C3×D412yes—
235C3×C22⋊C4—64no—
236C3×C22⋊C4—64no—
237C3×C42—64no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—24no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3C2×C3×C412yes—
243C4⋊Dic3C3×D412yes—
244Dic3⋊C4—24no—
245C4×Dic3—24no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C22×C32C2×C22⋊C411yescharacteristic
250C22×C32#12011yescharacteristic
251C22×C32#12311yescharacteristic
252C22×C32#13511yescharacteristic
253C22×C32#13511yescharacteristic
254C22×C32#13711yescharacteristic
255C22×C32#13711yescharacteristic
256C2×C3×D3—22no—
257C2×C3×D3—22no—
258C2×C3×D3—22no—
259C2×C3×D3—41no—
260C2×C3×D3—41no—
261C2×C3×D3—41no—
262C2×C3×D3—41no—
263C32×C4—42no—
264C32×C4—42no—
265C32×C4C4×D411yescharacteristic
266C32×C4C4×D411yescharacteristic
267C32×C4#12111yescharacteristic
268C32×C4#12111yescharacteristic
269C3×Dic3—22no—
270C3×Dic3—42no—
271C3×Dic3—42no—
272C22×Q8C2×C3×D311yescharacteristic
273C23×C4—31no—
274C22×C8—31no—
275C22×C8C2×C3×D311yescharacteristic
276C22⋊Q8—34no—
277C22⋊Q8—62no—
278#117—62no—
279#117—62no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4C2×C3×D311yescharacteristic
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C42—61no—
287#103—22no—
288#103—62no—
289C22⋊C8—62no—
290C23×C3—21no—
291C23×C3—32no—
292C23×C3—61no—
293C23×C3—61no—
294C23×C3—61no—
295C23×C3—61no—
296C23×C3—61no—
297C23×C3—61no—
298C23×C3D3×D411yescharacteristic
299C23×C3C2×C3×D411yescharacteristic
300C22×D3—21no—
301C22×D3—21no—
302C22×D3—21no—
303C22×D3—21no—
304C22×D3—21no—
305C22×D3—21no—
306C22×D3C3×C22⋊C412yes—
307C3×Q8—24no—
308C3×Q8—22no—
309C3×Q8—22no—
310C3×Q8—22no—
311C3×Q8—22no—
312C3×Q8—42no—
313C3×Q8—42no—
314C3×Q8—42no—
315C3×Q8—41no—
316C3×Q8—41no—
317C3×Q8—41no—
318C3×Q8—41no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—32no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—62no—
345C2×C3×C4—62no—
346C2×C3×C4—62no—
347C2×C3×C4—62no—
348C2×C3×C4—62no—
349C2×C3×C4—62no—
350C2×C3×C4—81no—
351C2×C3×C4—81no—
352C2×C3×C4—121no—
353C2×C3×C4—121no—
354C2×C3×C4—121no—
355C2×C3×C4—121no—
356C2×C3×C4C2×C4×D311yescharacteristic
357C2×C3×C4#23511yescharacteristic
358C2×C3×C4D3×D411yescharacteristic
359C2×C3×C4D3×D411yescharacteristic
360C2×C3×C4#23711yescharacteristic
361C2×C3×C4C2×#6711yescharacteristic
362C2×C3×C4C22×C3×C411yescharacteristic
363C2×C3×C4C2×C3×D411yescharacteristic
364C2×C3×C4C2×C3×D411yescharacteristic
365C2×C3×C4C2×C3×D411yescharacteristic
366C2×C3×C4C3×#4111yescharacteristic
367C2×C3×C4C3×#4111yescharacteristic
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3C3×C22⋊C412yes—
377C4×D3—22no—
378C4×D3—22no—
379C4×D3—22no—
380C4×D3—22no—
381C3×C8—42no—
382C3×C8—41no—
383C3×C8—41no—
384C3×C8—121no—
385C3⋊C8—41no—
386C2×C32#38611yescharacteristic
387C2×C32#41611yescharacteristic
388C2×C32#41911yescharacteristic
389C2×C32#45211yescharacteristic
390C2×C32#45211yescharacteristic
391C2×C32#48511yescharacteristic
392C2×C32#48511yescharacteristic
393C3×D3—42no—
394C24—31no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8C3×C4×D312yes—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—31no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4C22×C3×D311yescharacteristic
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C4⋊C4—21no—
420C4⋊C4—32no—
421C4⋊C4—32no—
422C4⋊C4—64no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4C3×#6712yes—
426C22⋊C4—64no—
427C22⋊C4—64no—
428C42—64no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—62no—
437C22×C3—62no—
438C22×C3—62no—
439C22×C3—121no—
440C22×C3—121no—
441C22×C3—121no—
442C22×C3—121no—
443C22×C3D3×C22⋊C411yescharacteristic
444C22×C3#88311yescharacteristic
445C22×C3#91511yescharacteristic
446C22×C3#91611yescharacteristic
447C22×C3#91811yescharacteristic
448C22×C3#91911yescharacteristic
449C22×C3#93711yescharacteristic
450C22×C3C2×C3×C22⋊C411yescharacteristic
451C22×C3C3×#12011yescharacteristic
452C22×C3C3×#12311yescharacteristic
453C22×C3C3×#13511yescharacteristic
454C22×C3C3×#13511yescharacteristic
455C22×C3C3×#13711yescharacteristic
456C22×C3C3×#13711yescharacteristic
457C2×D3—22no—
458C2×D3—22no—
459C2×D3—22no—
460C2×D3—41no—
461C2×D3—41no—
462C2×D3—41no—
463C2×D3—41no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—44no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—42no—
473C3×C4—62no—
474C3×C4—82no—
475C3×C4—122no—
476C3×C4—122no—
477C3×C4#67⋊C411yescharacteristic
478C3×C4#88411yescharacteristic
479C3×C4C4×#6711yescharacteristic
480C3×C4#93811yescharacteristic
481C3×C4C3×C4×D411yescharacteristic
482C3×C4C3×C4×D411yescharacteristic
483C3×C4C3×#12111yescharacteristic
484C3×C4C3×#12111yescharacteristic
485Dic3—22no—
486Dic3—42no—
487Dic3—42no—
488C32#187311yescharacteristic, complemented, Sylow
489C23—32no—
490C23—61no—
491C23—61no—
492C23—61no—
493C23—61no—
494C23—61no—
495C23—61no—
496C23C3×D3×D411yescharacteristic
497Q8—22no—
498Q8—22no—
499Q8—41no—
500Q8—41no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—32no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4C2×C3×C4×D311yescharacteristic
521C2×C4C3×#23511yescharacteristic
522C2×C4C3×D3×D411yescharacteristic
523C2×C4C3×D3×D411yescharacteristic
524C2×C4C3×#23711yescharacteristic
525C2×C4C2×C3×#6711yescharacteristic
526C8—41no—
527C8—121no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—122no—
536C2×C3#480211yescharacteristic
537C2×C3#480411yescharacteristic
538C2×C3#481411yescharacteristic
539C2×C3#481511yescharacteristic
540C2×C3#495611yescharacteristic
541C2×C3#517311yescharacteristic
542C2×C3#518711yescharacteristic
543C2×C3C3×#38611yescharacteristic
544C2×C3C3×#41611yescharacteristic
545C2×C3C3×#41911yescharacteristic
546C2×C3C3×#45211yescharacteristic
547C2×C3C3×#45211yescharacteristic
548C2×C3C3×#48511yescharacteristic
549C2×C3C3×#48511yescharacteristic
550D3—42no—
551C22—62no—
552C22—62no—
553C22—62no—
554C22—121no—
555C22—121no—
556C22—121no—
557C22—121no—
558C22C3×D3×C22⋊C411yescharacteristic
559C22C3×#88311yescharacteristic
560C22C3×#91511yescharacteristic
561C22C3×#91611yescharacteristic
562C22C3×#91811yescharacteristic
563C22C3×#91911yescharacteristic
564C22C3×#93711yescharacteristic
565C4—42no—
566C4—42no—
567C4—62no—
568C4—122no—
569C4—122no—
570C4C3×#67⋊C411yescharacteristic
571C4C3×#88411yescharacteristic
572C4C3×C4×#6711yescharacteristic
573C4C3×#93811yescharacteristic
574C3—21no—
575C3#7026911yescharacteristic, direct product factor
576C3C3×#187311yescharacteristic, complemented
577C2—122no—
578C2C3×#480211yescharacteristic
579C2C3×#480411yescharacteristic
580C2C3×#481411yescharacteristic
581C2C3×#481511yescharacteristic
582C2C3×#495611yescharacteristic
583C2C3×#517311yescharacteristic
584C2C3×#518711yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#103C2
socleC23×C32C2×D4
prime radicalC23×C3×D3D4
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#103D3
hypocenterC3C3×#1873
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#103

Complements

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

NH ≅ G/Ntype
3. C2×C32×#103C2semidirect
4. C2×C3×#835C2semidirect
6. C3×#4551C2semidirect
7. C3×#4529C2semidirect
9. #70269C3direct (N × H)
15. C3×#946C4semidirect
34. C2×C3×#103D3semidirect
35. C2×C3×#103C2×C3semidirect
37. C2×#835C2×C3semidirect
41. #4551C2×C3semidirect
42. #4529C2×C3semidirect
102. #946C3×C4semidirect
161. C2×#103C3×D3semidirect
488. C32#1873semidirect
575. C3#70269direct (N × H)
576. C3C3×#1873semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9462
C2×C3×#8922
C2×C32×#1032
C2×C3×#8352
C2×C3×(C2×D3)⋊C82
C3×#45512
C3×#45292
C3×#18733
#702693