Finite Group Databasebeta

← C2×C3×#4647 #11801567

Subgroups — C2×C3×#4647 #11801567

Order 1152.

1765 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Dic3C211yesmaximal, characteristic
2C22×C3×Dic3⋊C4C211yesmaximal, characteristic
3C22×C3×Dic3⋊C4C211yesmaximal, characteristic
4C22×C3×Dic3⋊C4C211yesmaximal, characteristic
5C22×C3×C4×Dic3C211yesmaximal, characteristic
6C2×C32×#95C211yesmaximal, characteristic
7C2×C3×C3⋊#95C211yesmaximal, characteristic
8C3×#4647C218yesmaximal, direct product factor
9C2×C3×#389—31nomaximal
10C2×#4647C311yesmaximal, characteristic, direct product factor
11C23×C32×C4C2211yescharacteristic
12C23×C32×C4C2211yescharacteristic
13C23×C32×C4C2211yescharacteristic
14C23×C3×Dic3C2211yescharacteristic
15C23×C3×Dic3C2211yescharacteristic
16C23×C3×Dic3C2211yescharacteristic
17C23×C3×Dic3C2211yescharacteristic
18C2×C3×C4⋊Dic3—24no—
19C2×C3×C4⋊Dic3C2214yes—
20C2×C3×Dic3⋊C4—24no—
21C2×C3×Dic3⋊C4—24no—
22C2×C3×Dic3⋊C4C418yescomplemented
23C2×C3×Dic3⋊C4C2214yes—
24C2×C3×Dic3⋊C4C2214yes—
25C2×C3×Dic3⋊C4C2214yes—
26C2×C3×C4×Dic3—24no—
27C2×C3×C4×Dic3C2214yes—
28C32×#95C2214yes—
29C3×C3⋊#95C2214yes—
30C22×C3×C4⋊C4—31no—
31C22×C3×C4⋊C4—31no—
32C22×C3×C4⋊C4—31no—
33C22×C3×C4⋊C4—31no—
34C22×C3×C42—31no—
35C22×C4⋊Dic3C2×C311yescharacteristic
36C22×Dic3⋊C4C2×C311yescharacteristic
37C22×Dic3⋊C4C2×C311yescharacteristic
38C22×Dic3⋊C4C2×C311yescharacteristic
39C22×C4×Dic3C2×C311yescharacteristic
40C3×#389—38no—
41C2×C3×#95—21no—
42C2×C3×#95—31no—
43C2×C3×#95D311yescharacteristic
44C2×C3×#95C2×C311yescharacteristic
45C2×C3⋊#95C2×C311yescharacteristic
46#4647C2×C318yesdirect product factor
47C24×C32C2311yescharacteristic
48C22×C32×C4—22no—
49C22×C32×C4—22no—
50C22×C32×C4—22no—
51C22×C32×C4—22no—
52C22×C32×C4—22no—
53C22×C32×C4—22no—
54C22×C32×C4—22no—
55C22×C32×C4C2×C414yes—
56C22×C32×C4Q814yes—
57C22×C32×C4C2312yes—
58C22×C32×C4C2312yes—
59C22×C32×C4C2312yes—
60C22×C3×Dic3—22no—
61C22×C3×Dic3—22no—
62C22×C3×Dic3—22no—
63C22×C3×Dic3—22no—
64C22×C3×Dic3—22no—
65C22×C3×Dic3—22no—
66C22×C3×Dic3—22no—
67C22×C3×Dic3C2×C414yes—
68C22×C3×Dic3C2×C414yes—
69C22×C3×Dic3D414yes—
70C22×C3×Dic3D414yes—
71C22×C3×Dic3Q814yes—
72C22×C3×Dic3C2312yes—
73C22×C3×Dic3C2312yes—
74C22×C3×Dic3C2312yes—
75C22×C3×Dic3C2312yes—
76C3×C4⋊Dic3—28no—
77C3×Dic3⋊C4—28no—
78C3×Dic3⋊C4—28no—
79C3×Dic3⋊C4C2×C4116yescomplemented
80C3×C4×Dic3—28no—
81C2×#389—31noSylow
82C23×C3×C4—21no—
83C23×C3×C4—21no—
84C23×C3×C4—21no—
85C23×C3×C4—31no—
86C23×C3×C4—31no—
87C23×C3×C4—31no—
88C23×C3×C4—31no—
89C23×C3×C4C2×D311yescharacteristic
90C23×C3×C4C2×D311yescharacteristic
91C23×C3×C4C2×D311yescharacteristic
92C23×C3×C4C22×C311yescharacteristic
93C23×C3×C4C22×C311yescharacteristic
94C23×C3×C4C22×C311yescharacteristic
95C23×Dic3C22×C311yescharacteristic
96C23×Dic3C22×C311yescharacteristic
97C23×Dic3C22×C311yescharacteristic
98C23×Dic3C22×C311yescharacteristic
99C2×C3×C4⋊C4—38no—
100C2×C3×C4⋊C4—34no—
101C2×C3×C4⋊C4—34no—
102C2×C3×C4⋊C4—34no—
103C2×C3×C4⋊C4—34no—
104C2×C3×C4⋊C4—64no—
105C2×C3×C4⋊C4—64no—
106C2×C3×C4⋊C4—64no—
107C2×C3×C42—34no—
108C2×C3×C42—64no—
109C2×C4⋊Dic3—24no—
110C2×C4⋊Dic3C22×C314yes—
111C2×Dic3⋊C4—24no—
112C2×Dic3⋊C4—24no—
113C2×Dic3⋊C4C3×C418yescomplemented
114C2×Dic3⋊C4C22×C314yes—
115C2×Dic3⋊C4C22×C314yes—
116C2×Dic3⋊C4C22×C314yes—
117C2×C4×Dic3—24no—
118C2×C4×Dic3C22×C314yes—
119C3×#95—24no—
120C3×#95—34no—
121C3×#95C2×D314yes—
122C3×#95C22×C314yes—
123C3⋊#95C22×C314yes—
124C23×C32C22×C412yes—
125C23×C32C2×D412yes—
126C23×C32C2×D412yes—
127C23×C32C2×Q812yes—
128C23×C32C2×Q812yes—
129C23×C32#4112yes—
130C23×C32#4112yes—
131C23×C32C2411yescharacteristic
132C2×C32×C4—22no—
133C2×C32×C4—28no—
134C2×C32×C4—28no—
135C2×C32×C4—22no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4—44no—
142C2×C32×C4C22×C414yes—
143C2×C32×C4C2×Q814yes—
144C2×C3×Dic3—28no—
145C2×C3×Dic3—28no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—44no—
154C2×C3×Dic3C4⋊C4116yescomplemented
155C2×C3×Dic3C22×C414yes—
156C2×C3×Dic3C22×C414yes—
157C2×C3×Dic3C2×D414yes—
158C2×C3×Dic3C2×D414yes—
159C2×C3×Dic3C2×Q814yes—
160C22×C4⋊C4—31no—
161C22×C4⋊C4—31no—
162C22×C4⋊C4—31no—
163C22×C4⋊C4—31no—
164C22×C42—31no—
165#389—38no—
166C2×#95—31no—
167C2×#95C3×D311yescharacteristic
168C24×C3—21no—
169C24×C3C22×D311yescharacteristic
170C24×C3C23×C311yescharacteristic
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—24no—
187C22×C3×C4—24no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—34no—
192C22×C3×C4—34no—
193C22×C3×C4—34no—
194C22×C3×C4—32no—
195C22×C3×C4—34no—
196C22×C3×C4—34no—
197C22×C3×C4—32no—
198C22×C3×C4—32no—
199C22×C3×C4—32no—
200C22×C3×C4—42no—
201C22×C3×C4—42no—
202C22×C3×C4—42no—
203C22×C3×C4—42no—
204C22×C3×C4—42no—
205C22×C3×C4—42no—
206C22×C3×C4—62no—
207C22×C3×C4—62no—
208C22×C3×C4—62no—
209C22×C3×C4—62no—
210C22×C3×C4—62no—
211C22×C3×C4—62no—
212C22×C3×C4—62no—
213C22×C3×C4Dic614yes—
214C22×C3×C4C4×D314yes—
215C22×C3×C4C2×C3×C414yes—
216C22×C3×C4C3×Q814yes—
217C22×C3×C4C22×D312yes—
218C22×C3×C4C22×D312yes—
219C22×C3×C4C22×D312yes—
220C22×C3×C4C23×C312yes—
221C22×C3×C4C23×C312yes—
222C22×C3×C4C23×C312yes—
223C22×Dic3—22no—
224C22×Dic3—22no—
225C22×Dic3—22no—
226C22×Dic3—22no—
227C22×Dic3—22no—
228C22×Dic3—22no—
229C22×Dic3—22no—
230C22×Dic3C2×C3×C414yes—
231C22×Dic3C2×C3×C414yes—
232C22×Dic3C3×D414yes—
233C22×Dic3C3×D414yes—
234C22×Dic3C3×Q814yes—
235C22×Dic3C23×C312yes—
236C22×Dic3C23×C312yes—
237C22×Dic3C23×C312yes—
238C22×Dic3C23×C312yes—
239C3×C4⋊C4—316no—
240C3×C4⋊C4—68no—
241C3×C4⋊C4—68no—
242C3×C4⋊C4—68no—
243C3×C42—68no—
244C4⋊Dic3—28no—
245Dic3⋊C4—28no—
246Dic3⋊C4—28no—
247Dic3⋊C4C2×C3×C4116yescomplemented
248C4×Dic3—28no—
249C22×C32C2×C4⋊C414yes—
250C22×C32C4×D414yes—
251C22×C32C4×Q814yes—
252C22×C32#12114yes—
253C22×C32C22⋊Q814yes—
254C22×C32#12514yes—
255C22×C32C4⋊Q814yes—
256C22×C32C23×C411yescharacteristic
257C22×C32C22×D411yescharacteristic
258C22×C32C22×D411yescharacteristic
259C22×C32C22×Q811yescharacteristic
260C22×C32C22×Q811yescharacteristic
261C22×C32C2×#4111yescharacteristic
262C22×C32C2×#4111yescharacteristic
263C32×C4—24no—
264C32×C4—24no—
265C32×C4—42no—
266C3×Dic3—24no—
267C3×Dic3—24no—
268C3×Dic3—42no—
269C3×Dic3C2×C4⋊C418yescomplemented
270C23×C4—31no—
271C23×C4—31no—
272C23×C4—31no—
273C23×C4—31no—
274C23×C4C2×C3×D311yescharacteristic
275C23×C4C2×C3×D311yescharacteristic
276C23×C4C2×C3×D311yescharacteristic
277C2×C4⋊C4—38no—
278C2×C4⋊C4—34no—
279C2×C4⋊C4—34no—
280C2×C4⋊C4—34no—
281C2×C4⋊C4—34no—
282C2×C4⋊C4—64no—
283C2×C4⋊C4—64no—
284C2×C4⋊C4—64no—
285C2×C42—34no—
286C2×C42—64no—
287#95—34no—
288#95C2×C3×D314yes—
289C23×C3—21no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3C2×Dic612yes—
298C23×C3C2×C4×D312yes—
299C23×C3#23512yes—
300C23×C3D3×D412yes—
301C23×C3D3×D412yes—
302C23×C3#23712yes—
303C23×C3D3×Q812yes—
304C23×C3C22×C3×C412yes—
305C23×C3C2×C3×D412yes—
306C23×C3C2×C3×D412yes—
307C23×C3C2×C3×Q812yes—
308C23×C3C2×C3×Q812yes—
309C23×C3C3×#4112yes—
310C23×C3C3×#4112yes—
311C23×C3C23×D311yescharacteristic
312C23×C3C24×C311yescharacteristic
313C2×C3×C4—28no—
314C2×C3×C4—28no—
315C2×C3×C4—28no—
316C2×C3×C4—28no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—316no—
335C2×C3×C4—34no—
336C2×C3×C4—34no—
337C2×C3×C4—34no—
338C2×C3×C4—34no—
339C2×C3×C4—34no—
340C2×C3×C4—48no—
341C2×C3×C4—48no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—48no—
349C2×C3×C4—44no—
350C2×C3×C4—44no—
351C2×C3×C4—68no—
352C2×C3×C4—68no—
353C2×C3×C4—62no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—62no—
357C2×C3×C4—62no—
358C2×C3×C4—62no—
359C2×C3×C4—62no—
360C2×C3×C4—124no—
361C2×C3×C4C2×Dic614yes—
362C2×C3×C4C2×C4×D314yes—
363C2×C3×C4C22×C3×C414yes—
364C2×C3×C4C2×C3×Q814yes—
365C2×Dic3—28no—
366C2×Dic3—28no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—44no—
375C2×Dic3C3×C4⋊C4116yescomplemented
376C2×Dic3C22×C3×C414yes—
377C2×Dic3C22×C3×C414yes—
378C2×Dic3C2×C3×D414yes—
379C2×Dic3C2×C3×D414yes—
380C2×Dic3C2×C3×Q814yes—
381C2×C32#38918yescomplemented
382C2×C32C22×C4⋊C411yescharacteristic
383C2×C32C2×C4×D411yescharacteristic
384C2×C32C2×C4×Q811yescharacteristic
385C2×C32C2×#12111yescharacteristic
386C2×C32C2×C22⋊Q811yescharacteristic
387C2×C32C2×#12511yescharacteristic
388C2×C32C2×C4⋊Q811yescharacteristic
389C24C22×C3×D311yescharacteristic
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—32no—
398C22×C4—34no—
399C22×C4—34no—
400C22×C4—34no—
401C22×C4—32no—
402C22×C4—32no—
403C22×C4—34no—
404C22×C4—34no—
405C22×C4—32no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4C3×Dic614yes—
414C22×C4C3×C4×D314yes—
415C22×C4C22×C3×D312yes—
416C22×C4C22×C3×D312yes—
417C22×C4C22×C3×D312yes—
418C4⋊C4—316no—
419C4⋊C4—68no—
420C4⋊C4—68no—
421C4⋊C4—68no—
422C42—68no—
423C22×C3—24no—
424C22×C3—24no—
425C22×C3—24no—
426C22×C3—24no—
427C22×C3—24no—
428C22×C3—24no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—24no—
437C22×C3C4×Dic614yes—
438C22×C3#87814yes—
439C22×C3#67⋊C414yes—
440C22×C3#88414yes—
441C22×C3#88814yes—
442C22×C3#88914yes—
443C22×C3D3×C4⋊C414yes—
444C22×C3C2×C3×C4⋊C414yes—
445C22×C3C3×C4×D414yes—
446C22×C3C3×C4×Q814yes—
447C22×C3C3×#12114yes—
448C22×C3C3×C22⋊Q814yes—
449C22×C3C3×#12514yes—
450C22×C3C3×C4⋊Q814yes—
451C22×C3C22×Dic611yescharacteristic
452C22×C3C22×C4×D311yescharacteristic
453C22×C3C2×#23511yescharacteristic
454C22×C3C2×D3×D411yescharacteristic
455C22×C3C2×D3×D411yescharacteristic
456C22×C3C2×#23711yescharacteristic
457C22×C3C2×D3×Q811yescharacteristic
458C22×C3C23×C3×C411yescharacteristic
459C22×C3C22×C3×D411yescharacteristic
460C22×C3C22×C3×D411yescharacteristic
461C22×C3C22×C3×Q811yescharacteristic
462C22×C3C22×C3×Q811yescharacteristic
463C22×C3C2×C3×#4111yescharacteristic
464C22×C3C2×C3×#4111yescharacteristic
465C3×C4—24no—
466C3×C4—24no—
467C3×C4—24no—
468C3×C4—24no—
469C3×C4—38no—
470C3×C4—44no—
471C3×C4—44no—
472C3×C4—44no—
473C3×C4—42no—
474C3×C4—42no—
475C3×C4—64no—
476C3×C4—64no—
477C3×C4—122no—
478Dic3—24no—
479Dic3—24no—
480Dic3—42no—
481Dic3C2×C3×C4⋊C418yescomplemented
482C32C2×#38911yescharacteristic, complemented, Sylow
483C23C2×C3×Dic612yes—
484C23C2×C3×C4×D312yes—
485C23C3×#23512yes—
486C23C3×D3×D412yes—
487C23C3×D3×D412yes—
488C23C3×#23712yes—
489C23C3×D3×Q812yes—
490C23C23×C3×D311yescharacteristic
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—316no—
501C2×C4—34no—
502C2×C4—34no—
503C2×C4—34no—
504C2×C4—34no—
505C2×C4—34no—
506C2×C4—44no—
507C2×C4—68no—
508C2×C4—68no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—124no—
517C2×C4C2×C3×Dic614yes—
518C2×C4C2×C3×C4×D314yes—
519C2×C3—28no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3#464718yesdirect product factor
528C2×C3C3×#38918yescomplemented
529C2×C3C2×C4×Dic611yescharacteristic
530C2×C3C2×#87811yescharacteristic
531C2×C3C2×#67⋊C411yescharacteristic
532C2×C3C2×#88411yescharacteristic
533C2×C3C2×#88811yescharacteristic
534C2×C3C2×#88911yescharacteristic
535C2×C3C2×D3×C4⋊C411yescharacteristic
536C2×C3C22×C3×C4⋊C411yescharacteristic
537C2×C3C2×C3×C4×D411yescharacteristic
538C2×C3C2×C3×C4×Q811yescharacteristic
539C2×C3C2×C3×#12111yescharacteristic
540C2×C3C2×C3×C22⋊Q811yescharacteristic
541C2×C3C2×C3×#12511yescharacteristic
542C2×C3C2×C3×C4⋊Q811yescharacteristic
543C22C3×C4×Dic614yes—
544C22C3×#87814yes—
545C22C3×#67⋊C414yes—
546C22C3×#88414yes—
547C22C3×#88814yes—
548C22C3×#88914yes—
549C22C3×D3×C4⋊C414yes—
550C22C22×C3×Dic611yescharacteristic
551C22C22×C3×C4×D311yescharacteristic
552C22C2×C3×#23511yescharacteristic
553C22C2×C3×D3×D411yescharacteristic
554C22C2×C3×D3×D411yescharacteristic
555C22C2×C3×#23711yescharacteristic
556C22C2×C3×D3×Q811yescharacteristic
557C4—24no—
558C4—24no—
559C4—38no—
560C4—42no—
561C4—64no—
562C4—64no—
563C4—122no—
564C3—21no—
565C3C2×#464711yescharacteristic, direct product factor
566C3C2×C3×#38911yescharacteristic, complemented
567C2C3×#464718yesdirect product factor
568C2C2×C3×C4×Dic611yescharacteristic
569C2C2×C3×#87811yescharacteristic
570C2C2×C3×#67⋊C411yescharacteristic
571C2C2×C3×#88411yescharacteristic
572C2C2×C3×#88811yescharacteristic
573C2C2×C3×#88911yescharacteristic
574C2C2×C3×D3×C4⋊C411yescharacteristic
subgroupG/H
centerC24×C3C22×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC2×C32×#95C2
socle, prime radicalC24×C32C23
cosocleC23×C3C24×C3
hypercenterC2×C3×#95D3
hypocenterC3C2×C3×#389
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24×C3 ◁ C2×C3×#95

Complements

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

NH ≅ G/Ntype
8. C3×#4647C2direct (N × H)
10. C2×#4647C3direct (N × H)
22. C2×C3×Dic3⋊C4C4semidirect
46. #4647C2×C3direct (N × H)
79. C3×Dic3⋊C4C2×C4semidirect
113. C2×Dic3⋊C4C3×C4semidirect
154. C2×C3×Dic3C4⋊C4semidirect
247. Dic3⋊C4C2×C3×C4semidirect
269. C3×Dic3C2×C4⋊C4semidirect
375. C2×Dic3C3×C4⋊C4semidirect
381. C2×C32#389semidirect
481. Dic3C2×C3×C4⋊C4semidirect
482. C32C2×#389semidirect
527. C2×C3#4647direct (N × H)
528. C2×C3C3×#389semidirect
565. C3C2×#4647direct (N × H)
566. C3C2×C3×#389semidirect
567. C2C3×#4647direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Dic32
C22×C3×Dic3⋊C42
C22×C3×C4×Dic32
C2×C32×#952
C2×C3×C3⋊#952
C3×#46472
C2×C3×#3893
C2×#46473