Finite Group Databasebeta

← C2×C3×#5670 #11903301

Subgroups — C2×C3×#5670 #11903301

Order 1152.

2645 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×C32×#124C211yesmaximal, characteristic, complemented
2C2×C3×#947C211yesmaximal, characteristic, complemented
3C2×C3×#944C211yesmaximal, characteristic, complemented
4C2×C3×#939C212yesmaximal, complemented
5C3×#5670C2116yesmaximal, direct product factor
6C2×C3×#885C212yesmaximal, complemented
7C2×C3×#884C212yesmaximal, complemented
8C2×C3×#67⋊C4C212yesmaximal, complemented
9C2×C3×#877C212yesmaximal, complemented
10C2×C3×#872C212yesmaximal, complemented
11C2×C3×#555—31nomaximal
12C2×#5670C311yesmaximal, characteristic, direct product factor
13C22×C32×Q8C2211yescharacteristic
14C22×C32×D4C2211yescharacteristic
15C22×C3×#67C2212yes—
16C22×C3×#67C2212yes—
17C23×C3×Dic3C2212yes—
18C22×C3×D12C2211yescharacteristic
19C22×C3×C4×D3C2212yes—
20C22×C3×Dic6C2211yescharacteristic
21C32×#124C2218yescomplemented
22C2×C32×C22⋊C4C2212yes—
23C2×C32×C22⋊C4C2212yes—
24C2×C32×C42C2211yescharacteristic
25C2×C3×#217C2212yes—
26C3×#947C2218yescomplemented
27C3×#944C2218yescomplemented
28C3×#939C22116yescomplemented
29C2×C3×(C2×D3)⋊C4C2212yes—
30C2×C3×(C2×D3)⋊C4C2212yes—
31C2×C3×(C2×D3)⋊C4C2212yes—
32C2×C3×Dic3⋊C4C2212yes—
33C2×C3×Dic3⋊C4C2212yes—
34C2×C3×Dic3⋊C4C2212yes—
35C2×C3×C4×Dic3C2212yes—
36C2×C3×C4×Dic3C2211yescharacteristic
37C2×C3×C4×Dic3C2211yescharacteristic
38C3×#885C22116yescomplemented
39C3×#884C22116yescomplemented
40C3×#67⋊C4C22116yescomplemented
41C3×#877C22116yescomplemented
42C3×#872C22116yescomplemented
43C3×#555—316no—
44C2×C3×C4⋊Q8—31no—
45C2×C3×#124—21no—
46C2×C3×#124—31no—
47C2×C3×#124—32no—
48C2×C3×#124D311yescharacteristic, complemented
49C2×C3×#124C2×C311yescharacteristic, complemented
50C2×C3×#121—32no—
51C2×C3×#121—32no—
52C2×C3×C4×D4—32no—
53C2×C3×#117—32no—
54C2×C3×#117—32no—
55C2×#947C2×C311yescharacteristic, complemented
56C2×#944C2×C311yescharacteristic, complemented
57C2×#939C2×C312yescomplemented
58#5670C2×C3116yesdirect product factor
59C2×#885C2×C312yescomplemented
60C2×#884C2×C312yescomplemented
61C2×#67⋊C4C2×C312yescomplemented
62C2×#877C2×C312yescomplemented
63C2×#872C2×C312yescomplemented
64C24×C32C2312yes—
65C23×C3×D3C2312yes—
66C2×C32×Q8—42no—
67C2×C32×Q8C2314yes—
68C2×C32×D4—42no—
69C2×C32×D4C2314yes—
70C22×C32×C4C2311yescharacteristic
71C22×C32×C4C2312yes—
72C22×C32×C4C2311yescharacteristic
73C22×C32×C4C2311yescharacteristic
74C2×C3×#67—28no—
75C2×C3×#67—44no—
76C2×C3×#67C2318yes—
77C2×C3×#67C2318yes—
78C22×C3×Dic3—28no—
79C22×C3×Dic3C2318yes—
80C22×C3×Dic3C2312yes—
81C22×C3×Dic3C2312yes—
82C22×C3×Dic3C2312yes—
83C2×C3×D12—42no—
84C2×C3×D12C2314yes—
85C2×C3×C4×D3—28no—
86C2×C3×C4×D3C2318yes—
87C2×C3×Dic6—42no—
88C2×C3×Dic6C2314yes—
89C32×C22⋊C4C2318yes—
90C32×C22⋊C4C2318yes—
91C32×C42C2314yes—
92C3×#217C2318yes—
93C3×(C2×D3)⋊C4C2318yes—
94C3×(C2×D3)⋊C4C2318yes—
95C3×(C2×D3)⋊C4C2318yes—
96C3×Dic3⋊C4C2318yes—
97C3×Dic3⋊C4C2318yes—
98C3×Dic3⋊C4C2318yes—
99C3×C4×Dic3C2314yes—
100C3×C4×Dic3C2318yes—
101C3×C4×Dic3C2314yes—
102C2×#555—31noSylow
103C22×C3×Q8—21no—
104C22×C3×Q8—31no—
105C22×C3×Q8C2×D311yescharacteristic
106C22×C3×Q8C22×C311yescharacteristic
107C22×C3×D4—21no—
108C22×C3×D4—32no—
109C22×C3×D4—32no—
110C22×C3×D4—31no—
111C22×C3×D4C2×D311yescharacteristic
112C22×C3×D4C22×C311yescharacteristic
113C23×C3×C4—32no—
114C23×C3×C4—32no—
115C22×#67C22×C312yes—
116C22×#67C22×C312yes—
117C23×Dic3C22×C312yes—
118C22×D12C22×C311yescharacteristic
119C22×C4×D3C22×C312yes—
120C22×Dic6C22×C311yescharacteristic
121C3×C4⋊Q8—38no—
122C3×#124—28no—
123C3×#124—38no—
124C3×#124—316no—
125C3×#124C2×D318yescomplemented
126C3×#124C22×C318yescomplemented
127C3×#121—316no—
128C3×#121—316no—
129C3×C4×D4—316no—
130C3×#117—316no—
131C3×#117—316no—
132C2×C3×C4⋊C4—32no—
133C2×C3×C4⋊C4—32no—
134C2×C3×C4⋊C4—32no—
135C2×C3×C22⋊C4—22no—
136C2×C3×C22⋊C4—22no—
137C2×C3×C22⋊C4—32no—
138C2×C3×C22⋊C4—32no—
139C2×C3×C22⋊C4—32no—
140C2×C3×C22⋊C4—32no—
141C2×C3×C22⋊C4C2×D312yes—
142C2×C3×C22⋊C4C2×D312yes—
143C2×C3×C22⋊C4C22×C312yes—
144C2×C3×C22⋊C4C22×C312yes—
145C2×C3×C42—21no—
146C2×C3×C42—31no—
147C2×C3×C42—32no—
148C2×C3×C42—31no—
149C2×C3×C42C2×D311yescharacteristic
150C2×C3×C42C22×C311yescharacteristic
151C2×#217C22×C312yes—
152#947C22×C318yescomplemented
153#944C22×C318yescomplemented
154#939C22×C3116yescomplemented
155C2×(C2×D3)⋊C4C22×C312yes—
156C2×(C2×D3)⋊C4C22×C312yes—
157C2×(C2×D3)⋊C4C22×C312yes—
158C2×Dic3⋊C4C22×C312yes—
159C2×Dic3⋊C4C22×C312yes—
160C2×Dic3⋊C4C22×C312yes—
161C2×C4×Dic3C22×C312yes—
162C2×C4×Dic3C22×C311yescharacteristic
163C2×C4×Dic3C22×C311yescharacteristic
164#885C22×C3116yescomplemented
165#884C22×C3116yescomplemented
166#67⋊C4C22×C3116yescomplemented
167#877C22×C3116yescomplemented
168#872C22×C3116yescomplemented
169C23×C32—24no—
170C23×C32—24no—
171C23×C32—24no—
172C23×C32C2414yes—
173C23×C32C2411yescharacteristic
174C22×C3×D3—24no—
175C22×C3×D3—24no—
176C22×C3×D3—24no—
177C22×C3×D3C2414yes—
178C32×Q8—44no—
179C32×D4—44no—
180C2×C32×C4—22no—
181C2×C32×C4—24no—
182C2×C32×C4—22no—
183C2×C32×C4—22no—
184C2×C32×C4C2412yes—
185C2×C32×C4C2414yes—
186C2×C32×C4C2412yes—
187C2×C32×C4C2412yes—
188C3×#67—216no—
189C3×#67—48no—
190C2×C3×Dic3—24no—
191C2×C3×Dic3—24no—
192C2×C3×Dic3—216no—
193C2×C3×Dic3#4118yes—
194C2×C3×Dic3C2414yes—
195C2×C3×Dic3C2414yes—
196C2×C3×Dic3C2414yes—
197C3×D12—44no—
198C3×C4×D3—216no—
199C3×Dic6—44no—
200#555—316no—
201C2×C4⋊Q8—31no—
202C2×#124—31no—
203C2×#124—32no—
204C2×#124C3×D311yescharacteristic, complemented
205C2×#121—32no—
206C2×#121—32no—
207C2×C4×D4—32no—
208C2×#117—32no—
209C2×#117—32no—
210C24×C3—22no—
211C24×C3—32no—
212C24×C3C22×D312yes—
213C24×C3C23×C312yes—
214C23×D3C23×C312yes—
215C2×C3×Q8—24no—
216C2×C3×Q8—34no—
217C2×C3×Q8—42no—
218C2×C3×Q8—42no—
219C2×C3×Q8—44no—
220C2×C3×Q8—122no—
221C2×C3×Q8C22×D314yes—
222C2×C3×Q8C23×C314yes—
223C2×C3×D4—24no—
224C2×C3×D4—38no—
225C2×C3×D4—38no—
226C2×C3×D4—34no—
227C2×C3×D4—42no—
228C2×C3×D4—42no—
229C2×C3×D4—44no—
230C2×C3×D4—68no—
231C2×C3×D4—122no—
232C2×C3×D4—124no—
233C2×C3×D4C22×D314yes—
234C2×C3×D4C23×C314yes—
235C22×C3×C4—21no—
236C22×C3×C4—22no—
237C22×C3×C4—21no—
238C22×C3×C4—21no—
239C22×C3×C4—32no—
240C22×C3×C4—38no—
241C22×C3×C4—32no—
242C22×C3×C4—32no—
243C22×C3×C4—38no—
244C22×C3×C4—68no—
245C22×C3×C4—68no—
246C22×C3×C4C22×D312yes—
247C22×C3×C4C22×D311yescharacteristic
248C22×C3×C4C22×D311yescharacteristic
249C22×C3×C4C22×D311yescharacteristic
250C22×C3×C4C23×C311yescharacteristic
251C22×C3×C4C23×C312yes—
252C22×C3×C4C23×C311yescharacteristic
253C22×C3×C4C23×C311yescharacteristic
254C2×#67—28no—
255C2×#67—44no—
256C2×#67C23×C318yes—
257C2×#67C23×C318yes—
258C22×Dic3—28no—
259C22×Dic3C23×C318yes—
260C22×Dic3C23×C312yes—
261C22×Dic3C23×C312yes—
262C22×Dic3C23×C312yes—
263C2×D12—42no—
264C2×D12C23×C314yes—
265C2×C4×D3—28no—
266C2×C4×D3C23×C318yes—
267C2×Dic6—42no—
268C2×Dic6C23×C314yes—
269C3×C4⋊C4—38no—
270C3×C4⋊C4—38no—
271C3×C4⋊C4—38no—
272C3×C22⋊C4—28no—
273C3×C22⋊C4—28no—
274C3×C22⋊C4—38no—
275C3×C22⋊C4—38no—
276C3×C22⋊C4—38no—
277C3×C22⋊C4—38no—
278C3×C22⋊C4C22×D318yes—
279C3×C22⋊C4C22×D318yes—
280C3×C22⋊C4C23×C318yes—
281C3×C22⋊C4C23×C318yes—
282C3×C42—24no—
283C3×C42—34no—
284C3×C42—38no—
285C3×C42—34no—
286C3×C42C22×D314yes—
287C3×C42C23×C314yes—
288#217C23×C318yes—
289(C2×D3)⋊C4C23×C318yes—
290(C2×D3)⋊C4C23×C318yes—
291(C2×D3)⋊C4C23×C318yes—
292Dic3⋊C4C23×C318yes—
293Dic3⋊C4C23×C318yes—
294Dic3⋊C4C23×C318yes—
295C4×Dic3C23×C318yes—
296C4×Dic3C23×C314yes—
297C4×Dic3C23×C314yes—
298C22×C32—24no—
299C22×C32—24no—
300C22×C32—24no—
301C22×C32—48no—
302C22×C32C2×#4114yes—
303C22×C32H(5,𝔽2)12yes—
304C22×C32C2511yescharacteristic
305C2×C3×D3—24no—
306C2×C3×D3—24no—
307C2×C3×D3—24no—
308C2×C3×D3—48no—
309C32×C4—22no—
310C32×C4—24no—
311C32×C4—22no—
312C32×C4—22no—
313C3×Dic3—24no—
314C3×Dic3—24no—
315C3×Dic3C2×#4118yes—
316C22×Q8—31no—
317C22×Q8C2×C3×D311yescharacteristic
318C22×D4—32no—
319C22×D4—32no—
320C22×D4—31no—
321C22×D4C2×C3×D311yescharacteristic
322C23×C4—32no—
323C23×C4—32no—
324C4⋊Q8—38no—
325#124—38no—
326#124—316no—
327#124C2×C3×D318yescomplemented
328#121—316no—
329#121—316no—
330C4×D4—316no—
331#117—316no—
332#117—316no—
333C2×C4⋊C4—32no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—32no—
336C2×C22⋊C4—32no—
337C2×C22⋊C4—32no—
338C2×C22⋊C4—32no—
339C2×C22⋊C4—32no—
340C2×C22⋊C4C2×C3×D312yes—
341C2×C22⋊C4C2×C3×D312yes—
342C2×C42—31no—
343C2×C42—32no—
344C2×C42—31no—
345C2×C42C2×C3×D311yescharacteristic
346C23×C3—24no—
347C23×C3—24no—
348C23×C3—24no—
349C23×C3—24no—
350C23×C3—24no—
351C23×C3—24no—
352C23×C3—21no—
353C23×C3—24no—
354C23×C3—34no—
355C23×C3—44no—
356C23×C3—44no—
357C23×C3—44no—
358C23×C3—64no—
359C23×C3—64no—
360C23×C3—64no—
361C23×C3C23×D314yes—
362C23×C3C23×D311yescharacteristic
363C23×C3C24×C314yes—
364C23×C3C24×C311yescharacteristic
365C22×D3—24no—
366C22×D3—24no—
367C22×D3—24no—
368C22×D3C24×C314yes—
369C3×Q8—44no—
370C3×Q8—44no—
371C3×Q8—48no—
372C3×Q8—124no—
373C3×D4—44no—
374C3×D4—44no—
375C3×D4—48no—
376C3×D4—616no—
377C3×D4—124no—
378C3×D4—128no—
379C2×C3×C4—24no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—24no—
384C2×C3×C4—24no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—24no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—38no—
393C2×C3×C4—34no—
394C2×C3×C4—34no—
395C2×C3×C4—34no—
396C2×C3×C4—44no—
397C2×C3×C4—42no—
398C2×C3×C4—42no—
399C2×C3×C4—616no—
400C2×C3×C4—616no—
401C2×C3×C4—64no—
402C2×C3×C4—64no—
403C2×C3×C4C23×D314yes—
404C2×C3×C4C23×D312yes—
405C2×C3×C4C23×D312yes—
406C2×C3×C4C23×D312yes—
407C2×C3×C4C24×C312yes—
408C2×C3×C4C24×C314yes—
409C2×C3×C4C24×C312yes—
410C2×C3×C4C24×C312yes—
411#67—216no—
412#67—48no—
413C2×Dic3—24no—
414C2×Dic3—24no—
415C2×Dic3—216no—
416C2×Dic3C3×#4118yes—
417C2×Dic3C24×C314yes—
418C2×Dic3C24×C314yes—
419C2×Dic3C24×C314yes—
420D12—44no—
421C4×D3—216no—
422Dic6—44no—
423C2×C32—44no—
424C2×C32#55514yescomplemented
425C2×C32C22×#4112yes—
426C2×C32C2×H(5,𝔽2)11yescharacteristic
427C3×D3—44no—
428C24—32no—
429C24C22×C3×D312yes—
430C2×Q8—34no—
431C2×Q8—42no—
432C2×Q8—122no—
433C2×Q8C22×C3×D314yes—
434C2×D4—38no—
435C2×D4—38no—
436C2×D4—34no—
437C2×D4—42no—
438C2×D4—68no—
439C2×D4—122no—
440C2×D4—124no—
441C2×D4C22×C3×D314yes—
442C22×C4—32no—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—38no—
446C22×C4—38no—
447C22×C4—68no—
448C22×C4—68no—
449C22×C4C22×C3×D312yes—
450C22×C4C22×C3×D311yescharacteristic
451C22×C4C22×C3×D311yescharacteristic
452C22×C4C22×C3×D311yescharacteristic
453C4⋊C4—38no—
454C4⋊C4—38no—
455C4⋊C4—38no—
456C22⋊C4—38no—
457C22⋊C4—38no—
458C22⋊C4—38no—
459C22⋊C4—38no—
460C22⋊C4C22×C3×D318yes—
461C22⋊C4C22×C3×D318yes—
462C42—34no—
463C42—38no—
464C42—34no—
465C42C22×C3×D314yes—
466C22×C3—24no—
467C22×C3—24no—
468C22×C3—24no—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—22no—
473C22×C3—24no—
474C22×C3—21no—
475C22×C3—44no—
476C22×C3—44no—
477C22×C3—44no—
478C22×C3—48no—
479C22×C3—48no—
480C22×C3—64no—
481C22×C3—64no—
482C22×C3—64no—
483C22×C3—88no—
484C22×C3—128no—
485C22×C3#100412yes—
486C22×C3D3×#4114yes—
487C22×C3C2×C3×#4114yes—
488C22×C3C3×H(5,𝔽2)12yes—
489C22×C3C24×D311yescharacteristic
490C22×C3C25×C311yescharacteristic
491C2×D3—24no—
492C2×D3—24no—
493C2×D3—24no—
494C2×D3—48no—
495C3×C4—24no—
496C3×C4—22no—
497C3×C4—22no—
498C3×C4—22no—
499C3×C4—24no—
500C3×C4—24no—
501C3×C4—22no—
502C3×C4—22no—
503C3×C4—22no—
504C3×C4—38no—
505C3×C4—42no—
506C3×C4—42no—
507C3×C4—44no—
508C3×C4—64no—
509C3×C4—64no—
510Dic3—24no—
511Dic3—24no—
512Dic3C2×C3×#4118yes—
513C32C2×#55511yescharacteristic, complemented, Sylow
514C23—24no—
515C23—24no—
516C23—24no—
517C23—34no—
518C23—64no—
519C23—64no—
520C23—64no—
521C23C23×C3×D314yes—
522C23C23×C3×D311yescharacteristic
523Q8—44no—
524Q8—124no—
525D4—44no—
526D4—616no—
527D4—124no—
528D4—128no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—38no—
534C2×C4—34no—
535C2×C4—34no—
536C2×C4—34no—
537C2×C4—616no—
538C2×C4—616no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4C23×C3×D314yes—
542C2×C4C23×C3×D312yes—
543C2×C4C23×C3×D312yes—
544C2×C4C23×C3×D312yes—
545C2×C3—21no—
546C2×C3—22no—
547C2×C3—24no—
548C2×C3—44no—
549C2×C3—44no—
550C2×C3—84no—
551C2×C3—124no—
552C2×C3#567014yesdirect product factor
553C2×C3C3×#55514yescomplemented
554C2×C3C2×#100411yescharacteristic
555C2×C3C2×D3×#4112yes—
556C2×C3C22×C3×#4112yes—
557C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
558D3—44no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—48no—
563C22—64no—
564C22—64no—
565C22—64no—
566C22—128no—
567C22C3×#100412yes—
568C22C3×D3×#4114yes—
569C22C24×C3×D311yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—24no—
574C4—38no—
575C4—64no—
576C4—64no—
577C3—21no—
578C3C2×#567011yescharacteristic, direct product factor
579C3C2×C3×#55511yescharacteristic, complemented
580C2—44no—
581C2—124no—
582C2C3×#567014yesdirect product factor
583C2C2×C3×#100411yescharacteristic
584C2C2×C3×D3×#4112yes—
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×#124C2
socleC23×C32C24
hypercenterC2×C3×#124D3
hypocenterC3C2×C3×#555
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×#124

Complements

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

NH ≅ G/Ntype
1. C2×C32×#124C2semidirect
2. C2×C3×#947C2semidirect
3. C2×C3×#944C2semidirect
4. C2×C3×#939C2semidirect
5. C3×#5670C2direct (N × H)
6. C2×C3×#885C2semidirect
7. C2×C3×#884C2semidirect
8. C2×C3×#67⋊C4C2semidirect
9. C2×C3×#877C2semidirect
10. C2×C3×#872C2semidirect
12. C2×#5670C3direct (N × H)
21. C32×#124C22semidirect
26. C3×#947C22semidirect
27. C3×#944C22semidirect
28. C3×#939C22semidirect
38. C3×#885C22semidirect
39. C3×#884C22semidirect
40. C3×#67⋊C4C22semidirect
41. C3×#877C22semidirect
42. C3×#872C22semidirect
48. C2×C3×#124D3semidirect
49. C2×C3×#124C2×C3semidirect
55. C2×#947C2×C3semidirect
56. C2×#944C2×C3semidirect
57. C2×#939C2×C3semidirect
58. #5670C2×C3direct (N × H)
59. C2×#885C2×C3semidirect
60. C2×#884C2×C3semidirect
61. C2×#67⋊C4C2×C3semidirect
62. C2×#877C2×C3semidirect
63. C2×#872C2×C3semidirect
125. C3×#124C2×D3semidirect
126. C3×#124C22×C3semidirect
152. #947C22×C3semidirect
153. #944C22×C3semidirect
154. #939C22×C3semidirect
164. #885C22×C3semidirect
165. #884C22×C3semidirect
166. #67⋊C4C22×C3semidirect
167. #877C22×C3semidirect
168. #872C22×C3semidirect
204. C2×#124C3×D3semidirect
327. #124C2×C3×D3semidirect
424. C2×C32#555semidirect
513. C32C2×#555semidirect
552. C2×C3#5670direct (N × H)
553. C2×C3C3×#555semidirect
578. C3C2×#5670direct (N × H)
579. C3C2×C3×#555semidirect
582. C2C3×#5670direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1242
C2×C3×#9472
C2×C3×#9442
C2×C3×#9392
C3×#56702
C2×C3×#8852
C2×C3×#8842
C2×C3×#67⋊C42
C2×C3×#8772
C2×C3×#8722
C2×C3×#5553
C2×#56703