Finite Group Databasebeta

← C2×C3×#5668 #11900752

Subgroups — C2×C3×#5668 #11900752

Order 1152.

3621 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
1C22×C3×#239C211yesmaximal, characteristic, complemented
2C22×C3×#237C211yesmaximal, characteristic, complemented
3C2×C32×#124C211yesmaximal, characteristic, complemented
4C2×C3×#944C211yesmaximal, characteristic, complemented
5C2×C3×#940C211yesmaximal, characteristic, complemented
6C3×#5668C2116yesmaximal, direct product factor
7C2×C3×#884C214yesmaximal, complemented
8C2×C3×#67⋊C4C214yesmaximal, complemented
9C2×C3×#875C211yesmaximal, characteristic, complemented
10C2×C3×C42×D3C211yesmaximal, characteristic, complemented
11C2×C3×#532—31nomaximal
12C2×#5668C311yesmaximal, characteristic, direct product factor
13C22×C32×Q8C2211yescharacteristic
14C22×C32×D4C2211yescharacteristic
15C22×C3×#67C2212yes—
16C22×C3×#67C2214yes—
17C23×C3×Dic3C2212yes—
18C2×C3×#239—28no—
19C2×C3×#239C2218yescomplemented
20C2×C3×#237—28no—
21C2×C3×#237C2218yescomplemented
22C22×C3×D12C2212yes—
23C22×C3×D12C2211yescharacteristic
24C22×C3×C4×D3C2212yes—
25C22×C3×C4×D3C2212yes—
26C22×C3×C4×D3C2211yescharacteristic
27C22×C3×Dic6C2211yescharacteristic
28C32×#124C2218yescomplemented
29C2×C32×C22⋊C4C2214yes—
30C2×C32×C42C2211yescharacteristic
31C3×#944C2218yescomplemented
32C3×#940C2218yescomplemented
33C2×C3×(C2×D3)⋊C4C2214yes—
34C2×C3×Dic3⋊C4C2214yes—
35C2×C3×C4×Dic3C2211yescharacteristic
36C2×C3×C4×Dic3C2212yes—
37C3×#884C22132yescomplemented
38C3×#67⋊C4C22132yescomplemented
39C3×#875C2218yescomplemented
40C3×C42×D3C2218yescomplemented
41C22×C3×#41—31no—
42C22×C3×#41—31no—
43C22×#239C2×C311yescharacteristic, complemented
44C22×#237C2×C311yescharacteristic, complemented
45C3×#532—316no—
46C2×C3×C4⋊Q8—31no—
47C2×C3×#127—31no—
48C2×C3×#124—21no—
49C2×C3×#124—31no—
50C2×C3×#124D311yescharacteristic, complemented
51C2×C3×#124C2×C311yescharacteristic, complemented
52C2×C3×#121—34no—
53C2×C3×C4×D4—34no—
54C22×C3×C42—31no—
55C2×#944C2×C311yescharacteristic, complemented
56C2×#940C2×C311yescharacteristic, complemented
57#5668C2×C3116yesdirect product factor
58C2×#884C2×C314yescomplemented
59C2×#67⋊C4C2×C314yescomplemented
60C2×#875C2×C311yescharacteristic, complemented
61C2×C42×D3C2×C311yescharacteristic, complemented
62C24×C32C2312yes—
63C23×C3×D3C2312yes—
64C23×C3×D3C2311yescharacteristic
65C2×C32×Q8—24no—
66C2×C32×Q8C2314yes—
67C2×C32×D4—24no—
68C2×C32×D4C2314yes—
69C22×C32×C4C2312yes—
70C22×C32×C4C2312yes—
71C22×C32×C4C2311yescharacteristic
72C2×C3×#67—28no—
73C2×C3×#67—216no—
74C2×C3×#67C23116yes—
75C2×C3×#67C2318yes—
76C22×C3×Dic3—28no—
77C22×C3×Dic3C2318yes—
78C22×C3×Dic3C2312yes—
79C22×C3×Dic3C2311yescharacteristic
80C22×C3×Dic3C2312yes—
81C3×#239—232no—
82C3×#237—232no—
83C2×C3×D12—28no—
84C2×C3×D12—24no—
85C2×C3×D12C2318yes—
86C2×C3×D12C2314yes—
87C2×C3×C4×D3—28no—
88C2×C3×C4×D3—28no—
89C2×C3×C4×D3D418yescomplemented
90C2×C3×C4×D3C2318yes—
91C2×C3×C4×D3C2318yes—
92C2×C3×C4×D3C2314yes—
93C2×C3×Dic6—24no—
94C2×C3×Dic6C2314yes—
95C32×C22⋊C4C23116yes—
96C32×C42C2314yes—
97C3×(C2×D3)⋊C4C23116yes—
98C3×Dic3⋊C4C23116yes—
99C3×C4×Dic3C2318yes—
100C3×C4×Dic3C2314yes—
101C2×#532—31noSylow
102C2×C3×#41—38no—
103C2×C3×#41—38no—
104C2×C3×#41—68no—
105C2×C3×#41—68no—
106C22×C3×Q8—21no—
107C22×C3×Q8—31no—
108C22×C3×Q8C2×D311yescharacteristic
109C22×C3×Q8C22×C311yescharacteristic
110C22×C3×D4—21no—
111C22×C3×D4—32no—
112C22×C3×D4—34no—
113C22×C3×D4—32no—
114C22×C3×D4—31no—
115C22×C3×D4C2×D311yescharacteristic
116C22×C3×D4C22×C311yescharacteristic
117C23×C3×C4—31no—
118C23×C3×C4—32no—
119C23×C3×C4—32no—
120C23×C3×C4—32no—
121C22×#67C22×C312yes—
122C22×#67C22×C314yes—
123C23×Dic3C22×C312yes—
124C2×#239—28no—
125C2×#239C22×C318yescomplemented
126C2×#237—28no—
127C2×#237C22×C318yescomplemented
128C22×D12C22×C311yescharacteristic
129C22×D12C22×C312yes—
130C22×C4×D3C22×C312yes—
131C22×C4×D3C22×C312yes—
132C22×C4×D3C22×C311yescharacteristic
133C22×Dic6C22×C311yescharacteristic
134C3×C4⋊Q8—38no—
135C3×#127—38no—
136C3×#124—28no—
137C3×#124—38no—
138C3×#124C2×D318yescomplemented
139C3×#124C22×C318yescomplemented
140C3×#121—332no—
141C3×C4×D4—332no—
142C2×C3×C4⋊C4—34no—
143C2×C3×C22⋊C4—24no—
144C2×C3×C22⋊C4—34no—
145C2×C3×C22⋊C4C2×D314yes—
146C2×C3×C22⋊C4C22×C314yes—
147C2×C3×C42—21no—
148C2×C3×C42—32no—
149C2×C3×C42—31no—
150C2×C3×C42—38no—
151C2×C3×C42C2×D311yescharacteristic
152C2×C3×C42C22×C311yescharacteristic
153#944C22×C318yescomplemented
154#940C22×C318yescomplemented
155C2×(C2×D3)⋊C4C22×C314yes—
156C2×Dic3⋊C4C22×C314yes—
157C2×C4×Dic3C22×C312yes—
158C2×C4×Dic3C22×C311yescharacteristic
159#884C22×C3132yescomplemented
160#67⋊C4C22×C3132yescomplemented
161#875C22×C318yescomplemented
162C42×D3C22×C318yescomplemented
163C23×C32—24no—
164C23×C32—28no—
165C23×C32C2414yes—
166C23×C32C2411yescharacteristic
167C22×C3×D3—24no—
168C22×C3×D3—24no—
169C22×C3×D3—28no—
170C22×C3×D3C2×D414yes—
171C22×C3×D3C2414yes—
172C22×C3×D3C2412yes—
173C32×Q8—28no—
174C32×D4—28no—
175C2×C32×C4—24no—
176C2×C32×C4—24no—
177C2×C32×C4C2×D414yes—
178C2×C32×C4C2414yes—
179C2×C32×C4C2414yes—
180C2×C32×C4C2412yes—
181C3×#67—216no—
182C3×#67—232no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—216no—
185C2×C3×Dic3C2×D414yes—
186C2×C3×Dic3#4118yescomplemented
187C2×C3×Dic3C2414yes—
188C2×C3×Dic3C2412yes—
189C2×C3×Dic3C2414yes—
190C3×D12—216no—
191C3×D12—28no—
192C3×C4×D3—216no—
193C3×C4×D3—216no—
194C3×C4×D3C2×D4116yescomplemented
195C3×Dic6—28no—
196C22×#41—31no—
197C22×#41—31no—
198#532—316no—
199C2×C4⋊Q8—31no—
200C2×#127—31no—
201C2×#124—31no—
202C2×#124C3×D311yescharacteristic, complemented
203C2×#121—34no—
204C2×C4×D4—34no—
205C22×C42—31no—
206C24×C3—22no—
207C24×C3—31no—
208C24×C3—32no—
209C24×C3C22×D312yes—
210C24×C3C23×C312yes—
211C23×D3C23×C312yes—
212C23×D3C23×C311yescharacteristic
213C3×#41—632no—
214C3×#41—632no—
215C2×C3×Q8—24no—
216C2×C3×Q8—24no—
217C2×C3×Q8—24no—
218C2×C3×Q8—34no—
219C2×C3×Q8—44no—
220C2×C3×Q8—64no—
221C2×C3×Q8C22×D314yes—
222C2×C3×Q8C23×C314yes—
223C2×C3×D4—24no—
224C2×C3×D4—24no—
225C2×C3×D4—24no—
226C2×C3×D4—316no—
227C2×C3×D4—38no—
228C2×C3×D4—38no—
229C2×C3×D4—34no—
230C2×C3×D4—44no—
231C2×C3×D4—68no—
232C2×C3×D4—616no—
233C2×C3×D4—68no—
234C2×C3×D4—64no—
235C2×C3×D4C22×D314yes—
236C2×C3×D4C23×C314yes—
237C22×C3×C4—21no—
238C22×C3×C4—22no—
239C22×C3×C4—22no—
240C22×C3×C4—32no—
241C22×C3×C4—31no—
242C22×C3×C4—38no—
243C22×C3×C4—34no—
244C22×C3×C4—38no—
245C22×C3×C4—32no—
246C22×C3×C4—38no—
247C22×C3×C4—38no—
248C22×C3×C4—68no—
249C22×C3×C4—68no—
250C22×C3×C4—68no—
251C22×C3×C4C22×D312yes—
252C22×C3×C4C22×D312yes—
253C22×C3×C4C22×D311yescharacteristic
254C22×C3×C4C23×C312yes—
255C22×C3×C4C23×C312yes—
256C22×C3×C4C23×C311yescharacteristic
257C2×#67—28no—
258C2×#67—216no—
259C2×#67C23×C3116yes—
260C2×#67C23×C318yes—
261C22×Dic3—28no—
262C22×Dic3C23×C318yes—
263C22×Dic3C23×C312yes—
264C22×Dic3C23×C311yescharacteristic
265C22×Dic3C23×C312yes—
266#239—232no—
267#237—232no—
268C2×D12—28no—
269C2×D12—24no—
270C2×D12C23×C314yes—
271C2×D12C23×C318yes—
272C2×C4×D3—28no—
273C2×C4×D3—28no—
274C2×C4×D3C3×D418yescomplemented
275C2×C4×D3C23×C318yes—
276C2×C4×D3C23×C318yes—
277C2×C4×D3C23×C314yes—
278C2×Dic6—24no—
279C2×Dic6C23×C314yes—
280C3×C4⋊C4—316no—
281C3×C22⋊C4—216no—
282C3×C22⋊C4—316no—
283C3×C22⋊C4C22×D3116yes—
284C3×C22⋊C4C23×C3116yes—
285C3×C42—24no—
286C3×C42—38no—
287C3×C42—34no—
288C3×C42C22×D314yes—
289C3×C42C23×C314yes—
290(C2×D3)⋊C4C23×C3116yes—
291Dic3⋊C4C23×C3116yes—
292C4×Dic3C23×C318yes—
293C4×Dic3C23×C314yes—
294C22×C32—24no—
295C22×C32—28no—
296C22×C32—48no—
297C22×C32C22×D412yes—
298C22×C32C2×#4114yes—
299C22×C32C2511yescharacteristic
300C2×C3×D3—28no—
301C2×C3×D3—24no—
302C2×C3×D3—24no—
303C2×C3×D3—28no—
304C2×C3×D3—48no—
305C2×C3×D3C22×D414yes—
306C32×C4—24no—
307C32×C4—24no—
308C32×C4C22×D414yes—
309C3×Dic3—24no—
310C3×Dic3C22×D414yes—
311C3×Dic3C2×#4118yescomplemented
312C2×#41—38no—
313C2×#41—38no—
314C2×#41—68no—
315C2×#41—68no—
316C22×Q8—31no—
317C22×Q8C2×C3×D311yescharacteristic
318C22×D4—34no—
319C22×D4—32no—
320C22×D4—32no—
321C22×D4—31no—
322C22×D4C2×C3×D311yescharacteristic
323C23×C4—31no—
324C23×C4—32no—
325C23×C4—32no—
326C23×C4—32no—
327C4⋊Q8—38no—
328#127—38no—
329#124—38no—
330#124C2×C3×D318yescomplemented
331#121—332no—
332C4×D4—332no—
333C2×C4⋊C4—34no—
334C2×C22⋊C4—34no—
335C2×C22⋊C4C2×C3×D314yes—
336C2×C42—32no—
337C2×C42—31no—
338C2×C42—38no—
339C2×C42C2×C3×D311yescharacteristic
340C23×C3—24no—
341C23×C3—28no—
342C23×C3—24no—
343C23×C3—28no—
344C23×C3—21no—
345C23×C3—24no—
346C23×C3—34no—
347C23×C3—32no—
348C23×C3—34no—
349C23×C3—44no—
350C23×C3—48no—
351C23×C3—64no—
352C23×C3—64no—
353C23×C3—68no—
354C23×C3C23×D314yes—
355C23×C3C23×D311yescharacteristic
356C23×C3C24×C314yes—
357C23×C3C24×C311yescharacteristic
358C22×D3—24no—
359C22×D3—24no—
360C22×D3—28no—
361C22×D3C2×C3×D414yes—
362C22×D3C24×C314yes—
363C22×D3C24×C312yes—
364C3×Q8—28no—
365C3×Q8—28no—
366C3×Q8—48no—
367C3×Q8—68no—
368C3×D4—28no—
369C3×D4—28no—
370C3×D4—48no—
371C3×D4—616no—
372C3×D4—632no—
373C3×D4—616no—
374C3×D4—68no—
375C2×C3×C4—24no—
376C2×C3×C4—24no—
377C2×C3×C4—24no—
378C2×C3×C4—24no—
379C2×C3×C4—22no—
380C2×C3×C4—24no—
381C2×C3×C4—24no—
382C2×C3×C4—24no—
383C2×C3×C4—34no—
384C2×C3×C4—38no—
385C2×C3×C4—34no—
386C2×C3×C4—32no—
387C2×C3×C4—316no—
388C2×C3×C4—34no—
389C2×C3×C4—44no—
390C2×C3×C4—44no—
391C2×C3×C4—616no—
392C2×C3×C4—616no—
393C2×C3×C4—616no—
394C2×C3×C4—64no—
395C2×C3×C4D3×D414yes—
396C2×C3×C4C2×C3×D414yes—
397C2×C3×C4C23×D314yes—
398C2×C3×C4C23×D314yes—
399C2×C3×C4C23×D312yes—
400C2×C3×C4C24×C314yes—
401C2×C3×C4C24×C314yes—
402C2×C3×C4C24×C312yes—
403#67—216no—
404#67—232no—
405C2×Dic3—24no—
406C2×Dic3—216no—
407C2×Dic3C2×C3×D414yes—
408C2×Dic3C3×#4118yescomplemented
409C2×Dic3C24×C314yes—
410C2×Dic3C24×C312yes—
411C2×Dic3C24×C314yes—
412D12—216no—
413D12—28no—
414C4×D3—216no—
415C4×D3—216no—
416C4×D3C2×C3×D4116yescomplemented
417Dic6—28no—
418C2×C32—44no—
419C2×C32#53214yescomplemented
420C2×C32C23×D411yescharacteristic
421C2×C32C22×#4112yes—
422C3×D3—24no—
423C3×D3—44no—
424C24—31no—
425C24—32no—
426C24C22×C3×D312yes—
427#41—632no—
428#41—632no—
429C2×Q8—24no—
430C2×Q8—34no—
431C2×Q8—64no—
432C2×Q8C22×C3×D314yes—
433C2×D4—24no—
434C2×D4—316no—
435C2×D4—38no—
436C2×D4—38no—
437C2×D4—34no—
438C2×D4—68no—
439C2×D4—616no—
440C2×D4—68no—
441C2×D4—64no—
442C2×D4C22×C3×D314yes—
443C22×C4—32no—
444C22×C4—31no—
445C22×C4—34no—
446C22×C4—38no—
447C22×C4—32no—
448C22×C4—38no—
449C22×C4—38no—
450C22×C4—38no—
451C22×C4—68no—
452C22×C4—68no—
453C22×C4—68no—
454C22×C4C22×C3×D312yes—
455C22×C4C22×C3×D312yes—
456C22×C4C22×C3×D311yescharacteristic
457C4⋊C4—316no—
458C22⋊C4—316no—
459C22⋊C4C22×C3×D3116yes—
460C42—38no—
461C42—34no—
462C42C22×C3×D314yes—
463C22×C3—24no—
464C22×C3—28no—
465C22×C3—24no—
466C22×C3—28no—
467C22×C3—22no—
468C22×C3—24no—
469C22×C3—21no—
470C22×C3—34no—
471C22×C3—44no—
472C22×C3—48no—
473C22×C3—48no—
474C22×C3—48no—
475C22×C3—68no—
476C22×C3—64no—
477C22×C3—64no—
478C22×C3—68no—
479C22×C3—88no—
480C22×C3—128no—
481C22×C3C2×D3×D412yes—
482C22×C3D3×#4114yes—
483C22×C3C22×C3×D412yes—
484C22×C3C2×C3×#4114yes—
485C22×C3C24×D311yescharacteristic
486C22×C3C25×C311yescharacteristic
487C2×D3—24no—
488C2×D3—28no—
489C2×D3—24no—
490C2×D3—28no—
491C2×D3—48no—
492C2×D3C22×C3×D414yes—
493C3×C4—24no—
494C3×C4—24no—
495C3×C4—24no—
496C3×C4—24no—
497C3×C4—24no—
498C3×C4—34no—
499C3×C4—38no—
500C3×C4—44no—
501C3×C4—44no—
502C3×C4—64no—
503C3×C4C2×D3×D414yes—
504C3×C4C22×C3×D414yes—
505Dic3—24no—
506Dic3C22×C3×D414yes—
507Dic3C2×C3×#4118yescomplemented
508C32C2×#53211yescharacteristic, complemented, Sylow
509C23—24no—
510C23—28no—
511C23—34no—
512C23—32no—
513C23—34no—
514C23—64no—
515C23—64no—
516C23—68no—
517C23C23×C3×D314yes—
518C23C23×C3×D311yescharacteristic
519Q8—28no—
520Q8—68no—
521D4—28no—
522D4—616no—
523D4—632no—
524D4—616no—
525D4—68no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—34no—
529C2×C4—38no—
530C2×C4—34no—
531C2×C4—32no—
532C2×C4—316no—
533C2×C4—34no—
534C2×C4—616no—
535C2×C4—616no—
536C2×C4—616no—
537C2×C4—64no—
538C2×C4C3×D3×D414yes—
539C2×C4C23×C3×D314yes—
540C2×C4C23×C3×D314yes—
541C2×C4C23×C3×D312yes—
542C2×C3—21no—
543C2×C3—22no—
544C2×C3—24no—
545C2×C3—44no—
546C2×C3—44no—
547C2×C3—64no—
548C2×C3—84no—
549C2×C3—124no—
550C2×C3#566814yesdirect product factor
551C2×C3C3×#53214yescomplemented
552C2×C3C22×D3×D411yescharacteristic
553C2×C3C2×D3×#4112yes—
554C2×C3C23×C3×D411yescharacteristic
555C2×C3C22×C3×#4112yes—
556D3—24no—
557D3—44no—
558C22—24no—
559C22—28no—
560C22—34no—
561C22—48no—
562C22—64no—
563C22—68no—
564C22—64no—
565C22—68no—
566C22—128no—
567C22C2×C3×D3×D412yes—
568C22C3×D3×#4114yes—
569C22C24×C3×D311yescharacteristic
570C4—24no—
571C4—24no—
572C4—34no—
573C4—38no—
574C4—64no—
575C4C2×C3×D3×D414yes—
576C3—21no—
577C3C2×#566811yescharacteristic, direct product factor
578C3C2×C3×#53211yescharacteristic, complemented
579C2—44no—
580C2—64no—
581C2—124no—
582C2C3×#566814yesdirect product factor
583C2C22×C3×D3×D411yescharacteristic
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×#532
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. C22×C3×#239C2semidirect
2. C22×C3×#237C2semidirect
3. C2×C32×#124C2semidirect
4. C2×C3×#944C2semidirect
5. C2×C3×#940C2semidirect
6. C3×#5668C2direct (N × H)
7. C2×C3×#884C2semidirect
8. C2×C3×#67⋊C4C2semidirect
9. C2×C3×#875C2semidirect
10. C2×C3×C42×D3C2semidirect
12. C2×#5668C3direct (N × H)
19. C2×C3×#239C22semidirect
21. C2×C3×#237C22semidirect
28. C32×#124C22semidirect
31. C3×#944C22semidirect
32. C3×#940C22semidirect
37. C3×#884C22semidirect
38. C3×#67⋊C4C22semidirect
39. C3×#875C22semidirect
40. C3×C42×D3C22semidirect
43. C22×#239C2×C3semidirect
44. C22×#237C2×C3semidirect
50. C2×C3×#124D3semidirect
51. C2×C3×#124C2×C3semidirect
55. C2×#944C2×C3semidirect
56. C2×#940C2×C3semidirect
57. #5668C2×C3direct (N × H)
58. C2×#884C2×C3semidirect
59. C2×#67⋊C4C2×C3semidirect
60. C2×#875C2×C3semidirect
61. C2×C42×D3C2×C3semidirect
89. C2×C3×C4×D3D4semidirect
125. C2×#239C22×C3semidirect
127. C2×#237C22×C3semidirect
138. C3×#124C2×D3semidirect
139. C3×#124C22×C3semidirect
153. #944C22×C3semidirect
154. #940C22×C3semidirect
159. #884C22×C3semidirect
160. #67⋊C4C22×C3semidirect
161. #875C22×C3semidirect
162. C42×D3C22×C3semidirect
186. C2×C3×Dic3#41semidirect
194. C3×C4×D3C2×D4semidirect
202. C2×#124C3×D3semidirect
274. C2×C4×D3C3×D4semidirect
311. C3×Dic3C2×#41semidirect
330. #124C2×C3×D3semidirect
408. C2×Dic3C3×#41semidirect
416. C4×D3C2×C3×D4semidirect
419. C2×C32#532semidirect
507. Dic3C2×C3×#41semidirect
508. C32C2×#532semidirect
550. C2×C3#5668direct (N × H)
551. C2×C3C3×#532semidirect
577. C3C2×#5668direct (N × H)
578. C3C2×C3×#532semidirect
582. C2C3×#5668direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2392
C22×C3×#2372
C2×C32×#1242
C2×C3×#9442
C2×C3×#9402
C3×#56682
C2×C3×#8842
C2×C3×#67⋊C42
C2×C3×#8752
C2×C3×C42×D32
C2×C3×#5323
C2×#56683