Finite Group Databasebeta

← C2×C3×#5223 #11808083

Subgroups — C2×C3×#5223 #11808083

Order 1152.

4305 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×D4C211yesmaximal, characteristic, complemented
2C23×C3×#67C212yesmaximal, complemented
3C22×C3×#217C212yesmaximal, complemented
4C22×C3×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
5C3×#5223C218yesmaximal, direct product factor
6C2×C3×C3⋊#95C211yesmaximal, characteristic, complemented
7C2×C3×#392—31nomaximal
8C2×#5223C311yesmaximal, characteristic, direct product factor
9C25×C32C2212yes—
10C24×C3×D3C2211yescharacteristic
11C22×C32×D4—24no—
12C22×C32×D4—28no—
13C22×C32×D4C2214yescomplemented
14C23×C32×C4C2211yescharacteristic
15C22×C3×#67—28no—
16C22×C3×#67—28no—
17C22×C3×#67—28no—
18C22×C3×#67C2218yescomplemented
19C23×C3×Dic3C2211yescharacteristic
20C23×C3×Dic3C2212yes—
21C2×C3×#217—28no—
22C2×C3×#217C2218yescomplemented
23C2×C3×(C2×D3)⋊C4—24no—
24C2×C3×(C2×D3)⋊C4C2214yescomplemented
25C3×C3⋊#95C2214yescomplemented
26C23×C3×D4—21no—
27C23×C3×D4—32no—
28C23×C3×D4D311yescharacteristic, complemented
29C23×C3×D4C2×C311yescharacteristic, complemented
30C23×#67C2×C312yescomplemented
31C22×C3×C22⋊C4—31no—
32C22×C3×C22⋊C4—32no—
33C22×#217C2×C312yescomplemented
34C22×(C2×D3)⋊C4C2×C311yescharacteristic, complemented
35C3×#392—38no—
36C2×C3×#95—31no—
37#5223C2×C318yesdirect product factor
38C2×C3⋊#95C2×C311yescharacteristic, complemented
39C24×C32—24no—
40C24×C32—24no—
41C24×C32—24no—
42C24×C32—24no—
43C24×C32—24no—
44C24×C32—24no—
45C24×C32D418yescomplemented
46C24×C32C2314yes—
47C24×C32C2311yescharacteristic
48C23×C3×D3—22no—
49C23×C3×D3—24no—
50C23×C3×D3—24no—
51C23×C3×D3—22no—
52C23×C3×D3D414yescomplemented
53C23×C3×D3C2312yes—
54C2×C32×D4—232no—
55C2×C32×D4—216no—
56C2×C32×D4—28no—
57C22×C32×C4—24no—
58C22×C32×C4D414yescomplemented
59C22×C32×C4C2312yes—
60C2×C3×#67—264no—
61C2×C3×#67—216no—
62C2×C3×#67—216no—
63C2×C3×#67—216no—
64C22×C3×Dic3—24no—
65C22×C3×Dic3—24no—
66C22×C3×Dic3—22no—
67C22×C3×Dic3—24no—
68C22×C3×Dic3D418yescomplemented
69C22×C3×Dic3C2312yes—
70C22×C3×Dic3C2314yes—
71C3×#217—216no—
72C3×(C2×D3)⋊C4—28no—
73C2×#392—31noSylow
74C25×C3—22no—
75C25×C3—31no—
76C25×C3C2×D312yes—
77C25×C3C22×C312yes—
78C24×D3C22×C311yescharacteristic
79C22×C3×D4—216no—
80C22×C3×D4—28no—
81C22×C3×D4—28no—
82C22×C3×D4—28no—
83C22×C3×D4—24no—
84C22×C3×D4—24no—
85C22×C3×D4—24no—
86C22×C3×D4—38no—
87C22×C3×D4—68no—
88C22×C3×D4—68no—
89C22×C3×D4—68no—
90C22×C3×D4C2×D314yescomplemented
91C22×C3×D4C22×C314yescomplemented
92C23×C3×C4—21no—
93C23×C3×C4—32no—
94C23×C3×C4—31no—
95C23×C3×C4C2×D311yescharacteristic
96C23×C3×C4C22×C311yescharacteristic
97C22×#67—28no—
98C22×#67—28no—
99C22×#67—28no—
100C22×#67C22×C318yescomplemented
101C23×Dic3C22×C311yescharacteristic
102C23×Dic3C22×C312yes—
103C2×C3×C22⋊C4—38no—
104C2×C3×C22⋊C4—34no—
105C2×C3×C22⋊C4—68no—
106C2×C3×C22⋊C4—64no—
107C2×#217—28no—
108C2×#217C22×C318yescomplemented
109C2×(C2×D3)⋊C4—24no—
110C2×(C2×D3)⋊C4C22×C314yescomplemented
111C3×#95—34no—
112C3⋊#95C22×C314yescomplemented
113C23×C32—216no—
114C23×C32—24no—
115C23×C32—24no—
116C23×C32—216no—
117C23×C32—216no—
118C23×C32—24no—
119C23×C32—24no—
120C23×C32—24no—
121C23×C32—24no—
122C23×C32—48no—
123C23×C32—48no—
124C23×C32—48no—
125C23×C32—48no—
126C23×C32C2×D412yes—
127C23×C32C2×D414yes—
128C23×C32C2×D414yes—
129C23×C32C2×D412yes—
130C23×C32C2×D418yescomplemented
131C23×C32#4112yes—
132C23×C32C2411yescharacteristic
133C22×C3×D3—216no—
134C22×C3×D3—28no—
135C22×C3×D3—22no—
136C22×C3×D3—24no—
137C22×C3×D3—24no—
138C22×C3×D3—22no—
139C22×C3×D3—44no—
140C22×C3×D3—48no—
141C22×C3×D3—44no—
142C22×C3×D3C2×D414yescomplemented
143C32×D4—232no—
144C2×C32×C4—28no—
145C2×C32×C4—24no—
146C2×C32×C4C2×D414yescomplemented
147C3×#67—264no—
148C2×C3×Dic3—216no—
149C2×C3×Dic3—24no—
150C2×C3×Dic3—24no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—24no—
153C2×C3×Dic3—44no—
154C2×C3×Dic3C2×D418yescomplemented
155C23×D4—32no—
156C23×D4C3×D311yescharacteristic, complemented
157C22×C22⋊C4—32no—
158C22×C22⋊C4—31no—
159#392—38no—
160C2×#95—31no—
161C24×C3—28no—
162C24×C3—28no—
163C24×C3—24no—
164C24×C3—24no—
165C24×C3—24no—
166C24×C3—24no—
167C24×C3—24no—
168C24×C3—24no—
169C24×C3—24no—
170C24×C3—24no—
171C24×C3—24no—
172C24×C3—24no—
173C24×C3—24no—
174C24×C3—24no—
175C24×C3—28no—
176C24×C3—21no—
177C24×C3—24no—
178C24×C3—34no—
179C24×C3—32no—
180C24×C3—44no—
181C24×C3—44no—
182C24×C3—44no—
183C24×C3—44no—
184C24×C3—64no—
185C24×C3—62no—
186C24×C3—62no—
187C24×C3—64no—
188C24×C3#6718yescomplemented
189C24×C3C3×D418yescomplemented
190C24×C3C22×D314yes—
191C24×C3C22×D311yescharacteristic
192C24×C3C23×C314yes—
193C24×C3C23×C311yescharacteristic
194C23×D3—22no—
195C23×D3—24no—
196C23×D3—24no—
197C23×D3—22no—
198C23×D3C3×D414yescomplemented
199C23×D3C23×C312yes—
200C2×C3×D4—264no—
201C2×C3×D4—232no—
202C2×C3×D4—232no—
203C2×C3×D4—232no—
204C2×C3×D4—216no—
205C2×C3×D4—216no—
206C2×C3×D4—216no—
207C2×C3×D4—28no—
208C2×C3×D4—28no—
209C2×C3×D4—616no—
210C2×C3×D4—616no—
211C2×C3×D4—664no—
212C2×C3×D4—616no—
213C22×C3×C4—28no—
214C22×C3×C4—24no—
215C22×C3×C4—24no—
216C22×C3×C4—24no—
217C22×C3×C4—22no—
218C22×C3×C4—38no—
219C22×C3×C4—34no—
220C22×C3×C4—32no—
221C22×C3×C4—64no—
222C22×C3×C4—64no—
223C22×C3×C4—64no—
224C22×C3×C4—62no—
225C22×C3×C4#6714yescomplemented
226C22×C3×C4C3×D414yescomplemented
227C22×C3×C4C22×D312yes—
228C22×C3×C4C23×C312yes—
229C2×#67—264no—
230C2×#67—216no—
231C2×#67—216no—
232C2×#67—216no—
233C22×Dic3—24no—
234C22×Dic3—24no—
235C22×Dic3—22no—
236C22×Dic3—24no—
237C22×Dic3C3×D418yescomplemented
238C22×Dic3C23×C312yes—
239C22×Dic3C23×C314yes—
240C3×C22⋊C4—616no—
241C3×C22⋊C4—68no—
242#217—216no—
243(C2×D3)⋊C4—28no—
244C22×C32—28no—
245C22×C32—28no—
246C22×C32—28no—
247C22×C32—44no—
248C22×C32—44no—
249C22×C32—44no—
250C22×C32—44no—
251C22×C32—432no—
252C22×C32#12018yes—
253C22×C32#12014yes—
254C22×C32#12114yes—
255C22×C32#12118yes—
256C22×C32#12714yes—
257C22×C32C22×D411yescharacteristic
258C22×C32C22×D412yes—
259C22×C32C22×D412yes—
260C22×C32C22×D411yescharacteristic
261C22×C32C2×#4111yescharacteristic
262C2×C3×D3—28no—
263C2×C3×D3—24no—
264C2×C3×D3—42no—
265C2×C3×D3—44no—
266C2×C3×D3—42no—
267C2×C3×D3—416no—
268C32×C4—24no—
269C3×Dic3—28no—
270C3×Dic3—42no—
271C25—31no—
272C25C2×C3×D312yes—
273C22×D4—28no—
274C22×D4—24no—
275C22×D4—38no—
276C22×D4—68no—
277C22×D4—68no—
278C22×D4—68no—
279C22×D4C2×C3×D314yescomplemented
280C23×C4—32no—
281C23×C4—31no—
282C23×C4C2×C3×D311yescharacteristic
283C2×C22⋊C4—38no—
284C2×C22⋊C4—34no—
285C2×C22⋊C4—68no—
286C2×C22⋊C4—64no—
287#95—34no—
288C23×C3—232no—
289C23×C3—28no—
290C23×C3—28no—
291C23×C3—216no—
292C23×C3—216no—
293C23×C3—216no—
294C23×C3—216no—
295C23×C3—216no—
296C23×C3—216no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—21no—
310C23×C3—22no—
311C23×C3—24no—
312C23×C3—22no—
313C23×C3—24no—
314C23×C3—22no—
315C23×C3—28no—
316C23×C3—34no—
317C23×C3—416no—
318C23×C3—416no—
319C23×C3—44no—
320C23×C3—44no—
321C23×C3—44no—
322C23×C3—44no—
323C23×C3—416no—
324C23×C3—416no—
325C23×C3—416no—
326C23×C3—416no—
327C23×C3—48no—
328C23×C3—48no—
329C23×C3—48no—
330C23×C3—48no—
331C23×C3—48no—
332C23×C3—48no—
333C23×C3—48no—
334C23×C3—48no—
335C23×C3—616no—
336C23×C3—68no—
337C23×C3—64no—
338C23×C3—62no—
339C23×C3—62no—
340C23×C3—64no—
341C23×C3—128no—
342C23×C3—124no—
343C23×C3—124no—
344C23×C3D3×D412yes—
345C23×C3D3×D414yes—
346C23×C3#23712yes—
347C23×C3C2×#6714yes—
348C23×C3C2×#6712yes—
349C23×C3C2×#6718yescomplemented
350C23×C3C2×C3×D412yes—
351C23×C3C2×C3×D414yes—
352C23×C3C2×C3×D414yes—
353C23×C3C2×C3×D412yes—
354C23×C3C2×C3×D418yescomplemented
355C23×C3C3×#4112yes—
356C23×C3C23×D311yescharacteristic
357C23×C3C24×C311yescharacteristic
358C22×D3—216no—
359C22×D3—28no—
360C22×D3—22no—
361C22×D3—24no—
362C22×D3—24no—
363C22×D3—22no—
364C22×D3—44no—
365C22×D3—48no—
366C22×D3—44no—
367C22×D3C2×C3×D414yescomplemented
368C3×D4—264no—
369C3×D4—232no—
370C3×D4—232no—
371C3×D4—664no—
372C2×C3×C4—216no—
373C2×C3×C4—28no—
374C2×C3×C4—28no—
375C2×C3×C4—28no—
376C2×C3×C4—24no—
377C2×C3×C4—24no—
378C2×C3×C4—24no—
379C2×C3×C4—38no—
380C2×C3×C4—616no—
381C2×C3×C4—64no—
382C2×C3×C4—64no—
383C2×C3×C4—64no—
384C2×C3×C4—62no—
385C2×C3×C4—124no—
386C2×C3×C4C2×#6714yescomplemented
387C2×C3×C4C2×C3×D414yescomplemented
388#67—264no—
389C2×Dic3—216no—
390C2×Dic3—24no—
391C2×Dic3—24no—
392C2×Dic3—22no—
393C2×Dic3—24no—
394C2×Dic3—44no—
395C2×Dic3C2×C3×D418yescomplemented
396C2×C32—48no—
397C2×C32#39218yescomplemented
398C2×C32C2×#12012yes—
399C2×C32C2×#12011yescharacteristic
400C2×C32C2×#12111yescharacteristic
401C2×C32C2×#12112yes—
402C2×C32C2×#12711yescharacteristic
403C3×D3—44no—
404C24—24no—
405C24—24no—
406C24—24no—
407C24—24no—
408C24—24no—
409C24—24no—
410C24—32no—
411C24—34no—
412C24—64no—
413C24—62no—
414C24—62no—
415C24—64no—
416C24C3×#6718yescomplemented
417C24C22×C3×D314yes—
418C24C22×C3×D311yescharacteristic
419C2×D4—232no—
420C2×D4—216no—
421C2×D4—28no—
422C2×D4—616no—
423C2×D4—616no—
424C2×D4—664no—
425C2×D4—616no—
426C22×C4—24no—
427C22×C4—34no—
428C22×C4—38no—
429C22×C4—32no—
430C22×C4—64no—
431C22×C4—64no—
432C22×C4—64no—
433C22×C4—62no—
434C22×C4C3×#6714yescomplemented
435C22×C4C22×C3×D312yes—
436C22⋊C4—616no—
437C22⋊C4—68no—
438C22×C3—216no—
439C22×C3—28no—
440C22×C3—28no—
441C22×C3—28no—
442C22×C3—28no—
443C22×C3—28no—
444C22×C3—28no—
445C22×C3—28no—
446C22×C3—24no—
447C22×C3—28no—
448C22×C3—24no—
449C22×C3—24no—
450C22×C3—21no—
451C22×C3—22no—
452C22×C3—21no—
453C22×C3—22no—
454C22×C3—21no—
455C22×C3—48no—
456C22×C3—48no—
457C22×C3—48no—
458C22×C3—48no—
459C22×C3—48no—
460C22×C3—48no—
461C22×C3—464no—
462C22×C3—432no—
463C22×C3—432no—
464C22×C3—44no—
465C22×C3—44no—
466C22×C3—44no—
467C22×C3—44no—
468C22×C3—44no—
469C22×C3—44no—
470C22×C3—44no—
471C22×C3—44no—
472C22×C3—68no—
473C22×C3—64no—
474C22×C3—1216no—
475C22×C3—124no—
476C22×C3—122no—
477C22×C3—122no—
478C22×C3#93718yes—
479C22×C3#93814yes—
480C22×C3#93918yes—
481C22×C3#94014yes—
482C22×C3#95314yes—
483C22×C3C3×#12018yes—
484C22×C3C3×#12014yes—
485C22×C3C3×#12114yes—
486C22×C3C3×#12118yes—
487C22×C3C3×#12714yes—
488C22×C3C2×D3×D411yescharacteristic
489C22×C3C2×D3×D412yes—
490C22×C3C2×#23711yescharacteristic
491C22×C3C22×#6712yes—
492C22×C3C22×#6711yescharacteristic
493C22×C3C22×C3×D411yescharacteristic
494C22×C3C22×C3×D412yes—
495C22×C3C22×C3×D412yes—
496C22×C3C22×C3×D411yescharacteristic
497C22×C3C2×C3×#4111yescharacteristic
498C2×D3—28no—
499C2×D3—24no—
500C2×D3—416no—
501C2×D3—42no—
502C2×D3—44no—
503C2×D3—42no—
504C3×C4—28no—
505C3×C4—24no—
506C3×C4—24no—
507C3×C4—68no—
508C3×C4—122no—
509Dic3—28no—
510Dic3—42no—
511C32C2×#39211yescharacteristic, complemented, Sylow
512C23—216no—
513C23—216no—
514C23—216no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—34no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23—616no—
527C23—68no—
528C23—64no—
529C23—62no—
530C23—62no—
531C23—64no—
532C23—128no—
533C23—124no—
534C23—124no—
535C23C3×D3×D412yes—
536C23C3×D3×D414yes—
537C23C3×#23712yes—
538C23C2×C3×#6714yes—
539C23C2×C3×#6712yes—
540C23C2×C3×#6718yescomplemented
541C23C23×C3×D311yescharacteristic
542D4—232no—
543D4—664no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—38no—
547C2×C4—616no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—62no—
552C2×C4—124no—
553C2×C4C2×C3×#6714yescomplemented
554C2×C3—22no—
555C2×C3—21no—
556C2×C3—22no—
557C2×C3—21no—
558C2×C3—28no—
559C2×C3—21no—
560C2×C3—416no—
561C2×C3—48no—
562C2×C3—48no—
563C2×C3—124no—
564C2×C3#522318yesdirect product factor
565C2×C3C3×#39218yescomplemented
566C2×C3C2×#93712yes—
567C2×C3C2×#93811yescharacteristic
568C2×C3C2×#93912yes—
569C2×C3C2×#94011yescharacteristic
570C2×C3C2×#95311yescharacteristic
571C2×C3C2×C3×#12012yes—
572C2×C3C2×C3×#12011yescharacteristic
573C2×C3C2×C3×#12111yescharacteristic
574C2×C3C2×C3×#12112yes—
575C2×C3C2×C3×#12711yescharacteristic
576D3—44no—
577C22—28no—
578C22—28no—
579C22—28no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—432no—
585C22—68no—
586C22—64no—
587C22—124no—
588C22—122no—
589C22—1216no—
590C22—122no—
591C22C3×#93718yes—
592C22C3×#93814yes—
593C22C3×#93918yes—
594C22C3×#94014yes—
595C22C3×#95314yes—
596C22C2×C3×D3×D411yescharacteristic
597C22C2×C3×D3×D412yes—
598C22C2×C3×#23711yescharacteristic
599C22C22×C3×#6712yes—
600C22C22×C3×#6711yescharacteristic
601C4—24no—
602C4—68no—
603C4—122no—
604C3—21no—
605C3C2×#522311yescharacteristic, direct product factor
606C3C2×C3×#39211yescharacteristic, complemented
607C2—48no—
608C2—124no—
609C2C3×#522318yesdirect product factor
610C2C2×C3×#93712yes—
611C2C2×C3×#93811yescharacteristic
612C2C2×C3×#93912yes—
613C2C2×C3×#94011yescharacteristic
614C2C2×C3×#95311yescharacteristic
subgroupG/H
centerC24×C3C22×D3
commutator, cosocleC23×C3C24×C3
FrattiniC23C23×C3×D3
FittingC23×C32×D4C2
socleC24×C32C23
hypercenterC23×C3×D4D3
hypocenterC3C2×C3×#392
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24×C3 ◁ C23×C3×D4

Complements

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

NH ≅ G/Ntype
1. C23×C32×D4C2semidirect
2. C23×C3×#67C2semidirect
3. C22×C3×#217C2semidirect
4. C22×C3×(C2×D3)⋊C4C2semidirect
5. C3×#5223C2direct (N × H)
6. C2×C3×C3⋊#95C2semidirect
8. C2×#5223C3direct (N × H)
13. C22×C32×D4C22semidirect
18. C22×C3×#67C22semidirect
22. C2×C3×#217C22semidirect
24. C2×C3×(C2×D3)⋊C4C22semidirect
25. C3×C3⋊#95C22semidirect
28. C23×C3×D4D3semidirect
29. C23×C3×D4C2×C3semidirect
30. C23×#67C2×C3semidirect
33. C22×#217C2×C3semidirect
34. C22×(C2×D3)⋊C4C2×C3semidirect
37. #5223C2×C3direct (N × H)
38. C2×C3⋊#95C2×C3semidirect
45. C24×C32D4semidirect
52. C23×C3×D3D4semidirect
58. C22×C32×C4D4semidirect
68. C22×C3×Dic3D4semidirect
90. C22×C3×D4C2×D3semidirect
91. C22×C3×D4C22×C3semidirect
100. C22×#67C22×C3semidirect
108. C2×#217C22×C3semidirect
110. C2×(C2×D3)⋊C4C22×C3semidirect
112. C3⋊#95C22×C3semidirect
130. C23×C32C2×D4semidirect
142. C22×C3×D3C2×D4semidirect
146. C2×C32×C4C2×D4semidirect
154. C2×C3×Dic3C2×D4semidirect
156. C23×D4C3×D3semidirect
188. C24×C3#67semidirect
189. C24×C3C3×D4semidirect
198. C23×D3C3×D4semidirect
225. C22×C3×C4#67semidirect
226. C22×C3×C4C3×D4semidirect
237. C22×Dic3C3×D4semidirect
279. C22×D4C2×C3×D3semidirect
349. C23×C3C2×#67semidirect
354. C23×C3C2×C3×D4semidirect
367. C22×D3C2×C3×D4semidirect
386. C2×C3×C4C2×#67semidirect
387. C2×C3×C4C2×C3×D4semidirect
395. C2×Dic3C2×C3×D4semidirect
397. C2×C32#392semidirect
416. C24C3×#67semidirect
434. C22×C4C3×#67semidirect
511. C32C2×#392semidirect
540. C23C2×C3×#67semidirect
553. C2×C4C2×C3×#67semidirect
564. C2×C3#5223direct (N × H)
565. C2×C3C3×#392semidirect
605. C3C2×#5223direct (N × H)
606. C3C2×C3×#392semidirect
609. C2C3×#5223direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×D42
C23×C3×#672
C22×C3×#2172
C22×C3×(C2×D3)⋊C42
C3×#52232
C2×C3×C3⋊#952
C2×C3×#3923
C2×#52233