Finite Group Databasebeta

← C2×C3×#63962 #13175077

Subgroups — C2×C3×#63962 #13175077

Order 1728.

2106 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C22×C32×#67C211yesmaximal, characteristic
3C2×C32×#217C211yesmaximal, characteristic, complemented
4C2×C3×Dic3⋊Dic3C212yesmaximal, complemented
5C3×#63962C218yesmaximal, direct product factor
6C2×C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
7C2×C3×#935—31nomaximal
8C2×C3×#886—31nomaximal
9C2×#63962C311yesmaximal, characteristic, direct product factor
10C24×C33C2211yescharacteristic
11C23×C32×D3C2211yescharacteristic
12C22×C3×C3⋊Dic3—24no—
13C22×C3×C3⋊Dic3C2212yes—
14C22×C3×C3⋊Dic3C2214yescomplemented
15C2×C32×#67—24no—
16C2×C32×#67C2214yes—
17C22×C32×Dic3C2212yes—
18C22×C32×Dic3C2211yescharacteristic
19C32×#217C2214yescomplemented
20C3×Dic3⋊Dic3C2218yescomplemented
21C3×(C2×D3)⋊Dic3C2218yescomplemented
22C22×C32×D4—31no—
23C23×C3⋊Dic3C2×C311yescharacteristic, complemented
24C22×C3×#67—21no—
25C22×C3×#67D311yescharacteristic
26C22×C3×#67C2×C311yescharacteristic
27C23×C3×Dic3—31no—
28C23×C3×Dic3—31no—
29C23×C3×Dic3—61no—
30C2×C32×C22⋊C4—31no—
31C2×C3×#217—21no—
32C2×C3×#217—32no—
33C2×C3×#217D311yescharacteristic, complemented
34C2×C3×#217C2×C311yescharacteristic, complemented
35C3×#935—38no—
36C2×C3×(C2×D3)⋊C4—32no—
37C2×C3×C4⋊Dic3—32no—
38C2×C3×Dic3⋊C4—32no—
39C3×#886—38no—
40C2×Dic3⋊Dic3C2×C312yescomplemented
41#63962C2×C318yesdirect product factor
42C2×(C2×D3)⋊Dic3C2×C312yescomplemented
43C23×C33—24no—
44C23×C33D414yes—
45C23×C33C2312yes—
46C23×C33C2311yescharacteristic
47C22×C32×D3—22no—
48C22×C32×D3—24no—
49C22×C32×D3C2312yes—
50C2×C3×C3⋊Dic3—24no—
51C2×C3×C3⋊Dic3—28no—
52C2×C3×C3⋊Dic3C2314yes—
53C32×#67—28no—
54C2×C32×Dic3—22no—
55C2×C32×Dic3—24no—
56C2×C32×Dic3C2314yes—
57C2×C32×Dic3C2312yes—
58C2×C3×#123—91no—
59C2×#935—31no—
60C2×#886—31no—
61C24×C32—21no—
62C24×C32—21no—
63C24×C32—21no—
64C24×C32—31no—
65C24×C32—41no—
66C24×C32C2×D311yescharacteristic
67C24×C32C2×D311yescharacteristic
68C24×C32C22×C311yescharacteristic
69C23×C3×D3—21no—
70C23×C3×D3C2×D311yescharacteristic
71C23×C3×D3C22×C311yescharacteristic
72C2×C32×D4—34no—
73C2×C32×D4—64no—
74C22×C32×C4—31no—
75C22×C32×C4—32no—
76C22×C3⋊Dic3—24no—
77C22×C3⋊Dic3C22×C312yes—
78C22×C3⋊Dic3C22×C314yescomplemented
79C2×C3×#67—28no—
80C2×C3×#67—24no—
81C2×C3×#67—24no—
82C2×C3×#67—24no—
83C2×C3×#67C2×D314yes—
84C2×C3×#67C22×C314yes—
85C22×C3×Dic3—21no—
86C22×C3×Dic3—22no—
87C22×C3×Dic3—34no—
88C22×C3×Dic3—32no—
89C22×C3×Dic3—32no—
90C22×C3×Dic3—34no—
91C22×C3×Dic3—62no—
92C22×C3×Dic3—64no—
93C22×C3×Dic3—68no—
94C22×C3×Dic3—64no—
95C22×C3×Dic3—64no—
96C22×C3×Dic3C2×D312yes—
97C22×C3×Dic3C2×D311yescharacteristic
98C22×C3×Dic3C22×C312yes—
99C22×C3×Dic3C22×C311yescharacteristic
100C32×C22⋊C4—34no—
101C3×#217—24no—
102C3×#217—38no—
103C3×#217C2×D314yescomplemented
104C3×#217C22×C314yescomplemented
105C3×(C2×D3)⋊C4—38no—
106C3×C4⋊Dic3—38no—
107C3×Dic3⋊C4—38no—
108Dic3⋊Dic3C22×C318yescomplemented
109(C2×D3)⋊Dic3C22×C318yescomplemented
110C22×C33—28no—
111C22×C33—24no—
112C22×C33C2×D414yes—
113C22×C33C2×D412yes—
114C22×C33#4114yes—
115C22×C33C2411yescharacteristic
116C2×C32×D3—22no—
117C2×C32×D3—24no—
118C2×C32×D3—44no—
119C3×C3⋊Dic3—24no—
120C32×Dic3—22no—
121C32×Dic3—24no—
122C22×C3×D4—31no—
123C22×C3×D4—31no—
124C22×C3×D4—61no—
125C23×C3×C4—91no—
126C22×#67C3×D311yescharacteristic
127C23×Dic3—31no—
128C23×Dic3—31no—
129C23×Dic3—61no—
130C3×#123—98no—
131C2×C3×C4⋊C4—92no—
132C2×C3×C22⋊C4—31no—
133C2×C3×C22⋊C4—31no—
134C2×C3×C22⋊C4—61no—
135C2×C3×C22⋊C4—92no—
136C2×#217—32no—
137C2×#217C3×D311yescharacteristic, complemented
138#935—38no—
139C2×(C2×D3)⋊C4—32no—
140C2×C4⋊Dic3—32no—
141C2×Dic3⋊C4—32no—
142#886—38no—
143C23×C32—24no—
144C23×C32—28no—
145C23×C32—24no—
146C23×C32—24no—
147C23×C32—24no—
148C23×C32—24no—
149C23×C32—24no—
150C23×C32—22no—
151C23×C32—21no—
152C23×C32—21no—
153C23×C32—21no—
154C23×C32—22no—
155C23×C32—22no—
156C23×C32—32no—
157C23×C32—44no—
158C23×C32—41no—
159C23×C32—42no—
160C23×C32—44no—
161C23×C32—44no—
162C23×C32—48no—
163C23×C32—62no—
164C23×C32—64no—
165C23×C32D1214yes—
166C23×C32#6714yes—
167C23×C32C3×D414yes—
168C23×C32C22×D312yes—
169C23×C32C22×D312yes—
170C23×C32C22×D311yescharacteristic
171C23×C32C22×D311yescharacteristic
172C23×C32C23×C312yes—
173C23×C32C23×C311yescharacteristic
174C22×C3×D3—24no—
175C22×C3×D3—24no—
176C22×C3×D3—22no—
177C22×C3×D3—22no—
178C22×C3×D3—24no—
179C22×C3×D3—22no—
180C22×C3×D3—44no—
181C22×C3×D3C22×D312yes—
182C22×C3×D3C23×C312yes—
183C32×D4—68no—
184C2×C32×C4—32no—
185C2×C32×C4—34no—
186C2×C32×C4—62no—
187C2×C32×C4—64no—
188C2×C3⋊Dic3—24no—
189C2×C3⋊Dic3—28no—
190C2×C3⋊Dic3C23×C314yes—
191C3×#67—216no—
192C3×#67—28no—
193C3×#67—28no—
194C2×C3×Dic3—24no—
195C2×C3×Dic3—24no—
196C2×C3×Dic3—22no—
197C2×C3×Dic3—22no—
198C2×C3×Dic3—24no—
199C2×C3×Dic3—22no—
200C2×C3×Dic3—24no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3—44no—
204C2×C3×Dic3—64no—
205C2×C3×Dic3—68no—
206C2×C3×Dic3—616no—
207C2×C3×Dic3—68no—
208C2×C3×Dic3—64no—
209C2×C3×Dic3—68no—
210C2×C3×Dic3—64no—
211C2×C3×Dic3C22×D314yes—
212C2×C3×Dic3C22×D312yes—
213C2×C3×Dic3C23×C314yes—
214C2×C3×Dic3C23×C312yes—
215C2×#123—91noSylow
216C2×C33—24no—
217C2×C33#12314yescomplemented
218C2×C33C22×D411yescharacteristic
219C2×C33C2×#4112yes—
220C32×D3—42no—
221C24×C3—21no—
222C24×C3—21no—
223C24×C3—21no—
224C24×C3—31no—
225C24×C3—31no—
226C24×C3—41no—
227C24×C3—61no—
228C24×C3D3211yescharacteristic
229C24×C3C2×C3×D311yescharacteristic
230C24×C3C2×C3×D311yescharacteristic
231C23×D3C2×C3×D311yescharacteristic
232C2×C3×D4—34no—
233C2×C3×D4—34no—
234C2×C3×D4—64no—
235C2×C3×D4—68no—
236C2×C3×D4—64no—
237C2×C3×D4—64no—
238C22×C3×C4—32no—
239C22×C3×C4—31no—
240C22×C3×C4—31no—
241C22×C3×C4—32no—
242C22×C3×C4—61no—
243C22×C3×C4—62no—
244C22×C3×C4—94no—
245C22×C3×C4—92no—
246C22×C3×C4—184no—
247C2×#67—24no—
248C2×#67C2×C3×D314yes—
249C22×Dic3—34no—
250C22×Dic3—32no—
251C22×Dic3—32no—
252C22×Dic3—34no—
253C22×Dic3—62no—
254C22×Dic3—64no—
255C22×Dic3—68no—
256C22×Dic3—64no—
257C22×Dic3—64no—
258C22×Dic3C2×C3×D312yes—
259C22×Dic3C2×C3×D311yescharacteristic
260C3×C4⋊C4—98no—
261C3×C22⋊C4—34no—
262C3×C22⋊C4—34no—
263C3×C22⋊C4—64no—
264C3×C22⋊C4—98no—
265#217—38no—
266#217C2×C3×D314yescomplemented
267(C2×D3)⋊C4—38no—
268C4⋊Dic3—38no—
269Dic3⋊C4—38no—
270C22×C32—28no—
271C22×C32—24no—
272C22×C32—28no—
273C22×C32—216no—
274C22×C32—28no—
275C22×C32—28no—
276C22×C32—24no—
277C22×C32—24no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—24no—
282C22×C32—24no—
283C22×C32—22no—
284C22×C32—22no—
285C22×C32—24no—
286C22×C32—24no—
287C22×C32—24no—
288C22×C32—22no—
289C22×C32—24no—
290C22×C32—42no—
291C22×C32—44no—
292C22×C32—44no—
293C22×C32—41no—
294C22×C32—48no—
295C22×C32—48no—
296C22×C32—44no—
297C22×C32—44no—
298C22×C32—416no—
299C22×C32—48no—
300C22×C32—62no—
301C22×C32—64no—
302C22×C32—124no—
303C22×C32C2×D1214yes—
304C22×C32C2×D1212yes—
305C22×C32#23714yes—
306C22×C32#23714yes—
307C22×C32C2×#6714yes—
308C22×C32C2×#6712yes—
309C22×C32C2×C3×D414yes—
310C22×C32C2×C3×D412yes—
311C22×C32C3×#4114yes—
312C22×C32C23×D311yescharacteristic
313C22×C32C23×D311yescharacteristic
314C22×C32C24×C311yescharacteristic
315C2×C3×D3—24no—
316C2×C3×D3—24no—
317C2×C3×D3—22no—
318C2×C3×D3—22no—
319C2×C3×D3—24no—
320C2×C3×D3—44no—
321C2×C3×D3—48no—
322C2×C3×D3—44no—
323C2×C3×D3—44no—
324C32×C4—62no—
325C32×C4—64no—
326C3⋊Dic3—24no—
327C3×Dic3—24no—
328C3×Dic3—24no—
329C3×Dic3—22no—
330C3×Dic3—22no—
331C3×Dic3—24no—
332C3×Dic3—44no—
333C3×Dic3—68no—
334C3×Dic3—64no—
335C3×Dic3—64no—
336C22×D4—31no—
337C23×C4—91no—
338#123—98no—
339C2×C4⋊C4—92no—
340C2×C22⋊C4—31no—
341C2×C22⋊C4—92no—
342C33C2×#12311yescharacteristic, complemented, Sylow
343C23×C3—28no—
344C23×C3—24no—
345C23×C3—24no—
346C23×C3—24no—
347C23×C3—24no—
348C23×C3—24no—
349C23×C3—24no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—22no—
353C23×C3—21no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—32no—
357C23×C3—32no—
358C23×C3—44no—
359C23×C3—44no—
360C23×C3—48no—
361C23×C3—44no—
362C23×C3—42no—
363C23×C3—41no—
364C23×C3—62no—
365C23×C3—64no—
366C23×C3—64no—
367C23×C3—62no—
368C23×C3—62no—
369C23×C3—64no—
370C23×C3—124no—
371C23×C3#62614yes—
372C23×C3C3×D1214yes—
373C23×C3C3×#6714yes—
374C23×C3C2×D3212yes—
375C23×C3C2×D3211yescharacteristic
376C23×C3C22×C3×D312yes—
377C23×C3C22×C3×D312yes—
378C23×C3C22×C3×D311yescharacteristic
379C23×C3C22×C3×D311yescharacteristic
380C22×D3—24no—
381C22×D3—22no—
382C22×D3C22×C3×D312yes—
383C3×D4—616no—
384C3×D4—68no—
385C3×D4—68no—
386C2×C3×C4—34no—
387C2×C3×C4—32no—
388C2×C3×C4—32no—
389C2×C3×C4—34no—
390C2×C3×C4—62no—
391C2×C3×C4—64no—
392C2×C3×C4—64no—
393C2×C3×C4—64no—
394C2×C3×C4—62no—
395C2×C3×C4—62no—
396C2×C3×C4—64no—
397C2×C3×C4—94no—
398C2×C3×C4—124no—
399C2×C3×C4—188no—
400C2×C3×C4—184no—
401#67—28no—
402C2×Dic3—24no—
403C2×Dic3—22no—
404C2×Dic3—34no—
405C2×Dic3—34no—
406C2×Dic3—64no—
407C2×Dic3—68no—
408C2×Dic3—616no—
409C2×Dic3—68no—
410C2×Dic3—64no—
411C2×Dic3—68no—
412C2×Dic3—64no—
413C2×Dic3C22×C3×D314yes—
414C2×Dic3C22×C3×D312yes—
415C2×C32—24no—
416C2×C32—28no—
417C2×C32—24no—
418C2×C32—24no—
419C2×C32—24no—
420C2×C32—24no—
421C2×C32—22no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—22no—
425C2×C32—22no—
426C2×C32—21no—
427C2×C32—24no—
428C2×C32—44no—
429C2×C32—41no—
430C2×C32—42no—
431C2×C32—44no—
432C2×C32—44no—
433C2×C32—48no—
434C2×C32—122no—
435C2×C32#88614yescomplemented
436C2×C32#93514yescomplemented
437C2×C32C3×#12314yescomplemented
438C2×C32C22×D1211yescharacteristic
439C2×C32C2×#23712yes—
440C2×C32C2×#23712yes—
441C2×C32C22×#6711yescharacteristic
442C2×C32C22×C3×D411yescharacteristic
443C2×C32C2×C3×#4112yes—
444C3×D3—44no—
445C3×D3—42no—
446C3×D3—42no—
447C24—31no—
448C24C3×D3211yescharacteristic
449C2×D4—34no—
450C2×D4—64no—
451C22×C4—32no—
452C22×C4—31no—
453C22×C4—92no—
454C22×C4—94no—
455C22×C4—184no—
456C4⋊C4—98no—
457C22⋊C4—34no—
458C22⋊C4—98no—
459C22×C3—28no—
460C22×C3—216no—
461C22×C3—28no—
462C22×C3—28no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—28no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—24no—
473C22×C3—24no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—24no—
478C22×C3—24no—
479C22×C3—44no—
480C22×C3—44no—
481C22×C3—48no—
482C22×C3—48no—
483C22×C3—48no—
484C22×C3—416no—
485C22×C3—41no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—42no—
489C22×C3—64no—
490C22×C3—64no—
491C22×C3—62no—
492C22×C3—62no—
493C22×C3—64no—
494C22×C3—124no—
495C22×C3—128no—
496C22×C3—124no—
497C22×C3—124no—
498C22×C3#380514yes—
499C22×C3C2×#62614yes—
500C22×C3C2×#62612yes—
501C22×C3C2×C3×D1214yes—
502C22×C3C2×C3×D1212yes—
503C22×C3C3×#23714yes—
504C22×C3C3×#23714yes—
505C22×C3C2×C3×#6714yes—
506C22×C3C2×C3×#6712yes—
507C22×C3C22×D3211yescharacteristic
508C22×C3C23×C3×D311yescharacteristic
509C22×C3C23×C3×D311yescharacteristic
510C2×D3—24no—
511C2×D3—22no—
512C2×D3—44no—
513C3×C4—64no—
514C3×C4—64no—
515C3×C4—62no—
516C3×C4—62no—
517C3×C4—64no—
518C3×C4—124no—
519C3×C4—184no—
520Dic3—24no—
521Dic3—22no—
522Dic3—68no—
523Dic3—64no—
524Dic3—64no—
525C32—21no—
526C32—21no—
527C32—21no—
528C32—41no—
529C32C2×#88611yescharacteristic, complemented
530C32C2×#93511yescharacteristic, complemented
531C32C2×C3×#12311yescharacteristic, complemented
532C23—24no—
533C23—32no—
534C23—64no—
535C23—62no—
536C23C3×#62614yes—
537C23C2×C3×D3212yes—
538C23C2×C3×D3211yescharacteristic
539D4—68no—
540C2×C4—34no—
541C2×C4—32no—
542C2×C4—64no—
543C2×C4—62no—
544C2×C4—94no—
545C2×C4—188no—
546C2×C4—184no—
547C2×C3—28no—
548C2×C3—24no—
549C2×C3—24no—
550C2×C3—24no—
551C2×C3—22no—
552C2×C3—22no—
553C2×C3—22no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—24no—
558C2×C3—24no—
559C2×C3—24no—
560C2×C3—44no—
561C2×C3—44no—
562C2×C3—48no—
563C2×C3—42no—
564C2×C3—41no—
565C2×C3—44no—
566C2×C3—124no—
567C2×C3—122no—
568C2×C3—122no—
569C2×C3#6396214yesdirect product factor
570C2×C3C3×#88614yescomplemented
571C2×C3C3×#93514yescomplemented
572C2×C3C2×#380512yes—
573C2×C3C22×#62611yescharacteristic
574C2×C3C22×C3×D1211yescharacteristic
575C2×C3C2×C3×#23712yes—
576C2×C3C2×C3×#23712yes—
577C2×C3C22×C3×#6711yescharacteristic
578D3—42no—
579C22—24no—
580C22—28no—
581C22—64no—
582C22—62no—
583C22—124no—
584C22C3×#380514yes—
585C22C2×C3×#62614yes—
586C22C2×C3×#62612yes—
587C22C22×C3×D3211yescharacteristic
588C4—64no—
589C4—62no—
590C4—184no—
591C3—21no—
592C3—21no—
593C3—21no—
594C3—41no—
595C3C2×#6396211yescharacteristic, direct product factor
596C3C2×C3×#88611yescharacteristic, complemented
597C3C2×C3×#93511yescharacteristic, complemented
598C2—24no—
599C2—122no—
600C2C3×#6396214yesdirect product factor
601C2C2×C3×#380512yes—
602C2C22×C3×#62611yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC24×C33C22
socleC23×C33C23
prime radicalC22×C32×#67C2
hypercenterC24×C3D32
hypocenterC32C2×C3×#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×C3⋊Dic3C2semidirect
3. C2×C32×#217C2semidirect
4. C2×C3×Dic3⋊Dic3C2semidirect
5. C3×#63962C2direct (N × H)
6. C2×C3×(C2×D3)⋊Dic3C2semidirect
9. C2×#63962C3direct (N × H)
14. C22×C3×C3⋊Dic3C22semidirect
19. C32×#217C22semidirect
20. C3×Dic3⋊Dic3C22semidirect
21. C3×(C2×D3)⋊Dic3C22semidirect
23. C23×C3⋊Dic3C2×C3semidirect
33. C2×C3×#217D3semidirect
34. C2×C3×#217C2×C3semidirect
40. C2×Dic3⋊Dic3C2×C3semidirect
41. #63962C2×C3direct (N × H)
42. C2×(C2×D3)⋊Dic3C2×C3semidirect
78. C22×C3⋊Dic3C22×C3semidirect
103. C3×#217C2×D3semidirect
104. C3×#217C22×C3semidirect
108. Dic3⋊Dic3C22×C3semidirect
109. (C2×D3)⋊Dic3C22×C3semidirect
137. C2×#217C3×D3semidirect
217. C2×C33#123semidirect
266. #217C2×C3×D3semidirect
342. C33C2×#123semidirect
435. C2×C32#886semidirect
436. C2×C32#935semidirect
437. C2×C32C3×#123semidirect
529. C32C2×#886semidirect
530. C32C2×#935semidirect
531. C32C2×C3×#123semidirect
569. C2×C3#63962direct (N × H)
570. C2×C3C3×#886semidirect
571. C2×C3C3×#935semidirect
595. C3C2×#63962direct (N × H)
596. C3C2×C3×#886semidirect
597. C3C2×C3×#935semidirect
600. C2C3×#63962direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C22×C32×#672
C2×C32×#2172
C2×C3×Dic3⋊Dic32
C3×#639622
C2×C3×(C2×D3)⋊Dic32
C2×C3×#9353
C2×C3×#8863
C2×#639623