Finite Group Databasebeta

← C3×#32913 #10651520

Subgroups — C3×#32913 #10651520

Order 768.

1506 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#520C211yesmaximal, characteristic, complemented
2C2×C3×#460C211yesmaximal, characteristic, complemented
3C2×C3×#459C211yesmaximal, characteristic, complemented
4C2×C3×#449C212yesmaximal, complemented
5C3×#2619C211yesmaximal, characteristic, complemented
6C3×#1880C212yesmaximal, complemented
7C3×#1879C212yesmaximal, complemented
8C3×#1874C212yesmaximal, complemented
9C3×#1805C211yesmaximal, characteristic, complemented
10C3×#1804C211yesmaximal, characteristic, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#32913C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C22×C3×D8C2212yes—
16C2×C3×#127C2211yescharacteristic
17C2×C3×#121C2212yes—
18C3×#520—24no—
19C3×#520—24no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×#117—21no—
27C2×C3×#117C2211yescharacteristic
28C22×C3×C4⋊C4—21no—
29C22×C3×C4⋊C4C2211yescharacteristic
30C3×#460—24no—
31C3×#459—24no—
32C3×#449—28no—
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×C4⋊C8C2211yescharacteristic
35C2×C3×#103C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×#102C2212yes—
39C2×C3×C22⋊C8C2212yes—
40C3×#394—21no—
41C3×#394—21no—
42C3×#392—21no—
43C3×#392—21no—
44C3×#390C2212yes—
45C3×#382C2211yescharacteristic
46C3×#340C2212yes—
47C3×#340C2212yes—
48C3×#336C2212yes—
49C2×#520C2×C311yescharacteristic, complemented
50C2×#460C2×C311yescharacteristic, complemented
51C2×#459C2×C311yescharacteristic, complemented
52C2×#449C2×C312yescomplemented
53#2619C2×C311yescharacteristic, complemented
54#1880C2×C312yescomplemented
55#1879C2×C312yescomplemented
56#1874C2×C312yescomplemented
57#1805C2×C311yescharacteristic, complemented
58#1804C2×C311yescharacteristic, complemented
59#1798C2×C311yescharacteristic, complemented
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41D414yescomplemented
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—21no—
68C22×C3×D4—22no—
69C22×C3×D4—21no—
70C22×C3×D4—21no—
71C22×C3×D4C2312yes—
72C22×C3×D4C2311yescharacteristic
73C23×C3×C4—21no—
74C23×C3×C4—21no—
75C23×C3×C4—21no—
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C2×C3×D8—24no—
82C2×C3×D8—24no—
83C2×C3×D8—24no—
84C22×C3×C8C2312yes—
85C22×C3×C8C2312yes—
86C3×#127—22no—
87C3×#127—22no—
88C3×#121—24no—
89C3×#121—24no—
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C3×C4×Q8D414yescomplemented
94C3×C4×D4—28no—
95C3×C4×D4—28no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×C4×D4D414yescomplemented
100C3×#117—28no—
101C3×#117—22no—
102C3×#117—22no—
103C2×C3×C4⋊C4—28no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4—22no—
108C2×C3×C4⋊C4—22no—
109C2×C3×C4⋊C4D411yescharacteristic
110C2×C3×C4⋊C4D411yescharacteristic
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4—22no—
117C2×C3×C22⋊C4—22no—
118C2×C3×C22⋊C4—22no—
119C2×C3×C22⋊C4D411yescharacteristic
120C2×C3×C22⋊C4D411yescharacteristic
121C2×C3×C42—21no—
122C2×C3×C42—21no—
123C2×C3×C42C2311yescharacteristic
124C2×C3×C42C2311yescharacteristic
125C3×C4⋊C8—22no—
126C3×C4⋊C8—22no—
127C3×#103—24no—
128C3×#102—24no—
129C3×#102—24no—
130C3×#102—24no—
131C3×C22⋊C8—24no—
132C3×#95—21no—
133C3×#95—21no—
134C22×#41C22×C311yescharacteristic
135C22×SD16C22×C312yes—
136C22×D8C22×C312yes—
137C2×#127C22×C311yescharacteristic
138C2×#121C22×C312yes—
139#520—24no—
140#520—24no—
141C2×C4×Q8C22×C311yescharacteristic
142C2×C4×Q8C22×C311yescharacteristic
143C2×C4×D4—21no—
144C2×C4×D4—21no—
145C2×C4×D4C22×C311yescharacteristic
146C2×C4×D4C22×C311yescharacteristic
147C2×#117—21no—
148C2×#117C22×C311yescharacteristic
149C22×C4⋊C4—21no—
150C22×C4⋊C4C22×C311yescharacteristic
151#460—24no—
152#459—24no—
153#449—28no—
154C2×C4⋊C8C22×C311yescharacteristic
155C2×C4⋊C8C22×C311yescharacteristic
156C2×#103C22×C312yes—
157C2×#102C22×C312yes—
158C2×#102C22×C312yes—
159C2×#102C22×C312yes—
160C2×C22⋊C8C22×C312yes—
161#394—21no—
162#394—21no—
163#392—21no—
164#392—21no—
165#390C22×C312yes—
166#382C22×C311yescharacteristic
167#340C22×C312yes—
168#340C22×C312yes—
169#336C22×C312yes—
170C24×C3—21no—
171C24×C3—22no—
172C24×C3C2×D411yescharacteristic
173C3×#41—28no—
174C3×#41—28no—
175C3×#41—28no—
176C3×#41—28no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8—21no—
180C2×C3×Q8C2×D412yes—
181C2×C3×Q8C2×D412yes—
182C2×C3×Q8#4112yes—
183C2×C3×D4—24no—
184C2×C3×D4—24no—
185C2×C3×D4—24no—
186C2×C3×D4—21no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—44no—
196C2×C3×D4—42no—
197C2×C3×D4—42no—
198C2×C3×D4—82no—
199C2×C3×D4—82no—
200C2×C3×D4—82no—
201C2×C3×D4—82no—
202C2×C3×D4C2×D412yes—
203C2×C3×D4C2×D412yes—
204C2×C3×D4#4112yes—
205C22×C3×C4—24no—
206C22×C3×C4—24no—
207C22×C3×C4—21no—
208C22×C3×C4—24no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—24no—
216C22×C3×C4—24no—
217C22×C3×C4—24no—
218C22×C3×C4—22no—
219C22×C3×C4—82no—
220C22×C3×C4—82no—
221C22×C3×C4—82no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D412yes—
228C22×C3×C4#4112yes—
229C22×C3×C4C2411yescharacteristic
230C3×SD16—48no—
231C3×D8—48no—
232C2×C3×C8—22no—
233C2×C3×C8—22no—
234C2×C3×C8—22no—
235C2×C3×C8—22no—
236C2×C3×C8—22no—
237C2×C3×C8—22no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4C2×D412yes—
248C3×C4⋊C4C2×D412yes—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—24no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—44no—
254C3×C22⋊C4—82no—
255C3×C22⋊C4—82no—
256C3×C42—24no—
257C3×C42—24no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42C2×D412yes—
261C3×C42C2×D412yes—
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41—22no—
267C2×#41C3×D414yescomplemented
268C22×Q8C23×C311yescharacteristic
269C22×D4—21no—
270C22×D4—22no—
271C22×D4—21no—
272C22×D4—21no—
273C22×D4C23×C312yes—
274C22×D4C23×C311yescharacteristic
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4C23×C311yescharacteristic
280C2×SD16—24no—
281C2×SD16—24no—
282C2×SD16—24no—
283C2×D8—24no—
284C2×D8—24no—
285C2×D8—24no—
286C22×C8C23×C312yes—
287C22×C8C23×C312yes—
288#127—22no—
289#127—22no—
290#121—24no—
291#121—24no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8C3×D414yescomplemented
296C4×D4—28no—
297C4×D4—28no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4C3×D414yescomplemented
302#117—28no—
303#117—22no—
304#117—22no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4C3×D411yescharacteristic
312C2×C4⋊C4C3×D411yescharacteristic
313C2×C4⋊C4C23×C312yes—
314C2×C4⋊C4C23×C311yescharacteristic
315C2×C4⋊C4C23×C311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4C3×D411yescharacteristic
322C2×C22⋊C4C3×D411yescharacteristic
323C2×C42—21no—
324C2×C42—21no—
325C2×C42C23×C311yescharacteristic
326C2×C42C23×C311yescharacteristic
327C4⋊C8—22no—
328C4⋊C8—22no—
329#103—24no—
330#102—24no—
331#102—24no—
332#102—24no—
333C22⋊C8—24no—
334#95—21no—
335#95—21no—
336C23×C3—21no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—21no—
340C23×C3—22no—
341C23×C3—22no—
342C23×C3—22no—
343C23×C3—42no—
344C23×C3—42no—
345C23×C3—42no—
346C23×C3—42no—
347C23×C3—42no—
348C23×C3—42no—
349C23×C3—42no—
350C23×C3—42no—
351C23×C3—42no—
352C23×C3#12112yes—
353C23×C3C22×D411yescharacteristic
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—22no—
357C3×Q8#12114yes—
358C3×D4—216no—
359C3×D4—22no—
360C3×D4—22no—
361C3×D4—22no—
362C3×D4—42no—
363C3×D4—42no—
364C3×D4—42no—
365C3×D4—42no—
366C3×D4—84no—
367C3×D4—84no—
368C3×D4—88no—
369C3×D4#12114yes—
370C2×C3×C4—216no—
371C2×C3×C4—22no—
372C2×C3×C4—21no—
373C2×C3×C4—22no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—42no—
389C2×C3×C4—42no—
390C2×C3×C4—42no—
391C2×C3×C4—42no—
392C2×C3×C4—42no—
393C2×C3×C4—84no—
394C2×C3×C4—84no—
395C2×C3×C4—84no—
396C2×C3×C4#12012yes—
397C2×C3×C4#12012yes—
398C2×C3×C4#12112yes—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C3×C8—42no—
406C3×C8—42no—
407C24—21no—
408C24—22no—
409C24C2×C3×D411yescharacteristic
410#41—28no—
411#41—28no—
412#41—28no—
413#41—28no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8C2×C3×D412yes—
418C2×Q8C2×C3×D412yes—
419C2×Q8C3×#4112yes—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4C2×C3×D412yes—
440C2×D4C2×C3×D412yes—
441C2×D4C3×#4112yes—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4C2×C3×D411yescharacteristic
460C22×C4C2×C3×D411yescharacteristic
461C22×C4C2×C3×D411yescharacteristic
462C22×C4C2×C3×D411yescharacteristic
463C22×C4C2×C3×D411yescharacteristic
464C22×C4C2×C3×D412yes—
465C22×C4C3×#4112yes—
466C22×C4C24×C311yescharacteristic
467SD16—48no—
468D8—48no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4C2×C3×D412yes—
485C4⋊C4C2×C3×D412yes—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—44no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C42—24no—
494C42—24no—
495C42—21no—
496C42—21no—
497C42C2×C3×D412yes—
498C42C2×C3×D412yes—
499C22×C3—22no—
500C22×C3—24no—
501C22×C3—24no—
502C22×C3—42no—
503C22×C3—44no—
504C22×C3—41no—
505C22×C3—42no—
506C22×C3—41no—
507C22×C3—42no—
508C22×C3—44no—
509C22×C3—82no—
510C22×C3—82no—
511C22×C3—82no—
512C22×C3—82no—
513C22×C3—82no—
514C22×C3—82no—
515C22×C3—82no—
516C22×C3C2×#12011yescharacteristic
517C22×C3C2×#12111yescharacteristic
518C22×C3#52511yescharacteristic
519C22×C3#57612yes—
520C22×C3#57712yes—
521C3×C4—22no—
522C3×C4—24no—
523C3×C4—22no—
524C3×C4—22no—
525C3×C4—41no—
526C3×C4—42no—
527C3×C4—42no—
528C3×C4—41no—
529C3×C4—82no—
530C3×C4C2×#12111yescharacteristic
531C3×C4C2×#12111yescharacteristic
532C3×C4#52511yescharacteristic
533C3×C4#52511yescharacteristic
534C23—22no—
535C23—22no—
536C23—21no—
537C23—22no—
538C23—22no—
539C23—21no—
540C23—22no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23C3×#12112yes—
551C23C22×C3×D411yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—22no—
555Q8C3×#12114yes—
556D4—216no—
557D4—22no—
558D4—22no—
559D4—22no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—84no—
565D4—84no—
566D4—88no—
567D4C3×#12114yes—
568C2×C4—216no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4C3×#12012yes—
595C2×C4C3×#12012yes—
596C2×C4C3×#12112yes—
597C2×C4C22×C3×D411yescharacteristic
598C2×C4C22×C3×D411yescharacteristic
599C2×C4C22×C3×D411yescharacteristic
600C2×C4C2×C3×#4111yescharacteristic
601C2×C4C2×C3×#4111yescharacteristic
602C2×C4C2×C3×#4111yescharacteristic
603C8—42no—
604C8—42no—
605C2×C3—42no—
606C2×C3—44no—
607C2×C3—162no—
608C2×C3#284711yescharacteristic
609C2×C3#300811yescharacteristic
610C2×C3#301011yescharacteristic
611C2×C3#304512yes—
612C2×C3#311912yes—
613C22—24no—
614C22—22no—
615C22—24no—
616C22—41no—
617C22—42no—
618C22—41no—
619C22—42no—
620C22—44no—
621C22—42no—
622C22—44no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22C2×C3×#12011yescharacteristic
631C22C2×C3×#12111yescharacteristic
632C22C3×#52511yescharacteristic
633C22C3×#57612yes—
634C22C3×#57712yes—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—24no—
639C4—41no—
640C4—42no—
641C4—42no—
642C4—41no—
643C4—82no—
644C4C2×C3×#12111yescharacteristic
645C4C2×C3×#12111yescharacteristic
646C4C3×#52511yescharacteristic
647C4C3×#52511yescharacteristic
648C3#3291311yescharacteristic, direct product factor, Sylow
649C2—42no—
650C2—44no—
651C2—162no—
652C2C3×#284711yescharacteristic
653C2C3×#300811yescharacteristic
654C2C3×#301011yescharacteristic
655C2C3×#304512yes—
656C2C3×#311912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#520C2semidirect
2. C2×C3×#460C2semidirect
3. C2×C3×#459C2semidirect
4. C2×C3×#449C2semidirect
5. C3×#2619C2semidirect
6. C3×#1880C2semidirect
7. C3×#1879C2semidirect
8. C3×#1874C2semidirect
9. C3×#1805C2semidirect
10. C3×#1804C2semidirect
11. C3×#1798C2semidirect
12. #32913C3direct (N × H)
49. C2×#520C2×C3semidirect
50. C2×#460C2×C3semidirect
51. C2×#459C2×C3semidirect
52. C2×#449C2×C3semidirect
53. #2619C2×C3semidirect
54. #1880C2×C3semidirect
55. #1879C2×C3semidirect
56. #1874C2×C3semidirect
57. #1805C2×C3semidirect
58. #1804C2×C3semidirect
59. #1798C2×C3semidirect
65. C2×C3×#41D4semidirect
93. C3×C4×Q8D4semidirect
99. C3×C4×D4D4semidirect
267. C2×#41C3×D4semidirect
295. C4×Q8C3×D4semidirect
301. C4×D4C3×D4semidirect
648. C3#32913direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5202
C2×C3×#4602
C2×C3×#4592
C2×C3×#4492
C3×#26192
C3×#18802
C3×#18792
C3×#18742
C3×#18052
C3×#18042
C3×#17982
#329133