Finite Group Databasebeta

← C3×#20945 #10639552

Subgroups — C3×#20945 #10639552

Order 768.

1862 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C3×#3428C211yesmaximal, characteristic, complemented
3C2×C3×#421C211yesmaximal, characteristic, complemented
4C2×C3×#418C211yesmaximal, characteristic, complemented
5C3×#2309C211yesmaximal, characteristic, complemented
6C3×#1883C218yesmaximal, complemented
7C3×#1742C211yesmaximal, characteristic, complemented
8C3×#1733C211yesmaximal, characteristic, complemented
9#20945C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×#41C2211yescharacteristic
11C23×C3×D4C2211yescharacteristic
12C2×C3×AGL(1,ℤ/8)—24no—
13C2×C3×AGL(1,ℤ/8)—22no—
14C2×C3×AGL(1,ℤ/8)C418yescomplemented
15C2×C3×AGL(1,ℤ/8)C2214yes—
16C22×C3×SD16—21no—
17C22×C3×D8—21no—
18C22×C3×M16C2211yescharacteristic
19C22×C3×M16C2211yescharacteristic
20C3×#520—22no—
21C3×#520C2214yescomplemented
22C3×#519—21no—
23C3×#518—21no—
24C3×#518—22no—
25C3×C4×#41—22no—
26C3×C4×#41—22no—
27C2×C3×#117—22no—
28C2×C3×#117C2211yescharacteristic
29C2×C3×#117C2211yescharacteristic
30C2×C3×#117C2211yescharacteristic
31C3×#421—24no—
32C2×C3×#104C2214yescomplemented
33C3×#418—22no—
34C3×#418C2214yescomplemented
35C2×C3×#102—22no—
36C3×#390—22no—
37C3×#390C2214yescomplemented
38C3×#379—22no—
39C3×#343C2214yescomplemented
40C3×#339C2214yescomplemented
41C22×AGL(1,ℤ/8)C2×C311yescharacteristic
42#3428C2×C311yescharacteristic, complemented
43C2×#421C2×C311yescharacteristic, complemented
44C2×#418C2×C311yescharacteristic, complemented
45#2309C2×C311yescharacteristic, complemented
46#1883C2×C318yescomplemented
47#1742C2×C311yescharacteristic, complemented
48#1733C2×C311yescharacteristic, complemented
49C25×C3—21no—
50C2×C3×#41—21no—
51C2×C3×#41—21no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41C2×C414yes—
55C2×C3×#41D414yes—
56C2×C3×#41D412yes—
57C2×C3×#41C2312yes—
58C22×C3×Q8D411yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4—21no—
66C22×C3×D4—22no—
67C22×C3×D4—22no—
68C22×C3×D4C2×C414yescomplemented
69C22×C3×D4D411yescharacteristic
70C22×C3×D4C2312yes—
71C23×C3×C4—21no—
72C23×C3×C4C2311yescharacteristic
73C3×AGL(1,ℤ/8)—216no—
74C3×AGL(1,ℤ/8)—28no—
75C3×AGL(1,ℤ/8)C2×C4116yescomplemented
76C2×C3×SD16—28no—
77C2×C3×SD16—24no—
78C2×C3×D8—28no—
79C2×C3×D8—24no—
80C2×C3×M16—22no—
81C2×C3×M16—21no—
82C2×C3×M16—21no—
83C2×C3×M16—22no—
84C2×C3×M16—22no—
85C2×C3×M16C2×C414yescomplemented
86C2×C3×M16C2312yes—
87C2×C3×M16C2312yes—
88C22×C3×C8—21no—
89C22×C3×C8—21no—
90C3×C4×Q8—24no—
91C3×C4×Q8—24no—
92C3×C4×D4—28no—
93C3×C4×D4—24no—
94C3×C4×D4—24no—
95C3×C4×D4—28no—
96C3×#117—24no—
97C3×#117—24no—
98C3×#117—21no—
99C3×#117—21no—
100C3×#117—21no—
101C3×#117—21no—
102C3×#117—21no—
103C3×#117—21no—
104C3×#117—24no—
105C3×#117—22no—
106C3×#117—22no—
107C3×#117D412yes—
108C3×#117C2312yes—
109C2×C3×C4⋊C4—24no—
110C2×C3×C4⋊C4—21no—
111C2×C3×C4⋊C4D412yes—
112C2×C3×C4⋊C4C2312yes—
113C2×C3×C22⋊C4—24no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4—28no—
117C2×C3×C22⋊C4D412yes—
118C2×C3×C42—24no—
119C2×C3×C42—21no—
120C2×C3×C42D412yes—
121C2×C3×C42C2312yes—
122C3×#104—28no—
123C3×#102—28no—
124C22×#41C22×C311yescharacteristic
125C23×D4C22×C311yescharacteristic
126C2×AGL(1,ℤ/8)—24no—
127C2×AGL(1,ℤ/8)—22no—
128C2×AGL(1,ℤ/8)C3×C418yescomplemented
129C2×AGL(1,ℤ/8)C22×C314yes—
130C22×SD16—21no—
131C22×D8—21no—
132C22×M16C22×C311yescharacteristic
133C22×M16C22×C311yescharacteristic
134#520—22no—
135#520C22×C314yescomplemented
136#519—21no—
137#518—21no—
138#518—22no—
139C4×#41—22no—
140C4×#41—22no—
141C2×#117—22no—
142C2×#117C22×C311yescharacteristic
143C2×#117C22×C311yescharacteristic
144C2×#117C22×C311yescharacteristic
145#421—24no—
146C2×#104C22×C314yescomplemented
147#418—22no—
148#418C22×C314yescomplemented
149C2×#102—22no—
150#390—22no—
151#390C22×C314yescomplemented
152#379—22no—
153#343C22×C314yescomplemented
154#339C22×C314yescomplemented
155C24×C3—21no—
156C24×C3—22no—
157C24×C3—24no—
158C24×C3—41no—
159C24×C3—41no—
160C24×C3—42no—
161C24×C3—84no—
162C24×C3C2×D411yescharacteristic
163C3×#41—24no—
164C3×#41—24no—
165C3×#41—24no—
166C3×#41—24no—
167C3×#41—24no—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41C22⋊C418yescomplemented
171C3×#41C22×C414yes—
172C3×#41C2×D414yes—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8C22⋊C414yescomplemented
176C2×C3×Q8C2×D412yes—
177C2×C3×D4—24no—
178C2×C3×D4—22no—
179C2×C3×D4—24no—
180C2×C3×D4—22no—
181C2×C3×D4—24no—
182C2×C3×D4—24no—
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—22no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—41no—
194C2×C3×D4—41no—
195C2×C3×D4—42no—
196C2×C3×D4—42no—
197C2×C3×D4—42no—
198C2×C3×D4—42no—
199C2×C3×D4—42no—
200C2×C3×D4C22⋊C414yescomplemented
201C2×C3×D4C22×C414yes—
202C2×C3×D4C2×D412yes—
203C22×C3×C4—21no—
204C22×C3×C4—24no—
205C22×C3×C4—22no—
206C22×C3×C4—24no—
207C22×C3×C4—22no—
208C22×C3×C4—28no—
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4C22×C412yes—
212C22×C3×C4C2×D414yes—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4#4112yes—
220C22×C3×C4#4112yes—
221C22×C3×C4C2411yescharacteristic
222C3×SD16—216no—
223C3×D8—216no—
224C3×M16—24no—
225C3×M16—22no—
226C3×M16—22no—
227C3×M16—22no—
228C3×M16—42no—
229C3×M16C22×C414yes—
230C2×C3×C8—24no—
231C2×C3×C8—22no—
232C2×C3×C8—22no—
233C2×C3×C8—42no—
234C3×C4⋊C4—28no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4C2×D414yes—
240C3×C22⋊C4—28no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—88no—
247C3×C42—28no—
248C3×C42—22no—
249C3×C42—22no—
250C3×C42—22no—
251C3×C42—22no—
252C3×C42—22no—
253C3×C42—22no—
254C25—21no—
255C2×#41—21no—
256C2×#41—21no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41C2×C3×C414yes—
260C2×#41C3×D414yes—
261C2×#41C3×D412yes—
262C2×#41C23×C312yes—
263C22×Q8C3×D411yescharacteristic
264C22×D4—21no—
265C22×D4—21no—
266C22×D4—21no—
267C22×D4—22no—
268C22×D4—21no—
269C22×D4—21no—
270C22×D4—21no—
271C22×D4—22no—
272C22×D4—22no—
273C22×D4C2×C3×C414yescomplemented
274C22×D4C3×D411yescharacteristic
275C22×D4C23×C312yes—
276C23×C4—21no—
277C23×C4C23×C311yescharacteristic
278AGL(1,ℤ/8)—216no—
279AGL(1,ℤ/8)—28no—
280AGL(1,ℤ/8)C2×C3×C4116yescomplemented
281C2×SD16—28no—
282C2×SD16—24no—
283C2×D8—28no—
284C2×D8—24no—
285C2×M16—22no—
286C2×M16—21no—
287C2×M16—21no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16C2×C3×C414yescomplemented
291C2×M16C23×C312yes—
292C2×M16C23×C312yes—
293C22×C8—21no—
294C22×C8—21no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×D4—28no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—28no—
301#117—24no—
302#117—24no—
303#117—21no—
304#117—21no—
305#117—21no—
306#117—21no—
307#117—21no—
308#117—21no—
309#117—24no—
310#117—22no—
311#117—22no—
312#117C3×D412yes—
313#117C23×C312yes—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4C3×D412yes—
317C2×C4⋊C4C23×C312yes—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4C3×D412yes—
323C2×C42—24no—
324C2×C42—21no—
325C2×C42C3×D412yes—
326C2×C42C23×C312yes—
327#104—28no—
328#102—28no—
329C23×C3—22no—
330C23×C3—24no—
331C23×C3—24no—
332C23×C3—42no—
333C23×C3—48no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—44no—
337C23×C3—44no—
338C23×C3—82no—
339C23×C3—84no—
340C23×C3—84no—
341C23×C3—88no—
342C23×C3—88no—
343C23×C3C4×D412yes—
344C23×C3#12011yescharacteristic
345C23×C3#12011yescharacteristic
346C23×C3#12112yes—
347C23×C3C22×D411yescharacteristic
348C3×Q8—22no—
349C3×Q8—24no—
350C3×Q8C2×C22⋊C414yes—
351C3×D4—24no—
352C3×D4—24no—
353C3×D4—28no—
354C3×D4—24no—
355C3×D4—24no—
356C3×D4—22no—
357C3×D4—24no—
358C3×D4—42no—
359C3×D4—42no—
360C3×D4—44no—
361C3×D4—44no—
362C3×D4C2×C22⋊C414yes—
363C2×C3×C4—22no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—28no—
368C2×C3×C4—24no—
369C2×C3×C4—24no—
370C2×C3×C4—44no—
371C2×C3×C4—44no—
372C2×C3×C4—42no—
373C2×C3×C4—82no—
374C2×C3×C4—88no—
375C2×C3×C4C2×C22⋊C412yes—
376C2×C3×C4C2×C22⋊C412yes—
377C2×C3×C4C4×D412yes—
378C2×C3×C4C4×D412yes—
379C2×C3×C4C4×D412yes—
380C2×C3×C4#12011yescharacteristic
381C2×C3×C4#12011yescharacteristic
382C2×C3×C4#12112yes—
383C2×C3×C4#12112yes—
384C2×C3×C4#12112yes—
385C2×C3×C4#12312yes—
386C2×C3×C4#12312yes—
387C2×C3×C4C23×C411yescharacteristic
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C3×C8—24no—
394C3×C8—42no—
395C24—21no—
396C24—22no—
397C24—24no—
398C24—41no—
399C24—41no—
400C24—42no—
401C24—84no—
402C24C2×C3×D411yescharacteristic
403#41—24no—
404#41—24no—
405#41—24no—
406#41—24no—
407#41—24no—
408#41—22no—
409#41—22no—
410#41C3×C22⋊C418yescomplemented
411#41C22×C3×C414yes—
412#41C2×C3×D414yes—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8C3×C22⋊C414yescomplemented
416C2×Q8C2×C3×D412yes—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4C3×C22⋊C414yescomplemented
441C2×D4C22×C3×C414yes—
442C2×D4C2×C3×D412yes—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4C22×C3×C412yes—
452C22×C4C2×C3×D414yes—
453C22×C4C2×C3×D411yescharacteristic
454C22×C4C2×C3×D411yescharacteristic
455C22×C4C2×C3×D412yes—
456C22×C4C2×C3×D411yescharacteristic
457C22×C4C2×C3×D411yescharacteristic
458C22×C4C2×C3×D411yescharacteristic
459C22×C4C3×#4112yes—
460C22×C4C3×#4112yes—
461C22×C4C24×C311yescharacteristic
462SD16—216no—
463D8—216no—
464M16—24no—
465M16—22no—
466M16—22no—
467M16—22no—
468M16—42no—
469M16C22×C3×C414yes—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—42no—
474C4⋊C4—28no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4C2×C3×D414yes—
480C22⋊C4—28no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—42no—
486C22⋊C4—88no—
487C42—28no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—24no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—48no—
501C22×C3—44no—
502C22×C3—44no—
503C22×C3—84no—
504C22×C3—84no—
505C22×C3—88no—
506C22×C3—88no—
507C22×C3—84no—
508C22×C3#38612yes—
509C22×C3#38612yes—
510C22×C3C2×C4×D411yescharacteristic
511C22×C3C2×#12011yescharacteristic
512C22×C3C2×#12111yescharacteristic
513C3×C4—24no—
514C3×C4—44no—
515C3×C4—44no—
516C3×C4—82no—
517C3×C4#38612yes—
518C3×C4#38612yes—
519C3×C4C22×C22⋊C411yescharacteristic
520C3×C4C2×C4×D411yescharacteristic
521C3×C4C2×#12111yescharacteristic
522C3×C4C2×#12311yescharacteristic
523C23—22no—
524C23—24no—
525C23—24no—
526C23—42no—
527C23—48no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—82no—
533C23—84no—
534C23—84no—
535C23—88no—
536C23—88no—
537C23C3×C4×D412yes—
538C23C3×#12011yescharacteristic
539C23C3×#12011yescharacteristic
540C23C3×#12112yes—
541C23C22×C3×D411yescharacteristic
542Q8—24no—
543Q8—22no—
544Q8C2×C3×C22⋊C414yes—
545D4—28no—
546D4—24no—
547D4—24no—
548D4—24no—
549D4—24no—
550D4—24no—
551D4—22no—
552D4—44no—
553D4—44no—
554D4—42no—
555D4—42no—
556D4C2×C3×C22⋊C414yes—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—82no—
568C2×C4—88no—
569C2×C4C2×C3×C22⋊C412yes—
570C2×C4C2×C3×C22⋊C412yes—
571C2×C4C3×C4×D412yes—
572C2×C4C3×C4×D412yes—
573C2×C4C3×C4×D412yes—
574C2×C4C3×#12011yescharacteristic
575C2×C4C3×#12011yescharacteristic
576C2×C4C3×#12112yes—
577C2×C4C3×#12112yes—
578C2×C4C3×#12112yes—
579C2×C4C3×#12312yes—
580C2×C4C3×#12312yes—
581C2×C4C23×C3×C411yescharacteristic
582C2×C4C22×C3×D411yescharacteristic
583C2×C4C22×C3×D411yescharacteristic
584C2×C4C22×C3×D411yescharacteristic
585C2×C4C2×C3×#4111yescharacteristic
586C2×C4C2×C3×#4111yescharacteristic
587C8—24no—
588C8—42no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—22no—
592C2×C3—22no—
593C2×C3—44no—
594C2×C3—84no—
595C2×C3C2×#38611yescharacteristic
596C2×C3#302112yes—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—24no—
601C22—42no—
602C22—42no—
603C22—48no—
604C22—44no—
605C22—44no—
606C22—84no—
607C22—84no—
608C22—88no—
609C22—88no—
610C22—84no—
611C22C3×#38612yes—
612C22C3×#38612yes—
613C22C2×C3×C4×D411yescharacteristic
614C22C2×C3×#12011yescharacteristic
615C22C2×C3×#12111yescharacteristic
616C4—24no—
617C4—44no—
618C4—44no—
619C4—82no—
620C4C3×#38612yes—
621C4C3×#38612yes—
622C4C22×C3×C22⋊C411yescharacteristic
623C4C2×C3×C4×D411yescharacteristic
624C4C2×C3×#12111yescharacteristic
625C4C2×C3×#12311yescharacteristic
626C3#2094511yescharacteristic, direct product factor, Sylow
627C2—21no—
628C2—21no—
629C2—22no—
630C2—22no—
631C2—44no—
632C2—84no—
633C2C2×C3×#38611yescharacteristic
634C2C3×#302112yes—
subgroupG/H
center, socleC22×C3C2×#120
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×AGL(1,ℤ/8)C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3428C2semidirect
3. C2×C3×#421C2semidirect
4. C2×C3×#418C2semidirect
5. C3×#2309C2semidirect
6. C3×#1883C2semidirect
7. C3×#1742C2semidirect
8. C3×#1733C2semidirect
9. #20945C3direct (N × H)
14. C2×C3×AGL(1,ℤ/8)C4semidirect
21. C3×#520C22semidirect
32. C2×C3×#104C22semidirect
34. C3×#418C22semidirect
37. C3×#390C22semidirect
39. C3×#343C22semidirect
40. C3×#339C22semidirect
42. #3428C2×C3semidirect
43. C2×#421C2×C3semidirect
44. C2×#418C2×C3semidirect
45. #2309C2×C3semidirect
46. #1883C2×C3semidirect
47. #1742C2×C3semidirect
48. #1733C2×C3semidirect
68. C22×C3×D4C2×C4semidirect
75. C3×AGL(1,ℤ/8)C2×C4semidirect
85. C2×C3×M16C2×C4semidirect
128. C2×AGL(1,ℤ/8)C3×C4semidirect
135. #520C22×C3semidirect
146. C2×#104C22×C3semidirect
148. #418C22×C3semidirect
151. #390C22×C3semidirect
153. #343C22×C3semidirect
154. #339C22×C3semidirect
170. C3×#41C22⋊C4semidirect
175. C2×C3×Q8C22⋊C4semidirect
200. C2×C3×D4C22⋊C4semidirect
273. C22×D4C2×C3×C4semidirect
280. AGL(1,ℤ/8)C2×C3×C4semidirect
290. C2×M16C2×C3×C4semidirect
410. #41C3×C22⋊C4semidirect
415. C2×Q8C3×C22⋊C4semidirect
440. C2×D4C3×C22⋊C4semidirect
626. C3#20945direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×AGL(1,ℤ/8)2
C3×#34282
C2×C3×#4212
C2×C3×#4182
C3×#23092
C3×#18832
C3×#17422
C3×#17332
#209453