Finite Group Databasebeta

← C3×#25765 #10644372

Subgroups — C3×#25765 #10644372

Order 768.

1634 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic, complemented
2C2×C3×#496C211yesmaximal, characteristic, complemented
3C2×C3×#464C211yesmaximal, characteristic, complemented
4C2×C3×#452C212yesmaximal, complemented
5C2×C3×#449C212yesmaximal, complemented
6C3×#2540C211yesmaximal, characteristic, complemented
7C3×#2000C212yesmaximal, complemented
8C3×#1999C212yesmaximal, complemented
9C3×#1982C211yesmaximal, characteristic, complemented
10C3×#1955C211yesmaximal, characteristic, complemented
11C3×#1763C211yesmaximal, characteristic, complemented
12#25765C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C22×C3×Q16C2211yescharacteristic
15C22×C3×SD16C2211yescharacteristic
16C22×C3×SD16C2212yes—
17C22×C3×D8C2212yes—
18C3×#550—24no—
19C3×#550—24no—
20C2×C3×#127—21no—
21C2×C3×#127C2211yescharacteristic
22C2×C3×#124C2211yescharacteristic
23C2×C3×C22⋊Q8C2212yes—
24C2×C3×#121—22no—
25C2×C3×#121C2212yes—
26C2×C3×#121C2212yes—
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×D4—21no—
29C2×C3×C4×D4C2211yescharacteristic
30C3×#496—24no—
31C3×#464—24no—
32C3×#452—28no—
33C3×#449—28no—
34C2×C3×C4⋊C8C2211yescharacteristic
35C2×C3×#103C2211yescharacteristic
36C2×C3×#103C2212yes—
37C2×C3×#102C2211yescharacteristic
38C2×C3×#102C2212yes—
39C2×C3×#102C2212yes—
40C2×C3×C22⋊C8C2212yes—
41C2×C3×C22⋊C8C2212yes—
42C2×C3×#97C2211yescharacteristic
43C3×#399—21no—
44C3×#392—21no—
45C3×#390—22no—
46C3×C4×C4⋊C4C2211yescharacteristic
47C3×#340C2212yes—
48C2×#550C2×C311yescharacteristic, complemented
49C2×#496C2×C311yescharacteristic, complemented
50C2×#464C2×C311yescharacteristic, complemented
51C2×#452C2×C312yescomplemented
52C2×#449C2×C312yescomplemented
53#2540C2×C311yescharacteristic, complemented
54#2000C2×C312yescomplemented
55#1999C2×C312yescomplemented
56#1982C2×C311yescharacteristic, complemented
57#1955C2×C311yescharacteristic, complemented
58#1763C2×C311yescharacteristic, complemented
59C2×C3×#41—24no—
60C2×C3×#41—24no—
61C2×C3×#41—24no—
62C2×C3×#41—24no—
63C2×C3×#41—42no—
64C2×C3×#41—42no—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—22no—
68C22×C3×D4—22no—
69C22×C3×D4—22no—
70C22×C3×D4—21no—
71C22×C3×D4—21no—
72C22×C3×D4D411yescharacteristic
73C22×C3×D4C2311yescharacteristic
74C22×C3×D4C2312yes—
75C23×C3×C4—22no—
76C23×C3×C4C2312yes—
77C2×C3×Q16—24no—
78C2×C3×Q16—22no—
79C2×C3×SD16—24no—
80C2×C3×SD16—22no—
81C2×C3×SD16—24no—
82C2×C3×SD16—24no—
83C2×C3×SD16—42no—
84C2×C3×D8—24no—
85C2×C3×D8—24no—
86C2×C3×D8—42no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2312yes—
90C3×#127—28no—
91C3×#127—22no—
92C3×#127D414yescomplemented
93C3×#124—22no—
94C3×#124—22no—
95C3×C22⋊Q8—24no—
96C3×C22⋊Q8—42no—
97C3×#121—216no—
98C3×#121—24no—
99C3×#121—24no—
100C3×#121—24no—
101C3×#121—42no—
102C3×C4×Q8—22no—
103C3×C4×Q8D414yescomplemented
104C3×C4×D4—28no—
105C3×C4×D4—22no—
106C3×C4×D4—22no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—22no—
109C2×C3×C4⋊C4D411yescharacteristic
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—22no—
116C2×C3×C22⋊C4—21no—
117C2×C3×C22⋊C4D412yes—
118C2×C3×C42—21no—
119C2×C3×C42—21no—
120C2×C3×C42C2311yescharacteristic
121C3×C4⋊C8—22no—
122C3×#103—22no—
123C3×#103D418yescomplemented
124C3×#102—24no—
125C3×#102—24no—
126C3×#102—22no—
127C3×C22⋊C8—24no—
128C3×C22⋊C8—24no—
129C3×#97—22no—
130C3×#95—21no—
131C22×#41C22×C312yes—
132C22×Q16C22×C311yescharacteristic
133C22×SD16C22×C311yescharacteristic
134C22×SD16C22×C312yes—
135C22×D8C22×C312yes—
136#550—24no—
137#550—24no—
138C2×#127—21no—
139C2×#127C22×C311yescharacteristic
140C2×#124C22×C311yescharacteristic
141C2×C22⋊Q8C22×C312yes—
142C2×#121—22no—
143C2×#121C22×C312yes—
144C2×#121C22×C312yes—
145C2×C4×Q8C22×C311yescharacteristic
146C2×C4×D4—21no—
147C2×C4×D4C22×C311yescharacteristic
148#496—24no—
149#464—24no—
150#452—28no—
151#449—28no—
152C2×C4⋊C8C22×C311yescharacteristic
153C2×#103C22×C311yescharacteristic
154C2×#103C22×C312yes—
155C2×#102C22×C312yes—
156C2×#102C22×C312yes—
157C2×#102C22×C311yescharacteristic
158C2×C22⋊C8C22×C312yes—
159C2×C22⋊C8C22×C312yes—
160C2×#97C22×C311yescharacteristic
161#399—21no—
162#392—21no—
163#390—22no—
164C4×C4⋊C4C22×C311yescharacteristic
165#340C22×C312yes—
166C24×C3—22no—
167C24×C3—21no—
168C24×C3C2×D412yes—
169C3×#41—48no—
170C3×#41—48no—
171C3×#41—48no—
172C3×#41—48no—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8—21no—
176C2×C3×Q8—22no—
177C2×C3×Q8—21no—
178C2×C3×Q8—22no—
179C2×C3×Q8C2×D414yes—
180C2×C3×Q8C2×D412yes—
181C2×C3×D4—28no—
182C2×C3×D4—28no—
183C2×C3×D4—22no—
184C2×C3×D4—24no—
185C2×C3×D4—24no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—21no—
190C2×C3×D4—22no—
191C2×C3×D4—21no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—42no—
196C2×C3×D4—44no—
197C2×C3×D4—44no—
198C2×C3×D4—44no—
199C2×C3×D4—44no—
200C2×C3×D4—44no—
201C2×C3×D4—81no—
202C2×C3×D4—81no—
203C2×C3×D4—82no—
204C2×C3×D4—82no—
205C2×C3×D4—82no—
206C2×C3×D4C2×D414yes—
207C22×C3×C4—21no—
208C22×C3×C4—28no—
209C22×C3×C4—22no—
210C22×C3×C4—21no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—22no—
214C22×C3×C4—22no—
215C22×C3×C4—22no—
216C22×C3×C4—22no—
217C22×C3×C4—21no—
218C22×C3×C4—44no—
219C22×C3×C4—44no—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C3×Q16—44no—
225C3×SD16—48no—
226C3×SD16—44no—
227C3×D8—48no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8C2×D414yes—
235C3×C4⋊C4—24no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—44no—
240C3×C4⋊C4C2×D414yes—
241C3×C4⋊C4C2×D412yes—
242C3×C22⋊C4—28no—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—44no—
245C3×C22⋊C4—44no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—42no—
248C3×C42—24no—
249C3×C42—81no—
250C3×C42C2×D412yes—
251C3×C42C2×D412yes—
252C2×#41—24no—
253C2×#41—24no—
254C2×#41—24no—
255C2×#41—24no—
256C2×#41—42no—
257C2×#41—42no—
258C22×Q8C23×C311yescharacteristic
259C22×Q8C23×C311yescharacteristic
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4C3×D411yescharacteristic
266C22×D4C23×C311yescharacteristic
267C22×D4C23×C312yes—
268C23×C4—22no—
269C23×C4C23×C312yes—
270C2×Q16—24no—
271C2×Q16—22no—
272C2×SD16—24no—
273C2×SD16—22no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—42no—
277C2×D8—24no—
278C2×D8—24no—
279C2×D8—42no—
280C22×C8C23×C311yescharacteristic
281C22×C8C23×C311yescharacteristic
282C22×C8C23×C312yes—
283#127—28no—
284#127—22no—
285#127C3×D414yescomplemented
286#124—22no—
287#124—22no—
288C22⋊Q8—24no—
289C22⋊Q8—42no—
290#121—216no—
291#121—24no—
292#121—24no—
293#121—24no—
294#121—42no—
295C4×Q8—22no—
296C4×Q8C3×D414yescomplemented
297C4×D4—28no—
298C4×D4—22no—
299C4×D4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4C3×D411yescharacteristic
303C2×C4⋊C4C23×C311yescharacteristic
304C2×C4⋊C4C23×C312yes—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4C3×D412yes—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42C23×C311yescharacteristic
314C4⋊C8—22no—
315#103—22no—
316#103C3×D418yescomplemented
317#102—24no—
318#102—24no—
319#102—22no—
320C22⋊C8—24no—
321C22⋊C8—24no—
322#97—22no—
323#95—21no—
324C23×C3—22no—
325C23×C3—24no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—41no—
339C23×C3—42no—
340C23×C3—42no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3C22×D411yescharacteristic
344C3×Q8—22no—
345C3×Q8—24no—
346C3×Q8—41no—
347C3×Q8—41no—
348C3×Q8—42no—
349C3×Q8#12714yes—
350C3×D4—24no—
351C3×D4—24no—
352C3×D4—44no—
353C3×D4—48no—
354C3×D4—416no—
355C3×D4—42no—
356C3×D4—41no—
357C3×D4—42no—
358C3×D4—41no—
359C3×D4—42no—
360C3×D4—84no—
361C3×D4—88no—
362C3×D4—88no—
363C2×C3×C4—22no—
364C2×C3×C4—24no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—24no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—416no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—42no—
382C2×C3×C4—41no—
383C2×C3×C4—42no—
384C2×C3×C4#12012yes—
385C2×C3×C4#12012yes—
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C22×D412yes—
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4H(5,𝔽2)12yes—
390C3×C8—24no—
391C3×C8—41no—
392C3×C8—41no—
393C24—22no—
394C24—21no—
395C24C2×C3×D412yes—
396#41—48no—
397#41—48no—
398#41—48no—
399#41—48no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8C2×C3×D414yes—
407C2×Q8C2×C3×D412yes—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—81no—
429C2×D4—81no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4C2×C3×D414yes—
434C22×C4—21no—
435C22×C4—28no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4C2×C3×D412yes—
448C22×C4C2×C3×D411yescharacteristic
449C22×C4C2×C3×D411yescharacteristic
450C22×C4C24×C311yescharacteristic
451Q16—44no—
452SD16—48no—
453SD16—44no—
454D8—48no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8C2×C3×D414yes—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—44no—
467C4⋊C4C2×C3×D414yes—
468C4⋊C4C2×C3×D412yes—
469C22⋊C4—28no—
470C22⋊C4—24no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C42—24no—
476C42—81no—
477C42C2×C3×D412yes—
478C42C2×C3×D412yes—
479C22×C3—44no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—44no—
483C22×C3—42no—
484C22×C3—44no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—82no—
490C22×C3—82no—
491C22×C3—82no—
492C22×C3—82no—
493C22×C3—81no—
494C22×C3—81no—
495C22×C3—81no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3C2×#12011yescharacteristic
499C22×C3D4212yes—
500C22×C3#57512yes—
501C22×C3#57611yescharacteristic
502C22×C3#57711yescharacteristic
503C3×C4—24no—
504C3×C4—22no—
505C3×C4—22no—
506C3×C4—42no—
507C3×C4—42no—
508C3×C4—44no—
509C3×C4—81no—
510C3×C4C2×#12711yescharacteristic
511C3×C4#53511yescharacteristic
512C3×C4D4212yes—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—22no—
520C23—22no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23C22×C3×D411yescharacteristic
533Q8—22no—
534Q8—24no—
535Q8—41no—
536Q8—41no—
537Q8—42no—
538Q8C3×#12714yes—
539D4—24no—
540D4—24no—
541D4—416no—
542D4—44no—
543D4—48no—
544D4—42no—
545D4—41no—
546D4—42no—
547D4—41no—
548D4—42no—
549D4—84no—
550D4—88no—
551D4—88no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—416no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4C3×#12012yes—
574C2×C4C3×#12012yes—
575C2×C4C22×C3×D411yescharacteristic
576C2×C4C22×C3×D412yes—
577C2×C4C22×C3×D411yescharacteristic
578C2×C4C3×H(5,𝔽2)12yes—
579C8—24no—
580C8—41no—
581C8—41no—
582C2×C3—82no—
583C2×C3—84no—
584C2×C3—161no—
585C2×C3#243311yescharacteristic
586C2×C3#300511yescharacteristic
587C2×C3#301211yescharacteristic
588C2×C3#326812yes—
589C2×C3#327112yes—
590C22—44no—
591C22—42no—
592C22—42no—
593C22—44no—
594C22—42no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—82no—
608C22—82no—
609C22C2×C3×#12011yescharacteristic
610C22C3×D4212yes—
611C22C3×#57512yes—
612C22C3×#57611yescharacteristic
613C22C3×#57711yescharacteristic
614C4—24no—
615C4—22no—
616C4—22no—
617C4—42no—
618C4—42no—
619C4—44no—
620C4—81no—
621C4C2×C3×#12711yescharacteristic
622C4C3×#53511yescharacteristic
623C4C3×D4212yes—
624C3#2576511yescharacteristic, direct product factor, Sylow
625C2—82no—
626C2—84no—
627C2—161no—
628C2C3×#243311yescharacteristic
629C2C3×#300511yescharacteristic
630C2C3×#301211yescharacteristic
631C2C3×#326812yes—
632C2C3×#327112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#550C2semidirect
2. C2×C3×#496C2semidirect
3. C2×C3×#464C2semidirect
4. C2×C3×#452C2semidirect
5. C2×C3×#449C2semidirect
6. C3×#2540C2semidirect
7. C3×#2000C2semidirect
8. C3×#1999C2semidirect
9. C3×#1982C2semidirect
10. C3×#1955C2semidirect
11. C3×#1763C2semidirect
12. #25765C3direct (N × H)
48. C2×#550C2×C3semidirect
49. C2×#496C2×C3semidirect
50. C2×#464C2×C3semidirect
51. C2×#452C2×C3semidirect
52. C2×#449C2×C3semidirect
53. #2540C2×C3semidirect
54. #2000C2×C3semidirect
55. #1999C2×C3semidirect
56. #1982C2×C3semidirect
57. #1955C2×C3semidirect
58. #1763C2×C3semidirect
92. C3×#127D4semidirect
103. C3×C4×Q8D4semidirect
123. C3×#103D4semidirect
285. #127C3×D4semidirect
296. C4×Q8C3×D4semidirect
316. #103C3×D4semidirect
624. C3#25765direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×C3×#4962
C2×C3×#4642
C2×C3×#4522
C2×C3×#4492
C3×#25402
C3×#20002
C3×#19992
C3×#19822
C3×#19552
C3×#17632
#257653