Finite Group Databasebeta

← C3×#19090 #10637697

Subgroups — C3×#19090 #10637697

Order 768.

888 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3561C211yesmaximal, characteristic, complemented
2C3×#3235C211yesmaximal, characteristic, complemented
3C3×#3219C211yesmaximal, characteristic, complemented
4C3×#3213C211yesmaximal, characteristic, complemented
5C3×#1664C212yesmaximal, complemented
6C3×#1661C212yesmaximal, complemented
7C3×#1657C212yesmaximal, complemented
8C3×#1656C212yesmaximal, complemented
9C3×#1537C211yesmaximal, characteristic, complemented
10C3×#1535C211yesmaximal, characteristic, complemented
11C3×#1533C211yesmaximal, characteristic, complemented
12#19090C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#563C2212yes—
14C3×#562C2212yes—
15C3×#556—21no—
16C3×#551—22no—
17C3×D4×Q8—21no—
18C3×#548—21no—
19C3×#546—21no—
20C3×#543—21no—
21C3×#542—21no—
22C3×#542—22no—
23C3×#538—21no—
24C3×#536—21no—
25C3×#533C2211yescharacteristic
26C3×#532C2211yescharacteristic
27C3×#529C2211yescharacteristic
28C2×C3×#124C2211yescharacteristic
29C3×#501C2212yescomplemented
30C3×#497C2211yescharacteristic, complemented
31C3×#496C2212yes—
32C3×#496C2211yescharacteristic, complemented
33C3×#491C2211yescharacteristic, complemented
34C3×#490C2211yescharacteristic, complemented
35C3×#490C2212yescomplemented
36C3×#489C2212yes—
37C3×#482—22no—
38C3×#476—21no—
39C3×#474—21no—
40C3×#464—21no—
41C3×#463—22no—
42C3×#463—21no—
43C3×#452—21no—
44C3×#449—21no—
45C3×#448—21no—
46C3×#447—21no—
47C3×#432C2211yescharacteristic
48C3×#432C2211yescharacteristic
49C3×#432C2211yescharacteristic
50C3×#330C2212yescomplemented
51C3×#330C2212yescomplemented
52C3×#330C2212yescomplemented
53C3×#329C2212yes—
54C3×#329C2212yes—
55C3×#329C2212yescomplemented
56#3561C2×C311yescharacteristic, complemented
57#3235C2×C311yescharacteristic, complemented
58#3219C2×C311yescharacteristic, complemented
59#3213C2×C311yescharacteristic, complemented
60#1664C2×C312yescomplemented
61#1661C2×C312yescomplemented
62#1657C2×C312yescomplemented
63#1656C2×C312yescomplemented
64#1537C2×C311yescharacteristic, complemented
65#1535C2×C311yescharacteristic, complemented
66#1533C2×C311yescharacteristic, complemented
67C2×C3×#41—21no—
68C2×C3×#41—21no—
69C22×C3×Q8—21no—
70C22×C3×Q8D411yescharacteristic
71C22×C3×D4D411yescharacteristic
72C2×C3×Q16—21no—
73C2×C3×SD16—22no—
74C2×C3×SD16—21no—
75C2×C3×SD16—21no—
76C2×C3×D8—22no—
77C2×C3×D8—21no—
78C3×C4⋊Q8—21no—
79C3×C4⋊Q8—22no—
80C3×C4⋊Q8D411yescharacteristic
81C3×C4⋊Q8D411yescharacteristic
82C3×C4⋊Q8C2311yescharacteristic
83C3×#127C2311yescharacteristic
84C3×#126—22no—
85C3×#126—22no—
86C3×#126—22no—
87C3×#126—21no—
88C3×#125C2312yes—
89C3×#125C2311yescharacteristic
90C3×#124—21no—
91C3×#124—22no—
92C3×#124—22no—
93C3×#124—21no—
94C3×#124—21no—
95C3×#124—21no—
96C3×#124D412yes—
97C3×#124C2311yescharacteristic
98C3×#124C2312yes—
99C3×#123—22no—
100C3×#123—22no—
101C3×#123—21no—
102C3×#123—22no—
103C3×#123—22no—
104C3×C22⋊Q8—22no—
105C3×C22⋊Q8—22no—
106C3×C22⋊Q8—21no—
107C3×C22⋊Q8—22no—
108C3×C22⋊Q8—22no—
109C3×C22⋊Q8—22no—
110C3×C22⋊Q8—21no—
111C3×C22⋊Q8—21no—
112C3×C22⋊Q8—22no—
113C3×C22⋊Q8—22no—
114C3×C22⋊Q8D411yescharacteristic
115C3×#121—21no—
116C3×#121—22no—
117C3×#121—22no—
118C3×#121—21no—
119C3×#121—21no—
120C3×#121D411yescharacteristic
121C3×#120—22no—
122C3×C4×Q8—21no—
123C3×C4×Q8—21no—
124C3×C4×Q8—21no—
125C3×C4×Q8D411yescharacteristic
126C3×C4×D4—22no—
127C3×C4×D4—21no—
128C3×C4×D4—22no—
129C3×C4×D4—21no—
130C3×C4×D4—21no—
131C3×C4×D4—21no—
132C3×C4×D4D411yescharacteristic
133C3×#117—22no—
134C3×#117—21no—
135C3×#117D412yes—
136C2×C3×C22⋊C4—22no—
137C2×C3×C22⋊C4—21no—
138C2×C3×C22⋊C4—21no—
139C2×C3×C42C2311yescharacteristic
140C3×#107—22no—
141C3×#106—22no—
142C3×C4⋊C8—21no—
143C3×C4⋊C8—21no—
144C3×C4⋊C8—21no—
145C3×#103—22no—
146C3×#103—21no—
147C3×#103—21no—
148C3×#102—22no—
149C3×#102—21no—
150C3×#102—21no—
151C3×#102—22no—
152C3×#102—22no—
153C3×C22⋊C8—21no—
154C3×C22⋊C8—21no—
155C3×C22⋊C8—21no—
156C3×#97C2312yes—
157C3×#97C2311yescharacteristic
158C3×#97C2311yescharacteristic
159C3×#97C2312yes—
160#563C22×C312yes—
161#562C22×C312yes—
162#556—21no—
163#551—22no—
164D4×Q8—21no—
165#548—21no—
166#546—21no—
167#543—21no—
168#542—21no—
169#542—22no—
170#538—21no—
171#536—21no—
172#533C22×C311yescharacteristic
173#532C22×C311yescharacteristic
174#529C22×C311yescharacteristic
175C2×#124C22×C311yescharacteristic
176#501C22×C312yescomplemented
177#497C22×C311yescharacteristic, complemented
178#496C22×C312yes—
179#496C22×C311yescharacteristic, complemented
180#491C22×C311yescharacteristic, complemented
181#490C22×C312yescomplemented
182#490C22×C311yescharacteristic, complemented
183#489C22×C312yes—
184#482—22no—
185#476—21no—
186#474—21no—
187#464—21no—
188#463—22no—
189#463—21no—
190#452—21no—
191#449—21no—
192#448—21no—
193#447—21no—
194#432C22×C311yescharacteristic
195#432C22×C311yescharacteristic
196#432C22×C311yescharacteristic
197#330C22×C312yescomplemented
198#330C22×C312yescomplemented
199#330C22×C312yescomplemented
200#329C22×C312yes—
201#329C22×C312yes—
202#329C22×C312yescomplemented
203C24×C3—21no—
204C3×#41—81no—
205C3×#41—81no—
206C3×#41—82no—
207C2×C3×Q8—22no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—82no—
213C2×C3×Q8—81no—
214C2×C3×Q8C2×D412yes—
215C2×C3×Q8C2×D411yescharacteristic
216C2×C3×D4—21no—
217C2×C3×D4—22no—
218C2×C3×D4—22no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—41no—
224C2×C3×D4—81no—
225C2×C3×D4C2×D412yes—
226C2×C3×D4C2×D411yescharacteristic
227C22×C3×C4—22no—
228C22×C3×C4—22no—
229C22×C3×C4—22no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—41no—
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2×D411yescharacteristic
236C3×Q16—81no—
237C3×SD16—82no—
238C3×SD16—81no—
239C3×SD16—81no—
240C3×D8—81no—
241C3×D8—82no—
242C2×C3×C8—22no—
243C2×C3×C8—21no—
244C2×C3×C8—22no—
245C2×C3×C8—21no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—22no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4C2×D411yescharacteristic
261C3×C4⋊C4C2×D411yescharacteristic
262C3×C4⋊C4C2×D412yes—
263C3×C4⋊C4C2×D412yes—
264C3×C22⋊C4—22no—
265C3×C22⋊C4—22no—
266C3×C22⋊C4—22no—
267C3×C22⋊C4—22no—
268C3×C22⋊C4—22no—
269C3×C22⋊C4—21no—
270C3×C22⋊C4—22no—
271C3×C22⋊C4—22no—
272C3×C22⋊C4—22no—
273C3×C22⋊C4—21no—
274C3×C22⋊C4—21no—
275C3×C22⋊C4—22no—
276C3×C22⋊C4—22no—
277C3×C22⋊C4—21no—
278C3×C22⋊C4—21no—
279C3×C22⋊C4—41no—
280C3×C22⋊C4—41no—
281C3×C42—22no—
282C3×C42—22no—
283C3×C42—21no—
284C3×C42C2×D411yescharacteristic
285C3×C42C2×D411yescharacteristic
286C3×C42C2×D411yescharacteristic
287C3×C42C2411yescharacteristic
288C2×#41—21no—
289C2×#41—21no—
290C22×Q8—21no—
291C22×Q8C3×D411yescharacteristic
292C22×D4C3×D411yescharacteristic
293C2×Q16—21no—
294C2×SD16—22no—
295C2×SD16—21no—
296C2×SD16—21no—
297C2×D8—22no—
298C2×D8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—22no—
301C4⋊Q8C3×D411yescharacteristic
302C4⋊Q8C3×D411yescharacteristic
303C4⋊Q8C23×C311yescharacteristic
304#127C23×C311yescharacteristic
305#126—22no—
306#126—22no—
307#126—22no—
308#126—21no—
309#125C23×C311yescharacteristic
310#125C23×C312yes—
311#124—21no—
312#124—22no—
313#124—22no—
314#124—21no—
315#124—21no—
316#124—21no—
317#124C3×D412yes—
318#124C23×C312yes—
319#124C23×C311yescharacteristic
320#123—22no—
321#123—22no—
322#123—21no—
323#123—22no—
324#123—22no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8—21no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8C3×D411yescharacteristic
336#121—21no—
337#121—22no—
338#121—22no—
339#121—21no—
340#121—21no—
341#121C3×D411yescharacteristic
342#120—22no—
343C4×Q8—21no—
344C4×Q8—21no—
345C4×Q8—21no—
346C4×Q8C3×D411yescharacteristic
347C4×D4—22no—
348C4×D4—21no—
349C4×D4—22no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4C3×D411yescharacteristic
354#117—22no—
355#117—21no—
356#117C3×D412yes—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C42C23×C311yescharacteristic
361#107—22no—
362#106—22no—
363C4⋊C8—21no—
364C4⋊C8—21no—
365C4⋊C8—21no—
366#103—22no—
367#103—21no—
368#103—21no—
369#102—22no—
370#102—21no—
371#102—21no—
372#102—22no—
373#102—22no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8—21no—
377#97C23×C312yes—
378#97C23×C311yescharacteristic
379#97C23×C311yescharacteristic
380#97C23×C312yes—
381C23×C3—22no—
382C23×C3—21no—
383C23×C3—42no—
384C23×C3—41no—
385C23×C3—81no—
386C23×C3—81no—
387C23×C3#12011yescharacteristic
388C3×Q8—41no—
389C3×Q8—42no—
390C3×Q8—42no—
391C3×Q8—42no—
392C3×Q8—41no—
393C3×Q8—41no—
394C3×Q8—81no—
395C3×Q8—82no—
396C3×D4—41no—
397C3×D4—42no—
398C3×D4—42no—
399C3×D4—42no—
400C3×D4—41no—
401C3×D4—41no—
402C3×D4—42no—
403C3×D4—42no—
404C3×D4—82no—
405C3×D4—82no—
406C3×D4—82no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—21no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—21no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—42no—
417C2×C3×C4—41no—
418C2×C3×C4—82no—
419C2×C3×C4—81no—
420C2×C3×C4—82no—
421C2×C3×C4—82no—
422C2×C3×C4—82no—
423C2×C3×C4#12011yescharacteristic
424C2×C3×C4#12011yescharacteristic
425C2×C3×C4#12011yescharacteristic
426C2×C3×C4C22×D411yescharacteristic
427C2×C3×C4C22×D411yescharacteristic
428C2×C3×C4C22×D411yescharacteristic
429C3×C8—42no—
430C3×C8—42no—
431C3×C8—41no—
432C3×C8—41no—
433C24—21no—
434#41—81no—
435#41—81no—
436#41—82no—
437C2×Q8—22no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—82no—
443C2×Q8—81no—
444C2×Q8C2×C3×D412yes—
445C2×Q8C2×C3×D411yescharacteristic
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—41no—
454C2×D4—81no—
455C2×D4C2×C3×D412yes—
456C2×D4C2×C3×D411yescharacteristic
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4C2×C3×D411yescharacteristic
464C22×C4C2×C3×D411yescharacteristic
465C22×C4C2×C3×D411yescharacteristic
466Q16—81no—
467SD16—82no—
468SD16—81no—
469SD16—81no—
470D8—81no—
471D8—82no—
472C2×C8—22no—
473C2×C8—21no—
474C2×C8—22no—
475C2×C8—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4C2×C3×D411yescharacteristic
491C4⋊C4C2×C3×D411yescharacteristic
492C4⋊C4C2×C3×D412yes—
493C4⋊C4C2×C3×D412yes—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—21no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C42—22no—
512C42—22no—
513C42—21no—
514C42C2×C3×D411yescharacteristic
515C42C2×C3×D411yescharacteristic
516C42C2×C3×D411yescharacteristic
517C42C24×C311yescharacteristic
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—21no—
521C22×C3—41no—
522C22×C3—42no—
523C22×C3—41no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—41no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3—162no—
531C22×C3C2×#12011yescharacteristic
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—41no—
539C3×C4—42no—
540C3×C4—42no—
541C3×C4—41no—
542C3×C4—41no—
543C3×C4—42no—
544C3×C4—81no—
545C23—22no—
546C23—21no—
547C23—42no—
548C23—41no—
549C23—81no—
550C23—81no—
551C23C3×#12011yescharacteristic
552Q8—41no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—41no—
557Q8—41no—
558Q8—81no—
559Q8—82no—
560D4—41no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—41no—
565D4—41no—
566D4—42no—
567D4—42no—
568D4—82no—
569D4—82no—
570D4—82no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—82no—
583C2×C4—81no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4C3×#12011yescharacteristic
588C2×C4C3×#12011yescharacteristic
589C2×C4C3×#12011yescharacteristic
590C2×C4C22×C3×D411yescharacteristic
591C2×C4C22×C3×D411yescharacteristic
592C2×C4C22×C3×D411yescharacteristic
593C8—42no—
594C8—41no—
595C8—41no—
596C8—42no—
597C2×C3—41no—
598C2×C3—81no—
599C2×C3—82no—
600C2×C3—161no—
601C2×C3#301911yescharacteristic
602C2×C3#302011yescharacteristic
603C2×C3#302111yescharacteristic
604C22—21no—
605C22—21no—
606C22—21no—
607C22—41no—
608C22—42no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—162no—
617C22C2×C3×#12011yescharacteristic
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—41no—
625C4—42no—
626C4—42no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—81no—
631C3#1909011yescharacteristic, direct product factor, Sylow
632C2—41no—
633C2—82no—
634C2—81no—
635C2—161no—
636C2C3×#301911yescharacteristic
637C2C3×#302011yescharacteristic
638C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3561C2semidirect
2. C3×#3235C2semidirect
3. C3×#3219C2semidirect
4. C3×#3213C2semidirect
5. C3×#1664C2semidirect
6. C3×#1661C2semidirect
7. C3×#1657C2semidirect
8. C3×#1656C2semidirect
9. C3×#1537C2semidirect
10. C3×#1535C2semidirect
11. C3×#1533C2semidirect
12. #19090C3direct (N × H)
29. C3×#501C22semidirect
30. C3×#497C22semidirect
32. C3×#496C22semidirect
33. C3×#491C22semidirect
34. C3×#490C22semidirect
35. C3×#490C22semidirect
50. C3×#330C22semidirect
51. C3×#330C22semidirect
52. C3×#330C22semidirect
55. C3×#329C22semidirect
56. #3561C2×C3semidirect
57. #3235C2×C3semidirect
58. #3219C2×C3semidirect
59. #3213C2×C3semidirect
60. #1664C2×C3semidirect
61. #1661C2×C3semidirect
62. #1657C2×C3semidirect
63. #1656C2×C3semidirect
64. #1537C2×C3semidirect
65. #1535C2×C3semidirect
66. #1533C2×C3semidirect
176. #501C22×C3semidirect
177. #497C22×C3semidirect
179. #496C22×C3semidirect
180. #491C22×C3semidirect
181. #490C22×C3semidirect
182. #490C22×C3semidirect
197. #330C22×C3semidirect
198. #330C22×C3semidirect
199. #330C22×C3semidirect
202. #329C22×C3semidirect
631. C3#19090direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35612
C3×#32352
C3×#32192
C3×#32132
C3×#16642
C3×#16612
C3×#16572
C3×#16562
C3×#15372
C3×#15352
C3×#15332
#190903