Finite Group Databasebeta

← C3×#19086 #10637693

Subgroups — C3×#19086 #10637693

Order 768.

964 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3558C211yesmaximal, characteristic, complemented
2C3×#3219C211yesmaximal, characteristic, complemented
3C3×#3214C211yesmaximal, characteristic, complemented
4C3×#3213C211yesmaximal, characteristic, complemented
5C3×#1662C212yesmaximal, complemented
6C3×#1659C212yesmaximal, complemented
7C3×#1656C212yesmaximal, complemented
8C3×#1655C212yesmaximal, complemented
9C3×#1536C211yesmaximal, characteristic, complemented
10C3×#1535C211yesmaximal, characteristic, complemented
11C3×#1533C211yesmaximal, characteristic, complemented
12#19086C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#562C2212yes—
14C3×H(3,𝔽4)C2212yes—
15C3×#559—21no—
16C3×#548—21no—
17C3×#546—22no—
18C3×#546—21no—
19C3×D42—21no—
20C3×#542—21no—
21C3×#538—22no—
22C3×#538—21no—
23C3×#535—21no—
24C3×#534—21no—
25C3×#532C2211yescharacteristic
26C3×#532C2211yescharacteristic, complemented
27C3×#529C2211yescharacteristic
28C2×C3×#124C2211yescharacteristic
29C3×#497C2211yescharacteristic, complemented
30C3×#496C2212yescomplemented
31C3×#496C2212yescomplemented
32C3×#496C2211yescharacteristic, complemented
33C3×#491C2211yescharacteristic, complemented
34C3×#490C2211yescharacteristic, complemented
35C3×#489C2212yescomplemented
36C3×#489C2212yescomplemented
37C3×#464—21no—
38C3×#463—22no—
39C3×#463—21no—
40C3×#461—21no—
41C3×#459—21no—
42C3×#452—21no—
43C3×#449—22no—
44C3×#449—21no—
45C3×#448—21no—
46C3×#447—21no—
47C3×#432C2211yescharacteristic, complemented
48C3×#432C2211yescharacteristic, complemented
49C3×#432C2211yescharacteristic, complemented
50C3×#330C2212yescomplemented
51C3×#329C2212yescomplemented
52C3×#329C2212yescomplemented
53C3×#329C2212yescomplemented
54C3×#329C2212yescomplemented
55C3×#329C2212yescomplemented
56#3558C2×C311yescharacteristic, complemented
57#3219C2×C311yescharacteristic, complemented
58#3214C2×C311yescharacteristic, complemented
59#3213C2×C311yescharacteristic, complemented
60#1662C2×C312yescomplemented
61#1659C2×C312yescomplemented
62#1656C2×C312yescomplemented
63#1655C2×C312yescomplemented
64#1536C2×C311yescharacteristic, complemented
65#1535C2×C311yescharacteristic, complemented
66#1533C2×C311yescharacteristic, complemented
67C2×C3×#41—21no—
68C2×C3×#41—21no—
69C2×C3×#41D412yes—
70C22×C3×Q8D411yescharacteristic
71C22×C3×D4—22no—
72C22×C3×D4—21no—
73C22×C3×D4D411yescharacteristic
74C2×C3×Q16—21no—
75C2×C3×SD16—22no—
76C2×C3×SD16—22no—
77C2×C3×SD16—21no—
78C2×C3×SD16—21no—
79C2×C3×D8—22no—
80C2×C3×D8—22no—
81C2×C3×D8—21no—
82C3×C4⋊Q8D411yescharacteristic
83C3×C4⋊Q8C2311yescharacteristic
84C3×#127—21no—
85C3×#127D411yescharacteristic
86C3×#127C2311yescharacteristic
87C3×#126—22no—
88C3×#126—21no—
89C3×#125C2311yescharacteristic
90C3×#124—21no—
91C3×#124—22no—
92C3×#124—22no—
93C3×#124—22no—
94C3×#124—21no—
95C3×#124—21no—
96C3×#124—21no—
97C3×#124D412yes—
98C3×#124C2311yescharacteristic
99C3×#124C2312yes—
100C3×#124C2312yes—
101C3×#123—22no—
102C3×#123—22no—
103C3×#123—21no—
104C3×#123—22no—
105C3×#123—22no—
106C3×C22⋊Q8—21no—
107C3×C22⋊Q8—22no—
108C3×C22⋊Q8—22no—
109C3×C22⋊Q8D411yescharacteristic
110C3×#121—22no—
111C3×#121—22no—
112C3×#121—22no—
113C3×#121—21no—
114C3×#121—22no—
115C3×#121—22no—
116C3×#121—21no—
117C3×#121—21no—
118C3×#121—21no—
119C3×#121—21no—
120C3×#121D411yescharacteristic
121C3×#120—22no—
122C3×#120—22no—
123C3×#120—22no—
124C3×#120—22no—
125C3×#120—22no—
126C3×C4×Q8—21no—
127C3×C4×Q8D411yescharacteristic
128C3×C4×D4—22no—
129C3×C4×D4—21no—
130C3×C4×D4—22no—
131C3×C4×D4—21no—
132C3×C4×D4—21no—
133C3×C4×D4—21no—
134C3×C4×D4—21no—
135C3×C4×D4—21no—
136C3×C4×D4D411yescharacteristic
137C3×#117—21no—
138C2×C3×C22⋊C4—22no—
139C2×C3×C22⋊C4—21no—
140C2×C3×C22⋊C4—21no—
141C2×C3×C42C2311yescharacteristic
142C3×C4⋊C8—21no—
143C3×C4⋊C8—21no—
144C3×C4⋊C8—21no—
145C3×#103—22no—
146C3×#103—22no—
147C3×#103—21no—
148C3×#103—21no—
149C3×#102—22no—
150C3×#102—22no—
151C3×#102—21no—
152C3×#102—21no—
153C3×C22⋊C8—21no—
154C3×C22⋊C8—21no—
155C3×C22⋊C8—21no—
156C3×#97C2311yescharacteristic
157C3×#97C2311yescharacteristic
158C3×#97C2312yes—
159C3×#97C2312yes—
160#562C22×C312yes—
161H(3,𝔽4)C22×C312yes—
162#559—21no—
163#548—21no—
164#546—22no—
165#546—21no—
166D42—21no—
167#542—21no—
168#538—22no—
169#538—21no—
170#535—21no—
171#534—21no—
172#532C22×C311yescharacteristic
173#532C22×C311yescharacteristic, complemented
174#529C22×C311yescharacteristic
175C2×#124C22×C311yescharacteristic
176#497C22×C311yescharacteristic, complemented
177#496C22×C311yescharacteristic, complemented
178#496C22×C312yescomplemented
179#496C22×C312yescomplemented
180#491C22×C311yescharacteristic, complemented
181#490C22×C311yescharacteristic, complemented
182#489C22×C312yescomplemented
183#489C22×C312yescomplemented
184#464—21no—
185#463—22no—
186#463—21no—
187#461—21no—
188#459—21no—
189#452—21no—
190#449—22no—
191#449—21no—
192#448—21no—
193#447—21no—
194#432C22×C311yescharacteristic, complemented
195#432C22×C311yescharacteristic, complemented
196#432C22×C311yescharacteristic, complemented
197#330C22×C312yescomplemented
198#329C22×C312yescomplemented
199#329C22×C312yescomplemented
200#329C22×C312yescomplemented
201#329C22×C312yescomplemented
202#329C22×C312yescomplemented
203C24×C3—22no—
204C24×C3—21no—
205C3×#41—81no—
206C3×#41—81no—
207C3×#41—82no—
208C3×#41—82no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—81no—
212C2×C3×Q8C2×D412yes—
213C2×C3×Q8C2×D412yes—
214C2×C3×Q8C2×D411yescharacteristic
215C2×C3×D4—22no—
216C2×C3×D4—21no—
217C2×C3×D4—22no—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C2×C3×D4—22no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—21no—
228C2×C3×D4—41no—
229C2×C3×D4—81no—
230C2×C3×D4—82no—
231C2×C3×D4—84no—
232C2×C3×D4C2×D412yes—
233C2×C3×D4C2×D412yes—
234C2×C3×D4C2×D411yescharacteristic
235C22×C3×C4—22no—
236C22×C3×C4—22no—
237C22×C3×C4—21no—
238C22×C3×C4—21no—
239C22×C3×C4—22no—
240C22×C3×C4—41no—
241C22×C3×C4C2×D411yescharacteristic
242C22×C3×C4C2×D411yescharacteristic
243C22×C3×C4C2×D411yescharacteristic
244C3×Q16—81no—
245C3×SD16—82no—
246C3×SD16—82no—
247C3×SD16—81no—
248C3×SD16—81no—
249C3×D8—81no—
250C3×D8—82no—
251C3×D8—82no—
252C2×C3×C8—21no—
253C2×C3×C8—22no—
254C2×C3×C8—22no—
255C2×C3×C8—21no—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—22no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4C2×D411yescharacteristic
267C3×C4⋊C4C2×D411yescharacteristic
268C3×C22⋊C4—22no—
269C3×C22⋊C4—22no—
270C3×C22⋊C4—22no—
271C3×C22⋊C4—22no—
272C3×C22⋊C4—22no—
273C3×C22⋊C4—21no—
274C3×C22⋊C4—22no—
275C3×C22⋊C4—21no—
276C3×C22⋊C4—22no—
277C3×C22⋊C4—21no—
278C3×C22⋊C4—22no—
279C3×C22⋊C4—22no—
280C3×C22⋊C4—21no—
281C3×C22⋊C4—21no—
282C3×C22⋊C4—22no—
283C3×C22⋊C4—22no—
284C3×C22⋊C4—41no—
285C3×C22⋊C4—41no—
286C3×C42—22no—
287C3×C42—21no—
288C3×C42C2×D411yescharacteristic
289C3×C42C2×D411yescharacteristic
290C3×C42C2×D411yescharacteristic
291C3×C42C2411yescharacteristic
292C2×#41—21no—
293C2×#41—21no—
294C2×#41C3×D412yes—
295C22×Q8C3×D411yescharacteristic
296C22×D4—22no—
297C22×D4—21no—
298C22×D4C3×D411yescharacteristic
299C2×Q16—21no—
300C2×SD16—22no—
301C2×SD16—22no—
302C2×SD16—21no—
303C2×SD16—21no—
304C2×D8—22no—
305C2×D8—22no—
306C2×D8—21no—
307C4⋊Q8C3×D411yescharacteristic
308C4⋊Q8C23×C311yescharacteristic
309#127—21no—
310#127C3×D411yescharacteristic
311#127C23×C311yescharacteristic
312#126—22no—
313#126—21no—
314#125C23×C311yescharacteristic
315#124—21no—
316#124—22no—
317#124—22no—
318#124—22no—
319#124—21no—
320#124—21no—
321#124—21no—
322#124C3×D412yes—
323#124C23×C312yes—
324#124C23×C311yescharacteristic
325#124C23×C312yes—
326#123—22no—
327#123—22no—
328#123—21no—
329#123—22no—
330#123—22no—
331C22⋊Q8—21no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8C3×D411yescharacteristic
335#121—22no—
336#121—22no—
337#121—22no—
338#121—21no—
339#121—22no—
340#121—22no—
341#121—21no—
342#121—21no—
343#121—21no—
344#121—21no—
345#121C3×D411yescharacteristic
346#120—22no—
347#120—22no—
348#120—22no—
349#120—22no—
350#120—22no—
351C4×Q8—21no—
352C4×Q8C3×D411yescharacteristic
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4C3×D411yescharacteristic
362#117—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C42C23×C311yescharacteristic
367C4⋊C8—21no—
368C4⋊C8—21no—
369C4⋊C8—21no—
370#103—22no—
371#103—22no—
372#103—21no—
373#103—21no—
374#102—22no—
375#102—22no—
376#102—21no—
377#102—21no—
378C22⋊C8—21no—
379C22⋊C8—21no—
380C22⋊C8—21no—
381#97C23×C311yescharacteristic
382#97C23×C311yescharacteristic
383#97C23×C312yes—
384#97C23×C312yes—
385C23×C3—21no—
386C23×C3—22no—
387C23×C3—22no—
388C23×C3—42no—
389C23×C3—41no—
390C23×C3—81no—
391C23×C3—81no—
392C23×C3—82no—
393C23×C3—82no—
394C23×C3—82no—
395C23×C3#12011yescharacteristic
396C3×Q8—41no—
397C3×Q8—42no—
398C3×Q8—42no—
399C3×Q8—42no—
400C3×Q8—41no—
401C3×Q8—41no—
402C3×D4—41no—
403C3×D4—42no—
404C3×D4—42no—
405C3×D4—42no—
406C3×D4—42no—
407C3×D4—41no—
408C3×D4—42no—
409C3×D4—41no—
410C3×D4—41no—
411C3×D4—41no—
412C3×D4—42no—
413C3×D4—44no—
414C3×D4—81no—
415C3×D4—82no—
416C3×D4—82no—
417C3×D4—82no—
418C3×D4—82no—
419C3×D4—82no—
420C3×D4—82no—
421C2×C3×C4—21no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—21no—
425C2×C3×C4—21no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4—41no—
429C2×C3×C4—42no—
430C2×C3×C4—41no—
431C2×C3×C4—82no—
432C2×C3×C4—82no—
433C2×C3×C4—81no—
434C2×C3×C4—82no—
435C2×C3×C4—82no—
436C2×C3×C4#12011yescharacteristic
437C2×C3×C4#12011yescharacteristic
438C2×C3×C4#12011yescharacteristic
439C2×C3×C4C22×D411yescharacteristic
440C2×C3×C4C22×D411yescharacteristic
441C2×C3×C4C22×D411yescharacteristic
442C3×C8—42no—
443C3×C8—42no—
444C3×C8—41no—
445C3×C8—41no—
446C24—22no—
447C24—21no—
448#41—81no—
449#41—81no—
450#41—82no—
451#41—82no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—81no—
455C2×Q8C2×C3×D412yes—
456C2×Q8C2×C3×D412yes—
457C2×Q8C2×C3×D411yescharacteristic
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—41no—
472C2×D4—81no—
473C2×D4—82no—
474C2×D4—84no—
475C2×D4C2×C3×D412yes—
476C2×D4C2×C3×D412yes—
477C2×D4C2×C3×D411yescharacteristic
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—41no—
484C22×C4C2×C3×D411yescharacteristic
485C22×C4C2×C3×D411yescharacteristic
486C22×C4C2×C3×D411yescharacteristic
487Q16—81no—
488SD16—82no—
489SD16—82no—
490SD16—81no—
491SD16—81no—
492D8—81no—
493D8—82no—
494D8—82no—
495C2×C8—21no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4C2×C3×D411yescharacteristic
510C4⋊C4C2×C3×D411yescharacteristic
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—21no—
517C22⋊C4—22no—
518C22⋊C4—21no—
519C22⋊C4—22no—
520C22⋊C4—21no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C42—22no—
530C42—21no—
531C42C2×C3×D411yescharacteristic
532C42C2×C3×D411yescharacteristic
533C42C2×C3×D411yescharacteristic
534C42C24×C311yescharacteristic
535C22×C3—21no—
536C22×C3—21no—
537C22×C3—21no—
538C22×C3—41no—
539C22×C3—42no—
540C22×C3—42no—
541C22×C3—42no—
542C22×C3—42no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3—42no—
546C22×C3—42no—
547C22×C3—81no—
548C22×C3—81no—
549C22×C3—81no—
550C22×C3—162no—
551C22×C3—164no—
552C22×C3C2×#12011yescharacteristic
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—21no—
559C3×C4—41no—
560C3×C4—41no—
561C3×C4—42no—
562C3×C4—42no—
563C3×C4—41no—
564C3×C4—81no—
565C23—21no—
566C23—22no—
567C23—22no—
568C23—42no—
569C23—41no—
570C23—81no—
571C23—81no—
572C23—82no—
573C23—82no—
574C23—82no—
575C23C3×#12011yescharacteristic
576Q8—41no—
577Q8—42no—
578Q8—42no—
579Q8—42no—
580Q8—41no—
581Q8—41no—
582D4—41no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—41no—
588D4—42no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—42no—
593D4—44no—
594D4—81no—
595D4—82no—
596D4—82no—
597D4—82no—
598D4—82no—
599D4—82no—
600D4—82no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—81no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4C3×#12011yescharacteristic
617C2×C4C3×#12011yescharacteristic
618C2×C4C3×#12011yescharacteristic
619C2×C4C22×C3×D411yescharacteristic
620C2×C4C22×C3×D411yescharacteristic
621C2×C4C22×C3×D411yescharacteristic
622C8—42no—
623C8—42no—
624C8—41no—
625C8—41no—
626C2×C3—41no—
627C2×C3—82no—
628C2×C3—81no—
629C2×C3—82no—
630C2×C3—161no—
631C2×C3#301911yescharacteristic
632C2×C3#301911yescharacteristic
633C2×C3#302111yescharacteristic
634C22—21no—
635C22—21no—
636C22—21no—
637C22—41no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—41no—
643C22—41no—
644C22—42no—
645C22—42no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—162no—
650C22—164no—
651C22C2×C3×#12011yescharacteristic
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—21no—
657C4—21no—
658C4—41no—
659C4—41no—
660C4—42no—
661C4—42no—
662C4—41no—
663C4—81no—
664C3#1908611yescharacteristic, direct product factor, Sylow
665C2—41no—
666C2—81no—
667C2—82no—
668C2—82no—
669C2—161no—
670C2C3×#301911yescharacteristic
671C2C3×#301911yescharacteristic
672C2C3×#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×#3558C2semidirect
2. C3×#3219C2semidirect
3. C3×#3214C2semidirect
4. C3×#3213C2semidirect
5. C3×#1662C2semidirect
6. C3×#1659C2semidirect
7. C3×#1656C2semidirect
8. C3×#1655C2semidirect
9. C3×#1536C2semidirect
10. C3×#1535C2semidirect
11. C3×#1533C2semidirect
12. #19086C3direct (N × H)
26. C3×#532C22semidirect
29. C3×#497C22semidirect
30. C3×#496C22semidirect
31. C3×#496C22semidirect
32. C3×#496C22semidirect
33. C3×#491C22semidirect
34. C3×#490C22semidirect
35. C3×#489C22semidirect
36. C3×#489C22semidirect
47. C3×#432C22semidirect
48. C3×#432C22semidirect
49. C3×#432C22semidirect
50. C3×#330C22semidirect
51. C3×#329C22semidirect
52. C3×#329C22semidirect
53. C3×#329C22semidirect
54. C3×#329C22semidirect
55. C3×#329C22semidirect
56. #3558C2×C3semidirect
57. #3219C2×C3semidirect
58. #3214C2×C3semidirect
59. #3213C2×C3semidirect
60. #1662C2×C3semidirect
61. #1659C2×C3semidirect
62. #1656C2×C3semidirect
63. #1655C2×C3semidirect
64. #1536C2×C3semidirect
65. #1535C2×C3semidirect
66. #1533C2×C3semidirect
173. #532C22×C3semidirect
176. #497C22×C3semidirect
177. #496C22×C3semidirect
178. #496C22×C3semidirect
179. #496C22×C3semidirect
180. #491C22×C3semidirect
181. #490C22×C3semidirect
182. #489C22×C3semidirect
183. #489C22×C3semidirect
194. #432C22×C3semidirect
195. #432C22×C3semidirect
196. #432C22×C3semidirect
197. #330C22×C3semidirect
198. #329C22×C3semidirect
199. #329C22×C3semidirect
200. #329C22×C3semidirect
201. #329C22×C3semidirect
202. #329C22×C3semidirect
664. C3#19086direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35582
C3×#32192
C3×#32142
C3×#32132
C3×#16622
C3×#16592
C3×#16562
C3×#16552
C3×#15362
C3×#15352
C3×#15332
#190863