Finite Group Databasebeta

← C3×#19087 #10637694

Subgroups — C3×#19087 #10637694

Order 768.

930 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3558C211yesmaximal, characteristic
2C3×#3240C211yesmaximal, characteristic, complemented
3C3×#3216C211yesmaximal, characteristic, complemented
4C3×#3215C211yesmaximal, characteristic, complemented
5C3×#1661C212yesmaximal, complemented
6C3×#1661C212yesmaximal, complemented
7C3×#1659C212yesmaximal, complemented
8C3×#1659C212yesmaximal, complemented
9C3×#1536C211yesmaximal, characteristic, complemented
10C3×#1535C211yesmaximal, characteristic, complemented
11C3×#1533C211yesmaximal, characteristic, complemented
12#19087C311yesmaximal, 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×#533C2211yescharacteristic, complemented
26C3×#532C2211yescharacteristic
27C3×#529C2211yescharacteristic
28C2×C3×#124C2211yescharacteristic
29C3×#501C2211yescharacteristic, complemented
30C3×#501C2211yescharacteristic, complemented
31C3×#497C2212yescomplemented
32C3×#497C2212yescomplemented
33C3×#489C2212yescomplemented
34C3×#489C2212yescomplemented
35C3×#489C2211yescharacteristic, complemented
36C3×#489C2211yescharacteristic, complemented
37C3×#485—21no—
38C3×#482—21no—
39C3×#479—21no—
40C3×#478—21no—
41C3×#464—22no—
42C3×#462—21no—
43C3×#460—21no—
44C3×#452—22no—
45C3×#451—21no—
46C3×#450—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
57#3240C2×C311yescharacteristic, complemented
58#3216C2×C311yescharacteristic, complemented
59#3215C2×C311yescharacteristic, complemented
60#1661C2×C312yescomplemented
61#1661C2×C312yescomplemented
62#1659C2×C312yescomplemented
63#1659C2×C312yescomplemented
64#1536C2×C311yescharacteristic, complemented
65#1535C2×C311yescharacteristic, complemented
66#1533C2×C311yescharacteristic, complemented
67C2×C3×#41—21no—
68C2×C3×#41D412yes—
69C22×C3×Q8D411yescharacteristic
70C22×C3×D4—22no—
71C22×C3×D4—21no—
72C22×C3×D4D411yescharacteristic
73C2×C3×Q16—22no—
74C2×C3×Q16—22no—
75C2×C3×SD16—22no—
76C2×C3×SD16—22no—
77C3×C4⋊Q8D411yescharacteristic
78C3×C4⋊Q8C2311yescharacteristic
79C3×C4⋊Q8C2311yescharacteristic
80C3×#127—21no—
81C3×#127D411yescharacteristic
82C3×#126—22no—
83C3×#126—21no—
84C3×#125—21no—
85C3×#125C2311yescharacteristic
86C3×#124—21no—
87C3×#124—22no—
88C3×#124—22no—
89C3×#124—22no—
90C3×#124—21no—
91C3×#124—21no—
92C3×#124D412yes—
93C3×#124C2311yescharacteristic
94C3×#124C2312yes—
95C3×#124C2312yes—
96C3×#123—22no—
97C3×#123—22no—
98C3×#123—21no—
99C3×#123—22no—
100C3×#123—22no—
101C3×C22⋊Q8—21no—
102C3×C22⋊Q8—22no—
103C3×C22⋊Q8—22no—
104C3×C22⋊Q8—21no—
105C3×C22⋊Q8—21no—
106C3×C22⋊Q8D411yescharacteristic
107C3×#121—22no—
108C3×#121—22no—
109C3×#121—22no—
110C3×#121—21no—
111C3×#121—22no—
112C3×#121—22no—
113C3×#121—21no—
114C3×#121—21no—
115C3×#121D411yescharacteristic
116C3×#120—22no—
117C3×#120—22no—
118C3×#120—22no—
119C3×#120—22no—
120C3×#120—22no—
121C3×C4×Q8—21no—
122C3×C4×Q8—21no—
123C3×C4×Q8—21no—
124C3×C4×Q8D411yescharacteristic
125C3×C4×D4—22no—
126C3×C4×D4—21no—
127C3×C4×D4—22no—
128C3×C4×D4—21no—
129C3×C4×D4—21no—
130C3×C4×D4—21no—
131C3×C4×D4D411yescharacteristic
132C3×#117—21no—
133C3×#117—21no—
134C2×C3×C22⋊C4—22no—
135C2×C3×C22⋊C4—21no—
136C2×C3×C22⋊C4—21no—
137C2×C3×C42C2311yescharacteristic
138C3×#107—21no—
139C3×#107—21no—
140C3×#106—21no—
141C3×#106—21no—
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#533C22×C311yescharacteristic, complemented
173#532C22×C311yescharacteristic
174#529C22×C311yescharacteristic
175C2×#124C22×C311yescharacteristic
176#501C22×C311yescharacteristic, complemented
177#501C22×C311yescharacteristic, complemented
178#497C22×C312yescomplemented
179#497C22×C312yescomplemented
180#489C22×C311yescharacteristic, complemented
181#489C22×C311yescharacteristic, complemented
182#489C22×C312yescomplemented
183#489C22×C312yescomplemented
184#485—21no—
185#482—21no—
186#479—21no—
187#478—21no—
188#464—22no—
189#462—21no—
190#460—21no—
191#452—22no—
192#451—21no—
193#450—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—82no—
206C3×#41—82no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—81no—
211C2×C3×Q8C2×D411yescharacteristic
212C2×C3×Q8C2×D412yes—
213C2×C3×Q8C2×D412yes—
214C2×C3×D4—22no—
215C2×C3×D4—21no—
216C2×C3×D4—22no—
217C2×C3×D4—22no—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—81no—
225C2×C3×D4—82no—
226C2×C3×D4—84no—
227C2×C3×D4C2×D411yescharacteristic
228C2×C3×D4C2×D412yes—
229C2×C3×D4C2×D412yes—
230C22×C3×C4—22no—
231C22×C3×C4—22no—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4—22no—
235C22×C3×C4C2×D411yescharacteristic
236C22×C3×C4C2×D411yescharacteristic
237C22×C3×C4C2×D411yescharacteristic
238C3×Q16—82no—
239C3×Q16—82no—
240C3×SD16—82no—
241C3×SD16—82no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—22no—
245C2×C3×C8—22no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
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—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—41no—
259C3×C4⋊C4—41no—
260C3×C4⋊C4C2×D411yescharacteristic
261C3×C4⋊C4C2×D411yescharacteristic
262C3×C22⋊C4—22no—
263C3×C22⋊C4—22no—
264C3×C22⋊C4—22no—
265C3×C22⋊C4—22no—
266C3×C22⋊C4—22no—
267C3×C22⋊C4—21no—
268C3×C22⋊C4—22no—
269C3×C22⋊C4—21no—
270C3×C22⋊C4—22no—
271C3×C22⋊C4—21no—
272C3×C22⋊C4—22no—
273C3×C22⋊C4—22no—
274C3×C22⋊C4—21no—
275C3×C22⋊C4—21no—
276C3×C22⋊C4—22no—
277C3×C22⋊C4—22no—
278C3×C22⋊C4—41no—
279C3×C42—22no—
280C3×C42—21no—
281C3×C42—41no—
282C3×C42C2×D411yescharacteristic
283C3×C42C2×D411yescharacteristic
284C3×C42C2×D411yescharacteristic
285C3×C42C2411yescharacteristic
286C2×#41—21no—
287C2×#41C3×D412yes—
288C22×Q8C3×D411yescharacteristic
289C22×D4—22no—
290C22×D4—21no—
291C22×D4C3×D411yescharacteristic
292C2×Q16—22no—
293C2×Q16—22no—
294C2×SD16—22no—
295C2×SD16—22no—
296C4⋊Q8C3×D411yescharacteristic
297C4⋊Q8C23×C311yescharacteristic
298C4⋊Q8C23×C311yescharacteristic
299#127—21no—
300#127C3×D411yescharacteristic
301#126—22no—
302#126—21no—
303#125—21no—
304#125C23×C311yescharacteristic
305#124—21no—
306#124—22no—
307#124—22no—
308#124—22no—
309#124—21no—
310#124—21no—
311#124C3×D412yes—
312#124C23×C312yes—
313#124C23×C311yescharacteristic
314#124C23×C312yes—
315#123—22no—
316#123—22no—
317#123—21no—
318#123—22no—
319#123—22no—
320C22⋊Q8—21no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325C22⋊Q8C3×D411yescharacteristic
326#121—22no—
327#121—22no—
328#121—22no—
329#121—21no—
330#121—22no—
331#121—22no—
332#121—21no—
333#121—21no—
334#121C3×D411yescharacteristic
335#120—22no—
336#120—22no—
337#120—22no—
338#120—22no—
339#120—22no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8—21no—
343C4×Q8C3×D411yescharacteristic
344C4×D4—22no—
345C4×D4—22no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4—21no—
350C4×D4C3×D411yescharacteristic
351#117—21no—
352#117—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C42C23×C311yescharacteristic
357#107—21no—
358#107—21no—
359#106—21no—
360#106—21no—
361C4⋊C8—21no—
362C4⋊C8—21no—
363C4⋊C8—21no—
364#103—22no—
365#103—22no—
366#103—21no—
367#103—21no—
368#102—22no—
369#102—22no—
370#102—21no—
371#102—21no—
372C22⋊C8—21no—
373C22⋊C8—21no—
374C22⋊C8—21no—
375#97C23×C311yescharacteristic
376#97C23×C311yescharacteristic
377#97C23×C312yes—
378#97C23×C312yes—
379C23×C3—21no—
380C23×C3—22no—
381C23×C3—22no—
382C23×C3—42no—
383C23×C3—81no—
384C23×C3—81no—
385C23×C3—82no—
386C23×C3—82no—
387C23×C3—82no—
388C23×C3#12011yescharacteristic
389C3×Q8—42no—
390C3×Q8—42no—
391C3×Q8—41no—
392C3×Q8—42no—
393C3×Q8—41no—
394C3×Q8—42no—
395C3×Q8—41no—
396C3×D4—42no—
397C3×D4—42no—
398C3×D4—41no—
399C3×D4—42no—
400C3×D4—42no—
401C3×D4—42no—
402C3×D4—44no—
403C3×D4—81no—
404C3×D4—82no—
405C3×D4—82no—
406C3×D4—82no—
407C3×D4—82no—
408C3×D4—82no—
409C2×C3×C4—21no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—42no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4—82no—
421C2×C3×C4—81no—
422C2×C3×C4—82no—
423C2×C3×C4—82no—
424C2×C3×C4#12011yescharacteristic
425C2×C3×C4#12011yescharacteristic
426C2×C3×C4#12011yescharacteristic
427C2×C3×C4C22×D411yescharacteristic
428C2×C3×C4C22×D411yescharacteristic
429C2×C3×C4C22×D411yescharacteristic
430C3×C8—41no—
431C3×C8—41no—
432C3×C8—42no—
433C3×C8—42no—
434C24—22no—
435C24—21no—
436#41—82no—
437#41—82no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—81no—
442C2×Q8C2×C3×D411yescharacteristic
443C2×Q8C2×C3×D412yes—
444C2×Q8C2×C3×D412yes—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—81no—
456C2×D4—82no—
457C2×D4—84no—
458C2×D4C2×C3×D411yescharacteristic
459C2×D4C2×C3×D412yes—
460C2×D4C2×C3×D412yes—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4C2×C3×D411yescharacteristic
467C22×C4C2×C3×D411yescharacteristic
468C22×C4C2×C3×D411yescharacteristic
469Q16—82no—
470Q16—82no—
471SD16—82no—
472SD16—82no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—22no—
476C2×C8—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4C2×C3×D411yescharacteristic
492C4⋊C4C2×C3×D411yescharacteristic
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—21no—
501C22⋊C4—22no—
502C22⋊C4—21no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—41no—
510C42—22no—
511C42—21no—
512C42—41no—
513C42C2×C3×D411yescharacteristic
514C42C2×C3×D411yescharacteristic
515C42C2×C3×D411yescharacteristic
516C42C24×C311yescharacteristic
517C22×C3—21no—
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—41no—
521C22×C3—42no—
522C22×C3—42no—
523C22×C3—42no—
524C22×C3—42no—
525C22×C3—41no—
526C22×C3—41no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—162no—
530C22×C3—164no—
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—41no—
540C3×C4—42no—
541C3×C4—42no—
542C3×C4—41no—
543C3×C4—81no—
544C3×C4—81no—
545C23—21no—
546C23—22no—
547C23—22no—
548C23—42no—
549C23—81no—
550C23—81no—
551C23—82no—
552C23—82no—
553C23—82no—
554C23C3×#12011yescharacteristic
555Q8—42no—
556Q8—42no—
557Q8—41no—
558Q8—42no—
559Q8—41no—
560Q8—42no—
561Q8—41no—
562D4—42no—
563D4—42no—
564D4—41no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—44no—
569D4—81no—
570D4—82no—
571D4—82no—
572D4—82no—
573D4—82no—
574D4—82no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—82no—
587C2×C4—81no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4C3×#12011yescharacteristic
591C2×C4C3×#12011yescharacteristic
592C2×C4C3×#12011yescharacteristic
593C2×C4C22×C3×D411yescharacteristic
594C2×C4C22×C3×D411yescharacteristic
595C2×C4C22×C3×D411yescharacteristic
596C8—42no—
597C8—42no—
598C8—41no—
599C8—41no—
600C2×C3—41no—
601C2×C3—82no—
602C2×C3—81no—
603C2×C3—82no—
604C2×C3#301911yescharacteristic
605C2×C3#301911yescharacteristic
606C2×C3#302211yescharacteristic
607C22—21no—
608C22—21no—
609C22—21no—
610C22—41no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—162no—
620C22—164no—
621C22C2×C3×#12011yescharacteristic
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—81no—
634C4—81no—
635C3#1908711yescharacteristic, direct product factor, Sylow
636C2—41no—
637C2—81no—
638C2—82no—
639C2—82no—
640C2C3×#301911yescharacteristic
641C2C3×#301911yescharacteristic
642C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3558C2
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
2. C3×#3240C2semidirect
3. C3×#3216C2semidirect
4. C3×#3215C2semidirect
5. C3×#1661C2semidirect
6. C3×#1661C2semidirect
7. C3×#1659C2semidirect
8. C3×#1659C2semidirect
9. C3×#1536C2semidirect
10. C3×#1535C2semidirect
11. C3×#1533C2semidirect
12. #19087C3direct (N × H)
25. C3×#533C22semidirect
29. C3×#501C22semidirect
30. C3×#501C22semidirect
31. C3×#497C22semidirect
32. C3×#497C22semidirect
33. C3×#489C22semidirect
34. C3×#489C22semidirect
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
57. #3240C2×C3semidirect
58. #3216C2×C3semidirect
59. #3215C2×C3semidirect
60. #1661C2×C3semidirect
61. #1661C2×C3semidirect
62. #1659C2×C3semidirect
63. #1659C2×C3semidirect
64. #1536C2×C3semidirect
65. #1535C2×C3semidirect
66. #1533C2×C3semidirect
172. #533C22×C3semidirect
176. #501C22×C3semidirect
177. #501C22×C3semidirect
178. #497C22×C3semidirect
179. #497C22×C3semidirect
180. #489C22×C3semidirect
181. #489C22×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
635. C3#19087direct (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×#32402
C3×#32162
C3×#32152
C3×#16612
C3×#16592
C3×#15362
C3×#15352
C3×#15332
#190873