Finite Group Databasebeta

← C3×#32890 #10651497

Subgroups — C3×#32890 #10651497

Order 768.

3688 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#585C211yesmaximal, characteristic, complemented
2C2×C3×#454C212yesmaximal, complemented
3C2×C3×#453C212yesmaximal, complemented
4C3×#3006C212yesmaximal, complemented
5C3×#2431C211yesmaximal, characteristic, complemented
6C3×#1793C212yesmaximal, complemented
7C3×#1792C212yesmaximal, complemented
8C3×#1789C212yesmaximal, complemented
9C3×#1785C211yesmaximal, characteristic, complemented
10#32890C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#585—28no—
12C3×#585—28no—
13C2×C3×2−1+4—22no—
14C2×C3×2−1+4C2212yes—
15C2×C3×H(5,𝔽2)—22no—
16C2×C3×H(5,𝔽2)—22no—
17C2×C3×H(5,𝔽2)C2212yes—
18C22×C3×#41—24no—
19C22×C3×#41—22no—
20C22×C3×#41C2212yes—
21C22×C3×#41C2211yescharacteristic
22C2×C3×AGL(1,ℤ/8)C2214yescomplemented
23C2×C3×#127C2211yescharacteristic, complemented
24C2×C3×#124C2211yescharacteristic, complemented
25C2×C3×#121C2211yescharacteristic, complemented
26C3×#454—28no—
27C3×#453—28no—
28C3×#449—24no—
29C3×#448—22no—
30C3×#447—22no—
31C2×C3×#104C2214yescomplemented
32C2×C3×#104C2214yescomplemented
33C3×#417C2214yescomplemented
34C2×C3×#101C2212yescomplemented
35C2×C3×#100C2212yescomplemented
36C3×#407C2212yescomplemented
37C3×#392—22no—
38C3×#391C2211yescharacteristic, complemented
39C3×#390C2211yescharacteristic, complemented
40C3×#386—22no—
41C3×C4×C22⋊C4C2211yescharacteristic, complemented
42C2×#585C2×C311yescharacteristic, complemented
43C2×#454C2×C312yescomplemented
44C2×#453C2×C312yescomplemented
45#3006C2×C312yescomplemented
46#2431C2×C311yescharacteristic, complemented
47#1793C2×C312yescomplemented
48#1792C2×C312yescomplemented
49#1789C2×C312yescomplemented
50#1785C2×C311yescharacteristic, complemented
51C3×2−1+4—232no—
52C3×2−1+4—28no—
53C3×2−1+4—24no—
54C3×2−1+4D418yescomplemented
55C3×H(5,𝔽2)—216no—
56C3×H(5,𝔽2)—232no—
57C3×H(5,𝔽2)—216no—
58C3×H(5,𝔽2)—28no—
59C3×H(5,𝔽2)—24no—
60C3×H(5,𝔽2)D418yescomplemented
61C2×C3×#41—24no—
62C2×C3×#41—28no—
63C2×C3×#41—24no—
64C2×C3×#41—232no—
65C2×C3×#41—232no—
66C2×C3×#41—216no—
67C2×C3×#41—216no—
68C2×C3×#41—28no—
69C2×C3×#41—28no—
70C2×C3×#41—24no—
71C2×C3×#41—24no—
72C2×C3×#41—24no—
73C2×C3×#41—24no—
74C2×C3×#41—22no—
75C2×C3×#41—22no—
76C2×C3×#41—24no—
77C2×C3×#41—22no—
78C2×C3×#41—22no—
79C2×C3×#41D414yescomplemented
80C2×C3×#41C2314yes—
81C22×C3×Q8—22no—
82C22×C3×Q8—24no—
83C22×C3×Q8D412yescomplemented
84C22×C3×Q8C2311yescharacteristic
85C22×C3×D4—28no—
86C22×C3×D4—22no—
87C22×C3×D4—24no—
88C22×C3×D4—22no—
89C22×C3×D4—24no—
90C22×C3×D4—22no—
91C22×C3×D4—21no—
92C22×C3×D4D412yes—
93C22×C3×D4D412yescomplemented
94C22×C3×D4C2311yescharacteristic
95C22×C3×D4C2311yescharacteristic
96C23×C3×C4—24no—
97C23×C3×C4—22no—
98C23×C3×C4D412yes—
99C23×C3×C4C2311yescharacteristic
100C3×AGL(1,ℤ/8)—28no—
101C3×AGL(1,ℤ/8)—28no—
102C2×C3×SD16—24no—
103C2×C3×D8—24no—
104C2×C3×M16C2314yes—
105C3×#127—24no—
106C3×#124—24no—
107C3×#121—22no—
108C3×#121—22no—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C3×#104—28no—
117C3×#104—28no—
118C3×#103—24no—
119C3×#102—24no—
120C3×#101—24no—
121C3×#100—24no—
122C3×C22⋊C8—22no—
123C3×C22⋊C8—22no—
124C3×#95—21no—
125C3×#95—22no—
126#585—28no—
127#585—28no—
128C2×2−1+4—22no—
129C2×2−1+4C22×C312yes—
130C2×H(5,𝔽2)—22no—
131C2×H(5,𝔽2)—22no—
132C2×H(5,𝔽2)C22×C312yes—
133C22×#41—24no—
134C22×#41—22no—
135C22×#41C22×C312yes—
136C22×#41C22×C311yescharacteristic
137C2×AGL(1,ℤ/8)C22×C314yescomplemented
138C2×#127C22×C311yescharacteristic, complemented
139C2×#124C22×C311yescharacteristic, complemented
140C2×#121C22×C311yescharacteristic, complemented
141#454—28no—
142#453—28no—
143#449—24no—
144#448—22no—
145#447—22no—
146C2×#104C22×C314yescomplemented
147C2×#104C22×C314yescomplemented
148#417C22×C314yescomplemented
149C2×#101C22×C312yescomplemented
150C2×#100C22×C312yescomplemented
151#407C22×C312yescomplemented
152#392—22no—
153#391C22×C311yescharacteristic, complemented
154#390C22×C311yescharacteristic, complemented
155#386—22no—
156C4×C22⋊C4C22×C311yescharacteristic, complemented
157C24×C3—24no—
158C24×C3—22no—
159C24×C3—21no—
160C24×C3C2×D412yes—
161C24×C3C2×D411yescharacteristic
162C3×#41—232no—
163C3×#41—28no—
164C3×#41—28no—
165C3×#41—216no—
166C3×#41—264no—
167C3×#41—232no—
168C3×#41—216no—
169C3×#41—216no—
170C3×#41—216no—
171C3×#41—216no—
172C3×#41—216no—
173C3×#41—28no—
174C3×#41—216no—
175C3×#41—28no—
176C3×#41—216no—
177C3×#41—216no—
178C3×#41—24no—
179C3×#41—24no—
180C3×#41C2×D418yes—
181C3×#41C2×D418yes—
182C2×C3×Q8—24no—
183C2×C3×Q8—216no—
184C2×C3×Q8—216no—
185C2×C3×Q8—216no—
186C2×C3×Q8—28no—
187C2×C3×Q8—22no—
188C2×C3×Q8—22no—
189C2×C3×Q8—22no—
190C2×C3×Q8C2×D414yes—
191C2×C3×Q8C2×D414yes—
192C2×C3×D4—28no—
193C2×C3×D4—24no—
194C2×C3×D4—264no—
195C2×C3×D4—216no—
196C2×C3×D4—216no—
197C2×C3×D4—28no—
198C2×C3×D4—28no—
199C2×C3×D4—216no—
200C2×C3×D4—24no—
201C2×C3×D4—24no—
202C2×C3×D4—216no—
203C2×C3×D4—24no—
204C2×C3×D4—24no—
205C2×C3×D4—216no—
206C2×C3×D4—28no—
207C2×C3×D4—22no—
208C2×C3×D4—24no—
209C2×C3×D4—22no—
210C2×C3×D4—22no—
211C2×C3×D4—22no—
212C2×C3×D4—22no—
213C2×C3×D4—22no—
214C2×C3×D4—44no—
215C2×C3×D4—42no—
216C2×C3×D4—44no—
217C2×C3×D4—82no—
218C2×C3×D4—82no—
219C2×C3×D4C2×D414yes—
220C2×C3×D4C2×D414yes—
221C22×C3×C4—24no—
222C22×C3×C4—28no—
223C22×C3×C4—232no—
224C22×C3×C4—216no—
225C22×C3×C4—22no—
226C22×C3×C4—24no—
227C22×C3×C4—24no—
228C22×C3×C4—22no—
229C22×C3×C4—22no—
230C22×C3×C4—24no—
231C22×C3×C4—24no—
232C22×C3×C4—22no—
233C22×C3×C4—22no—
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—21no—
237C22×C3×C4C2×D411yescharacteristic
238C22×C3×C4C2×D412yes—
239C22×C3×C4C2×D412yes—
240C22×C3×C4C2×D412yes—
241C22×C3×C4C2411yescharacteristic
242C3×SD16—48no—
243C3×D8—48no—
244C3×M16—24no—
245C3×M16—24no—
246C2×C3×C8—24no—
247C3×C4⋊C4—22no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—41no—
250C3×C22⋊C4—41no—
251C3×C22⋊C4—44no—
252C3×C22⋊C4—82no—
253C3×C42—22no—
254C3×C42—22no—
2552−1+4—232no—
2562−1+4—28no—
2572−1+4—24no—
2582−1+4C3×D418yescomplemented
259H(5,𝔽2)—216no—
260H(5,𝔽2)—232no—
261H(5,𝔽2)—216no—
262H(5,𝔽2)—28no—
263H(5,𝔽2)—24no—
264H(5,𝔽2)C3×D418yescomplemented
265C2×#41—24no—
266C2×#41—28no—
267C2×#41—24no—
268C2×#41—232no—
269C2×#41—232no—
270C2×#41—216no—
271C2×#41—216no—
272C2×#41—28no—
273C2×#41—28no—
274C2×#41—24no—
275C2×#41—24no—
276C2×#41—24no—
277C2×#41—24no—
278C2×#41—22no—
279C2×#41—22no—
280C2×#41—24no—
281C2×#41—22no—
282C2×#41—22no—
283C2×#41C3×D414yescomplemented
284C2×#41C23×C314yes—
285C22×Q8—22no—
286C22×Q8—24no—
287C22×Q8C3×D412yescomplemented
288C22×Q8C23×C311yescharacteristic
289C22×D4—28no—
290C22×D4—22no—
291C22×D4—24no—
292C22×D4—22no—
293C22×D4—24no—
294C22×D4—22no—
295C22×D4—21no—
296C22×D4C3×D412yes—
297C22×D4C3×D412yescomplemented
298C22×D4C23×C311yescharacteristic
299C22×D4C23×C311yescharacteristic
300C23×C4—24no—
301C23×C4—22no—
302C23×C4C3×D412yes—
303C23×C4C23×C311yescharacteristic
304AGL(1,ℤ/8)—28no—
305AGL(1,ℤ/8)—28no—
306C2×SD16—24no—
307C2×D8—24no—
308C2×M16C23×C314yes—
309#127—24no—
310#124—24no—
311#121—22no—
312#121—22no—
313C2×C4⋊C4C23×C311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C42C23×C311yescharacteristic
319C2×C42C23×C311yescharacteristic
320#104—28no—
321#104—28no—
322#103—24no—
323#102—24no—
324#101—24no—
325#100—24no—
326C22⋊C8—22no—
327C22⋊C8—22no—
328#95—21no—
329#95—22no—
330C23×C3—28no—
331C23×C3—216no—
332C23×C3—22no—
333C23×C3—24no—
334C23×C3—21no—
335C23×C3—22no—
336C23×C3—21no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3—42no—
340C23×C3—42no—
341C23×C3—42no—
342C23×C3—42no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—82no—
347C23×C3—82no—
348C23×C3—84no—
349C23×C3—84no—
350C23×C3#12012yes—
351C23×C3#12014yes—
352C23×C3C22×D411yescharacteristic
353C3×Q8—216no—
354C3×Q8—28no—
355C3×Q8—24no—
356C3×Q8—28no—
357C3×Q8#12018yes—
358C3×D4—232no—
359C3×D4—216no—
360C3×D4—216no—
361C3×D4—216no—
362C3×D4—28no—
363C3×D4—28no—
364C3×D4—28no—
365C3×D4—28no—
366C3×D4—24no—
367C3×D4—44no—
368C3×D4—88no—
369C3×D4—84no—
370C3×D4#12018yes—
371C2×C3×C4—216no—
372C2×C3×C4—28no—
373C2×C3×C4—216no—
374C2×C3×C4—28no—
375C2×C3×C4—232no—
376C2×C3×C4—28no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—24no—
381C2×C3×C4—22no—
382C2×C3×C4—24no—
383C2×C3×C4—28no—
384C2×C3×C4—28no—
385C2×C3×C4—24no—
386C2×C3×C4—22no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—42no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4#12014yes—
395C2×C3×C4#12011yescharacteristic
396C2×C3×C4#12011yescharacteristic
397C2×C3×C4C22×D412yes—
398C2×C3×C4C22×D412yes—
399C2×C3×C4C22×D412yes—
400C3×C8—44no—
401C24—24no—
402C24—22no—
403C24—21no—
404C24C2×C3×D412yes—
405C24C2×C3×D411yescharacteristic
406#41—232no—
407#41—28no—
408#41—28no—
409#41—216no—
410#41—264no—
411#41—232no—
412#41—216no—
413#41—216no—
414#41—216no—
415#41—216no—
416#41—216no—
417#41—28no—
418#41—216no—
419#41—28no—
420#41—216no—
421#41—216no—
422#41—24no—
423#41—24no—
424#41C2×C3×D418yes—
425#41C2×C3×D418yes—
426C2×Q8—24no—
427C2×Q8—216no—
428C2×Q8—216no—
429C2×Q8—216no—
430C2×Q8—28no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8C2×C3×D414yes—
435C2×Q8C2×C3×D414yes—
436C2×D4—28no—
437C2×D4—24no—
438C2×D4—264no—
439C2×D4—216no—
440C2×D4—216no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—216no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—216no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—216no—
450C2×D4—28no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—44no—
459C2×D4—42no—
460C2×D4—44no—
461C2×D4—82no—
462C2×D4—82no—
463C2×D4C2×C3×D414yes—
464C2×D4C2×C3×D414yes—
465C22×C4—24no—
466C22×C4—28no—
467C22×C4—232no—
468C22×C4—216no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4C2×C3×D411yescharacteristic
482C22×C4C2×C3×D412yes—
483C22×C4C2×C3×D412yes—
484C22×C4C2×C3×D412yes—
485C22×C4C24×C311yescharacteristic
486SD16—48no—
487D8—48no—
488M16—24no—
489M16—24no—
490C2×C8—24no—
491C4⋊C4—22no—
492C22⋊C4—42no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—44no—
496C22⋊C4—82no—
497C42—22no—
498C42—22no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—22no—
502C22×C3—24no—
503C22×C3—28no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—44no—
507C22×C3—42no—
508C22×C3—44no—
509C22×C3—42no—
510C22×C3—44no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—82no—
514C22×C3—82no—
515C22×C3—82no—
516C22×C3—88no—
517C22×C3—816no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3C2×#12011yescharacteristic
522C22×C3C2×#12011yescharacteristic
523C22×C3C2×#12011yescharacteristic
524C3×C4—21no—
525C3×C4—28no—
526C3×C4—24no—
527C3×C4—21no—
528C3×C4—42no—
529C3×C4—81no—
530C3×C4—81no—
531C3×C4C2×#12014yes—
532C23—28no—
533C23—216no—
534C23—22no—
535C23—24no—
536C23—21no—
537C23—22no—
538C23—21no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—82no—
549C23—82no—
550C23—84no—
551C23—84no—
552C23C3×#12012yes—
553C23C3×#12014yes—
554C23C22×C3×D411yescharacteristic
555Q8—216no—
556Q8—28no—
557Q8—24no—
558Q8—28no—
559Q8C3×#12018yes—
560D4—232no—
561D4—216no—
562D4—216no—
563D4—216no—
564D4—28no—
565D4—28no—
566D4—28no—
567D4—28no—
568D4—24no—
569D4—44no—
570D4—88no—
571D4—84no—
572D4C3×#12018yes—
573C2×C4—216no—
574C2×C4—28no—
575C2×C4—216no—
576C2×C4—28no—
577C2×C4—232no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4C3×#12011yescharacteristic
597C2×C4C3×#12011yescharacteristic
598C2×C4C3×#12014yes—
599C2×C4C22×C3×D412yes—
600C2×C4C22×C3×D412yes—
601C2×C4C22×C3×D412yes—
602C8—44no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—42no—
606C2×C3—44no—
607C2×C3—48no—
608C2×C3—161no—
609C2×C3#284611yescharacteristic
610C2×C3#301811yescharacteristic
611C2×C3#302111yescharacteristic
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—24no—
618C22—28no—
619C22—44no—
620C22—42no—
621C22—44no—
622C22—42no—
623C22—44no—
624C22—81no—
625C22—81no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—88no—
630C22—816no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22C2×C3×#12011yescharacteristic
635C22C2×C3×#12011yescharacteristic
636C22C2×C3×#12011yescharacteristic
637C4—21no—
638C4—28no—
639C4—24no—
640C4—21no—
641C4—42no—
642C4—81no—
643C4—81no—
644C4C2×C3×#12014yes—
645C3#3289011yescharacteristic, direct product factor, Sylow
646C2—21no—
647C2—21no—
648C2—42no—
649C2—44no—
650C2—48no—
651C2—161no—
652C2C3×#284611yescharacteristic
653C2C3×#301811yescharacteristic
654C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#585C2semidirect
2. C2×C3×#454C2semidirect
3. C2×C3×#453C2semidirect
4. C3×#3006C2semidirect
5. C3×#2431C2semidirect
6. C3×#1793C2semidirect
7. C3×#1792C2semidirect
8. C3×#1789C2semidirect
9. C3×#1785C2semidirect
10. #32890C3direct (N × H)
22. C2×C3×AGL(1,ℤ/8)C22semidirect
23. C2×C3×#127C22semidirect
24. C2×C3×#124C22semidirect
25. C2×C3×#121C22semidirect
31. C2×C3×#104C22semidirect
32. C2×C3×#104C22semidirect
33. C3×#417C22semidirect
34. C2×C3×#101C22semidirect
35. C2×C3×#100C22semidirect
36. C3×#407C22semidirect
38. C3×#391C22semidirect
39. C3×#390C22semidirect
41. C3×C4×C22⋊C4C22semidirect
42. C2×#585C2×C3semidirect
43. C2×#454C2×C3semidirect
44. C2×#453C2×C3semidirect
45. #3006C2×C3semidirect
46. #2431C2×C3semidirect
47. #1793C2×C3semidirect
48. #1792C2×C3semidirect
49. #1789C2×C3semidirect
50. #1785C2×C3semidirect
54. C3×2−1+4D4semidirect
60. C3×H(5,𝔽2)D4semidirect
79. C2×C3×#41D4semidirect
83. C22×C3×Q8D4semidirect
93. C22×C3×D4D4semidirect
137. C2×AGL(1,ℤ/8)C22×C3semidirect
138. C2×#127C22×C3semidirect
139. C2×#124C22×C3semidirect
140. C2×#121C22×C3semidirect
146. C2×#104C22×C3semidirect
147. C2×#104C22×C3semidirect
148. #417C22×C3semidirect
149. C2×#101C22×C3semidirect
150. C2×#100C22×C3semidirect
151. #407C22×C3semidirect
153. #391C22×C3semidirect
154. #390C22×C3semidirect
156. C4×C22⋊C4C22×C3semidirect
258. 2−1+4C3×D4semidirect
264. H(5,𝔽2)C3×D4semidirect
283. C2×#41C3×D4semidirect
287. C22×Q8C3×D4semidirect
297. C22×D4C3×D4semidirect
645. C3#32890direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5852
C2×C3×#4542
C2×C3×#4532
C3×#30062
C3×#24312
C3×#17932
C3×#17922
C3×#17892
C3×#17852
#328903