Finite Group Databasebeta

← C3×#32893 #10651500

Subgroups — C3×#32893 #10651500

Order 768.

3630 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#585C211yesmaximal, characteristic
2C2×C3×#456C212yesmaximal, complemented
3C2×C3×#455C212yesmaximal, complemented
4C3×#3007C212yesmaximal, complemented
5C3×#2432C211yesmaximal, characteristic, complemented
6C3×#1793C212yesmaximal, complemented
7C3×#1792C212yesmaximal, complemented
8C3×#1789C212yesmaximal, complemented
9C3×#1785C211yesmaximal, characteristic, complemented
10#32893C311yesmaximal, 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×#137C2214yescomplemented
23C2×C3×C4⋊Q8C2211yescharacteristic, complemented
24C2×C3×#124C2211yescharacteristic, complemented
25C2×C3×C22⋊Q8C2211yescharacteristic, complemented
26C3×#456—28no—
27C3×#455—28no—
28C3×#452—24no—
29C3×#451—22no—
30C3×#450—22no—
31C2×C3×#104C2214yescomplemented
32C2×C3×#104C2214yescomplemented
33C3×#417C2214yescomplemented
34C2×C3×#101C2212yescomplemented
35C2×C3×#100C2212yescomplemented
36C3×#407C2212yescomplemented
37C3×#394—22no—
38C3×#391C2211yescharacteristic, complemented
39C3×#390C2211yescharacteristic, complemented
40C3×#385—22no—
41C3×C4×C22⋊C4C2211yescharacteristic, complemented
42C2×#585C2×C311yescharacteristic
43C2×#456C2×C312yescomplemented
44C2×#455C2×C312yescomplemented
45#3007C2×C312yescomplemented
46#2432C2×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+4D418yes—
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)D418yes—
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×#41D414yes—
80C2×C3×#41C2314yes—
81C22×C3×Q8—22no—
82C22×C3×Q8—24no—
83C22×C3×Q8D412yes—
84C22×C3×Q8C2311yescharacteristic
85C22×C3×Q8C2311yescharacteristic
86C22×C3×D4—28no—
87C22×C3×D4—22no—
88C22×C3×D4—24no—
89C22×C3×D4—22no—
90C22×C3×D4—24no—
91C22×C3×D4D412yes—
92C22×C3×D4D412yes—
93C22×C3×D4C2311yescharacteristic
94C23×C3×C4—24no—
95C23×C3×C4—22no—
96C23×C3×C4D412yes—
97C23×C3×C4C2311yescharacteristic
98C3×#137—28no—
99C3×#137—28no—
100C2×C3×Q16—24no—
101C2×C3×SD16—24no—
102C2×C3×M16C2314yes—
103C3×C4⋊Q8—24no—
104C3×#124—24no—
105C3×C22⋊Q8—22no—
106C3×C22⋊Q8—22no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—22no—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—22no—
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—21no—
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×#137C22×C314yescomplemented
138C2×C4⋊Q8C22×C311yescharacteristic, complemented
139C2×#124C22×C311yescharacteristic, complemented
140C2×C22⋊Q8C22×C311yescharacteristic, complemented
141#456—28no—
142#455—28no—
143#452—24no—
144#451—22no—
145#450—22no—
146C2×#104C22×C314yescomplemented
147C2×#104C22×C314yescomplemented
148#417C22×C314yescomplemented
149C2×#101C22×C312yescomplemented
150C2×#100C22×C312yescomplemented
151#407C22×C312yescomplemented
152#394—22no—
153#391C22×C311yescharacteristic, complemented
154#390C22×C311yescharacteristic, complemented
155#385—22no—
156C4×C22⋊C4C22×C311yescharacteristic, complemented
157C24×C3—24no—
158C24×C3—22no—
159C24×C3C2×D412yes—
160C24×C3C2×D411yescharacteristic
161C3×#41—232no—
162C3×#41—28no—
163C3×#41—28no—
164C3×#41—216no—
165C3×#41—264no—
166C3×#41—232no—
167C3×#41—216no—
168C3×#41—216no—
169C3×#41—216no—
170C3×#41—216no—
171C3×#41—216no—
172C3×#41—28no—
173C3×#41—216no—
174C3×#41—28no—
175C3×#41—216no—
176C3×#41—216no—
177C3×#41—24no—
178C3×#41—24no—
179C3×#41C2×D418yes—
180C3×#41C2×D418yes—
181C2×C3×Q8—24no—
182C2×C3×Q8—216no—
183C2×C3×Q8—216no—
184C2×C3×Q8—216no—
185C2×C3×Q8—28no—
186C2×C3×Q8—22no—
187C2×C3×Q8—22no—
188C2×C3×Q8—22no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8—22no—
192C2×C3×Q8C2×D414yes—
193C2×C3×Q8C2×D414yes—
194C2×C3×D4—28no—
195C2×C3×D4—24no—
196C2×C3×D4—264no—
197C2×C3×D4—216no—
198C2×C3×D4—216no—
199C2×C3×D4—28no—
200C2×C3×D4—28no—
201C2×C3×D4—216no—
202C2×C3×D4—24no—
203C2×C3×D4—24no—
204C2×C3×D4—216no—
205C2×C3×D4—24no—
206C2×C3×D4—24no—
207C2×C3×D4—216no—
208C2×C3×D4—28no—
209C2×C3×D4—22no—
210C2×C3×D4—24no—
211C2×C3×D4—22no—
212C2×C3×D4—22no—
213C2×C3×D4C2×D414yes—
214C2×C3×D4C2×D414yes—
215C22×C3×C4—24no—
216C22×C3×C4—28no—
217C22×C3×C4—232no—
218C22×C3×C4—216no—
219C22×C3×C4—22no—
220C22×C3×C4—24no—
221C22×C3×C4—24no—
222C22×C3×C4—22no—
223C22×C3×C4—22no—
224C22×C3×C4—24no—
225C22×C3×C4—24no—
226C22×C3×C4—22no—
227C22×C3×C4—22no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4C2×D411yescharacteristic
233C22×C3×C4C2×D412yes—
234C22×C3×C4C2×D412yes—
235C22×C3×C4C2×D412yes—
236C22×C3×C4C2411yescharacteristic
237C3×Q16—48no—
238C3×SD16—48no—
239C3×M16—24no—
240C3×M16—24no—
241C2×C3×C8—24no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—44no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—44no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—44no—
250C3×C42—22no—
251C3×C42—22no—
2522−1+4—232no—
2532−1+4—28no—
2542−1+4—24no—
2552−1+4C3×D418yes—
256H(5,𝔽2)—216no—
257H(5,𝔽2)—232no—
258H(5,𝔽2)—216no—
259H(5,𝔽2)—28no—
260H(5,𝔽2)—24no—
261H(5,𝔽2)C3×D418yes—
262C2×#41—24no—
263C2×#41—28no—
264C2×#41—24no—
265C2×#41—232no—
266C2×#41—232no—
267C2×#41—216no—
268C2×#41—216no—
269C2×#41—28no—
270C2×#41—28no—
271C2×#41—24no—
272C2×#41—24no—
273C2×#41—24no—
274C2×#41—24no—
275C2×#41—22no—
276C2×#41—22no—
277C2×#41—24no—
278C2×#41—22no—
279C2×#41—22no—
280C2×#41C3×D414yes—
281C2×#41C23×C314yes—
282C22×Q8—22no—
283C22×Q8—24no—
284C22×Q8C3×D412yes—
285C22×Q8C23×C311yescharacteristic
286C22×Q8C23×C311yescharacteristic
287C22×D4—28no—
288C22×D4—22no—
289C22×D4—24no—
290C22×D4—22no—
291C22×D4—24no—
292C22×D4C3×D412yes—
293C22×D4C3×D412yes—
294C22×D4C23×C311yescharacteristic
295C23×C4—24no—
296C23×C4—22no—
297C23×C4C3×D412yes—
298C23×C4C23×C311yescharacteristic
299#137—28no—
300#137—28no—
301C2×Q16—24no—
302C2×SD16—24no—
303C2×M16C23×C314yes—
304C4⋊Q8—24no—
305#124—24no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4C23×C311yescharacteristic
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C42C23×C311yescharacteristic
316C2×C42C23×C311yescharacteristic
317#104—28no—
318#104—28no—
319#103—24no—
320#102—24no—
321#101—24no—
322#100—24no—
323C22⋊C8—22no—
324C22⋊C8—22no—
325#95—21no—
326#95—22no—
327C23×C3—28no—
328C23×C3—216no—
329C23×C3—22no—
330C23×C3—24no—
331C23×C3—21no—
332C23×C3—22no—
333C23×C3—21no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—82no—
339C23×C3—82no—
340C23×C3—84no—
341C23×C3—84no—
342C23×C3#12012yes—
343C23×C3#12014yes—
344C23×C3C22×D411yescharacteristic
345C3×Q8—216no—
346C3×Q8—28no—
347C3×Q8—24no—
348C3×Q8—28no—
349C3×Q8—44no—
350C3×Q8#12018yes—
351C3×D4—232no—
352C3×D4—216no—
353C3×D4—216no—
354C3×D4—216no—
355C3×D4—28no—
356C3×D4—28no—
357C3×D4—28no—
358C3×D4—28no—
359C3×D4—24no—
360C3×D4#12018yes—
361C2×C3×C4—216no—
362C2×C3×C4—28no—
363C2×C3×C4—216no—
364C2×C3×C4—28no—
365C2×C3×C4—232no—
366C2×C3×C4—28no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—24no—
371C2×C3×C4—22no—
372C2×C3×C4—24no—
373C2×C3×C4—28no—
374C2×C3×C4—28no—
375C2×C3×C4—24no—
376C2×C3×C4—22no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—42no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4#12014yes—
388C2×C3×C4#12011yescharacteristic
389C2×C3×C4#12011yescharacteristic
390C2×C3×C4C22×D412yes—
391C2×C3×C4C22×D412yes—
392C2×C3×C4C22×D412yes—
393C3×C8—44no—
394C24—24no—
395C24—22no—
396C24C2×C3×D412yes—
397C24C2×C3×D411yescharacteristic
398#41—232no—
399#41—28no—
400#41—28no—
401#41—216no—
402#41—264no—
403#41—232no—
404#41—216no—
405#41—216no—
406#41—216no—
407#41—216no—
408#41—216no—
409#41—28no—
410#41—216no—
411#41—28no—
412#41—216no—
413#41—216no—
414#41—24no—
415#41—24no—
416#41C2×C3×D418yes—
417#41C2×C3×D418yes—
418C2×Q8—24no—
419C2×Q8—216no—
420C2×Q8—216no—
421C2×Q8—216no—
422C2×Q8—28no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8C2×C3×D414yes—
430C2×Q8C2×C3×D414yes—
431C2×D4—28no—
432C2×D4—24no—
433C2×D4—264no—
434C2×D4—216no—
435C2×D4—216no—
436C2×D4—28no—
437C2×D4—28no—
438C2×D4—216no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—216no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—216no—
445C2×D4—28no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4C2×C3×D414yes—
451C2×D4C2×C3×D414yes—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—232no—
455C22×C4—216no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4C2×C3×D411yescharacteristic
470C22×C4C2×C3×D412yes—
471C22×C4C2×C3×D412yes—
472C22×C4C2×C3×D412yes—
473C22×C4C24×C311yescharacteristic
474Q16—48no—
475SD16—48no—
476M16—24no—
477M16—24no—
478C2×C8—24no—
479C4⋊C4—22no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—44no—
483C22⋊C4—42no—
484C22⋊C4—44no—
485C22⋊C4—42no—
486C22⋊C4—44no—
487C42—22no—
488C42—22no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—22no—
492C22×C3—24no—
493C22×C3—28no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—44no—
497C22×C3—42no—
498C22×C3—44no—
499C22×C3—42no—
500C22×C3—44no—
501C22×C3—81no—
502C22×C3—82no—
503C22×C3—81no—
504C22×C3—88no—
505C22×C3—816no—
506C22×C3C2×#12011yescharacteristic
507C22×C3C2×#12011yescharacteristic
508C22×C3C2×#12011yescharacteristic
509C3×C4—21no—
510C3×C4—28no—
511C3×C4—24no—
512C3×C4—21no—
513C3×C4—42no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4—81no—
517C3×C4C2×#12014yes—
518C23—28no—
519C23—216no—
520C23—22no—
521C23—24no—
522C23—21no—
523C23—22no—
524C23—21no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—82no—
530C23—82no—
531C23—84no—
532C23—84no—
533C23C3×#12012yes—
534C23C3×#12014yes—
535C23C22×C3×D411yescharacteristic
536Q8—216no—
537Q8—28no—
538Q8—24no—
539Q8—28no—
540Q8—44no—
541Q8C3×#12018yes—
542D4—232no—
543D4—216no—
544D4—216no—
545D4—216no—
546D4—28no—
547D4—28no—
548D4—28no—
549D4—28no—
550D4—24no—
551D4C3×#12018yes—
552C2×C4—216no—
553C2×C4—28no—
554C2×C4—216no—
555C2×C4—28no—
556C2×C4—232no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4C3×#12011yescharacteristic
579C2×C4C3×#12011yescharacteristic
580C2×C4C3×#12014yes—
581C2×C4C22×C3×D412yes—
582C2×C4C22×C3×D412yes—
583C2×C4C22×C3×D412yes—
584C8—44no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—42no—
588C2×C3—44no—
589C2×C3—48no—
590C2×C3#284611yescharacteristic
591C2×C3#301811yescharacteristic
592C2×C3#302211yescharacteristic
593C22—21no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—24no—
599C22—28no—
600C22—44no—
601C22—42no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—81no—
606C22—82no—
607C22—81no—
608C22—88no—
609C22—816no—
610C22C2×C3×#12011yescharacteristic
611C22C2×C3×#12011yescharacteristic
612C22C2×C3×#12011yescharacteristic
613C4—21no—
614C4—28no—
615C4—24no—
616C4—21no—
617C4—42no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4C2×C3×#12014yes—
622C3#3289311yescharacteristic, direct product factor, Sylow
623C2—21no—
624C2—21no—
625C2—42no—
626C2—44no—
627C2—48no—
628C2C3×#284611yescharacteristic
629C2C3×#301811yescharacteristic
630C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#585C2
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
2. C2×C3×#456C2semidirect
3. C2×C3×#455C2semidirect
4. C3×#3007C2semidirect
5. C3×#2432C2semidirect
6. C3×#1793C2semidirect
7. C3×#1792C2semidirect
8. C3×#1789C2semidirect
9. C3×#1785C2semidirect
10. #32893C3direct (N × H)
22. C2×C3×#137C22semidirect
23. C2×C3×C4⋊Q8C22semidirect
24. C2×C3×#124C22semidirect
25. C2×C3×C22⋊Q8C22semidirect
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
43. C2×#456C2×C3semidirect
44. C2×#455C2×C3semidirect
45. #3007C2×C3semidirect
46. #2432C2×C3semidirect
47. #1793C2×C3semidirect
48. #1792C2×C3semidirect
49. #1789C2×C3semidirect
50. #1785C2×C3semidirect
137. C2×#137C22×C3semidirect
138. C2×C4⋊Q8C22×C3semidirect
139. C2×#124C22×C3semidirect
140. C2×C22⋊Q8C22×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
622. C3#32893direct (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×#4562
C2×C3×#4552
C3×#30072
C3×#24322
C3×#17932
C3×#17922
C3×#17892
C3×#17852
#328933