Finite Group Databasebeta

← C3×#24213 #10642820

Subgroups — C3×#24213 #10642820

Order 768.

1482 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic, complemented
3C2×C3×#460C211yesmaximal, characteristic, complemented
4C2×C3×#452C212yesmaximal, complemented
5C3×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
6C3×#2042C212yesmaximal, complemented
7C3×#2015C212yesmaximal, complemented
8C3×#1982C211yesmaximal, characteristic, complemented
9C3×#1981C211yesmaximal, characteristic, complemented
10C3×#1797C211yesmaximal, characteristic, complemented
11C3×#1788C212yesmaximal, complemented
12#24213C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C3×#532—24no—
17C3×#532—24no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×C3×#124C2212yes—
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×#121—21no—
23C2×C3×#121—21no—
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4—21no—
27C22×C3×C42C2211yescharacteristic
28C3×#462—24no—
29C3×#460—24no—
30C3×#452—28no—
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×#102C2212yes—
39C2×C3×C22⋊C8C2212yes—
40C3×#391C2211yescharacteristic
41C3×#389C2211yescharacteristic
42C3×#387—21no—
43C3×#385—21no—
44C3×#380C2211yescharacteristic
45C3×C4×C4⋊C4—21no—
46C3×C4×C4⋊C4C2211yescharacteristic
47C3×C4×C22⋊C4—21no—
48C3×#336C2212yes—
49C2×#532C2×C311yescharacteristic
50C2×#462C2×C311yescharacteristic, complemented
51C2×#460C2×C311yescharacteristic, complemented
52C2×#452C2×C312yescomplemented
53C4×C22⋊Q8C2×C311yescharacteristic, complemented
54#2042C2×C312yescomplemented
55#2015C2×C312yescomplemented
56#1982C2×C311yescharacteristic, complemented
57#1981C2×C311yescharacteristic, complemented
58#1797C2×C311yescharacteristic, complemented
59#1788C2×C312yescomplemented
60C2×C3×#41—24no—
61C2×C3×#41—24no—
62C2×C3×#41—24no—
63C2×C3×#41—24no—
64C2×C3×#41—42no—
65C2×C3×#41—42no—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×Q8C2312yes—
68C22×C3×D4—22no—
69C22×C3×D4—21no—
70C22×C3×D4—21no—
71C22×C3×D4C2312yes—
72C23×C3×C4—22no—
73C23×C3×C4D411yescharacteristic
74C23×C3×C4D411yescharacteristic
75C23×C3×C4C2311yescharacteristic
76C2×C3×Q16—24no—
77C2×C3×Q16—24no—
78C2×C3×Q16—42no—
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C2×C3×SD16—42no—
82C22×C3×C8C2312yes—
83C22×C3×C8C2312yes—
84C3×C4⋊Q8—22no—
85C3×C4⋊Q8D414yes—
86C3×#127—22no—
87C3×#127D414yes—
88C3×#124—24no—
89C3×#124—24no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8D414yescomplemented
92C3×#121—28no—
93C3×#121—28no—
94C3×C4×Q8—22no—
95C3×C4×Q8—22no—
96C3×C4×D4—28no—
97C3×C4×D4—28no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C42—22no—
111C2×C3×C42—22no—
112C2×C3×C42—21no—
113C2×C3×C42—21no—
114C2×C3×C42D411yescharacteristic
115C2×C3×C42D411yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C3×C4⋊C8—22no—
119C3×C4⋊C8—22no—
120C3×#103—24no—
121C3×#103—24no—
122C3×#103—42no—
123C3×#102—24no—
124C3×#102—24no—
125C3×#102—42no—
126C3×C22⋊C8—24no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C22×#41C22×C312yes—
133C22×Q16C22×C312yes—
134C22×SD16C22×C312yes—
135#532—24no—
136#532—24no—
137C2×C4⋊Q8C22×C311yescharacteristic
138C2×#127C22×C311yescharacteristic
139C2×#124C22×C312yes—
140C2×C22⋊Q8C22×C311yescharacteristic
141C2×#121—21no—
142C2×#121—21no—
143C2×C4×Q8C22×C311yescharacteristic
144C2×C4×D4—21no—
145C2×C4×D4—21no—
146C22×C42C22×C311yescharacteristic
147#462—24no—
148#460—24no—
149#452—28no—
150C2×C4⋊C8C22×C311yescharacteristic
151C2×C4⋊C8C22×C311yescharacteristic
152C2×#103C22×C312yes—
153C2×#103C22×C312yes—
154C2×#103C22×C312yes—
155C2×#102C22×C312yes—
156C2×#102C22×C312yes—
157C2×#102C22×C312yes—
158C2×C22⋊C8C22×C312yes—
159#391C22×C311yescharacteristic
160#389C22×C311yescharacteristic
161#387—21no—
162#385—21no—
163#380C22×C311yescharacteristic
164C4×C4⋊C4—21no—
165C4×C4⋊C4C22×C311yescharacteristic
166C4×C22⋊C4—21no—
167#336C22×C312yes—
168C24×C3—22no—
169C24×C3C2×D411yescharacteristic
170C3×#41—48no—
171C3×#41—48no—
172C3×#41—48no—
173C3×#41—48no—
174C2×C3×Q8—22no—
175C2×C3×Q8—21no—
176C2×C3×Q8—22no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8—22no—
180C2×C3×Q8—22no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—22no—
184C2×C3×Q8C2×D414yes—
185C2×C3×Q8C2×D412yes—
186C2×C3×D4—28no—
187C2×C3×D4—24no—
188C2×C3×D4—24no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—42no—
195C2×C3×D4—42no—
196C2×C3×D4—44no—
197C2×C3×D4—44no—
198C2×C3×D4—42no—
199C2×C3×D4—42no—
200C2×C3×D4C2×D414yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—28no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—41no—
216C22×C3×C4—41no—
217C22×C3×C4—44no—
218C22×C3×C4—44no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D412yes—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4#4112yes—
226C22×C3×C4C2411yescharacteristic
227C3×Q16—48no—
228C3×SD16—48no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8—22no—
233C2×C3×C8—22no—
234C2×C3×C8—22no—
235C3×C4⋊C4—24no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—42no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4C2×D412yes—
246C3×C22⋊C4—28no—
247C3×C22⋊C4—28no—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—42no—
250C3×C42—22no—
251C3×C42—22no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—42no—
255C3×C42—42no—
256C3×C42C2×D412yes—
257C3×C42C2×D412yes—
258C2×#41—24no—
259C2×#41—24no—
260C2×#41—24no—
261C2×#41—24no—
262C2×#41—42no—
263C2×#41—42no—
264C22×Q8C23×C311yescharacteristic
265C22×Q8C23×C312yes—
266C22×D4—22no—
267C22×D4—21no—
268C22×D4—21no—
269C22×D4C23×C312yes—
270C23×C4—22no—
271C23×C4C3×D411yescharacteristic
272C23×C4C3×D411yescharacteristic
273C23×C4C23×C311yescharacteristic
274C2×Q16—24no—
275C2×Q16—24no—
276C2×Q16—42no—
277C2×SD16—24no—
278C2×SD16—24no—
279C2×SD16—42no—
280C22×C8C23×C312yes—
281C22×C8C23×C312yes—
282C4⋊Q8—22no—
283C4⋊Q8C3×D414yes—
284#127—22no—
285#127C3×D414yes—
286#124—24no—
287#124—24no—
288C22⋊Q8—22no—
289C22⋊Q8C3×D414yescomplemented
290#121—28no—
291#121—28no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×D4—28no—
295C4×D4—28no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4C23×C311yescharacteristic
302C2×C4⋊C4C23×C311yescharacteristic
303C2×C4⋊C4C23×C311yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42C3×D411yescharacteristic
313C2×C42C3×D411yescharacteristic
314C2×C42C23×C311yescharacteristic
315C2×C42C23×C311yescharacteristic
316C4⋊C8—22no—
317C4⋊C8—22no—
318#103—24no—
319#103—24no—
320#103—42no—
321#102—24no—
322#102—24no—
323#102—42no—
324C22⋊C8—24no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—24no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—42no—
340C23×C3—42no—
341C23×C3—42no—
342C23×C3—42no—
343C23×C3#12112yes—
344C23×C3C22×D411yescharacteristic
345C3×Q8—22no—
346C3×Q8—22no—
347C3×Q8—24no—
348C3×Q8—24no—
349C3×Q8—42no—
350C3×Q8—41no—
351C3×Q8—41no—
352C3×Q8—42no—
353C3×D4—24no—
354C3×D4—24no—
355C3×D4—48no—
356C3×D4—416no—
357C3×D4—42no—
358C3×D4—42no—
359C3×D4—48no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—24no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—44no—
378C2×C3×C4—416no—
379C2×C3×C4—41no—
380C2×C3×C4—42no—
381C2×C3×C4—42no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—44no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4#12012yes—
393C2×C3×C4#12012yes—
394C2×C3×C4#12112yes—
395C2×C3×C4#13512yes—
396C2×C3×C4#13712yes—
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4H(5,𝔽2)11yescharacteristic
402C2×C3×C4H(5,𝔽2)11yescharacteristic
403C3×C8—42no—
404C3×C8—42no—
405C24—22no—
406C24C2×C3×D411yescharacteristic
407#41—48no—
408#41—48no—
409#41—48no—
410#41—48no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8C2×C3×D414yes—
422C2×Q8C2×C3×D412yes—
423C2×D4—28no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4C2×C3×D414yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—28no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4C2×C3×D411yescharacteristic
457C22×C4C2×C3×D411yescharacteristic
458C22×C4C2×C3×D411yescharacteristic
459C22×C4C2×C3×D411yescharacteristic
460C22×C4C2×C3×D412yes—
461C22×C4C2×C3×D411yescharacteristic
462C22×C4C3×#4112yes—
463C22×C4C24×C311yescharacteristic
464Q16—48no—
465SD16—48no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4C2×C3×D412yes—
483C22⋊C4—28no—
484C22⋊C4—28no—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C42—22no—
488C42—22no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—42no—
493C42C2×C3×D412yes—
494C42C2×C3×D412yes—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—44no—
503C22×C3—44no—
504C22×C3—44no—
505C22×C3—82no—
506C22×C3—82no—
507C22×C3—82no—
508C22×C3—82no—
509C22×C3C2×#12011yescharacteristic
510C22×C3C2×#12111yescharacteristic
511C22×C3#53511yescharacteristic
512C22×C3C2×#13511yescharacteristic
513C22×C3C2×#13711yescharacteristic
514C22×C3#57511yescharacteristic
515C22×C3#57511yescharacteristic
516C3×C4—22no—
517C3×C4—22no—
518C3×C4—44no—
519C3×C4—42no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4—42no—
523C3×C4—81no—
524C3×C4—81no—
525C3×C4#45214yescomplemented
526C3×C4D4211yescharacteristic
527C3×C4D4211yescharacteristic
528C3×C4#54611yescharacteristic
529C3×C4#54611yescharacteristic
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—24no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23C3×#12112yes—
544C23C22×C3×D411yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—24no—
548Q8—24no—
549Q8—42no—
550Q8—42no—
551Q8—41no—
552Q8—41no—
553D4—24no—
554D4—24no—
555D4—48no—
556D4—416no—
557D4—48no—
558D4—42no—
559D4—42no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—416no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4C3×#12012yes—
593C2×C4C3×#12012yes—
594C2×C4C3×#12112yes—
595C2×C4C3×#13512yes—
596C2×C4C3×#13712yes—
597C2×C4C22×C3×D411yescharacteristic
598C2×C4C22×C3×D411yescharacteristic
599C2×C4C22×C3×D411yescharacteristic
600C2×C4C2×C3×#4111yescharacteristic
601C2×C4C3×H(5,𝔽2)11yescharacteristic
602C2×C4C3×H(5,𝔽2)11yescharacteristic
603C8—42no—
604C8—42no—
605C2×C3—42no—
606C2×C3—84no—
607C2×C3#240811yescharacteristic
608C2×C3C2×#45211yescharacteristic
609C2×C3#300511yescharacteristic
610C2×C3#303511yescharacteristic
611C2×C3#303711yescharacteristic
612C2×C3#321211yescharacteristic
613C2×C3#321611yescharacteristic
614C22—22no—
615C22—22no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C2×C3×#12011yescharacteristic
629C22C2×C3×#12111yescharacteristic
630C22C3×#53511yescharacteristic
631C22C2×C3×#13511yescharacteristic
632C22C2×C3×#13711yescharacteristic
633C22C3×#57511yescharacteristic
634C22C3×#57511yescharacteristic
635C4—22no—
636C4—22no—
637C4—44no—
638C4—42no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—81no—
643C4—81no—
644C4C3×#45214yescomplemented
645C4C3×D4211yescharacteristic
646C4C3×D4211yescharacteristic
647C4C3×#54611yescharacteristic
648C4C3×#54611yescharacteristic
649C3#2421311yescharacteristic, direct product factor, Sylow
650C2—42no—
651C2—84no—
652C2C3×#240811yescharacteristic
653C2C2×C3×#45211yescharacteristic
654C2C3×#300511yescharacteristic
655C2C3×#303511yescharacteristic
656C2C3×#303711yescharacteristic
657C2C3×#321211yescharacteristic
658C2C3×#321611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#532C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#462C2semidirect
3. C2×C3×#460C2semidirect
4. C2×C3×#452C2semidirect
5. C3×C4×C22⋊Q8C2semidirect
6. C3×#2042C2semidirect
7. C3×#2015C2semidirect
8. C3×#1982C2semidirect
9. C3×#1981C2semidirect
10. C3×#1797C2semidirect
11. C3×#1788C2semidirect
12. #24213C3direct (N × H)
50. C2×#462C2×C3semidirect
51. C2×#460C2×C3semidirect
52. C2×#452C2×C3semidirect
53. C4×C22⋊Q8C2×C3semidirect
54. #2042C2×C3semidirect
55. #2015C2×C3semidirect
56. #1982C2×C3semidirect
57. #1981C2×C3semidirect
58. #1797C2×C3semidirect
59. #1788C2×C3semidirect
91. C3×C22⋊Q8D4semidirect
289. C22⋊Q8C3×D4semidirect
525. C3×C4#452semidirect
644. C4C3×#452semidirect
649. C3#24213direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4622
C2×C3×#4602
C2×C3×#4522
C3×C4×C22⋊Q82
C3×#20422
C3×#20152
C3×#19822
C3×#19812
C3×#17972
C3×#17882
#242133