Finite Group Databasebeta

← C3×#24214 #10642821

Subgroups — C3×#24214 #10642821

Order 768.

1566 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C2×C3×#463C212yesmaximal, complemented
3C2×C3×#449C212yesmaximal, complemented
4C3×C4×#121C211yesmaximal, characteristic, complemented
5C3×#2043C211yesmaximal, characteristic, complemented
6C3×#2041C211yesmaximal, characteristic, complemented
7C3×#2013C211yesmaximal, characteristic, complemented
8C3×#2011C211yesmaximal, characteristic, complemented
9C3×#1983C212yesmaximal, complemented
10C3×#1797C211yesmaximal, characteristic, complemented
11C3×#1788C212yesmaximal, complemented
12#24214C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C22×C3×SD16C2212yes—
15C22×C3×D8C2212yes—
16C3×#532—28no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#127C2211yescharacteristic
19C2×C3×#124C2212yes—
20C2×C3×#121—21no—
21C2×C3×#121—21no—
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4C2211yescharacteristic
27C22×C3×C42C2211yescharacteristic
28C3×#463—28no—
29C3×#449—28no—
30C2×C3×C4⋊C8C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C3×#390C2211yescharacteristic
39C3×#389C2211yescharacteristic
40C3×#388—21no—
41C3×#386—21no—
42C3×#380C2211yescharacteristic
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×C4×C22⋊C4—21no—
45C3×#336C2211yescharacteristic
46C3×#336C2211yescharacteristic
47C2×#532C2×C311yescharacteristic, complemented
48C2×#463C2×C312yescomplemented
49C2×#449C2×C312yescomplemented
50C4×#121C2×C311yescharacteristic, complemented
51#2043C2×C311yescharacteristic, complemented
52#2041C2×C311yescharacteristic, complemented
53#2013C2×C311yescharacteristic, complemented
54#2011C2×C311yescharacteristic, complemented
55#1983C2×C312yescomplemented
56#1797C2×C311yescharacteristic, complemented
57#1788C2×C312yescomplemented
58C2×C3×#41—24no—
59C2×C3×#41—24no—
60C2×C3×#41—24no—
61C2×C3×#41—24no—
62C2×C3×#41—42no—
63C2×C3×#41—42no—
64C22×C3×Q8C2312yes—
65C22×C3×D4—22no—
66C22×C3×D4—21no—
67C22×C3×D4—21no—
68C22×C3×D4—21no—
69C22×C3×D4C2311yescharacteristic
70C22×C3×D4C2312yes—
71C23×C3×C4—22no—
72C23×C3×C4—21no—
73C23×C3×C4D411yescharacteristic
74C23×C3×C4D411yescharacteristic
75C23×C3×C4C2311yescharacteristic
76C2×C3×SD16—24no—
77C2×C3×SD16—24no—
78C2×C3×SD16—42no—
79C2×C3×D8—24no—
80C2×C3×D8—24no—
81C2×C3×D8—42no—
82C22×C3×C8C2312yes—
83C22×C3×C8C2312yes—
84C3×C4⋊Q8—24no—
85C3×#127—24no—
86C3×#124—24no—
87C3×#124D418yescomplemented
88C3×#121—28no—
89C3×#121—28no—
90C3×#121—22no—
91C3×#121D414yescomplemented
92C3×C4×D4—28no—
93C3×C4×D4—28no—
94C3×C4×D4—24no—
95C3×C4×D4—44no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C42—24no—
110C2×C3×C42—21no—
111C2×C3×C42D411yescharacteristic
112C2×C3×C42D411yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C3×C4⋊C8—24no—
116C3×#103—24no—
117C3×#103—24no—
118C3×#103—42no—
119C3×#102—24no—
120C3×#102—24no—
121C3×#102—42no—
122C3×C22⋊C8—24no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C22×#41C22×C312yes—
127C22×SD16C22×C312yes—
128C22×D8C22×C312yes—
129#532—28no—
130C2×C4⋊Q8C22×C311yescharacteristic
131C2×#127C22×C311yescharacteristic
132C2×#124C22×C312yes—
133C2×#121—21no—
134C2×#121—21no—
135C2×#121C22×C311yescharacteristic
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×C4×D4C22×C311yescharacteristic
140C22×C42C22×C311yescharacteristic
141#463—28no—
142#449—28no—
143C2×C4⋊C8C22×C312yes—
144C2×#103C22×C312yes—
145C2×#103C22×C312yes—
146C2×#103C22×C312yes—
147C2×#102C22×C312yes—
148C2×#102C22×C312yes—
149C2×#102C22×C312yes—
150C2×C22⋊C8C22×C312yes—
151#390C22×C311yescharacteristic
152#389C22×C311yescharacteristic
153#388—21no—
154#386—21no—
155#380C22×C311yescharacteristic
156C4×C4⋊C4C22×C311yescharacteristic
157C4×C22⋊C4—21no—
158#336C22×C311yescharacteristic
159#336C22×C311yescharacteristic
160C24×C3—21no—
161C24×C3—22no—
162C24×C3C2×D411yescharacteristic
163C3×#41—48no—
164C3×#41—48no—
165C3×#41—48no—
166C3×#41—48no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8C2×D414yes—
173C2×C3×D4—24no—
174C2×C3×D4—24no—
175C2×C3×D4—28no—
176C2×C3×D4—24no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—44no—
190C2×C3×D4—44no—
191C2×C3×D4—42no—
192C2×C3×D4—42no—
193C2×C3×D4C2×D414yes—
194C2×C3×D4C2×D412yes—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—28no—
198C22×C3×C4—21no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—41no—
207C22×C3×C4—41no—
208C22×C3×C4—44no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4—44no—
212C22×C3×C4—42no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4#4112yes—
220C22×C3×C4C2411yescharacteristic
221C3×SD16—48no—
222C3×D8—48no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4C2×D412yes—
236C3×C22⋊C4—28no—
237C3×C22⋊C4—28no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—42no—
240C3×C22⋊C4—42no—
241C3×C22⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C42—22no—
244C3×C42—22no—
245C3×C42—22no—
246C3×C42—42no—
247C3×C42C2×D414yes—
248C2×#41—24no—
249C2×#41—24no—
250C2×#41—24no—
251C2×#41—24no—
252C2×#41—42no—
253C2×#41—42no—
254C22×Q8C23×C312yes—
255C22×D4—21no—
256C22×D4—22no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4C23×C311yescharacteristic
260C22×D4C23×C312yes—
261C23×C4—22no—
262C23×C4—21no—
263C23×C4C3×D411yescharacteristic
264C23×C4C3×D411yescharacteristic
265C23×C4C23×C311yescharacteristic
266C2×SD16—24no—
267C2×SD16—24no—
268C2×SD16—42no—
269C2×D8—24no—
270C2×D8—24no—
271C2×D8—42no—
272C22×C8C23×C312yes—
273C22×C8C23×C312yes—
274C4⋊Q8—24no—
275#127—24no—
276#124—24no—
277#124C3×D418yescomplemented
278#121—28no—
279#121—28no—
280#121—22no—
281#121C3×D414yescomplemented
282C4×D4—28no—
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—44no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C4⋊C4C23×C311yescharacteristic
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C42—24no—
300C2×C42—21no—
301C2×C42C3×D411yescharacteristic
302C2×C42C3×D411yescharacteristic
303C2×C42C23×C311yescharacteristic
304C2×C42C23×C311yescharacteristic
305C4⋊C8—24no—
306#103—24no—
307#103—24no—
308#103—42no—
309#102—24no—
310#102—24no—
311#102—42no—
312C22⋊C8—24no—
313#95—21no—
314#95—21no—
315#95—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—24no—
323C23×C3—22no—
324C23×C3—41no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—42no—
330C23×C3—41no—
331C23×C3—42no—
332C23×C3—41no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—41no—
336C23×C3#12112yes—
337C23×C3C22×D411yescharacteristic
338C3×Q8—24no—
339C3×Q8—24no—
340C3×Q8—42no—
341C3×Q8—42no—
342C3×D4—24no—
343C3×D4—24no—
344C3×D4—24no—
345C3×D4—48no—
346C3×D4—48no—
347C3×D4—48no—
348C3×D4—416no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—42no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—22no—
366C2×C3×C4—44no—
367C2×C3×C4—44no—
368C2×C3×C4—416no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—84no—
380C2×C3×C4#12012yes—
381C2×C3×C4#12012yes—
382C2×C3×C4#12112yes—
383C2×C3×C4#13512yes—
384C2×C3×C4AGL(1,ℤ/8)12yes—
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C22×D412yes—
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)12yes—
389C3×C8—42no—
390C3×C8—42no—
391C24—21no—
392C24—22no—
393C24C2×C3×D411yescharacteristic
394#41—48no—
395#41—48no—
396#41—48no—
397#41—48no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8C2×C3×D414yes—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4C2×C3×D414yes—
425C2×D4C2×C3×D412yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—28no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4C2×C3×D411yescharacteristic
445C22×C4C2×C3×D411yescharacteristic
446C22×C4C2×C3×D411yescharacteristic
447C22×C4C2×C3×D412yes—
448C22×C4C2×C3×D411yescharacteristic
449C22×C4C2×C3×D411yescharacteristic
450C22×C4C3×#4112yes—
451C22×C4C24×C311yescharacteristic
452SD16—48no—
453D8—48no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4C2×C3×D412yes—
467C22⋊C4—28no—
468C22⋊C4—28no—
469C22⋊C4—24no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—42no—
478C42C2×C3×D414yes—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—44no—
487C22×C3—42no—
488C22×C3—44no—
489C22×C3—44no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—82no—
493C22×C3—81no—
494C22×C3—82no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—82no—
499C22×C3—82no—
500C22×C3C2×#12011yescharacteristic
501C22×C3C2×#12111yescharacteristic
502C22×C3#53511yescharacteristic
503C22×C3C2×#13511yescharacteristic
504C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
505C22×C3#57511yescharacteristic
506C22×C3#57511yescharacteristic
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—22no—
512C3×C4—44no—
513C3×C4—42no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4D4212yes—
517C3×C4#54612yes—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—24no—
525C23—22no—
526C23—41no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—41no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23C3×#12112yes—
539C23C22×C3×D411yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—42no—
543Q8—42no—
544D4—24no—
545D4—24no—
546D4—24no—
547D4—48no—
548D4—48no—
549D4—416no—
550D4—48no—
551D4—42no—
552D4—42no—
553D4—42no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—416no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—84no—
582C2×C4C3×#12012yes—
583C2×C4C3×#12012yes—
584C2×C4C3×#12112yes—
585C2×C4C3×#13512yes—
586C2×C4C3×AGL(1,ℤ/8)12yes—
587C2×C4C22×C3×D411yescharacteristic
588C2×C4C22×C3×D412yes—
589C2×C4C2×C3×#4111yescharacteristic
590C2×C4C3×H(5,𝔽2)12yes—
591C8—42no—
592C8—42no—
593C2×C3—42no—
594C2×C3—84no—
595C2×C3—82no—
596C2×C3#240811yescharacteristic
597C2×C3#300211yescharacteristic
598C2×C3#300611yescharacteristic
599C2×C3#303511yescharacteristic
600C2×C3#303611yescharacteristic
601C2×C3#320711yescharacteristic
602C2×C3#321911yescharacteristic
603C22—22no—
604C22—22no—
605C22—22no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—44no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—82no—
618C22—81no—
619C22—82no—
620C22—81no—
621C22—81no—
622C22—82no—
623C22—82no—
624C22C2×C3×#12011yescharacteristic
625C22C2×C3×#12111yescharacteristic
626C22C3×#53511yescharacteristic
627C22C2×C3×#13511yescharacteristic
628C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
629C22C3×#57511yescharacteristic
630C22C3×#57511yescharacteristic
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—44no—
637C4—42no—
638C4—81no—
639C4—81no—
640C4C3×D4212yes—
641C4C3×#54612yes—
642C3#2421411yescharacteristic, direct product factor, Sylow
643C2—42no—
644C2—84no—
645C2—82no—
646C2C3×#240811yescharacteristic
647C2C3×#300211yescharacteristic
648C2C3×#300611yescharacteristic
649C2C3×#303511yescharacteristic
650C2C3×#303611yescharacteristic
651C2C3×#320711yescharacteristic
652C2C3×#321911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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
1. C2×C3×#532C2semidirect
2. C2×C3×#463C2semidirect
3. C2×C3×#449C2semidirect
4. C3×C4×#121C2semidirect
5. C3×#2043C2semidirect
6. C3×#2041C2semidirect
7. C3×#2013C2semidirect
8. C3×#2011C2semidirect
9. C3×#1983C2semidirect
10. C3×#1797C2semidirect
11. C3×#1788C2semidirect
12. #24214C3direct (N × H)
47. C2×#532C2×C3semidirect
48. C2×#463C2×C3semidirect
49. C2×#449C2×C3semidirect
50. C4×#121C2×C3semidirect
51. #2043C2×C3semidirect
52. #2041C2×C3semidirect
53. #2013C2×C3semidirect
54. #2011C2×C3semidirect
55. #1983C2×C3semidirect
56. #1797C2×C3semidirect
57. #1788C2×C3semidirect
87. C3×#124D4semidirect
91. C3×#121D4semidirect
277. #124C3×D4semidirect
281. #121C3×D4semidirect
642. C3#24214direct (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×#4632
C2×C3×#4492
C3×C4×#1212
C3×#20432
C3×#20412
C3×#20132
C3×#20112
C3×#19832
C3×#17972
C3×#17882
#242143