Finite Group Databasebeta

← C3×#24217 #10642824

Subgroups — C3×#24217 #10642824

Order 768.

1558 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
1C22×C3×#124C211yesmaximal, characteristic
2C2×C3×#464C212yesmaximal, complemented
3C2×C3×#451C211yesmaximal, characteristic, complemented
4C2×C3×#450C211yesmaximal, characteristic
5C3×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
6C3×#2042C212yesmaximal, complemented
7C3×#2015C212yesmaximal, complemented
8C3×#1983C212yesmaximal, complemented
9C3×#1797C211yesmaximal, characteristic, complemented
10C3×#1787C211yesmaximal, characteristic, complemented
11C3×#1786C211yesmaximal, characteristic
12#24217C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×Q16C2212yescomplemented
16C22×C3×SD16C2212yes—
17C2×C3×#124—28no—
18C2×C3×#124—21no—
19C2×C3×#124—21no—
20C2×C3×#124C2212yes—
21C2×C3×#124C2212yes—
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic, complemented
24C22×C3×C22⋊C4—21no—
25C22×C3×C22⋊C4—21no—
26C22×C3×C42C2211yescharacteristic
27C3×#464—28no—
28C3×#451—24no—
29C3×#450—24no—
30C2×C3×C4⋊C8C2212yescomplemented
31C2×C3×#103C2212yescomplemented
32C2×C3×#103C2212yescomplemented
33C2×C3×#103C2212yescomplemented
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×C22⋊C8C2211yescharacteristic
38C2×C3×C22⋊C8C2211yescharacteristic
39C3×#391C2211yescharacteristic, complemented
40C3×#389C2211yescharacteristic, complemented
41C3×#387—21no—
42C3×#385—21no—
43C3×#380C2211yescharacteristic
44C3×C4×C4⋊C4—21no—
45C3×C4×C4⋊C4C2211yescharacteristic, complemented
46C3×C4×C22⋊C4—21no—
47C3×#336C2212yescomplemented
48C22×#124C2×C311yescharacteristic
49C2×#464C2×C312yescomplemented
50C2×#451C2×C311yescharacteristic, complemented
51C2×#450C2×C311yescharacteristic
52C4×C22⋊Q8C2×C311yescharacteristic, complemented
53#2042C2×C312yescomplemented
54#2015C2×C312yescomplemented
55#1983C2×C312yescomplemented
56#1797C2×C311yescharacteristic, complemented
57#1787C2×C311yescharacteristic, complemented
58#1786C2×C311yescharacteristic
59C25×C3—21no—
60C22×C3×Q8—24no—
61C22×C3×Q8—21no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8—42no—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×Q8C2312yes—
67C22×C3×D4—24no—
68C22×C3×D4—21no—
69C22×C3×D4—22no—
70C22×C3×D4—22no—
71C22×C3×D4—42no—
72C22×C3×D4C2312yes—
73C23×C3×C4—21no—
74C23×C3×C4D411yescharacteristic
75C23×C3×C4D411yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C2×C3×Q16—24no—
78C2×C3×Q16—24no—
79C2×C3×Q16—42no—
80C2×C3×SD16—24no—
81C2×C3×SD16—24no—
82C2×C3×SD16—42no—
83C22×C3×C8C2312yes—
84C22×C3×C8C2312yes—
85C3×#124—28no—
86C3×#124—28no—
87C3×#124—24no—
88C3×#124—24no—
89C3×#124—24no—
90C3×#124D418yes—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8D414yes—
93C3×C4×Q8—24no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—28no—
101C2×C3×C22⋊C4—28no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C42—24no—
109C2×C3×C42—21no—
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—22no—
123C3×C22⋊C8—22no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C23×Q8C22×C311yescharacteristic
130C23×D4C22×C311yescharacteristic
131C22×Q16C22×C312yescomplemented
132C22×SD16C22×C312yes—
133C2×#124—28no—
134C2×#124—21no—
135C2×#124—21no—
136C2×#124C22×C312yes—
137C2×#124C22×C312yes—
138C2×C22⋊Q8C22×C311yescharacteristic
139C2×C4×Q8C22×C311yescharacteristic, complemented
140C22×C22⋊C4—21no—
141C22×C22⋊C4—21no—
142C22×C42C22×C311yescharacteristic
143#464—28no—
144#451—24no—
145#450—24no—
146C2×C4⋊C8C22×C312yescomplemented
147C2×#103C22×C312yescomplemented
148C2×#103C22×C312yescomplemented
149C2×#103C22×C312yescomplemented
150C2×#102C22×C312yes—
151C2×#102C22×C312yes—
152C2×#102C22×C312yes—
153C2×C22⋊C8C22×C311yescharacteristic
154C2×C22⋊C8C22×C311yescharacteristic
155#391C22×C311yescharacteristic, complemented
156#389C22×C311yescharacteristic, complemented
157#387—21no—
158#385—21no—
159#380C22×C311yescharacteristic
160C4×C4⋊C4—21no—
161C4×C4⋊C4C22×C311yescharacteristic, complemented
162C4×C22⋊C4—21no—
163#336C22×C312yescomplemented
164C24×C3—24no—
165C24×C3—22no—
166C24×C3—42no—
167C24×C3—42no—
168C24×C3—42no—
169C24×C3—42no—
170C24×C3—42no—
171C24×C3—42no—
172C24×C3C2×D411yescharacteristic
173C2×C3×Q8—24no—
174C2×C3×Q8—22no—
175C2×C3×Q8—21no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8—22no—
179C2×C3×Q8—22no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8—48no—
183C2×C3×Q8—48no—
184C2×C3×Q8—42no—
185C2×C3×Q8—42no—
186C2×C3×Q8C2×D414yes—
187C2×C3×Q8C2×D412yes—
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—48no—
195C2×C3×D4—48no—
196C2×C3×D4—42no—
197C2×C3×D4—42no—
198C2×C3×D4C2×D414yes—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—24no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—41no—
212C22×C3×C4—41no—
213C22×C3×C4—42no—
214C22×C3×C4—42no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D412yes—
221C22×C3×C4#4112yes—
222C22×C3×C4C2411yescharacteristic
223C3×Q16—48no—
224C3×SD16—48no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4C2×D412yes—
238C3×C22⋊C4—28no—
239C3×C22⋊C4—28no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—28no—
242C3×C22⋊C4—28no—
243C3×C22⋊C4—42no—
244C3×C42—22no—
245C3×C42—22no—
246C3×C42—22no—
247C3×C42—42no—
248C3×C42—42no—
249C3×C42C2×D414yes—
250C25—21no—
251C22×Q8—24no—
252C22×Q8—21no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—42no—
256C22×Q8C23×C311yescharacteristic
257C22×Q8C23×C312yes—
258C22×D4—24no—
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—42no—
263C22×D4C23×C312yes—
264C23×C4—21no—
265C23×C4C3×D411yescharacteristic
266C23×C4C3×D411yescharacteristic
267C23×C4C23×C311yescharacteristic
268C2×Q16—24no—
269C2×Q16—24no—
270C2×Q16—42no—
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—42no—
274C22×C8C23×C312yes—
275C22×C8C23×C312yes—
276#124—28no—
277#124—28no—
278#124—24no—
279#124—24no—
280#124—24no—
281#124C3×D418yes—
282C22⋊Q8—22no—
283C22⋊Q8C3×D414yes—
284C4×Q8—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C42—24no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42C3×D411yescharacteristic
303C2×C42C3×D411yescharacteristic
304C2×C42C23×C311yescharacteristic
305C2×C42C23×C311yescharacteristic
306C4⋊C8—24no—
307#103—24no—
308#103—24no—
309#103—42no—
310#102—24no—
311#102—24no—
312#102—42no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—24no—
325C23×C3—48no—
326C23×C3—48no—
327C23×C3—48no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—84no—
335C23×C3—84no—
336C23×C3—84no—
337C23×C3—84no—
338C23×C3#12112yes—
339C23×C3#13512yes—
340C23×C3#13712yes—
341C23×C3C22×D411yescharacteristic
342C3×Q8—24no—
343C3×Q8—24no—
344C3×Q8—24no—
345C3×Q8—48no—
346C3×Q8—42no—
347C3×Q8—42no—
348C3×Q8—42no—
349C3×D4—24no—
350C3×D4—24no—
351C3×D4—48no—
352C3×D4—42no—
353C3×D4—42no—
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—24no—
361C2×C3×C4—24no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—24no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—44no—
372C2×C3×C4—44no—
373C2×C3×C4—48no—
374C2×C3×C4—41no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4#12012yes—
387C2×C3×C4#12012yes—
388C2×C3×C4#12112yes—
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×D412yes—
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4H(5,𝔽2)12yes—
393C3×C8—42no—
394C3×C8—42no—
395C24—24no—
396C24—22no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24C2×C3×D411yescharacteristic
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8C2×C3×D414yes—
418C2×Q8C2×C3×D412yes—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4C2×C3×D414yes—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4C2×C3×D411yescharacteristic
447C22×C4C2×C3×D411yescharacteristic
448C22×C4C2×C3×D411yescharacteristic
449C22×C4C2×C3×D411yescharacteristic
450C22×C4C2×C3×D411yescharacteristic
451C22×C4C2×C3×D412yes—
452C22×C4C3×#4112yes—
453C22×C4C24×C311yescharacteristic
454Q16—48no—
455SD16—48no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4C2×C3×D412yes—
469C22⋊C4—28no—
470C22⋊C4—28no—
471C22⋊C4—24no—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—42no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—42no—
479C42—42no—
480C42C2×C3×D414yes—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3—44no—
490C22×C3—816no—
491C22×C3—82no—
492C22×C3—82no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3#46414yescomplemented
496C22×C3C2×#12011yescharacteristic
497C22×C3C2×#12111yescharacteristic
498C22×C3#53511yescharacteristic
499C22×C3C2×#13511yescharacteristic
500C22×C3C2×#13711yescharacteristic
501C22×C3#57511yescharacteristic
502C22×C3#57511yescharacteristic
503C3×C4—22no—
504C3×C4—22no—
505C3×C4—22no—
506C3×C4—44no—
507C3×C4—42no—
508C3×C4—41no—
509C3×C4—41no—
510C3×C4—42no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4D4212yes—
514C3×C4#54612yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—24no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23C3×#12112yes—
534C23C3×#13512yes—
535C23C3×#13712yes—
536C23C22×C3×D411yescharacteristic
537Q8—24no—
538Q8—24no—
539Q8—24no—
540Q8—48no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544D4—24no—
545D4—24no—
546D4—48no—
547D4—42no—
548D4—42no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4C3×#12012yes—
582C2×C4C3×#12012yes—
583C2×C4C3×#12112yes—
584C2×C4C22×C3×D411yescharacteristic
585C2×C4C22×C3×D412yes—
586C2×C4C2×C3×#4111yescharacteristic
587C2×C4C3×H(5,𝔽2)12yes—
588C8—42no—
589C8—42no—
590C2×C3—24no—
591C2×C3—84no—
592C2×C3#240811yescharacteristic
593C2×C3#300211yescharacteristic
594C2×C3#300711yescharacteristic
595C2×C3C2×#46411yescharacteristic
596C2×C3#304011yescharacteristic
597C2×C3#321111yescharacteristic
598C2×C3#321511yescharacteristic
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—816no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22C3×#46414yescomplemented
614C22C2×C3×#12011yescharacteristic
615C22C2×C3×#12111yescharacteristic
616C22C3×#53511yescharacteristic
617C22C2×C3×#13511yescharacteristic
618C22C2×C3×#13711yescharacteristic
619C22C3×#57511yescharacteristic
620C22C3×#57511yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—44no—
625C4—42no—
626C4—41no—
627C4—41no—
628C4—42no—
629C4—81no—
630C4—81no—
631C4C3×D4212yes—
632C4C3×#54612yes—
633C3#2421711yescharacteristic, direct product factor, Sylow
634C2—24no—
635C2—84no—
636C2C3×#240811yescharacteristic
637C2C3×#300211yescharacteristic
638C2C3×#300711yescharacteristic
639C2C2×C3×#46411yescharacteristic
640C2C3×#304011yescharacteristic
641C2C3×#321111yescharacteristic
642C2C3×#321511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C22
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×#464C2semidirect
3. C2×C3×#451C2semidirect
5. C3×C4×C22⋊Q8C2semidirect
6. C3×#2042C2semidirect
7. C3×#2015C2semidirect
8. C3×#1983C2semidirect
9. C3×#1797C2semidirect
10. C3×#1787C2semidirect
12. #24217C3direct (N × H)
15. C22×C3×Q16C22semidirect
23. C2×C3×C4×Q8C22semidirect
30. C2×C3×C4⋊C8C22semidirect
31. C2×C3×#103C22semidirect
32. C2×C3×#103C22semidirect
33. C2×C3×#103C22semidirect
39. C3×#391C22semidirect
40. C3×#389C22semidirect
45. C3×C4×C4⋊C4C22semidirect
47. C3×#336C22semidirect
49. C2×#464C2×C3semidirect
50. C2×#451C2×C3semidirect
52. C4×C22⋊Q8C2×C3semidirect
53. #2042C2×C3semidirect
54. #2015C2×C3semidirect
55. #1983C2×C3semidirect
56. #1797C2×C3semidirect
57. #1787C2×C3semidirect
131. C22×Q16C22×C3semidirect
139. C2×C4×Q8C22×C3semidirect
146. C2×C4⋊C8C22×C3semidirect
147. C2×#103C22×C3semidirect
148. C2×#103C22×C3semidirect
149. C2×#103C22×C3semidirect
155. #391C22×C3semidirect
156. #389C22×C3semidirect
161. C4×C4⋊C4C22×C3semidirect
163. #336C22×C3semidirect
495. C22×C3#464semidirect
613. C22C3×#464semidirect
633. C3#24217direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1242
C2×C3×#4642
C2×C3×#4512
C2×C3×#4502
C3×C4×C22⋊Q82
C3×#20422
C3×#20152
C3×#19832
C3×#17972
C3×#17872
C3×#17862
#242173