Finite Group Databasebeta

← C3×#24239 #10642846

Subgroups — C3×#24239 #10642846

Order 768.

1490 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#461C211yesmaximal, characteristic, complemented
3C2×C3×#448C212yesmaximal, complemented
4C2×C3×C4×SD16C211yesmaximal, characteristic, complemented
5C3×#2537C211yesmaximal, characteristic
6C3×#2035C212yesmaximal, complemented
7C3×#2000C212yesmaximal, complemented
8C3×#1956C211yesmaximal, characteristic, complemented
9C3×#1954C211yesmaximal, characteristic, complemented
10C3×#1803C211yesmaximal, characteristic, complemented
11C3×#1787C212yesmaximal, complemented
12#24239C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2212yes—
14C22×C3×SD16C2212yes—
15C22×C3×SD16C2211yescharacteristic
16C3×D4×Q8—24no—
17C3×D4×Q8—24no—
18C2×C3×C4⋊Q8—21no—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8—22no—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×#121—21no—
23C2×C3×#121C2212yes—
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4C2211yescharacteristic
27C2×C3×C4×D4C2211yescharacteristic
28C3×#461—24no—
29C3×#448—28no—
30C3×C4×SD16—24no—
31C2×C3×#106C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2211yescharacteristic
36C2×C3×#103C2212yes—
37C2×C3×#103C2212yes—
38C2×C3×#102C2211yescharacteristic
39C2×C3×C22⋊C8C2212yes—
40C2×C3×C22⋊C8C2212yes—
41C2×C3×C4×C8C2211yescharacteristic
42C3×#394—22no—
43C3×#384C2211yescharacteristic
44C3×#380—21no—
45C3×#380C2212yes—
46C3×#340C2212yes—
47C2×D4×Q8C2×C311yescharacteristic, complemented
48C2×#461C2×C311yescharacteristic, complemented
49C2×#448C2×C312yescomplemented
50C2×C4×SD16C2×C311yescharacteristic, complemented
51#2537C2×C311yescharacteristic
52#2035C2×C312yescomplemented
53#2000C2×C312yescomplemented
54#1956C2×C311yescharacteristic, complemented
55#1954C2×C311yescharacteristic, complemented
56#1803C2×C311yescharacteristic, complemented
57#1787C2×C312yescomplemented
58C22×C3×Q8—24no—
59C22×C3×Q8—24no—
60C22×C3×Q8—22no—
61C22×C3×Q8—24no—
62C22×C3×Q8—24no—
63C22×C3×Q8—42no—
64C22×C3×Q8—42no—
65C22×C3×Q8C2312yes—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—22no—
68C22×C3×D4D411yescharacteristic
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—22no—
71C23×C3×C4—21no—
72C23×C3×C4C2312yes—
73C2×C3×SD16—22no—
74C2×C3×SD16—22no—
75C2×C3×SD16—24no—
76C2×C3×SD16—24no—
77C2×C3×SD16—42no—
78C2×C3×SD16D414yescomplemented
79C22×C3×C8C2312yes—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4⋊Q8—28no—
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8D414yescomplemented
85C3×C22⋊Q8—216no—
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—24no—
88C3×#121—24no—
89C3×#121—42no—
90C3×#121—44no—
91C3×C4×Q8—22no—
92C3×C4×Q8D414yescomplemented
93C3×C4×D4—28no—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4D411yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—22no—
110C2×C3×C22⋊C4—22no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4D412yes—
113C2×C3×C42—21no—
114C2×C3×C42C2311yescharacteristic
115C3×#106—22no—
116C3×C4⋊C8—22no—
117C3×#103—22no—
118C3×#103—24no—
119C3×#103—24no—
120C3×#103—24no—
121C3×#103—42no—
122C3×#102—22no—
123C3×C22⋊C8—24no—
124C3×C22⋊C8—42no—
125C3×C4×C8—22no—
126C3×#95—22no—
127C23×Q8C22×C312yes—
128C22×SD16C22×C312yes—
129C22×SD16C22×C311yescharacteristic
130D4×Q8—24no—
131D4×Q8—24no—
132C2×C4⋊Q8—21no—
133C2×C4⋊Q8C22×C311yescharacteristic
134C2×C22⋊Q8—22no—
135C2×C22⋊Q8C22×C312yes—
136C2×#121—21no—
137C2×#121C22×C312yes—
138C2×C4×Q8C22×C311yescharacteristic
139C2×C4×D4—21no—
140C2×C4×D4C22×C311yescharacteristic
141C2×C4×D4C22×C311yescharacteristic
142#461—24no—
143#448—28no—
144C4×SD16—24no—
145C2×#106C22×C311yescharacteristic
146C2×C4⋊C8C22×C311yescharacteristic
147C2×#103C22×C312yes—
148C2×#103C22×C311yescharacteristic
149C2×#103C22×C312yes—
150C2×#103C22×C312yes—
151C2×#103C22×C312yes—
152C2×#102C22×C311yescharacteristic
153C2×C22⋊C8C22×C312yes—
154C2×C22⋊C8C22×C312yes—
155C2×C4×C8C22×C311yescharacteristic
156#394—22no—
157#384C22×C311yescharacteristic
158#380—21no—
159#380C22×C312yes—
160#340C22×C312yes—
161C24×C3—21no—
162C24×C3C2×D412yes—
163C2×C3×Q8—28no—
164C2×C3×Q8—21no—
165C2×C3×Q8—22no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—44no—
173C2×C3×Q8—44no—
174C2×C3×Q8—48no—
175C2×C3×Q8—48no—
176C2×C3×Q8—48no—
177C2×C3×Q8—48no—
178C2×C3×Q8C2×D414yes—
179C2×C3×Q8C2×D412yes—
180C2×C3×Q8C2×D412yes—
181C2×C3×Q8#4112yes—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—44no—
190C2×C3×D4—44no—
191C2×C3×D4—82no—
192C2×C3×D4SD1614yescomplemented
193C2×C3×D4C2×D412yes—
194C22×C3×C4—21no—
195C22×C3×C4—28no—
196C22×C3×C4—21no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
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—22no—
206C22×C3×C4—44no—
207C22×C3×C4—44no—
208C22×C3×C4—42no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4C2×D412yes—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C3×SD16—28no—
216C3×SD16—48no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—21no—
223C2×C3×C8—22no—
224C2×C3×C8—21no—
225C2×C3×C8C2×D412yes—
226C3×C4⋊C4—28no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—81no—
238C3×C4⋊C4C2×D412yes—
239C3×C22⋊C4—28no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—44no—
242C3×C22⋊C4—44no—
243C3×C22⋊C4—42no—
244C3×C42—24no—
245C3×C42C2×D412yes—
246C3×C42C2×D412yes—
247C22×Q8—24no—
248C22×Q8—24no—
249C22×Q8—22no—
250C22×Q8—24no—
251C22×Q8—24no—
252C22×Q8—42no—
253C22×Q8—42no—
254C22×Q8C23×C312yes—
255C22×Q8C23×C311yescharacteristic
256C22×D4—22no—
257C22×D4C3×D411yescharacteristic
258C22×D4C23×C311yescharacteristic
259C23×C4—22no—
260C23×C4—21no—
261C23×C4C23×C312yes—
262C2×SD16—22no—
263C2×SD16—22no—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—42no—
267C2×SD16C3×D414yescomplemented
268C22×C8C23×C312yes—
269C22×C8C23×C311yescharacteristic
270C22×C8C23×C311yescharacteristic
271C4⋊Q8—28no—
272C4⋊Q8—22no—
273C4⋊Q8C3×D414yescomplemented
274C22⋊Q8—216no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277#121—24no—
278#121—42no—
279#121—44no—
280C4×Q8—22no—
281C4×Q8C3×D414yescomplemented
282C4×D4—28no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4C3×D411yescharacteristic
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C4⋊C4C23×C312yes—
296C2×C4⋊C4C23×C311yescharacteristic
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4C3×D412yes—
302C2×C42—21no—
303C2×C42C23×C311yescharacteristic
304#106—22no—
305C4⋊C8—22no—
306#103—22no—
307#103—24no—
308#103—24no—
309#103—24no—
310#103—42no—
311#102—22no—
312C22⋊C8—24no—
313C22⋊C8—42no—
314C4×C8—22no—
315#95—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3C2×SD1614yes—
330C23×C3C22×D411yescharacteristic
331C3×Q8—22no—
332C3×Q8—22no—
333C3×Q8—22no—
334C3×Q8—24no—
335C3×Q8—24no—
336C3×Q8—416no—
337C3×Q8—42no—
338C3×Q8—42no—
339C3×Q8#12114yescomplemented
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—28no—
343C3×D4—41no—
344C3×D4—41no—
345C3×D4—88no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—416no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—41no—
368C2×C3×C4—42no—
369C2×C3×C4—41no—
370C2×C3×C4—42no—
371C2×C3×C4—84no—
372C2×C3×C4#12012yes—
373C2×C3×C4#12012yes—
374C2×C3×C4C2×SD1612yes—
375C2×C3×C4#13712yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382C3×C8—22no—
383C3×C8—41no—
384C3×C8—42no—
385C24—21no—
386C24C2×C3×D412yes—
387C2×Q8—28no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—48no—
402C2×Q8C2×C3×D414yes—
403C2×Q8C2×C3×D412yes—
404C2×Q8C2×C3×D412yes—
405C2×Q8C3×#4112yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—82no—
416C2×D4C3×SD1614yescomplemented
417C2×D4C2×C3×D412yes—
418C22×C4—21no—
419C22×C4—28no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4C2×C3×D412yes—
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C2×C3×D411yescharacteristic
438C22×C4C24×C311yescharacteristic
439SD16—28no—
440SD16—48no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—21no—
447C2×C8—22no—
448C2×C8—21no—
449C2×C8C2×C3×D412yes—
450C4⋊C4—28no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—42no—
461C4⋊C4—81no—
462C4⋊C4C2×C3×D412yes—
463C22⋊C4—28no—
464C22⋊C4—24no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—42no—
468C42—24no—
469C42C2×C3×D412yes—
470C42C2×C3×D412yes—
471C22×C3—24no—
472C22×C3—24no—
473C22×C3—24no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3C2×#12011yescharacteristic
486C22×C3D4211yescharacteristic
487C22×C3#54611yescharacteristic
488C22×C3C22×SD1611yescharacteristic
489C22×C3C2×#13711yescharacteristic
490C22×C3#57711yescharacteristic
491C22×C3#57811yescharacteristic
492C3×C4—24no—
493C3×C4—22no—
494C3×C4—42no—
495C3×C4—44no—
496C3×C4—42no—
497C3×C4—81no—
498C3×C4—82no—
499C3×C4#44814yescomplemented
500C3×C4C2×#12111yescharacteristic
501C3×C4#53511yescharacteristic
502C3×C4D4211yescharacteristic
503C3×C4#54611yescharacteristic
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C2×C3×SD1614yes—
518C23C22×C3×D411yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—22no—
522Q8—24no—
523Q8—24no—
524Q8—416no—
525Q8—42no—
526Q8—42no—
527Q8C3×#12114yescomplemented
528D4—28no—
529D4—22no—
530D4—22no—
531D4—41no—
532D4—41no—
533D4—88no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—416no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—84no—
560C2×C4C3×#12012yes—
561C2×C4C3×#12012yes—
562C2×C4C2×C3×SD1612yes—
563C2×C4C3×#13712yes—
564C2×C4C22×C3×D411yescharacteristic
565C2×C4C22×C3×D411yescharacteristic
566C2×C4C22×C3×D411yescharacteristic
567C2×C4C2×C3×#4111yescharacteristic
568C2×C4C3×H(5,𝔽2)11yescharacteristic
569C2×C4C3×H(5,𝔽2)11yescharacteristic
570C8—22no—
571C8—41no—
572C8—42no—
573C2×C3—44no—
574C2×C3—82no—
575C2×C3#240811yescharacteristic
576C2×C3C2×#44811yescharacteristic
577C2×C3#301311yescharacteristic
578C2×C3#327611yescharacteristic
579C2×C3D4×SD1611yescharacteristic
580C2×C3#329811yescharacteristic
581C2×C3#332911yescharacteristic
582C22—24no—
583C22—24no—
584C22—24no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22C2×C3×#12011yescharacteristic
597C22C3×D4211yescharacteristic
598C22C3×#54611yescharacteristic
599C22C22×C3×SD1611yescharacteristic
600C22C2×C3×#13711yescharacteristic
601C22C3×#57711yescharacteristic
602C22C3×#57811yescharacteristic
603C4—24no—
604C4—22no—
605C4—42no—
606C4—44no—
607C4—42no—
608C4—81no—
609C4—82no—
610C4C3×#44814yescomplemented
611C4C2×C3×#12111yescharacteristic
612C4C3×#53511yescharacteristic
613C4C3×D4211yescharacteristic
614C4C3×#54611yescharacteristic
615C3#2423911yescharacteristic, direct product factor, Sylow
616C2—44no—
617C2—82no—
618C2C3×#240811yescharacteristic
619C2C2×C3×#44811yescharacteristic
620C2C3×#301311yescharacteristic
621C2C3×#327611yescharacteristic
622C2C3×D4×SD1611yescharacteristic
623C2C3×#329811yescharacteristic
624C2C3×#332911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×Q8C2semidirect
2. C2×C3×#461C2semidirect
3. C2×C3×#448C2semidirect
4. C2×C3×C4×SD16C2semidirect
6. C3×#2035C2semidirect
7. C3×#2000C2semidirect
8. C3×#1956C2semidirect
9. C3×#1954C2semidirect
10. C3×#1803C2semidirect
11. C3×#1787C2semidirect
12. #24239C3direct (N × H)
47. C2×D4×Q8C2×C3semidirect
48. C2×#461C2×C3semidirect
49. C2×#448C2×C3semidirect
50. C2×C4×SD16C2×C3semidirect
52. #2035C2×C3semidirect
53. #2000C2×C3semidirect
54. #1956C2×C3semidirect
55. #1954C2×C3semidirect
56. #1803C2×C3semidirect
57. #1787C2×C3semidirect
78. C2×C3×SD16D4semidirect
84. C3×C4⋊Q8D4semidirect
92. C3×C4×Q8D4semidirect
192. C2×C3×D4SD16semidirect
267. C2×SD16C3×D4semidirect
273. C4⋊Q8C3×D4semidirect
281. C4×Q8C3×D4semidirect
339. C3×Q8#121semidirect
416. C2×D4C3×SD16semidirect
499. C3×C4#448semidirect
527. Q8C3×#121semidirect
610. C4C3×#448semidirect
615. C3#24239direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×C3×#4612
C2×C3×#4482
C2×C3×C4×SD162
C3×#25372
C3×#20352
C3×#20002
C3×#19562
C3×#19542
C3×#18032
C3×#17872
#242393