Finite Group Databasebeta

← C3×#32909 #10651516

Subgroups — C3×#32909 #10651516

Order 768.

1462 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#464C212yesmaximal, complemented
3C2×C3×#451C211yesmaximal, characteristic, complemented
4C2×C3×#448C211yesmaximal, characteristic
5C3×#2850C211yesmaximal, characteristic
6C3×#1880C212yesmaximal, complemented
7C3×#1878C212yesmaximal, complemented
8C3×#1874C211yesmaximal, characteristic
9C3×#1872C211yesmaximal, characteristic, complemented
10C3×#1805C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#32909C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C2×C3×#124C2211yescharacteristic
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×C3×C4×Q8—28no—
20C2×C3×C4×Q8—21no—
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8C2212yes—
23C2×C3×C4×Q8C2212yes—
24C22×C3×C4⋊C4—21no—
25C22×C3×C4⋊C4C2211yescharacteristic
26C22×C3×C42—21no—
27C22×C3×C42C2211yescharacteristic
28C3×#464—28no—
29C3×#451—24no—
30C3×#448—24no—
31C2×C3×C4⋊C8C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×C22⋊C8C2211yescharacteristic
38C3×#391C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#388—21no—
41C3×#388—21no—
42C3×#386—21no—
43C3×#386—21no—
44C3×#382C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C22×C4×Q8C2×C311yescharacteristic, complemented
49C2×#464C2×C312yescomplemented
50C2×#451C2×C311yescharacteristic, complemented
51C2×#448C2×C311yescharacteristic
52#2850C2×C311yescharacteristic
53#1880C2×C312yescomplemented
54#1878C2×C312yescomplemented
55#1874C2×C311yescharacteristic
56#1872C2×C311yescharacteristic, complemented
57#1805C2×C312yescomplemented
58#1797C2×C311yescharacteristic, complemented
59C22×C3×Q8—24no—
60C22×C3×Q8—24no—
61C22×C3×Q8—21no—
62C22×C3×Q8—22no—
63C22×C3×Q8D414yes—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×Q8C2312yes—
66C22×C3×D4—21no—
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C23×C3×C4D411yescharacteristic
71C23×C3×C4D411yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C2×C3×Q16—24no—
74C2×C3×Q16—24no—
75C2×C3×Q16—24no—
76C2×C3×SD16—24no—
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C22×C3×C8C2312yes—
80C22×C3×C8C2312yes—
81C3×#124—24no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C3×#121—22no—
85C3×#121—22no—
86C3×C4×Q8—28no—
87C3×C4×Q8—28no—
88C3×C4×Q8—24no—
89C3×C4×Q8—24no—
90C3×C4×Q8—24no—
91C3×C4×Q8D418yescomplemented
92C2×C3×C4⋊C4—28no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—24no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C42—28no—
108C2×C3×C42—22no—
109C2×C3×C42—22no—
110C2×C3×C42—24no—
111C2×C3×C42D411yescharacteristic
112C2×C3×C42D411yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C3×C4⋊C8—24no—
116C3×#103—24no—
117C3×#103—24no—
118C3×#103—24no—
119C3×#102—24no—
120C3×C22⋊C8—22no—
121C3×C22⋊C8—22no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C23×Q8C22×C311yescharacteristic
127C22×Q16C22×C312yes—
128C22×SD16C22×C312yes—
129C2×#124C22×C311yescharacteristic
130C2×C22⋊Q8C22×C311yescharacteristic
131C2×#121C22×C311yescharacteristic
132C2×C4×Q8—28no—
133C2×C4×Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×Q8C22×C312yes—
136C2×C4×Q8C22×C312yes—
137C22×C4⋊C4—21no—
138C22×C4⋊C4C22×C311yescharacteristic
139C22×C42—21no—
140C22×C42C22×C311yescharacteristic
141#464—28no—
142#451—24no—
143#448—24no—
144C2×C4⋊C8C22×C312yes—
145C2×#103C22×C312yes—
146C2×#103C22×C312yes—
147C2×#103C22×C312yes—
148C2×#102C22×C312yes—
149C2×C22⋊C8C22×C311yescharacteristic
150C2×C22⋊C8C22×C311yescharacteristic
151#391C22×C311yescharacteristic
152#390C22×C311yescharacteristic
153#388—21no—
154#388—21no—
155#386—21no—
156#386—21no—
157#382C22×C311yescharacteristic
158#340C22×C312yes—
159#340C22×C312yes—
160#336C22×C312yes—
161C24×C3—21no—
162C24×C3C2×D411yescharacteristic
163C2×C3×Q8—24no—
164C2×C3×Q8—216no—
165C2×C3×Q8—24no—
166C2×C3×Q8—216no—
167C2×C3×Q8—24no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8C2×D414yes—
176C2×C3×Q8C2×D414yes—
177C2×C3×Q8#4114yes—
178C2×C3×D4—22no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—22no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—24no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—24no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—41no—
202C22×C3×C4—42no—
203C22×C3×C4—42no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4#4112yes—
211C22×C3×C4C2411yescharacteristic
212C3×Q16—48no—
213C3×SD16—48no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C3×C4⋊C4—28no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—28no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4C2×D414yes—
230C3×C22⋊C4—42no—
231C3×C22⋊C4—42no—
232C3×C22⋊C4—42no—
233C3×C22⋊C4—42no—
234C3×C22⋊C4—42no—
235C3×C22⋊C4—42no—
236C3×C42—28no—
237C3×C42—28no—
238C3×C42—22no—
239C3×C42—22no—
240C3×C42—22no—
241C3×C42C2×D414yes—
242C22×Q8—24no—
243C22×Q8—24no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8C3×D414yes—
247C22×Q8C23×C311yescharacteristic
248C22×Q8C23×C312yes—
249C22×D4—21no—
250C22×D4C23×C311yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4C3×D411yescharacteristic
254C23×C4C3×D411yescharacteristic
255C23×C4C23×C311yescharacteristic
256C2×Q16—24no—
257C2×Q16—24no—
258C2×Q16—24no—
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—24no—
262C22×C8C23×C312yes—
263C22×C8C23×C312yes—
264#124—24no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267#121—22no—
268#121—22no—
269C4×Q8—28no—
270C4×Q8—28no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8C3×D418yescomplemented
275C2×C4⋊C4—28no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C312yes—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C42—28no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42C3×D411yescharacteristic
295C2×C42C3×D411yescharacteristic
296C2×C42C23×C311yescharacteristic
297C2×C42C23×C311yescharacteristic
298C4⋊C8—24no—
299#103—24no—
300#103—24no—
301#103—24no—
302#102—24no—
303C22⋊C8—22no—
304C22⋊C8—22no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3#12112yes—
321C23×C3#13512yes—
322C23×C3#13712yes—
323C23×C3C22×D411yescharacteristic
324C3×Q8—216no—
325C3×Q8—24no—
326C3×Q8—24no—
327C3×Q8—24no—
328C3×Q8—42no—
329C3×Q8—42no—
330C3×Q8#12118yes—
331C3×D4—42no—
332C3×D4—42no—
333C3×D4—84no—
334C2×C3×C4—28no—
335C2×C3×C4—216no—
336C2×C3×C4—21no—
337C2×C3×C4—24no—
338C2×C3×C4—21no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—21no—
350C2×C3×C4—24no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—44no—
354C2×C3×C4—48no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4#12012yes—
367C2×C3×C4#12012yes—
368C2×C3×C4#12112yes—
369C2×C3×C4#13712yes—
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D412yes—
372C2×C3×C4C2×#4112yes—
373C2×C3×C4C2×#4111yescharacteristic
374C3×C8—42no—
375C3×C8—42no—
376C24—21no—
377C24C2×C3×D411yescharacteristic
378C2×Q8—24no—
379C2×Q8—216no—
380C2×Q8—24no—
381C2×Q8—216no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8C2×C3×D414yes—
391C2×Q8C2×C3×D414yes—
392C2×Q8C3×#4114yes—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C2×C3×D411yescharacteristic
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C2×C3×D412yes—
424C22×C4C2×C3×D411yescharacteristic
425C22×C4C3×#4112yes—
426C22×C4C24×C311yescharacteristic
427Q16—48no—
428SD16—48no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C4⋊C4—28no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—42no—
444C4⋊C4C2×C3×D414yes—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C42—28no—
452C42—28no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42C2×C3×D414yes—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—81no—
464C22×C3—81no—
465C22×C3—81no—
466C22×C3—81no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3#46414yescomplemented
471C22×C3C2×#12011yescharacteristic
472C22×C3C2×#12111yescharacteristic
473C22×C3#52511yescharacteristic
474C22×C3C2×#13511yescharacteristic
475C22×C3C2×#13711yescharacteristic
476C22×C3C2×#13711yescharacteristic
477C22×C3#57511yescharacteristic
478C3×C4—28no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—41no—
484C3×C4—44no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4—81no—
488C3×C4C2×#12112yes—
489C3×C4#52512yes—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23C3×#12112yes—
502C23C3×#13512yes—
503C23C3×#13712yes—
504C23C22×C3×D411yescharacteristic
505Q8—216no—
506Q8—24no—
507Q8—24no—
508Q8—24no—
509Q8—42no—
510Q8—42no—
511Q8C3×#12118yes—
512D4—42no—
513D4—42no—
514D4—84no—
515C2×C4—216no—
516C2×C4—28no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—44no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4C3×#12012yes—
548C2×C4C3×#12012yes—
549C2×C4C3×#12112yes—
550C2×C4C3×#13712yes—
551C2×C4C22×C3×D411yescharacteristic
552C2×C4C22×C3×D412yes—
553C2×C4C2×C3×#4112yes—
554C2×C4C2×C3×#4111yescharacteristic
555C8—42no—
556C8—42no—
557C2×C3—24no—
558C2×C3—161no—
559C2×C3#284711yescharacteristic
560C2×C3#300211yescharacteristic
561C2×C3#300411yescharacteristic
562C2×C3C2×#46411yescharacteristic
563C2×C3#303711yescharacteristic
564C2×C3#310811yescharacteristic
565C2×C3#311711yescharacteristic
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22C3×#46414yescomplemented
580C22C2×C3×#12011yescharacteristic
581C22C2×C3×#12111yescharacteristic
582C22C3×#52511yescharacteristic
583C22C2×C3×#13511yescharacteristic
584C22C2×C3×#13711yescharacteristic
585C22C2×C3×#13711yescharacteristic
586C22C3×#57511yescharacteristic
587C4—22no—
588C4—22no—
589C4—22no—
590C4—28no—
591C4—22no—
592C4—41no—
593C4—44no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4C2×C3×#12112yes—
598C4C3×#52512yes—
599C3#3290911yescharacteristic, direct product factor, Sylow
600C2—24no—
601C2—161no—
602C2C3×#284711yescharacteristic
603C2C3×#300211yescharacteristic
604C2C3×#300411yescharacteristic
605C2C2×C3×#46411yescharacteristic
606C2C3×#303711yescharacteristic
607C2C3×#310811yescharacteristic
608C2C3×#311711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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. C22×C3×C4×Q8C2semidirect
2. C2×C3×#464C2semidirect
3. C2×C3×#451C2semidirect
6. C3×#1880C2semidirect
7. C3×#1878C2semidirect
9. C3×#1872C2semidirect
10. C3×#1805C2semidirect
11. C3×#1797C2semidirect
12. #32909C3direct (N × H)
48. C22×C4×Q8C2×C3semidirect
49. C2×#464C2×C3semidirect
50. C2×#451C2×C3semidirect
53. #1880C2×C3semidirect
54. #1878C2×C3semidirect
56. #1872C2×C3semidirect
57. #1805C2×C3semidirect
58. #1797C2×C3semidirect
91. C3×C4×Q8D4semidirect
274. C4×Q8C3×D4semidirect
470. C22×C3#464semidirect
579. C22C3×#464semidirect
599. C3#32909direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C2×C3×#4642
C2×C3×#4512
C2×C3×#4482
C3×#28502
C3×#18802
C3×#18782
C3×#18742
C3×#18722
C3×#18052
C3×#17972
#329093