Finite Group Databasebeta

← C3×#25767 #10644374

Subgroups — C3×#25767 #10644374

Order 768.

1534 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×#548C211yesmaximal, characteristic
2C2×C3×#497C211yesmaximal, characteristic, complemented
3C2×C3×#462C211yesmaximal, characteristic, complemented
4C2×C3×#452C212yesmaximal, complemented
5C2×C3×#451C212yesmaximal, complemented
6C3×#2542C211yesmaximal, characteristic, complemented
7C3×#2001C212yesmaximal, complemented
8C3×#2000C212yesmaximal, complemented
9C3×#1983C211yesmaximal, characteristic, complemented
10C3×#1955C211yesmaximal, characteristic, complemented
11C3×#1763C211yesmaximal, characteristic, complemented
12#25767C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×C3×Q16C2212yes—
16C22×C3×Q16C2212yes—
17C22×C3×SD16C2212yes—
18C3×#548—24no—
19C3×#548—24no—
20C2×C3×C4⋊Q8C2211yescharacteristic
21C2×C3×#124—21no—
22C2×C3×#124C2211yescharacteristic
23C2×C3×C22⋊Q8—21no—
24C2×C3×C22⋊Q8C2211yescharacteristic
25C2×C3×C22⋊Q8C2212yes—
26C2×C3×C22⋊Q8C2212yes—
27C2×C3×#121—21no—
28C2×C3×#121C2211yescharacteristic
29C2×C3×C4×Q8C2211yescharacteristic
30C2×C3×C4×D4—21no—
31C2×C3×C4×D4C2211yescharacteristic
32C3×#497—24no—
33C3×#462—24no—
34C3×#452—28no—
35C3×#451—28no—
36C2×C3×C4⋊C8C2211yescharacteristic
37C2×C3×#103C2212yes—
38C2×C3×#103C2212yes—
39C2×C3×#103C2212yes—
40C2×C3×#102C2212yes—
41C2×C3×C22⋊C8C2212yes—
42C2×C3×C22⋊C8C2212yes—
43C2×C3×#97C2211yescharacteristic
44C3×#399—21no—
45C3×#399—21no—
46C3×#390—21no—
47C3×#384—21no—
48C3×C4×C4⋊C4C2211yescharacteristic
49C3×#340C2212yes—
50C2×#548C2×C311yescharacteristic
51C2×#497C2×C311yescharacteristic, complemented
52C2×#462C2×C311yescharacteristic, complemented
53C2×#452C2×C312yescomplemented
54C2×#451C2×C312yescomplemented
55#2542C2×C311yescharacteristic, complemented
56#2001C2×C312yescomplemented
57#2000C2×C312yescomplemented
58#1983C2×C311yescharacteristic, complemented
59#1955C2×C311yescharacteristic, complemented
60#1763C2×C311yescharacteristic, complemented
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—24no—
64C2×C3×#41—42no—
65C22×C3×Q8—22no—
66C22×C3×Q8—22no—
67C22×C3×Q8—21no—
68C22×C3×Q8—24no—
69C22×C3×Q8—42no—
70C22×C3×Q8C2312yes—
71C22×C3×Q8C2311yescharacteristic
72C22×C3×Q8C2311yescharacteristic
73C22×C3×D4—21no—
74C22×C3×D4—21no—
75C22×C3×D4D411yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C23×C3×C4C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C2×C3×Q16—24no—
82C2×C3×Q16—24no—
83C2×C3×Q16—24no—
84C2×C3×Q16—24no—
85C2×C3×Q16—24no—
86C2×C3×Q16—42no—
87C2×C3×SD16—24no—
88C2×C3×SD16—24no—
89C2×C3×SD16—42no—
90C22×C3×C8C2312yes—
91C22×C3×C8C2312yes—
92C3×C4⋊Q8—22no—
93C3×C4⋊Q8—22no—
94C3×#124—28no—
95C3×#124—22no—
96C3×#124D414yes—
97C3×C22⋊Q8—28no—
98C3×C22⋊Q8—22no—
99C3×C22⋊Q8—22no—
100C3×C22⋊Q8—24no—
101C3×C22⋊Q8—24no—
102C3×C22⋊Q8—42no—
103C3×C22⋊Q8—42no—
104C3×#121—28no—
105C3×#121—22no—
106C3×#121—22no—
107C3×C4×Q8—22no—
108C3×C4×Q8D414yes—
109C3×C4×D4—28no—
110C3×C4×D4—22no—
111C3×C4×D4—22no—
112C2×C3×C4⋊C4—21no—
113C2×C3×C4⋊C4—21no—
114C2×C3×C4⋊C4—21no—
115C2×C3×C4⋊C4—22no—
116C2×C3×C4⋊C4—22no—
117C2×C3×C4⋊C4—21no—
118C2×C3×C4⋊C4D411yescharacteristic
119C2×C3×C4⋊C4C2312yes—
120C2×C3×C4⋊C4C2311yescharacteristic
121C2×C3×C22⋊C4—21no—
122C2×C3×C22⋊C4—21no—
123C2×C3×C22⋊C4—21no—
124C2×C3×C22⋊C4—21no—
125C2×C3×C22⋊C4—22no—
126C2×C3×C22⋊C4—22no—
127C2×C3×C22⋊C4D411yescharacteristic
128C2×C3×C22⋊C4D411yescharacteristic
129C2×C3×C42—21no—
130C2×C3×C42—21no—
131C2×C3×C42C2311yescharacteristic
132C3×C4⋊C8—22no—
133C3×#103—24no—
134C3×#103—24no—
135C3×#103D418yescomplemented
136C3×#102—24no—
137C3×C22⋊C8—24no—
138C3×C22⋊C8—24no—
139C3×#97—22no—
140C3×#95—21no—
141C22×#41C22×C311yescharacteristic
142C23×Q8C22×C311yescharacteristic
143C22×Q16C22×C312yes—
144C22×Q16C22×C312yes—
145C22×SD16C22×C312yes—
146#548—24no—
147#548—24no—
148C2×C4⋊Q8C22×C311yescharacteristic
149C2×#124—21no—
150C2×#124C22×C311yescharacteristic
151C2×C22⋊Q8—21no—
152C2×C22⋊Q8C22×C311yescharacteristic
153C2×C22⋊Q8C22×C312yes—
154C2×C22⋊Q8C22×C312yes—
155C2×#121—21no—
156C2×#121C22×C311yescharacteristic
157C2×C4×Q8C22×C311yescharacteristic
158C2×C4×D4—21no—
159C2×C4×D4C22×C311yescharacteristic
160#497—24no—
161#462—24no—
162#452—28no—
163#451—28no—
164C2×C4⋊C8C22×C311yescharacteristic
165C2×#103C22×C312yes—
166C2×#103C22×C312yes—
167C2×#103C22×C312yes—
168C2×#102C22×C312yes—
169C2×C22⋊C8C22×C312yes—
170C2×C22⋊C8C22×C312yes—
171C2×#97C22×C311yescharacteristic
172#399—21no—
173#399—21no—
174#390—21no—
175#384—21no—
176C4×C4⋊C4C22×C311yescharacteristic
177#340C22×C312yes—
178C24×C3—21no—
179C24×C3C2×D411yescharacteristic
180C24×C3C2×D411yescharacteristic
181C3×#41—48no—
182C3×#41—44no—
183C3×#41—44no—
184C2×C3×Q8—24no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—22no—
188C2×C3×Q8—22no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8—22no—
192C2×C3×Q8—22no—
193C2×C3×Q8—22no—
194C2×C3×Q8—22no—
195C2×C3×Q8—22no—
196C2×C3×Q8—44no—
197C2×C3×Q8—48no—
198C2×C3×Q8—44no—
199C2×C3×Q8—44no—
200C2×C3×Q8C2×D412yes—
201C2×C3×Q8C2×D414yes—
202C2×C3×Q8C2×D412yes—
203C2×C3×D4—24no—
204C2×C3×D4—24no—
205C2×C3×D4—22no—
206C2×C3×D4—22no—
207C2×C3×D4—22no—
208C2×C3×D4—21no—
209C2×C3×D4—22no—
210C2×C3×D4—41no—
211C2×C3×D4—44no—
212C2×C3×D4—42no—
213C2×C3×D4—42no—
214C2×C3×D4C2×D412yes—
215C22×C3×C4—21no—
216C22×C3×C4—24no—
217C22×C3×C4—24no—
218C22×C3×C4—22no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—22no—
222C22×C3×C4—22no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—22no—
228C22×C3×C4—22no—
229C22×C3×C4—22no—
230C22×C3×C4—44no—
231C22×C3×C4—44no—
232C22×C3×C4C2×D411yescharacteristic
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2×D411yescharacteristic
236C22×C3×C4C2411yescharacteristic
237C3×Q16—48no—
238C3×Q16—48no—
239C3×SD16—48no—
240C2×C3×C8—22no—
241C2×C3×C8—22no—
242C2×C3×C8—22no—
243C2×C3×C8—22no—
244C2×C3×C8—22no—
245C2×C3×C8C2×D414yes—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—44no—
252C3×C4⋊C4—44no—
253C3×C4⋊C4—42no—
254C3×C4⋊C4—42no—
255C3×C4⋊C4C2×D414yes—
256C3×C4⋊C4C2×D412yes—
257C3×C22⋊C4—24no—
258C3×C22⋊C4—24no—
259C3×C22⋊C4—24no—
260C3×C22⋊C4—24no—
261C3×C22⋊C4—22no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—44no—
264C3×C22⋊C4—44no—
265C3×C42—24no—
266C3×C42—42no—
267C3×C42C2×D412yes—
268C3×C42C2×D412yes—
269C2×#41—22no—
270C2×#41—22no—
271C2×#41—24no—
272C2×#41—42no—
273C22×Q8—22no—
274C22×Q8—22no—
275C22×Q8—21no—
276C22×Q8—24no—
277C22×Q8—42no—
278C22×Q8C23×C312yes—
279C22×Q8C23×C311yescharacteristic
280C22×Q8C23×C311yescharacteristic
281C22×D4—21no—
282C22×D4—21no—
283C22×D4C3×D411yescharacteristic
284C22×D4C23×C311yescharacteristic
285C23×C4—21no—
286C23×C4—21no—
287C23×C4C23×C311yescharacteristic
288C23×C4C23×C311yescharacteristic
289C2×Q16—24no—
290C2×Q16—24no—
291C2×Q16—24no—
292C2×Q16—24no—
293C2×Q16—24no—
294C2×Q16—42no—
295C2×SD16—24no—
296C2×SD16—24no—
297C2×SD16—42no—
298C22×C8C23×C312yes—
299C22×C8C23×C312yes—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302#124—28no—
303#124—22no—
304#124C3×D414yes—
305C22⋊Q8—28no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310C22⋊Q8—42no—
311C22⋊Q8—42no—
312#121—28no—
313#121—22no—
314#121—22no—
315C4×Q8—22no—
316C4×Q8C3×D414yes—
317C4×D4—28no—
318C4×D4—22no—
319C4×D4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4C3×D411yescharacteristic
327C2×C4⋊C4C23×C311yescharacteristic
328C2×C4⋊C4C23×C312yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C3×D411yescharacteristic
336C2×C22⋊C4C3×D411yescharacteristic
337C2×C42—21no—
338C2×C42—21no—
339C2×C42C23×C311yescharacteristic
340C4⋊C8—22no—
341#103—24no—
342#103—24no—
343#103C3×D418yescomplemented
344#102—24no—
345C22⋊C8—24no—
346C22⋊C8—24no—
347#97—22no—
348#95—21no—
349C23×C3—21no—
350C23×C3—22no—
351C23×C3—21no—
352C23×C3—22no—
353C23×C3—21no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—22no—
359C23×C3—42no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—42no—
363C23×C3#13712yes—
364C23×C3C22×D411yescharacteristic
365C3×Q8—22no—
366C3×Q8—24no—
367C3×Q8—24no—
368C3×Q8—48no—
369C3×Q8—41no—
370C3×Q8—41no—
371C3×Q8—42no—
372C3×Q8—42no—
373C3×Q8—42no—
374C3×Q8—42no—
375C3×Q8#12714yes—
376C3×D4—24no—
377C3×D4—44no—
378C3×D4—48no—
379C3×D4—48no—
380C3×D4—41no—
381C3×D4—41no—
382C2×C3×C4—22no—
383C2×C3×C4—24no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—24no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—48no—
395C2×C3×C4—48no—
396C2×C3×C4—42no—
397C2×C3×C4—41no—
398C2×C3×C4—42no—
399C2×C3×C4—41no—
400C2×C3×C4—42no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—42no—
406C2×C3×C4—42no—
407C2×C3×C4—42no—
408C2×C3×C4—42no—
409C2×C3×C4—42no—
410C2×C3×C4#12012yes—
411C2×C3×C4#12012yes—
412C2×C3×C4C22×D411yescharacteristic
413C2×C3×C4C22×D412yes—
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4H(5,𝔽2)12yes—
416C3×C8—24no—
417C3×C8—42no—
418C24—21no—
419C24C2×C3×D411yescharacteristic
420C24C2×C3×D411yescharacteristic
421#41—48no—
422#41—44no—
423#41—44no—
424C2×Q8—24no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—44no—
437C2×Q8—48no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8C2×C3×D412yes—
441C2×Q8C2×C3×D414yes—
442C2×Q8C2×C3×D412yes—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—41no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4C2×C3×D412yes—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4C2×C3×D411yescharacteristic
473C22×C4C2×C3×D411yescharacteristic
474C22×C4C2×C3×D411yescharacteristic
475C22×C4C2×C3×D411yescharacteristic
476C22×C4C24×C311yescharacteristic
477Q16—48no—
478Q16—48no—
479SD16—48no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8C2×C3×D414yes—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4C2×C3×D414yes—
496C4⋊C4C2×C3×D412yes—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C42—24no—
506C42—42no—
507C42C2×C3×D412yes—
508C42C2×C3×D412yes—
509C22×C3—22no—
510C22×C3—24no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—42no—
514C22×C3—42no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—42no—
518C22×C3—41no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—82no—
523C22×C3—82no—
524C22×C3C2×#12011yescharacteristic
525C22×C3D4212yes—
526C22×C3C2×#13711yescharacteristic
527C22×C3#57511yescharacteristic
528C22×C3#57812yes—
529C3×C4—24no—
530C3×C4—22no—
531C3×C4—22no—
532C3×C4—41no—
533C3×C4—41no—
534C3×C4—42no—
535C3×C4—42no—
536C3×C4—44no—
537C3×C4—82no—
538C3×C4C2×#12711yescharacteristic
539C3×C4#53511yescharacteristic
540C3×C4D4212yes—
541C23—21no—
542C23—22no—
543C23—21no—
544C23—22no—
545C23—21no—
546C23—21no—
547C23—22no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23C3×#13712yes—
556C23C22×C3×D411yescharacteristic
557Q8—22no—
558Q8—24no—
559Q8—24no—
560Q8—48no—
561Q8—41no—
562Q8—41no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—42no—
567Q8C3×#12714yes—
568D4—24no—
569D4—44no—
570D4—48no—
571D4—48no—
572D4—41no—
573D4—41no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4C3×#12012yes—
603C2×C4C3×#12012yes—
604C2×C4C22×C3×D411yescharacteristic
605C2×C4C22×C3×D412yes—
606C2×C4C22×C3×D411yescharacteristic
607C2×C4C3×H(5,𝔽2)12yes—
608C8—24no—
609C8—42no—
610C2×C3—42no—
611C2×C3—81no—
612C2×C3—82no—
613C2×C3#243311yescharacteristic
614C2×C3#300711yescharacteristic
615C2×C3#301111yescharacteristic
616C2×C3#326912yes—
617C2×C3#327012yes—
618C22—22no—
619C22—24no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22—42no—
624C22—41no—
625C22—41no—
626C22—42no—
627C22—41no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—82no—
632C22—82no—
633C22C2×C3×#12011yescharacteristic
634C22C3×D4212yes—
635C22C2×C3×#13711yescharacteristic
636C22C3×#57511yescharacteristic
637C22C3×#57812yes—
638C4—24no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—42no—
644C4—42no—
645C4—44no—
646C4—82no—
647C4C2×C3×#12711yescharacteristic
648C4C3×#53511yescharacteristic
649C4C3×D4212yes—
650C3#2576711yescharacteristic, direct product factor, Sylow
651C2—42no—
652C2—81no—
653C2—82no—
654C2C3×#243311yescharacteristic
655C2C3×#300711yescharacteristic
656C2C3×#301111yescharacteristic
657C2C3×#326912yes—
658C2C3×#327012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#548C2
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
2. C2×C3×#497C2semidirect
3. C2×C3×#462C2semidirect
4. C2×C3×#452C2semidirect
5. C2×C3×#451C2semidirect
6. C3×#2542C2semidirect
7. C3×#2001C2semidirect
8. C3×#2000C2semidirect
9. C3×#1983C2semidirect
10. C3×#1955C2semidirect
11. C3×#1763C2semidirect
12. #25767C3direct (N × H)
51. C2×#497C2×C3semidirect
52. C2×#462C2×C3semidirect
53. C2×#452C2×C3semidirect
54. C2×#451C2×C3semidirect
55. #2542C2×C3semidirect
56. #2001C2×C3semidirect
57. #2000C2×C3semidirect
58. #1983C2×C3semidirect
59. #1955C2×C3semidirect
60. #1763C2×C3semidirect
135. C3×#103D4semidirect
343. #103C3×D4semidirect
650. C3#25767direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5482
C2×C3×#4972
C2×C3×#4622
C2×C3×#4522
C2×C3×#4512
C3×#25422
C3×#20012
C3×#20002
C3×#19832
C3×#19552
C3×#17632
#257673