Finite Group Databasebeta

← C3×#25766 #10644373

Subgroups — C3×#25766 #10644373

Order 768.

1550 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#497C211yesmaximal, characteristic, complemented
3C2×C3×#464C211yesmaximal, characteristic, complemented
4C2×C3×#451C212yesmaximal, complemented
5C2×C3×#448C212yesmaximal, complemented
6C3×#2540C211yesmaximal, characteristic
7C3×#2001C212yesmaximal, complemented
8C3×#2000C212yesmaximal, complemented
9C3×#1981C211yesmaximal, characteristic, complemented
10C3×#1955C211yesmaximal, characteristic, complemented
11C3×#1763C211yesmaximal, characteristic, complemented
12#25766C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2212yes—
14C22×C3×Q16C2211yescharacteristic
15C22×C3×Q16C2212yes—
16C22×C3×SD16C2211yescharacteristic
17C22×C3×SD16C2212yes—
18C3×D4×Q8—24no—
19C3×D4×Q8—24no—
20C2×C3×C4⋊Q8—21no—
21C2×C3×C4⋊Q8C2211yescharacteristic
22C2×C3×#124C2211yescharacteristic
23C2×C3×C22⋊Q8—22no—
24C2×C3×C22⋊Q8C2212yes—
25C2×C3×C22⋊Q8C2212yes—
26C2×C3×#121C2212yes—
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×D4—21no—
29C2×C3×C4×D4C2211yescharacteristic
30C3×#497—24no—
31C3×#464—24no—
32C3×#451—28no—
33C3×#448—28no—
34C2×C3×C4⋊C8C2211yescharacteristic
35C2×C3×#103C2212yes—
36C2×C3×#103C2211yescharacteristic
37C2×C3×#103C2212yes—
38C2×C3×#103C2212yes—
39C2×C3×#102C2211yescharacteristic
40C2×C3×C22⋊C8C2212yes—
41C2×C3×C22⋊C8C2212yes—
42C2×C3×#97C2211yescharacteristic
43C3×#399—21no—
44C3×#392—21no—
45C3×#390—22no—
46C3×C4×C4⋊C4C2211yescharacteristic
47C3×#340C2212yes—
48C2×D4×Q8C2×C311yescharacteristic, complemented
49C2×#497C2×C311yescharacteristic, complemented
50C2×#464C2×C311yescharacteristic, complemented
51C2×#451C2×C312yescomplemented
52C2×#448C2×C312yescomplemented
53#2540C2×C311yescharacteristic
54#2001C2×C312yescomplemented
55#2000C2×C312yescomplemented
56#1981C2×C311yescharacteristic, complemented
57#1955C2×C311yescharacteristic, complemented
58#1763C2×C311yescharacteristic, complemented
59C22×C3×Q8—24no—
60C22×C3×Q8—24no—
61C22×C3×Q8—22no—
62C22×C3×Q8—24no—
63C22×C3×Q8—24no—
64C22×C3×Q8—42no—
65C22×C3×Q8—42no—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×Q8C2312yes—
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4—22no—
70C22×C3×D4D411yescharacteristic
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—22no—
73C23×C3×C4C2312yes—
74C2×C3×Q16—24no—
75C2×C3×Q16—22no—
76C2×C3×Q16—24no—
77C2×C3×Q16—24no—
78C2×C3×Q16—42no—
79C2×C3×SD16—24no—
80C2×C3×SD16—22no—
81C2×C3×SD16—24no—
82C2×C3×SD16—24no—
83C2×C3×SD16—42no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2312yes—
87C3×C4⋊Q8—28no—
88C3×C4⋊Q8—22no—
89C3×C4⋊Q8D414yescomplemented
90C3×#124—22no—
91C3×#124—22no—
92C3×C22⋊Q8—216no—
93C3×C22⋊Q8—24no—
94C3×C22⋊Q8—24no—
95C3×C22⋊Q8—24no—
96C3×C22⋊Q8—42no—
97C3×#121—24no—
98C3×#121—42no—
99C3×C4×Q8—22no—
100C3×C4×Q8D414yescomplemented
101C3×C4×D4—28no—
102C3×C4×D4—22no—
103C3×C4×D4—22no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4—22no—
108C2×C3×C4⋊C4—22no—
109C2×C3×C4⋊C4D411yescharacteristic
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—22no—
116C2×C3×C22⋊C4—21no—
117C2×C3×C22⋊C4D412yes—
118C2×C3×C42—21no—
119C2×C3×C42—21no—
120C2×C3×C42C2311yescharacteristic
121C3×C4⋊C8—22no—
122C3×#103—24no—
123C3×#103—24no—
124C3×#103—22no—
125C3×#103D418yescomplemented
126C3×#102—22no—
127C3×C22⋊C8—24no—
128C3×C22⋊C8—24no—
129C3×#97—22no—
130C3×#95—21no—
131C23×Q8C22×C312yes—
132C22×Q16C22×C311yescharacteristic
133C22×Q16C22×C312yes—
134C22×SD16C22×C311yescharacteristic
135C22×SD16C22×C312yes—
136D4×Q8—24no—
137D4×Q8—24no—
138C2×C4⋊Q8—21no—
139C2×C4⋊Q8C22×C311yescharacteristic
140C2×#124C22×C311yescharacteristic
141C2×C22⋊Q8—22no—
142C2×C22⋊Q8C22×C312yes—
143C2×C22⋊Q8C22×C312yes—
144C2×#121C22×C312yes—
145C2×C4×Q8C22×C311yescharacteristic
146C2×C4×D4—21no—
147C2×C4×D4C22×C311yescharacteristic
148#497—24no—
149#464—24no—
150#451—28no—
151#448—28no—
152C2×C4⋊C8C22×C311yescharacteristic
153C2×#103C22×C312yes—
154C2×#103C22×C312yes—
155C2×#103C22×C311yescharacteristic
156C2×#103C22×C312yes—
157C2×#102C22×C311yescharacteristic
158C2×C22⋊C8C22×C312yes—
159C2×C22⋊C8C22×C312yes—
160C2×#97C22×C311yescharacteristic
161#399—21no—
162#392—21no—
163#390—22no—
164C4×C4⋊C4C22×C311yescharacteristic
165#340C22×C312yes—
166C24×C3—21no—
167C24×C3C2×D412yes—
168C2×C3×Q8—28no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—21no—
177C2×C3×Q8—22no—
178C2×C3×Q8—22no—
179C2×C3×Q8—22no—
180C2×C3×Q8—44no—
181C2×C3×Q8—44no—
182C2×C3×Q8—48no—
183C2×C3×Q8—48no—
184C2×C3×Q8—48no—
185C2×C3×Q8—48no—
186C2×C3×Q8C2×D414yes—
187C2×C3×Q8C2×D414yes—
188C2×C3×Q8C2×D412yes—
189C2×C3×D4—24no—
190C2×C3×D4—22no—
191C2×C3×D4—21no—
192C2×C3×D4—22no—
193C2×C3×D4—21no—
194C2×C3×D4—22no—
195C2×C3×D4—44no—
196C2×C3×D4—44no—
197C2×C3×D4—82no—
198C22×C3×C4—21no—
199C22×C3×C4—28no—
200C22×C3×C4—22no—
201C22×C3×C4—21no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—21no—
210C22×C3×C4—44no—
211C22×C3×C4—44no—
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C3×Q16—48no—
217C3×Q16—44no—
218C3×SD16—48no—
219C3×SD16—44no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8C2×D414yes—
227C3×C4⋊C4—28no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—44no—
235C3×C4⋊C4C2×D412yes—
236C3×C4⋊C4C2×D414yes—
237C3×C22⋊C4—28no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—44no—
240C3×C22⋊C4—44no—
241C3×C22⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C42—24no—
244C3×C42—81no—
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×C311yescharacteristic
255C22×Q8C23×C312yes—
256C22×Q8C23×C311yescharacteristic
257C22×D4—22no—
258C22×D4C3×D411yescharacteristic
259C22×D4C23×C311yescharacteristic
260C23×C4—22no—
261C23×C4C23×C312yes—
262C2×Q16—24no—
263C2×Q16—22no—
264C2×Q16—24no—
265C2×Q16—24no—
266C2×Q16—42no—
267C2×SD16—24no—
268C2×SD16—22no—
269C2×SD16—24no—
270C2×SD16—24no—
271C2×SD16—42no—
272C22×C8C23×C311yescharacteristic
273C22×C8C23×C311yescharacteristic
274C22×C8C23×C312yes—
275C4⋊Q8—28no—
276C4⋊Q8—22no—
277C4⋊Q8C3×D414yescomplemented
278#124—22no—
279#124—22no—
280C22⋊Q8—216no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—42no—
285#121—24no—
286#121—42no—
287C4×Q8—22no—
288C4×Q8C3×D414yescomplemented
289C4×D4—28no—
290C4×D4—22no—
291C4×D4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4C3×D411yescharacteristic
298C2×C4⋊C4C23×C311yescharacteristic
299C2×C4⋊C4C23×C312yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4C3×D412yes—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42C23×C311yescharacteristic
309C4⋊C8—22no—
310#103—24no—
311#103—24no—
312#103—22no—
313#103C3×D418yescomplemented
314#102—22no—
315C22⋊C8—24no—
316C22⋊C8—24no—
317#97—22no—
318#95—21no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—41no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3#13714yes—
331C23×C3C22×D411yescharacteristic
332C3×Q8—22no—
333C3×Q8—24no—
334C3×Q8—24no—
335C3×Q8—24no—
336C3×Q8—416no—
337C3×Q8—42no—
338C3×Q8—42no—
339C3×Q8—41no—
340C3×Q8—41no—
341C3×Q8—42no—
342C3×Q8#12714yes—
343C3×D4—28no—
344C3×D4—41no—
345C3×D4—41no—
346C3×D4—42no—
347C3×D4—88no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—416no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—41no—
371C2×C3×C4—42no—
372C2×C3×C4#12012yes—
373C2×C3×C4#12012yes—
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×D412yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4H(5,𝔽2)12yes—
378C3×C8—24no—
379C3×C8—41no—
380C3×C8—41no—
381C24—21no—
382C24C2×C3×D412yes—
383C2×Q8—28no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—48no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8C2×C3×D414yes—
402C2×Q8C2×C3×D414yes—
403C2×Q8C2×C3×D412yes—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—82no—
413C22×C4—21no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4C2×C3×D412yes—
428C22×C4C2×C3×D411yescharacteristic
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C24×C311yescharacteristic
431Q16—48no—
432Q16—44no—
433SD16—48no—
434SD16—44no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8C2×C3×D414yes—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—44no—
450C4⋊C4C2×C3×D412yes—
451C4⋊C4C2×C3×D414yes—
452C22⋊C4—28no—
453C22⋊C4—24no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C42—24no—
459C42—81no—
460C42C2×C3×D412yes—
461C42C2×C3×D412yes—
462C22×C3—24no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3—82no—
472C22×C3—81no—
473C22×C3—82no—
474C22×C3C2×#12011yescharacteristic
475C22×C3D4212yes—
476C22×C3C2×#13712yes—
477C22×C3#57711yescharacteristic
478C22×C3#57811yescharacteristic
479C3×C4—24no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—42no—
483C3×C4—44no—
484C3×C4—44no—
485C3×C4—42no—
486C3×C4—81no—
487C3×C4C2×#12711yescharacteristic
488C3×C4#53511yescharacteristic
489C3×C4D4212yes—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—42no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23C3×#13714yes—
502C23C22×C3×D411yescharacteristic
503Q8—22no—
504Q8—24no—
505Q8—24no—
506Q8—24no—
507Q8—416no—
508Q8—42no—
509Q8—42no—
510Q8—41no—
511Q8—41no—
512Q8—42no—
513Q8C3×#12714yes—
514D4—28no—
515D4—41no—
516D4—41no—
517D4—42no—
518D4—88no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—416no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4C3×#12012yes—
544C2×C4C3×#12012yes—
545C2×C4C22×C3×D411yescharacteristic
546C2×C4C22×C3×D412yes—
547C2×C4C22×C3×D411yescharacteristic
548C2×C4C3×H(5,𝔽2)12yes—
549C8—24no—
550C8—41no—
551C8—41no—
552C2×C3—44no—
553C2×C3—161no—
554C2×C3#243311yescharacteristic
555C2×C3#300411yescharacteristic
556C2×C3#301311yescharacteristic
557C2×C3#326612yes—
558C2×C3#327012yes—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—81no—
567C22—81no—
568C22—82no—
569C22—81no—
570C22—82no—
571C22C2×C3×#12011yescharacteristic
572C22C3×D4212yes—
573C22C2×C3×#13712yes—
574C22C3×#57711yescharacteristic
575C22C3×#57811yescharacteristic
576C4—24no—
577C4—22no—
578C4—22no—
579C4—42no—
580C4—44no—
581C4—44no—
582C4—42no—
583C4—81no—
584C4C2×C3×#12711yescharacteristic
585C4C3×#53511yescharacteristic
586C4C3×D4212yes—
587C3#2576611yescharacteristic, direct product factor, Sylow
588C2—44no—
589C2—161no—
590C2C3×#243311yescharacteristic
591C2C3×#300411yescharacteristic
592C2C3×#301311yescharacteristic
593C2C3×#326612yes—
594C2C3×#327012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2540C2
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×#497C2semidirect
3. C2×C3×#464C2semidirect
4. C2×C3×#451C2semidirect
5. C2×C3×#448C2semidirect
7. C3×#2001C2semidirect
8. C3×#2000C2semidirect
9. C3×#1981C2semidirect
10. C3×#1955C2semidirect
11. C3×#1763C2semidirect
12. #25766C3direct (N × H)
48. C2×D4×Q8C2×C3semidirect
49. C2×#497C2×C3semidirect
50. C2×#464C2×C3semidirect
51. C2×#451C2×C3semidirect
52. C2×#448C2×C3semidirect
54. #2001C2×C3semidirect
55. #2000C2×C3semidirect
56. #1981C2×C3semidirect
57. #1955C2×C3semidirect
58. #1763C2×C3semidirect
89. C3×C4⋊Q8D4semidirect
100. C3×C4×Q8D4semidirect
125. C3×#103D4semidirect
277. C4⋊Q8C3×D4semidirect
288. C4×Q8C3×D4semidirect
313. #103C3×D4semidirect
587. C3#25766direct (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×#4972
C2×C3×#4642
C2×C3×#4512
C2×C3×#4482
C3×#25402
C3×#20012
C3×#20002
C3×#19812
C3×#19552
C3×#17632
#257663