Finite Group Databasebeta

← C3×#25764 #10644371

Subgroups — C3×#25764 #10644371

Order 768.

1582 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic, complemented
2C2×C3×#497C211yesmaximal, characteristic, complemented
3C2×C3×#463C211yesmaximal, characteristic, complemented
4C2×C3×#452C212yesmaximal, complemented
5C2×C3×#449C212yesmaximal, complemented
6C3×#2540C211yesmaximal, characteristic, complemented
7C3×#2001C212yesmaximal, complemented
8C3×#2000C212yesmaximal, complemented
9C3×#1981C211yesmaximal, characteristic, complemented
10C3×#1955C211yesmaximal, characteristic, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25764C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2211yescharacteristic
16C22×C3×SD16C2212yes—
17C22×C3×D8C2211yescharacteristic
18C3×#547—24no—
19C3×#547—24no—
20C2×C3×C4⋊Q8C2211yescharacteristic
21C2×C3×#124C2211yescharacteristic
22C2×C3×#123—22no—
23C2×C3×C22⋊Q8C2212yes—
24C2×C3×C22⋊Q8C2212yes—
25C2×C3×#121—21no—
26C2×C3×#121C2212yes—
27C2×C3×C4×D4C2211yescharacteristic
28C2×C3×C4×D4C2211yescharacteristic
29C22×C3×C4⋊C4—21no—
30C3×#497—24no—
31C3×#463—24no—
32C3×#452—28no—
33C3×#449—28no—
34C2×C3×C4⋊C8C2211yescharacteristic
35C2×C3×#103C2212yes—
36C2×C3×#103C2211yescharacteristic
37C2×C3×#103C2212yes—
38C2×C3×#102C2211yescharacteristic
39C2×C3×#102C2212yes—
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×#547C2×C311yescharacteristic, complemented
49C2×#497C2×C311yescharacteristic, complemented
50C2×#463C2×C311yescharacteristic, complemented
51C2×#452C2×C312yescomplemented
52C2×#449C2×C312yescomplemented
53#2540C2×C311yescharacteristic, complemented
54#2001C2×C312yescomplemented
55#2000C2×C312yescomplemented
56#1981C2×C311yescharacteristic, complemented
57#1955C2×C311yescharacteristic, complemented
58#1762C2×C311yescharacteristic, complemented
59C2×C3×#41—24no—
60C2×C3×#41—24no—
61C2×C3×#41—24no—
62C2×C3×#41—24no—
63C2×C3×#41—42no—
64C2×C3×#41—42no—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×Q8C2312yes—
67C22×C3×D4—22no—
68C22×C3×D4—22no—
69C22×C3×D4D411yescharacteristic
70C22×C3×D4C2311yescharacteristic
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—22no—
73C23×C3×C4—21no—
74C23×C3×C4C2312yes—
75C2×C3×Q16—24no—
76C2×C3×Q16—24no—
77C2×C3×Q16—42no—
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
80C2×C3×SD16—22no—
81C2×C3×SD16—24no—
82C2×C3×SD16—42no—
83C2×C3×D8—24no—
84C2×C3×D8—22no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2312yes—
88C3×C4⋊Q8—22no—
89C3×C4⋊Q8D414yescomplemented
90C3×#124—22no—
91C3×#124—22no—
92C3×#123—216no—
93C3×C22⋊Q8—24no—
94C3×C22⋊Q8—24no—
95C3×C22⋊Q8—24no—
96C3×C22⋊Q8—42no—
97C3×#121—28no—
98C3×#121—24no—
99C3×#121—42no—
100C3×C4×D4—22no—
101C3×C4×D4—22no—
102C3×C4×D4—22no—
103C3×C4×D4D414yescomplemented
104C2×C3×C4⋊C4—28no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—21no—
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—21no—
114C2×C3×C22⋊C4—22no—
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4—22no—
117C2×C3×C22⋊C4—21no—
118C2×C3×C22⋊C4D412yes—
119C2×C3×C42—21no—
120C2×C3×C42C2311yescharacteristic
121C3×C4⋊C8—22no—
122C3×#103—24no—
123C3×#103—24no—
124C3×#103—22no—
125C3×#102—22no—
126C3×#102D418yescomplemented
127C3×C22⋊C8—24no—
128C3×C22⋊C8—24no—
129C3×#97—22no—
130C3×#95—21no—
131C22×#41C22×C312yes—
132C22×Q16C22×C312yes—
133C22×SD16C22×C311yescharacteristic
134C22×SD16C22×C312yes—
135C22×D8C22×C311yescharacteristic
136#547—24no—
137#547—24no—
138C2×C4⋊Q8C22×C311yescharacteristic
139C2×#124C22×C311yescharacteristic
140C2×#123—22no—
141C2×C22⋊Q8C22×C312yes—
142C2×C22⋊Q8C22×C312yes—
143C2×#121—21no—
144C2×#121C22×C312yes—
145C2×C4×D4C22×C311yescharacteristic
146C2×C4×D4C22×C311yescharacteristic
147C22×C4⋊C4—21no—
148#497—24no—
149#463—24no—
150#452—28no—
151#449—28no—
152C2×C4⋊C8C22×C311yescharacteristic
153C2×#103C22×C312yes—
154C2×#103C22×C312yes—
155C2×#103C22×C311yescharacteristic
156C2×#102C22×C311yescharacteristic
157C2×#102C22×C312yes—
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×C3—21no—
168C24×C3C2×D412yes—
169C3×#41—48no—
170C3×#41—48no—
171C3×#41—48no—
172C3×#41—48no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—21no—
177C2×C3×Q8—22no—
178C2×C3×Q8—21no—
179C2×C3×Q8—22no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8C2×D414yes—
183C2×C3×D4—28no—
184C2×C3×D4—22no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—22no—
188C2×C3×D4—21no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—42no—
192C2×C3×D4—44no—
193C2×C3×D4—44no—
194C2×C3×D4—44no—
195C2×C3×D4—44no—
196C2×C3×D4—82no—
197C2×C3×D4C2×D414yes—
198C2×C3×D4C2×D412yes—
199C22×C3×C4—28no—
200C22×C3×C4—22no—
201C22×C3×C4—21no—
202C22×C3×C4—22no—
203C22×C3×C4—24no—
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—21no—
211C22×C3×C4—44no—
212C22×C3×C4—44no—
213C22×C3×C4—44no—
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C3×Q16—48no—
219C3×SD16—48no—
220C3×SD16—44no—
221C3×D8—44no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8C2×D414yes—
229C3×C4⋊C4—28no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—44no—
236C3×C4⋊C4C2×D412yes—
237C3×C4⋊C4C2×D414yes—
238C3×C22⋊C4—28no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—44no—
242C3×C22⋊C4—44no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—42no—
245C3×C42—81no—
246C3×C42C2×D412yes—
247C3×C42C2×D412yes—
248C2×#41—24no—
249C2×#41—24no—
250C2×#41—24no—
251C2×#41—24no—
252C2×#41—42no—
253C2×#41—42no—
254C22×Q8C23×C311yescharacteristic
255C22×Q8C23×C312yes—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4C3×D411yescharacteristic
259C22×D4C23×C311yescharacteristic
260C22×D4C23×C311yescharacteristic
261C23×C4—21no—
262C23×C4—22no—
263C23×C4C23×C312yes—
264C2×Q16—24no—
265C2×Q16—24no—
266C2×Q16—42no—
267C2×SD16—24no—
268C2×SD16—24no—
269C2×SD16—22no—
270C2×SD16—24no—
271C2×SD16—42no—
272C2×D8—24no—
273C2×D8—22no—
274C22×C8C23×C311yescharacteristic
275C22×C8C23×C311yescharacteristic
276C22×C8C23×C312yes—
277C4⋊Q8—22no—
278C4⋊Q8C3×D414yescomplemented
279#124—22no—
280#124—22no—
281#123—216no—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—42no—
286#121—28no—
287#121—24no—
288#121—42no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4C3×D414yescomplemented
293C2×C4⋊C4—28no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4C3×D411yescharacteristic
299C2×C4⋊C4C23×C311yescharacteristic
300C2×C4⋊C4C23×C312yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4C3×D412yes—
308C2×C42—21no—
309C2×C42C23×C311yescharacteristic
310C4⋊C8—22no—
311#103—24no—
312#103—24no—
313#103—22no—
314#102—22no—
315#102C3×D418yescomplemented
316C22⋊C8—24no—
317C22⋊C8—24no—
318#97—22no—
319#95—21no—
320C23×C3—22no—
321C23×C3—24no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—22no—
329C23×C3—42no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3C22×D411yescharacteristic
338C3×Q8—24no—
339C3×Q8—24no—
340C3×Q8—42no—
341C3×Q8—42no—
342C3×Q8—41no—
343C3×Q8—41no—
344C3×Q8—42no—
345C3×D4—22no—
346C3×D4—24no—
347C3×D4—44no—
348C3×D4—416no—
349C3×D4—41no—
350C3×D4—41no—
351C3×D4—42no—
352C3×D4—88no—
353C3×D4#12714yes—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—48no—
364C2×C3×C4—416no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—41no—
375C2×C3×C4—42no—
376C2×C3×C4#12012yes—
377C2×C3×C4#12012yes—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D412yes—
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4H(5,𝔽2)12yes—
382C3×C8—24no—
383C3×C8—41no—
384C3×C8—41no—
385C24—21no—
386C24—21no—
387C24C2×C3×D412yes—
388#41—48no—
389#41—48no—
390#41—48no—
391#41—48no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8C2×C3×D414yes—
402C2×D4—28no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—82no—
416C2×D4C2×C3×D414yes—
417C2×D4C2×C3×D412yes—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4C2×C3×D412yes—
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C24×C311yescharacteristic
437Q16—48no—
438SD16—48no—
439SD16—44no—
440D8—44no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8C2×C3×D414yes—
448C4⋊C4—28no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—44no—
455C4⋊C4C2×C3×D412yes—
456C4⋊C4C2×C3×D414yes—
457C22⋊C4—28no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C42—81no—
465C42C2×C3×D412yes—
466C42C2×C3×D412yes—
467C22×C3—24no—
468C22×C3—42no—
469C22×C3—44no—
470C22×C3—42no—
471C22×C3—44no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—82no—
482C22×C3—81no—
483C22×C3—82no—
484C22×C3C2×#12011yescharacteristic
485C22×C3D4212yes—
486C22×C3#57512yes—
487C22×C3#57711yescharacteristic
488C22×C3#57811yescharacteristic
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—42no—
492C3×C4—44no—
493C3×C4—42no—
494C3×C4—44no—
495C3×C4—81no—
496C3×C4C2×#12711yescharacteristic
497C3×C4#53511yescharacteristic
498C3×C4D4212yes—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23C22×C3×D411yescharacteristic
517Q8—24no—
518Q8—24no—
519Q8—42no—
520Q8—42no—
521Q8—41no—
522Q8—41no—
523Q8—42no—
524D4—22no—
525D4—24no—
526D4—44no—
527D4—416no—
528D4—41no—
529D4—41no—
530D4—42no—
531D4—88no—
532D4C3×#12714yes—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—48no—
543C2×C4—416no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
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×C4C3×#12012yes—
556C2×C4C3×#12012yes—
557C2×C4C22×C3×D411yescharacteristic
558C2×C4C22×C3×D412yes—
559C2×C4C22×C3×D411yescharacteristic
560C2×C4C3×H(5,𝔽2)12yes—
561C8—24no—
562C8—41no—
563C8—41no—
564C2×C3—44no—
565C2×C3—82no—
566C2×C3—161no—
567C2×C3#243311yescharacteristic
568C2×C3#300511yescharacteristic
569C2×C3#301311yescharacteristic
570C2×C3#326712yes—
571C2×C3#326912yes—
572C22—24no—
573C22—42no—
574C22—44no—
575C22—42no—
576C22—44no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—81no—
585C22—81no—
586C22—82no—
587C22—81no—
588C22—82no—
589C22C2×C3×#12011yescharacteristic
590C22C3×D4212yes—
591C22C3×#57512yes—
592C22C3×#57711yescharacteristic
593C22C3×#57811yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—44no—
598C4—42no—
599C4—44no—
600C4—81no—
601C4C2×C3×#12711yescharacteristic
602C4C3×#53511yescharacteristic
603C4C3×D4212yes—
604C3#2576411yescharacteristic, direct product factor, Sylow
605C2—44no—
606C2—82no—
607C2—161no—
608C2C3×#243311yescharacteristic
609C2C3×#300511yescharacteristic
610C2C3×#301311yescharacteristic
611C2C3×#326712yes—
612C2C3×#326912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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×#547C2semidirect
2. C2×C3×#497C2semidirect
3. C2×C3×#463C2semidirect
4. C2×C3×#452C2semidirect
5. C2×C3×#449C2semidirect
6. C3×#2540C2semidirect
7. C3×#2001C2semidirect
8. C3×#2000C2semidirect
9. C3×#1981C2semidirect
10. C3×#1955C2semidirect
11. C3×#1762C2semidirect
12. #25764C3direct (N × H)
48. C2×#547C2×C3semidirect
49. C2×#497C2×C3semidirect
50. C2×#463C2×C3semidirect
51. C2×#452C2×C3semidirect
52. C2×#449C2×C3semidirect
53. #2540C2×C3semidirect
54. #2001C2×C3semidirect
55. #2000C2×C3semidirect
56. #1981C2×C3semidirect
57. #1955C2×C3semidirect
58. #1762C2×C3semidirect
89. C3×C4⋊Q8D4semidirect
103. C3×C4×D4D4semidirect
126. C3×#102D4semidirect
278. C4⋊Q8C3×D4semidirect
292. C4×D4C3×D4semidirect
315. #102C3×D4semidirect
604. C3#25764direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×C3×#4972
C2×C3×#4632
C2×C3×#4522
C2×C3×#4492
C3×#25402
C3×#20012
C3×#20002
C3×#19812
C3×#19552
C3×#17622
#257643