Finite Group Databasebeta

← C3×#18694 #10637301

Subgroups — C3×#18694 #10637301

Order 768.

942 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3552C211yesmaximal, characteristic, complemented
2C3×#3217C211yesmaximal, characteristic, complemented
3C3×#3217C211yesmaximal, characteristic, complemented
4C2×C3×#486C211yesmaximal, characteristic, complemented
5C3×#1628C212yesmaximal, complemented
6C3×#1627C212yesmaximal, complemented
7C3×#1626C212yesmaximal, complemented
8C3×#1625C212yesmaximal, complemented
9C3×#1556C211yesmaximal, characteristic, complemented
10C3×#1494C211yesmaximal, characteristic, complemented
11C3×#1494C211yesmaximal, characteristic, complemented
12#18694C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#563—22no—
14C3×#557C2211yescharacteristic, complemented
15C3×#556C2211yescharacteristic, complemented
16C3×#554—21no—
17C3×#553—21no—
18C3×D4×Q8—21no—
19C3×D4×Q8C2211yescharacteristic, complemented
20C3×#548—21no—
21C3×#547C2211yescharacteristic, complemented
22C3×#544—21no—
23C3×#542—21no—
24C3×#541—21no—
25C3×#539—22no—
26C3×#536—21no—
27C2×C3×C4⋊Q8C2211yescharacteristic
28C2×C3×#127C2211yescharacteristic
29C3×#529C2211yescharacteristic
30C3×#529C2211yescharacteristic
31C3×#496—21no—
32C3×#496—21no—
33C3×#490—21no—
34C3×#490—21no—
35C3×#486—22no—
36C3×#486C2212yes—
37C3×#486C2212yes—
38C3×#460C2212yes—
39C3×#460C2212yes—
40C3×#459C2212yes—
41C3×#459C2212yes—
42C3×#450—21no—
43C3×#450—21no—
44C3×#447—21no—
45C3×#447—21no—
46C3×#432C2211yescharacteristic
47C3×#432C2211yescharacteristic
48C2×C3×#102—22no—
49C2×C3×C4×C8C2211yescharacteristic
50C3×#332C2212yescomplemented
51C3×#332C2212yescomplemented
52C3×Q8⋊C8C2212yescomplemented
53C3×Q8⋊C8C2212yescomplemented
54C3×D4⋊C8C2212yescomplemented
55C3×D4⋊C8C2212yescomplemented
56#3552C2×C311yescharacteristic, complemented
57#3217C2×C311yescharacteristic, complemented
58#3217C2×C311yescharacteristic, complemented
59C2×#486C2×C311yescharacteristic, complemented
60#1628C2×C312yescomplemented
61#1627C2×C312yescomplemented
62#1626C2×C312yescomplemented
63#1625C2×C312yescomplemented
64#1556C2×C311yescharacteristic, complemented
65#1494C2×C311yescharacteristic, complemented
66#1494C2×C311yescharacteristic, complemented
67C2×C3×#41—21no—
68C22×C3×Q8—21no—
69C22×C3×Q8—21no—
70C22×C3×D4—21no—
71C22×C3×D4—21no—
72C22×C3×D4—21no—
73C2×C3×SD16—22no—
74C2×C3×SD16—22no—
75C2×C3×D8—22no—
76C2×C3×D8—22no—
77C22×C3×C8—21no—
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8—21no—
80C3×C4⋊Q8—21no—
81C3×C4⋊Q8—21no—
82C3×C4⋊Q8—21no—
83C3×C4⋊Q8D412yes—
84C3×C4⋊Q8C2311yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×#127—21no—
87C3×#127—21no—
88C3×#127—21no—
89C3×#127C2312yes—
90C3×#126—22no—
91C3×#126—22no—
92C3×#126—22no—
93C3×#126—21no—
94C3×#126—21no—
95C3×#125—22no—
96C3×#125—22no—
97C3×#125—22no—
98C3×#125—21no—
99C3×#125D411yescharacteristic
100C3×#125D411yescharacteristic
101C3×#124—22no—
102C3×#124—21no—
103C3×#124D411yescharacteristic
104C3×#124D411yescharacteristic
105C3×#123—22no—
106C3×#123—22no—
107C3×#123—21no—
108C3×#123—21no—
109C3×#123—22no—
110C3×C22⋊Q8—22no—
111C3×C22⋊Q8—22no—
112C3×C22⋊Q8—22no—
113C3×C22⋊Q8—22no—
114C3×C22⋊Q8—22no—
115C3×C22⋊Q8—21no—
116C3×C22⋊Q8—22no—
117C3×C22⋊Q8—21no—
118C3×C22⋊Q8—21no—
119C3×C22⋊Q8D411yescharacteristic
120C3×C22⋊Q8D411yescharacteristic
121C3×#121—22no—
122C3×#121—21no—
123C3×#121D411yescharacteristic
124C3×#121D411yescharacteristic
125C3×C4×Q8—21no—
126C3×C4×Q8—21no—
127C3×C4×Q8—21no—
128C3×C4×Q8C2311yescharacteristic
129C3×C4×Q8C2311yescharacteristic
130C3×C4×D4—22no—
131C3×C4×D4—22no—
132C3×C4×D4—21no—
133C3×C4×D4—21no—
134C3×C4×D4—21no—
135C3×C4×D4C2311yescharacteristic
136C3×C4×D4C2311yescharacteristic
137C3×#117—21no—
138C3×#117—21no—
139C3×#117—21no—
140C2×C3×C4⋊C4—21no—
141C2×C3×C4⋊C4—21no—
142C2×C3×C4⋊C4D412yes—
143C2×C3×C42C2311yescharacteristic
144C3×C4⋊C8C2312yes—
145C3×C4⋊C8C2312yes—
146C3×#102—24no—
147C3×#102—24no—
148C3×#102—22no—
149C3×#102—22no—
150C3×#102—22no—
151C3×#102—22no—
152C3×C22⋊C8—21no—
153C3×C22⋊C8—21no—
154C3×#97—21no—
155C3×#97—21no—
156C3×C4×C8—21no—
157C3×C4×C8C2312yes—
158#563—22no—
159#557C22×C311yescharacteristic, complemented
160#556C22×C311yescharacteristic, complemented
161#554—21no—
162#553—21no—
163D4×Q8—21no—
164D4×Q8C22×C311yescharacteristic, complemented
165#548—21no—
166#547C22×C311yescharacteristic, complemented
167#544—21no—
168#542—21no—
169#541—21no—
170#539—22no—
171#536—21no—
172C2×C4⋊Q8C22×C311yescharacteristic
173C2×#127C22×C311yescharacteristic
174#529C22×C311yescharacteristic
175#529C22×C311yescharacteristic
176#496—21no—
177#496—21no—
178#490—21no—
179#490—21no—
180#486—22no—
181#486C22×C312yes—
182#486C22×C312yes—
183#460C22×C312yes—
184#460C22×C312yes—
185#459C22×C312yes—
186#459C22×C312yes—
187#450—21no—
188#450—21no—
189#447—21no—
190#447—21no—
191#432C22×C311yescharacteristic
192#432C22×C311yescharacteristic
193C2×#102—22no—
194C2×C4×C8C22×C311yescharacteristic
195#332C22×C312yescomplemented
196#332C22×C312yescomplemented
197Q8⋊C8C22×C312yescomplemented
198Q8⋊C8C22×C312yescomplemented
199D4⋊C8C22×C312yescomplemented
200D4⋊C8C22×C312yescomplemented
201C24×C3—41no—
202C3×#41—82no—
203C2×C3×Q8—22no—
204C2×C3×Q8—22no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—82no—
208C2×C3×Q8—82no—
209C2×C3×Q8C2×D411yescharacteristic
210C2×C3×Q8C2×D411yescharacteristic
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—22no—
214C2×C3×D4—22no—
215C2×C3×D4—22no—
216C2×C3×D4—22no—
217C2×C3×D4—44no—
218C2×C3×D4—41no—
219C2×C3×D4—41no—
220C2×C3×D4—41no—
221C2×C3×D4—82no—
222C2×C3×D4—82no—
223C2×C3×D4C2×D411yescharacteristic
224C2×C3×D4C2×D411yescharacteristic
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—22no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4C2×D411yescharacteristic
231C22×C3×C4C2×D411yescharacteristic
232C22×C3×C4C2×D411yescharacteristic
233C3×SD16—82no—
234C3×SD16—82no—
235C3×D8—82no—
236C3×D8—82no—
237C2×C3×C8—22no—
238C2×C3×C8—22no—
239C2×C3×C8—22no—
240C2×C3×C8—42no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4C2×D411yescharacteristic
260C3×C4⋊C4C2×D412yes—
261C3×C4⋊C4C2×D412yes—
262C3×C4⋊C4C2×D411yescharacteristic
263C3×C4⋊C4C2×D411yescharacteristic
264C3×C4⋊C4C2×D411yescharacteristic
265C3×C22⋊C4—22no—
266C3×C22⋊C4—22no—
267C3×C22⋊C4—22no—
268C3×C22⋊C4—22no—
269C3×C22⋊C4—21no—
270C3×C22⋊C4—21no—
271C3×C22⋊C4—21no—
272C3×C22⋊C4—21no—
273C3×C22⋊C4—21no—
274C3×C22⋊C4—21no—
275C3×C22⋊C4—21no—
276C3×C22⋊C4—21no—
277C3×C22⋊C4—21no—
278C3×C22⋊C4—21no—
279C3×C42—22no—
280C3×C42—22no—
281C3×C42—21no—
282C3×C42—21no—
283C3×C42C2×D411yescharacteristic
284C3×C42C2×D411yescharacteristic
285C3×C42C2×D411yescharacteristic
286C3×C42C2411yescharacteristic
287C2×#41—21no—
288C22×Q8—21no—
289C22×Q8—21no—
290C22×D4—21no—
291C22×D4—21no—
292C22×D4—21no—
293C2×SD16—22no—
294C2×SD16—22no—
295C2×D8—22no—
296C2×D8—22no—
297C22×C8—21no—
298C4⋊Q8—22no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8C3×D412yes—
304C4⋊Q8C23×C311yescharacteristic
305C4⋊Q8C23×C311yescharacteristic
306#127—21no—
307#127—21no—
308#127—21no—
309#127C23×C312yes—
310#126—22no—
311#126—22no—
312#126—22no—
313#126—21no—
314#126—21no—
315#125—22no—
316#125—22no—
317#125—22no—
318#125—21no—
319#125C3×D411yescharacteristic
320#125C3×D411yescharacteristic
321#124—22no—
322#124—21no—
323#124C3×D411yescharacteristic
324#124C3×D411yescharacteristic
325#123—22no—
326#123—22no—
327#123—21no—
328#123—21no—
329#123—22no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—21no—
336C22⋊Q8—22no—
337C22⋊Q8—21no—
338C22⋊Q8—21no—
339C22⋊Q8C3×D411yescharacteristic
340C22⋊Q8C3×D411yescharacteristic
341#121—22no—
342#121—21no—
343#121C3×D411yescharacteristic
344#121C3×D411yescharacteristic
345C4×Q8—21no—
346C4×Q8—21no—
347C4×Q8—21no—
348C4×Q8C23×C311yescharacteristic
349C4×Q8C23×C311yescharacteristic
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4—21no—
355C4×D4C23×C311yescharacteristic
356C4×D4C23×C311yescharacteristic
357#117—21no—
358#117—21no—
359#117—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4C3×D412yes—
363C2×C42C23×C311yescharacteristic
364C4⋊C8C23×C312yes—
365C4⋊C8C23×C312yes—
366#102—24no—
367#102—24no—
368#102—22no—
369#102—22no—
370#102—22no—
371#102—22no—
372C22⋊C8—21no—
373C22⋊C8—21no—
374#97—21no—
375#97—21no—
376C4×C8—21no—
377C4×C8C23×C312yes—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—42no—
381C23×C3—82no—
382C23×C3—82no—
383C23×C3—82no—
384C23×C3#12011yescharacteristic
385C3×Q8—22no—
386C3×Q8—22no—
387C3×Q8—42no—
388C3×Q8—42no—
389C3×Q8—82no—
390C3×Q8—82no—
391C3×D4—22no—
392C3×D4—22no—
393C3×D4—44no—
394C3×D4—42no—
395C3×D4—42no—
396C3×D4—42no—
397C3×D4—42no—
398C3×D4—42no—
399C3×D4—42no—
400C3×D4—82no—
401C3×D4—82no—
402C3×D4—82no—
403C3×D4—82no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—21no—
415C2×C3×C4—42no—
416C2×C3×C4—42no—
417C2×C3×C4—82no—
418C2×C3×C4—82no—
419C2×C3×C4—82no—
420C2×C3×C4#12011yescharacteristic
421C2×C3×C4#12011yescharacteristic
422C2×C3×C4#12011yescharacteristic
423C2×C3×C4C22×D411yescharacteristic
424C2×C3×C4C22×D411yescharacteristic
425C2×C3×C4C22×D411yescharacteristic
426C3×C8—42no—
427C3×C8—42no—
428C3×C8—42no—
429C24—41no—
430#41—82no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—82no—
436C2×Q8—82no—
437C2×Q8C2×C3×D411yescharacteristic
438C2×Q8C2×C3×D411yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—44no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4C2×C3×D411yescharacteristic
452C2×D4C2×C3×D411yescharacteristic
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4C2×C3×D411yescharacteristic
459C22×C4C2×C3×D411yescharacteristic
460C22×C4C2×C3×D411yescharacteristic
461SD16—82no—
462SD16—82no—
463D8—82no—
464D8—82no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—42no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4C2×C3×D411yescharacteristic
488C4⋊C4C2×C3×D412yes—
489C4⋊C4C2×C3×D412yes—
490C4⋊C4C2×C3×D411yescharacteristic
491C4⋊C4C2×C3×D411yescharacteristic
492C4⋊C4C2×C3×D411yescharacteristic
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42C2×C3×D411yescharacteristic
512C42C2×C3×D411yescharacteristic
513C42C2×C3×D411yescharacteristic
514C42C24×C311yescharacteristic
515C22×C3—21no—
516C22×C3—21no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—82no—
524C22×C3—82no—
525C22×C3—82no—
526C22×C3—164no—
527C22×C3#45412yes—
528C22×C3C2×#12011yescharacteristic
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—41no—
534C3×C4—42no—
535C3×C4—41no—
536C3×C4—42no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—41no—
540C3×C4—41no—
541C3×C4#57612yes—
542C3×C4#57712yes—
543C23—21no—
544C23—21no—
545C23—42no—
546C23—82no—
547C23—82no—
548C23—82no—
549C23C3×#12011yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—42no—
553Q8—42no—
554Q8—82no—
555Q8—82no—
556D4—22no—
557D4—22no—
558D4—44no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—82no—
566D4—82no—
567D4—82no—
568D4—82no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4C3×#12011yescharacteristic
586C2×C4C3×#12011yescharacteristic
587C2×C4C3×#12011yescharacteristic
588C2×C4C22×C3×D411yescharacteristic
589C2×C4C22×C3×D411yescharacteristic
590C2×C4C22×C3×D411yescharacteristic
591C8—42no—
592C8—42no—
593C8—42no—
594C2×C3—22no—
595C2×C3—81no—
596C2×C3—81no—
597C2×C3—162no—
598C2×C3#300811yescharacteristic
599C2×C3#301011yescharacteristic
600C2×C3C2×#45411yescharacteristic
601C22—21no—
602C22—21no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—164no—
613C22C3×#45412yes—
614C22C2×C3×#12011yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—42no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4C3×#57612yes—
628C4C3×#57712yes—
629C3#1869411yescharacteristic, direct product factor, Sylow
630C2—22no—
631C2—81no—
632C2—81no—
633C2—162no—
634C2C3×#300811yescharacteristic
635C2C3×#301011yescharacteristic
636C2C2×C3×#45411yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3552C2semidirect
2. C3×#3217C2semidirect
3. C3×#3217C2semidirect
4. C2×C3×#486C2semidirect
5. C3×#1628C2semidirect
6. C3×#1627C2semidirect
7. C3×#1626C2semidirect
8. C3×#1625C2semidirect
9. C3×#1556C2semidirect
10. C3×#1494C2semidirect
11. C3×#1494C2semidirect
12. #18694C3direct (N × H)
14. C3×#557C22semidirect
15. C3×#556C22semidirect
19. C3×D4×Q8C22semidirect
21. C3×#547C22semidirect
50. C3×#332C22semidirect
51. C3×#332C22semidirect
52. C3×Q8⋊C8C22semidirect
53. C3×Q8⋊C8C22semidirect
54. C3×D4⋊C8C22semidirect
55. C3×D4⋊C8C22semidirect
56. #3552C2×C3semidirect
57. #3217C2×C3semidirect
58. #3217C2×C3semidirect
59. C2×#486C2×C3semidirect
60. #1628C2×C3semidirect
61. #1627C2×C3semidirect
62. #1626C2×C3semidirect
63. #1625C2×C3semidirect
64. #1556C2×C3semidirect
65. #1494C2×C3semidirect
66. #1494C2×C3semidirect
159. #557C22×C3semidirect
160. #556C22×C3semidirect
164. D4×Q8C22×C3semidirect
166. #547C22×C3semidirect
195. #332C22×C3semidirect
196. #332C22×C3semidirect
197. Q8⋊C8C22×C3semidirect
198. Q8⋊C8C22×C3semidirect
199. D4⋊C8C22×C3semidirect
200. D4⋊C8C22×C3semidirect
629. C3#18694direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35522
C3×#32172
C2×C3×#4862
C3×#16282
C3×#16272
C3×#16262
C3×#16252
C3×#15562
C3×#14942
#186943