Finite Group Databasebeta

← C3×#18689 #10637296

Subgroups — C3×#18689 #10637296

Order 768.

1006 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3543C211yesmaximal, characteristic
2C3×#3239C211yesmaximal, characteristic, complemented
3C3×#3218C211yesmaximal, characteristic, complemented
4C2×C3×#486C211yesmaximal, characteristic, complemented
5C3×#1642C212yesmaximal, complemented
6C3×#1641C212yesmaximal, complemented
7C3×#1640C212yesmaximal, complemented
8C3×#1639C212yesmaximal, complemented
9C3×#1555C211yesmaximal, characteristic, complemented
10C3×#1494C211yesmaximal, characteristic, complemented
11C3×#1494C211yesmaximal, characteristic, complemented
12#18689C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#560—22no—
14C3×#559C2211yescharacteristic
15C3×#550C2211yescharacteristic
16C3×#550C2211yescharacteristic
17C3×#546—21no—
18C3×#546—21no—
19C3×D42—21no—
20C3×D42—21no—
21C3×D42C2211yescharacteristic
22C3×#540—21no—
23C3×#540—21no—
24C3×#535—21no—
25C3×#535—21no—
26C3×#534—22no—
27C2×C3×C4⋊Q8C2211yescharacteristic, complemented
28C2×C3×#127C2211yescharacteristic
29C3×#529C2211yescharacteristic
30C3×#529C2211yescharacteristic
31C3×#501—21no—
32C3×#497—21no—
33C3×#491—21no—
34C3×#489—21no—
35C3×#486—22no—
36C3×#486C2212yes—
37C3×#486C2212yescomplemented
38C3×#484—21no—
39C3×#481—21no—
40C3×#475C2212yescomplemented
41C3×#474C2212yescomplemented
42C3×#462C2212yescomplemented
43C3×#461C2212yescomplemented
44C3×#451—21no—
45C3×#448—21no—
46C3×#432C2211yescharacteristic, complemented
47C3×#432C2211yescharacteristic, complemented
48C2×C3×#102—22no—
49C2×C3×C4×C8C2211yescharacteristic, complemented
50C3×#331C2212yes—
51C3×#331C2212yescomplemented
52C3×Q8⋊C8C2212yescomplemented
53C3×Q8⋊C8C2212yescomplemented
54C3×D4⋊C8C2212yescomplemented
55C3×D4⋊C8C2212yescomplemented
56#3543C2×C311yescharacteristic
57#3239C2×C311yescharacteristic, complemented
58#3218C2×C311yescharacteristic, complemented
59C2×#486C2×C311yescharacteristic, complemented
60#1642C2×C312yescomplemented
61#1641C2×C312yescomplemented
62#1640C2×C312yescomplemented
63#1639C2×C312yescomplemented
64#1555C2×C311yescharacteristic, complemented
65#1494C2×C311yescharacteristic, complemented
66#1494C2×C311yescharacteristic, complemented
67C2×C3×#41—21no—
68C2×C3×#41—21no—
69C22×C3×Q8—21no—
70C22×C3×D4—22no—
71C22×C3×D4—21no—
72C22×C3×D4—21no—
73C22×C3×D4—21no—
74C22×C3×D4—21no—
75C22×C3×D4D412yes—
76C2×C3×Q16—22no—
77C2×C3×SD16—22no—
78C22×C3×C8—21no—
79C3×C4⋊Q8—21no—
80C3×C4⋊Q8—21no—
81C3×C4⋊Q8—21no—
82C3×C4⋊Q8C2312yes—
83C3×#127—21no—
84C3×#127—21no—
85C3×#127—21no—
86C3×#127—21no—
87C3×#127D412yes—
88C3×#127C2311yescharacteristic
89C3×#127C2311yescharacteristic
90C3×#126—21no—
91C3×#126—21no—
92C3×#125D411yescharacteristic
93C3×#125D411yescharacteristic
94C3×#124—21no—
95C3×#124—21no—
96C3×#124D411yescharacteristic
97C3×#124D411yescharacteristic
98C3×#123—22no—
99C3×#123—22no—
100C3×#123—22no—
101C3×#123—22no—
102C3×#123—21no—
103C3×#123—21no—
104C3×C22⋊Q8D411yescharacteristic
105C3×C22⋊Q8D411yescharacteristic
106C3×#121—22no—
107C3×#121—22no—
108C3×#121—22no—
109C3×#121—22no—
110C3×#121—22no—
111C3×#121—22no—
112C3×#121—22no—
113C3×#121—22no—
114C3×#121—21no—
115C3×#121—21no—
116C3×#121—21no—
117C3×#121—21no—
118C3×#121D411yescharacteristic
119C3×#121D411yescharacteristic
120C3×#120—22no—
121C3×#120—22no—
122C3×#120—22no—
123C3×#120—22no—
124C3×#120—22no—
125C3×#120—22no—
126C3×C4×Q8C2311yescharacteristic
127C3×C4×Q8C2311yescharacteristic
128C3×C4×D4—21no—
129C3×C4×D4—21no—
130C3×C4×D4—21no—
131C3×C4×D4—21no—
132C3×C4×D4—21no—
133C3×C4×D4—21no—
134C3×C4×D4C2311yescharacteristic
135C3×C4×D4C2311yescharacteristic
136C3×#117—21no—
137C3×#117—21no—
138C2×C3×C4⋊C4—21no—
139C2×C3×C4⋊C4—21no—
140C2×C3×C22⋊C4—22no—
141C2×C3×C42C2311yescharacteristic
142C3×#107—22no—
143C3×#106—22no—
144C3×C4⋊C8C2312yes—
145C3×C4⋊C8C2312yes—
146C3×#103—22no—
147C3×#103—22no—
148C3×#103—22no—
149C3×#102—24no—
150C3×#102—24no—
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#560—22no—
159#559C22×C311yescharacteristic
160#550C22×C311yescharacteristic
161#550C22×C311yescharacteristic
162#546—21no—
163#546—21no—
164D42—21no—
165D42—21no—
166D42C22×C311yescharacteristic
167#540—21no—
168#540—21no—
169#535—21no—
170#535—21no—
171#534—22no—
172C2×C4⋊Q8C22×C311yescharacteristic, complemented
173C2×#127C22×C311yescharacteristic
174#529C22×C311yescharacteristic
175#529C22×C311yescharacteristic
176#501—21no—
177#497—21no—
178#491—21no—
179#489—21no—
180#486—22no—
181#486C22×C312yes—
182#486C22×C312yescomplemented
183#484—21no—
184#481—21no—
185#475C22×C312yescomplemented
186#474C22×C312yescomplemented
187#462C22×C312yescomplemented
188#461C22×C312yescomplemented
189#451—21no—
190#448—21no—
191#432C22×C311yescharacteristic, complemented
192#432C22×C311yescharacteristic, complemented
193C2×#102—22no—
194C2×C4×C8C22×C311yescharacteristic, complemented
195#331C22×C312yes—
196#331C22×C312yescomplemented
197Q8⋊C8C22×C312yescomplemented
198Q8⋊C8C22×C312yescomplemented
199D4⋊C8C22×C312yescomplemented
200D4⋊C8C22×C312yescomplemented
201C24×C3—22no—
202C24×C3—22no—
203C3×#41—82no—
204C3×#41—82no—
205C2×C3×Q8—22no—
206C2×C3×Q8—41no—
207C2×C3×Q8—82no—
208C2×C3×Q8C2×D411yescharacteristic
209C2×C3×Q8C2×D411yescharacteristic
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—22no—
213C2×C3×D4—22no—
214C2×C3×D4—22no—
215C2×C3×D4—22no—
216C2×C3×D4—22no—
217C2×C3×D4—22no—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—44no—
226C2×C3×D4—82no—
227C2×C3×D4—82no—
228C2×C3×D4—82no—
229C2×C3×D4—82no—
230C2×C3×D4—84no—
231C2×C3×D4C2×D411yescharacteristic
232C2×C3×D4C2×D411yescharacteristic
233C2×C3×D4C2×D412yes—
234C2×C3×D4C2×D412yes—
235C22×C3×C4—22no—
236C22×C3×C4—22no—
237C22×C3×C4—22no—
238C22×C3×C4—21no—
239C22×C3×C4—21no—
240C22×C3×C4—41no—
241C22×C3×C4C2×D411yescharacteristic
242C22×C3×C4C2×D411yescharacteristic
243C22×C3×C4C2×D411yescharacteristic
244C3×Q16—82no—
245C3×SD16—82no—
246C2×C3×C8—22no—
247C2×C3×C8—22no—
248C2×C3×C8—22no—
249C2×C3×C8—42no—
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—22no—
259C3×C4⋊C4—22no—
260C3×C4⋊C4—41no—
261C3×C4⋊C4—41no—
262C3×C4⋊C4C2×D411yescharacteristic
263C3×C4⋊C4C2×D411yescharacteristic
264C3×C4⋊C4C2×D411yescharacteristic
265C3×C4⋊C4C2×D411yescharacteristic
266C3×C22⋊C4—22no—
267C3×C22⋊C4—22no—
268C3×C22⋊C4—22no—
269C3×C22⋊C4—22no—
270C3×C22⋊C4—22no—
271C3×C22⋊C4—22no—
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×C22⋊C4—21no—
280C3×C22⋊C4—21no—
281C3×C22⋊C4—21no—
282C3×C42—21no—
283C3×C42—21no—
284C3×C42C2×D411yescharacteristic
285C3×C42C2×D411yescharacteristic
286C3×C42C2×D411yescharacteristic
287C3×C42C2411yescharacteristic
288C2×#41—21no—
289C2×#41—21no—
290C22×Q8—21no—
291C22×D4—22no—
292C22×D4—21no—
293C22×D4—21no—
294C22×D4—21no—
295C22×D4—21no—
296C22×D4C3×D412yes—
297C2×Q16—22no—
298C2×SD16—22no—
299C22×C8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8C23×C312yes—
304#127—21no—
305#127—21no—
306#127—21no—
307#127—21no—
308#127C3×D412yes—
309#127C23×C311yescharacteristic
310#127C23×C311yescharacteristic
311#126—21no—
312#126—21no—
313#125C3×D411yescharacteristic
314#125C3×D411yescharacteristic
315#124—21no—
316#124—21no—
317#124C3×D411yescharacteristic
318#124C3×D411yescharacteristic
319#123—22no—
320#123—22no—
321#123—22no—
322#123—22no—
323#123—21no—
324#123—21no—
325C22⋊Q8C3×D411yescharacteristic
326C22⋊Q8C3×D411yescharacteristic
327#121—22no—
328#121—22no—
329#121—22no—
330#121—22no—
331#121—22no—
332#121—22no—
333#121—22no—
334#121—22no—
335#121—21no—
336#121—21no—
337#121—21no—
338#121—21no—
339#121C3×D411yescharacteristic
340#121C3×D411yescharacteristic
341#120—22no—
342#120—22no—
343#120—22no—
344#120—22no—
345#120—22no—
346#120—22no—
347C4×Q8C23×C311yescharacteristic
348C4×Q8C23×C311yescharacteristic
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4—21no—
355C4×D4C23×C311yescharacteristic
356C4×D4C23×C311yescharacteristic
357#117—21no—
358#117—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C42C23×C311yescharacteristic
363#107—22no—
364#106—22no—
365C4⋊C8C23×C312yes—
366C4⋊C8C23×C312yes—
367#103—22no—
368#103—22no—
369#103—22no—
370#102—24no—
371#102—24no—
372#102—22no—
373C22⋊C8—21no—
374C22⋊C8—21no—
375#97—21no—
376#97—21no—
377C4×C8—21no—
378C4×C8C23×C312yes—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—22no—
382C23×C3—22no—
383C23×C3—44no—
384C23×C3—82no—
385C23×C3—82no—
386C23×C3—82no—
387C23×C3—82no—
388C23×C3—82no—
389C23×C3#12011yescharacteristic
390C3×Q8—22no—
391C3×Q8—22no—
392C3×Q8—42no—
393C3×Q8—42no—
394C3×Q8—82no—
395C3×D4—22no—
396C3×D4—22no—
397C3×D4—42no—
398C3×D4—42no—
399C3×D4—42no—
400C3×D4—42no—
401C3×D4—42no—
402C3×D4—42no—
403C3×D4—42no—
404C3×D4—42no—
405C3×D4—82no—
406C3×D4—82no—
407C3×D4—82no—
408C3×D4—82no—
409C3×D4—82no—
410C3×D4—82no—
411C3×D4—82no—
412C3×D4—82no—
413C3×D4—82no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—42no—
424C2×C3×C4—42no—
425C2×C3×C4—42no—
426C2×C3×C4—82no—
427C2×C3×C4—82no—
428C2×C3×C4—82no—
429C2×C3×C4—82no—
430C2×C3×C4#12011yescharacteristic
431C2×C3×C4#12011yescharacteristic
432C2×C3×C4#12011yescharacteristic
433C2×C3×C4C22×D411yescharacteristic
434C2×C3×C4C22×D411yescharacteristic
435C2×C3×C4C22×D411yescharacteristic
436C3×C8—42no—
437C3×C8—42no—
438C3×C8—42no—
439C24—22no—
440C24—22no—
441#41—82no—
442#41—82no—
443C2×Q8—22no—
444C2×Q8—41no—
445C2×Q8—82no—
446C2×Q8C2×C3×D411yescharacteristic
447C2×Q8C2×C3×D411yescharacteristic
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—44no—
464C2×D4—82no—
465C2×D4—82no—
466C2×D4—82no—
467C2×D4—82no—
468C2×D4—84no—
469C2×D4C2×C3×D411yescharacteristic
470C2×D4C2×C3×D411yescharacteristic
471C2×D4C2×C3×D412yes—
472C2×D4C2×C3×D412yes—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—41no—
479C22×C4C2×C3×D411yescharacteristic
480C22×C4C2×C3×D411yescharacteristic
481C22×C4C2×C3×D411yescharacteristic
482Q16—82no—
483SD16—82no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—42no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4C2×C3×D411yescharacteristic
501C4⋊C4C2×C3×D411yescharacteristic
502C4⋊C4C2×C3×D411yescharacteristic
503C4⋊C4C2×C3×D411yescharacteristic
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C42—21no—
521C42—21no—
522C42C2×C3×D411yescharacteristic
523C42C2×C3×D411yescharacteristic
524C42C2×C3×D411yescharacteristic
525C42C24×C311yescharacteristic
526C22×C3—21no—
527C22×C3—21no—
528C22×C3—41no—
529C22×C3—42no—
530C22×C3—41no—
531C22×C3—42no—
532C22×C3—41no—
533C22×C3—42no—
534C22×C3—41no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—42no—
540C22×C3—164no—
541C22×C3—164no—
542C22×C3#45512yes—
543C22×C3C2×#12011yescharacteristic
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—41no—
549C3×C4—41no—
550C3×C4—41no—
551C3×C4—41no—
552C3×C4—41no—
553C3×C4—41no—
554C3×C4—82no—
555C3×C4#57612yes—
556C3×C4#57712yes—
557C23—21no—
558C23—21no—
559C23—22no—
560C23—22no—
561C23—44no—
562C23—82no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23—82no—
567C23C3×#12011yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—42no—
571Q8—42no—
572Q8—82no—
573D4—22no—
574D4—22no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—82no—
584D4—82no—
585D4—82no—
586D4—82no—
587D4—82no—
588D4—82no—
589D4—82no—
590D4—82no—
591D4—82no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4C3×#12011yescharacteristic
609C2×C4C3×#12011yescharacteristic
610C2×C4C3×#12011yescharacteristic
611C2×C4C22×C3×D411yescharacteristic
612C2×C4C22×C3×D411yescharacteristic
613C2×C4C22×C3×D411yescharacteristic
614C8—42no—
615C8—42no—
616C8—42no—
617C2×C3—22no—
618C2×C3—82no—
619C2×C3—81no—
620C2×C3—81no—
621C2×C3—82no—
622C2×C3#300811yescharacteristic
623C2×C3#300911yescharacteristic
624C2×C3C2×#45511yescharacteristic
625C22—21no—
626C22—21no—
627C22—41no—
628C22—42no—
629C22—41no—
630C22—42no—
631C22—41no—
632C22—42no—
633C22—41no—
634C22—42no—
635C22—42no—
636C22—41no—
637C22—41no—
638C22—42no—
639C22—164no—
640C22—164no—
641C22C3×#45512yes—
642C22C2×C3×#12011yescharacteristic
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—82no—
654C4C3×#57612yes—
655C4C3×#57712yes—
656C3#1868911yescharacteristic, direct product factor, Sylow
657C2—22no—
658C2—81no—
659C2—81no—
660C2—82no—
661C2—82no—
662C2C3×#300811yescharacteristic
663C2C3×#300911yescharacteristic
664C2C2×C3×#45511yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3543C2
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
2. C3×#3239C2semidirect
3. C3×#3218C2semidirect
4. C2×C3×#486C2semidirect
5. C3×#1642C2semidirect
6. C3×#1641C2semidirect
7. C3×#1640C2semidirect
8. C3×#1639C2semidirect
9. C3×#1555C2semidirect
10. C3×#1494C2semidirect
11. C3×#1494C2semidirect
12. #18689C3direct (N × H)
27. C2×C3×C4⋊Q8C22semidirect
37. C3×#486C22semidirect
40. C3×#475C22semidirect
41. C3×#474C22semidirect
42. C3×#462C22semidirect
43. C3×#461C22semidirect
46. C3×#432C22semidirect
47. C3×#432C22semidirect
49. C2×C3×C4×C8C22semidirect
51. C3×#331C22semidirect
52. C3×Q8⋊C8C22semidirect
53. C3×Q8⋊C8C22semidirect
54. C3×D4⋊C8C22semidirect
55. C3×D4⋊C8C22semidirect
57. #3239C2×C3semidirect
58. #3218C2×C3semidirect
59. C2×#486C2×C3semidirect
60. #1642C2×C3semidirect
61. #1641C2×C3semidirect
62. #1640C2×C3semidirect
63. #1639C2×C3semidirect
64. #1555C2×C3semidirect
65. #1494C2×C3semidirect
66. #1494C2×C3semidirect
172. C2×C4⋊Q8C22×C3semidirect
182. #486C22×C3semidirect
185. #475C22×C3semidirect
186. #474C22×C3semidirect
187. #462C22×C3semidirect
188. #461C22×C3semidirect
191. #432C22×C3semidirect
192. #432C22×C3semidirect
194. C2×C4×C8C22×C3semidirect
196. #331C22×C3semidirect
197. Q8⋊C8C22×C3semidirect
198. Q8⋊C8C22×C3semidirect
199. D4⋊C8C22×C3semidirect
200. D4⋊C8C22×C3semidirect
656. C3#18689direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35432
C3×#32392
C3×#32182
C2×C3×#4862
C3×#16422
C3×#16412
C3×#16402
C3×#16392
C3×#15552
C3×#14942
#186893