Finite Group Databasebeta

← C3×#18127 #10636734

Subgroups — C3×#18127 #10636734

Order 768.

950 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3455C211yesmaximal, characteristic
2C3×#3104C211yesmaximal, characteristic, complemented
3C3×#3103C211yesmaximal, characteristic, complemented
4C3×#2396C211yesmaximal, characteristic, complemented
5C3×#1618C212yesmaximal, complemented
6C3×#1615C212yesmaximal, complemented
7C3×#1606C212yesmaximal, complemented
8C3×#1603C212yesmaximal, complemented
9C3×#1503C211yesmaximal, characteristic, complemented
10C3×#1502C211yesmaximal, characteristic, complemented
11C3×#1458C211yesmaximal, characteristic, complemented
12#18127C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#542—21no—
14C3×#542—21no—
15C3×#541—21no—
16C3×#540—21no—
17C3×#539—21no—
18C3×#539C2212yes—
19C3×#538—21no—
20C3×#538C2212yes—
21C3×#533C2211yescharacteristic
22C3×#532C2211yescharacteristic
23C3×#529—22no—
24C3×#529—22no—
25C3×#525—21no—
26C3×C4×#41—21no—
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×#117C2211yescharacteristic
29C3×#484C2212yes—
30C3×#481C2212yes—
31C3×#477—21no—
32C3×#475—21no—
33C3×#462—21no—
34C3×#460—21no—
35C3×#451C2212yes—
36C3×#448C2212yes—
37C3×C4×Q16—22no—
38C3×C4×SD16—22no—
39C3×#431C2211yescharacteristic
40C3×#431C2211yescharacteristic
41C3×#400C2211yescharacteristic
42C3×#399C2211yescharacteristic
43C3×#398—21no—
44C3×#396—21no—
45C3×#395C2211yescharacteristic
46C3×#393C2211yescharacteristic
47C3×C4×C4⋊C4—21no—
48C3×C4×C4⋊C4C2211yescharacteristic
49C3×C4×C22⋊C4—21no—
50C3×#328C2212yes—
51C3×#328C2212yes—
52C3×#327C2212yes—
53C3×#327C2212yes—
54C3×#324C2212yes—
55C3×#323C2212yes—
56#3455C2×C311yescharacteristic
57#3104C2×C311yescharacteristic, complemented
58#3103C2×C311yescharacteristic, complemented
59#2396C2×C311yescharacteristic, complemented
60#1618C2×C312yescomplemented
61#1615C2×C312yescomplemented
62#1606C2×C312yescomplemented
63#1603C2×C312yescomplemented
64#1503C2×C311yescharacteristic, complemented
65#1502C2×C311yescharacteristic, complemented
66#1458C2×C311yescharacteristic, complemented
67C2×C3×#41—21no—
68C22×C3×Q8C2311yescharacteristic
69C23×C3×C4D411yescharacteristic
70C2×C3×Q16—22no—
71C2×C3×SD16—22no—
72C3×C4⋊Q8D411yescharacteristic
73C3×C4⋊Q8D411yescharacteristic
74C3×C4⋊Q8D411yescharacteristic
75C3×#127D411yescharacteristic
76C3×#126—22no—
77C3×#126—22no—
78C3×#126—22no—
79C3×#126—22no—
80C3×#125—22no—
81C3×#125—22no—
82C3×#125—21no—
83C3×#124—22no—
84C3×#124—22no—
85C3×#124—21no—
86C3×#123—22no—
87C3×#123—22no—
88C3×#123—22no—
89C3×#123—22no—
90C3×#123—22no—
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8—22no—
95C3×C22⋊Q8—21no—
96C3×C22⋊Q8C2312yes—
97C3×#121—21no—
98C3×#121—21no—
99C3×#121—22no—
100C3×#121—22no—
101C3×#121—21no—
102C3×#121C2312yes—
103C3×#120—22no—
104C3×C4×Q8—21no—
105C3×C4×Q8—21no—
106C3×C4×Q8—21no—
107C3×C4×Q8—21no—
108C3×C4×Q8—21no—
109C3×C4×Q8—21no—
110C3×C4×Q8—21no—
111C3×C4×Q8D412yes—
112C3×C4×D4—21no—
113C3×C4×D4—21no—
114C3×C4×D4—22no—
115C3×C4×D4—22no—
116C3×C4×D4—22no—
117C3×C4×D4—22no—
118C3×C4×D4—21no—
119C3×C4×D4D412yes—
120C3×#117—22no—
121C3×#117—21no—
122C3×#117—21no—
123C3×#117—21no—
124C3×#117—21no—
125C3×#117—21no—
126C2×C3×C4⋊C4—21no—
127C2×C3×C4⋊C4—21no—
128C2×C3×C4⋊C4—21no—
129C2×C3×C4⋊C4—21no—
130C2×C3×C4⋊C4D411yescharacteristic
131C2×C3×C4⋊C4C2311yescharacteristic
132C2×C3×C4⋊C4C2311yescharacteristic
133C2×C3×C22⋊C4—21no—
134C2×C3×C22⋊C4—21no—
135C2×C3×C22⋊C4D411yescharacteristic
136C2×C3×C22⋊C4C2311yescharacteristic
137C2×C3×C42—22no—
138C2×C3×C42—21no—
139C2×C3×C42—21no—
140C2×C3×C42D411yescharacteristic
141C2×C3×C42C2311yescharacteristic
142C3×#107—22no—
143C3×#106—22no—
144C3×C4⋊C8—21no—
145C3×C4⋊C8—21no—
146C3×#103—22no—
147C3×#103—22no—
148C3×#103—22no—
149C3×#102—22no—
150C3×C22⋊C8C2312yes—
151C3×C22⋊C8C2312yes—
152C3×C4×C8—21no—
153C3×C4×C8—21no—
154C3×#95—21no—
155C3×#95—21no—
156C3×#95—21no—
157C3×#95—21no—
158C3×#95C2311yescharacteristic
159C3×#95C2311yescharacteristic
160#542—21no—
161#542—21no—
162#541—21no—
163#540—21no—
164#539—21no—
165#539C22×C312yes—
166#538—21no—
167#538C22×C312yes—
168#533C22×C311yescharacteristic
169#532C22×C311yescharacteristic
170#529—22no—
171#529—22no—
172#525—21no—
173C4×#41—21no—
174C2×C4×Q8C22×C311yescharacteristic
175C2×#117C22×C311yescharacteristic
176#484C22×C312yes—
177#481C22×C312yes—
178#477—21no—
179#475—21no—
180#462—21no—
181#460—21no—
182#451C22×C312yes—
183#448C22×C312yes—
184C4×Q16—22no—
185C4×SD16—22no—
186#431C22×C311yescharacteristic
187#431C22×C311yescharacteristic
188#400C22×C311yescharacteristic
189#399C22×C311yescharacteristic
190#398—21no—
191#396—21no—
192#395C22×C311yescharacteristic
193#393C22×C311yescharacteristic
194C4×C4⋊C4—21no—
195C4×C4⋊C4C22×C311yescharacteristic
196C4×C22⋊C4—21no—
197#328C22×C312yes—
198#328C22×C312yes—
199#327C22×C312yes—
200#327C22×C312yes—
201#324C22×C312yes—
202#323C22×C312yes—
203C24×C3C2×D411yescharacteristic
204C3×#41—42no—
205C3×#41—42no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—42no—
211C2×C3×Q8C2×D412yes—
212C2×C3×D4—22no—
213C2×C3×D4—22no—
214C2×C3×D4—22no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4C2×D412yes—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—22no—
221C22×C3×C4—22no—
222C22×C3×C4—21no—
223C22×C3×C4—22no—
224C22×C3×C4—21no—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2×D411yescharacteristic
231C22×C3×C4C2×D411yescharacteristic
232C22×C3×C4C2×D411yescharacteristic
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2411yescharacteristic
235C3×Q16—44no—
236C3×SD16—44no—
237C2×C3×C8—22no—
238C2×C3×C8—22no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—42no—
256C3×C4⋊C4—42no—
257C3×C4⋊C4—42no—
258C3×C4⋊C4—41no—
259C3×C4⋊C4—41no—
260C3×C4⋊C4C2×D412yes—
261C3×C4⋊C4C2×D412yes—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—22no—
264C3×C22⋊C4—22no—
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—22no—
272C3×C22⋊C4—21no—
273C3×C22⋊C4—21no—
274C3×C22⋊C4—22no—
275C3×C22⋊C4—42no—
276C3×C22⋊C4—42no—
277C3×C42—22no—
278C3×C42—22no—
279C3×C42—21no—
280C3×C42—21no—
281C3×C42—22no—
282C3×C42—41no—
283C3×C42—41no—
284C3×C42—42no—
285C3×C42C2×D411yescharacteristic
286C3×C42C2×D411yescharacteristic
287C3×C42C2×D411yescharacteristic
288C3×C42C2×D411yescharacteristic
289C2×#41—21no—
290C22×Q8C23×C311yescharacteristic
291C23×C4C3×D411yescharacteristic
292C2×Q16—22no—
293C2×SD16—22no—
294C4⋊Q8C3×D411yescharacteristic
295C4⋊Q8C3×D411yescharacteristic
296C4⋊Q8C3×D411yescharacteristic
297#127C3×D411yescharacteristic
298#126—22no—
299#126—22no—
300#126—22no—
301#126—22no—
302#125—22no—
303#125—22no—
304#125—21no—
305#124—22no—
306#124—22no—
307#124—21no—
308#123—22no—
309#123—22no—
310#123—22no—
311#123—22no—
312#123—22no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—21no—
318C22⋊Q8C23×C312yes—
319#121—21no—
320#121—21no—
321#121—22no—
322#121—22no—
323#121—21no—
324#121C23×C312yes—
325#120—22no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×Q8—21no—
333C4×Q8C3×D412yes—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—21no—
341C4×D4C3×D412yes—
342#117—22no—
343#117—21no—
344#117—21no—
345#117—21no—
346#117—21no—
347#117—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4C3×D411yescharacteristic
353C2×C4⋊C4C23×C311yescharacteristic
354C2×C4⋊C4C23×C311yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4C3×D411yescharacteristic
358C2×C22⋊C4C23×C311yescharacteristic
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42C3×D411yescharacteristic
363C2×C42C23×C311yescharacteristic
364#107—22no—
365#106—22no—
366C4⋊C8—21no—
367C4⋊C8—21no—
368#103—22no—
369#103—22no—
370#103—22no—
371#102—22no—
372C22⋊C8C23×C312yes—
373C22⋊C8C23×C312yes—
374C4×C8—21no—
375C4×C8—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95C23×C311yescharacteristic
381#95C23×C311yescharacteristic
382C23×C3—21no—
383C23×C3—22no—
384C23×C3—21no—
385C23×C3—21no—
386C23×C3—41no—
387C23×C3—41no—
388C23×C3C22×D411yescharacteristic
389C3×Q8—22no—
390C3×Q8—22no—
391C3×Q8—42no—
392C3×Q8—41no—
393C3×Q8—41no—
394C3×Q8—42no—
395C3×Q8—81no—
396C3×D4—42no—
397C3×D4—42no—
398C3×D4—42no—
399C3×D4—42no—
400C3×D4—81no—
401C3×D4—82no—
402C3×D4—82no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4—42no—
420C2×C3×C4—42no—
421C2×C3×C4—44no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—41no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4—41no—
429C2×C3×C4—41no—
430C2×C3×C4—82no—
431C2×C3×C4—82no—
432C2×C3×C4#12011yescharacteristic
433C2×C3×C4#12011yescharacteristic
434C2×C3×C4#12011yescharacteristic
435C2×C3×C4#12011yescharacteristic
436C2×C3×C4#13512yes—
437C2×C3×C4#13712yes—
438C2×C3×C4C22×D411yescharacteristic
439C2×C3×C4C22×D411yescharacteristic
440C3×C8—42no—
441C3×C8—42no—
442C24C2×C3×D411yescharacteristic
443#41—42no—
444#41—42no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—42no—
450C2×Q8C2×C3×D412yes—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4C2×C3×D412yes—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4C2×C3×D411yescharacteristic
469C22×C4C2×C3×D411yescharacteristic
470C22×C4C2×C3×D411yescharacteristic
471C22×C4C2×C3×D411yescharacteristic
472C22×C4C2×C3×D411yescharacteristic
473C22×C4C24×C311yescharacteristic
474Q16—44no—
475SD16—44no—
476C2×C8—22no—
477C2×C8—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4C2×C3×D412yes—
500C4⋊C4C2×C3×D412yes—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—22no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—22no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C42—22no—
517C42—22no—
518C42—21no—
519C42—22no—
520C42—21no—
521C42—41no—
522C42—41no—
523C42—42no—
524C42C2×C3×D411yescharacteristic
525C42C2×C3×D411yescharacteristic
526C42C2×C3×D411yescharacteristic
527C42C2×C3×D411yescharacteristic
528C22×C3—21no—
529C22×C3—21no—
530C22×C3—42no—
531C22×C3—41no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—41no—
535C22×C3—42no—
536C22×C3—41no—
537C22×C3—81no—
538C22×C3C2×#12011yescharacteristic
539C22×C3C2×#13511yescharacteristic
540C22×C3C2×#13711yescharacteristic
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—42no—
546C3×C4—41no—
547C3×C4—42no—
548C3×C4—41no—
549C3×C4—42no—
550C3×C4—42no—
551C3×C4—41no—
552C3×C4—42no—
553C3×C4—81no—
554C3×C4—81no—
555C3×C4#45212yes—
556C3×C4#45212yes—
557C23—21no—
558C23—21no—
559C23—22no—
560C23—21no—
561C23—41no—
562C23—41no—
563C23C22×C3×D411yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—42no—
567Q8—41no—
568Q8—41no—
569Q8—42no—
570Q8—81no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—81no—
576D4—82no—
577D4—82no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4C3×#12011yescharacteristic
608C2×C4C3×#12011yescharacteristic
609C2×C4C3×#12011yescharacteristic
610C2×C4C3×#12011yescharacteristic
611C2×C4C3×#13512yes—
612C2×C4C3×#13712yes—
613C2×C4C22×C3×D411yescharacteristic
614C2×C4C22×C3×D411yescharacteristic
615C8—42no—
616C8—42no—
617C2×C3—21no—
618C2×C3—21no—
619C2×C3—82no—
620C2×C3—81no—
621C2×C3C2×#45211yescharacteristic
622C2×C3#301811yescharacteristic
623C2×C3#302511yescharacteristic
624C22—21no—
625C22—21no—
626C22—42no—
627C22—41no—
628C22—42no—
629C22—42no—
630C22—41no—
631C22—42no—
632C22—41no—
633C22—81no—
634C22C2×C3×#12011yescharacteristic
635C22C2×C3×#13511yescharacteristic
636C22C2×C3×#13711yescharacteristic
637C4—21no—
638C4—21no—
639C4—21no—
640C4—21no—
641C4—42no—
642C4—41no—
643C4—42no—
644C4—41no—
645C4—42no—
646C4—42no—
647C4—41no—
648C4—42no—
649C4—81no—
650C4—81no—
651C4C3×#45212yes—
652C4C3×#45212yes—
653C3#1812711yescharacteristic, direct product factor, Sylow
654C2—21no—
655C2—21no—
656C2—81no—
657C2—82no—
658C2C2×C3×#45211yescharacteristic
659C2C3×#301811yescharacteristic
660C2C3×#302511yescharacteristic
subgroupG/H
centerC2×C3×C4#120
commutator, Frattini, cosocleC22×C4C24×C3
socleC22×C3C2×#120
prime radicalC3×#3455C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3104C2semidirect
3. C3×#3103C2semidirect
4. C3×#2396C2semidirect
5. C3×#1618C2semidirect
6. C3×#1615C2semidirect
7. C3×#1606C2semidirect
8. C3×#1603C2semidirect
9. C3×#1503C2semidirect
10. C3×#1502C2semidirect
11. C3×#1458C2semidirect
12. #18127C3direct (N × H)
57. #3104C2×C3semidirect
58. #3103C2×C3semidirect
59. #2396C2×C3semidirect
60. #1618C2×C3semidirect
61. #1615C2×C3semidirect
62. #1606C2×C3semidirect
63. #1603C2×C3semidirect
64. #1503C2×C3semidirect
65. #1502C2×C3semidirect
66. #1458C2×C3semidirect
653. C3#18127direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34552
C3×#31042
C3×#31032
C3×#23962
C3×#16182
C3×#16152
C3×#16062
C3×#16032
C3×#15032
C3×#15022
C3×#14582
#181273