Finite Group Databasebeta

← C3×#18157 #10636764

Subgroups — C3×#18157 #10636764

Order 768.

926 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3455C211yesmaximal, characteristic
2C3×#3113C211yesmaximal, characteristic, complemented
3C3×#3112C211yesmaximal, characteristic, complemented
4C3×#2599C211yesmaximal, characteristic, complemented
5C3×#1612C212yesmaximal, complemented
6C3×#1609C212yesmaximal, complemented
7C3×#1606C212yesmaximal, complemented
8C3×#1603C212yesmaximal, complemented
9C3×#1515C211yesmaximal, characteristic, complemented
10C3×#1514C211yesmaximal, characteristic, complemented
11C3×#1464C211yesmaximal, characteristic, complemented
12#18157C311yesmaximal, 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×#483C2212yes—
30C3×#480C2212yes—
31C3×#477—21no—
32C3×#475—21no—
33C3×#462—21no—
34C3×#460—21no—
35C3×#451C2212yes—
36C3×#448C2212yes—
37C3×#441—22no—
38C3×#440—22no—
39C3×#432C2211yescharacteristic
40C3×#432C2211yescharacteristic
41C3×#398C2211yescharacteristic
42C3×#397C2211yescharacteristic
43C3×#395C2211yescharacteristic
44C3×#393C2211yescharacteristic
45C3×#388—21no—
46C3×#387—21no—
47C3×#387—21no—
48C3×#385—21no—
49C3×#383C2211yescharacteristic
50C3×#328C2212yes—
51C3×#328C2212yes—
52C3×#327C2212yes—
53C3×#327C2212yes—
54C3×#324C2212yes—
55C3×#323C2212yes—
56#3455C2×C311yescharacteristic
57#3113C2×C311yescharacteristic, complemented
58#3112C2×C311yescharacteristic, complemented
59#2599C2×C311yescharacteristic, complemented
60#1612C2×C312yescomplemented
61#1609C2×C312yescomplemented
62#1606C2×C312yescomplemented
63#1603C2×C312yescomplemented
64#1515C2×C311yescharacteristic, complemented
65#1514C2×C311yescharacteristic, complemented
66#1464C2×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⋊C4D411yescharacteristic
130C2×C3×C4⋊C4C2311yescharacteristic
131C2×C3×C4⋊C4C2311yescharacteristic
132C2×C3×C22⋊C4—21no—
133C2×C3×C22⋊C4—21no—
134C2×C3×C22⋊C4D411yescharacteristic
135C2×C3×C22⋊C4C2311yescharacteristic
136C2×C3×C42—22no—
137C2×C3×C42—21no—
138C2×C3×C42D411yescharacteristic
139C2×C3×C42C2311yescharacteristic
140C3×#107—22no—
141C3×#106—22no—
142C3×C4⋊C8—21no—
143C3×C4⋊C8—21no—
144C3×#103—22no—
145C3×#103—22no—
146C3×#103—22no—
147C3×#102—22no—
148C3×C22⋊C8C2312yes—
149C3×C22⋊C8C2312yes—
150C3×#97—21no—
151C3×#97—21no—
152C3×#95—21no—
153C3×#95—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#483C22×C312yes—
177#480C22×C312yes—
178#477—21no—
179#475—21no—
180#462—21no—
181#460—21no—
182#451C22×C312yes—
183#448C22×C312yes—
184#441—22no—
185#440—22no—
186#432C22×C311yescharacteristic
187#432C22×C311yescharacteristic
188#398C22×C311yescharacteristic
189#397C22×C311yescharacteristic
190#395C22×C311yescharacteristic
191#393C22×C311yescharacteristic
192#388—21no—
193#387—21no—
194#387—21no—
195#385—21no—
196#383C22×C311yescharacteristic
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—22no—
220C22×C3×C4—22no—
221C22×C3×C4—22no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—41no—
228C22×C3×C4—41no—
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—41no—
258C3×C4⋊C4—41no—
259C3×C4⋊C4C2×D412yes—
260C3×C4⋊C4C2×D412yes—
261C3×C22⋊C4—22no—
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—21no—
269C3×C22⋊C4—21no—
270C3×C22⋊C4—22no—
271C3×C22⋊C4—21no—
272C3×C22⋊C4—21no—
273C3×C22⋊C4—22no—
274C3×C22⋊C4—42no—
275C3×C22⋊C4—42no—
276C3×C42—22no—
277C3×C42—22no—
278C3×C42—21no—
279C3×C42—22no—
280C3×C42—21no—
281C3×C42—41no—
282C3×C42—41no—
283C3×C42C2×D411yescharacteristic
284C3×C42C2×D411yescharacteristic
285C3×C42C2×D411yescharacteristic
286C3×C42C2×D411yescharacteristic
287C2×#41—21no—
288C22×Q8C23×C311yescharacteristic
289C23×C4C3×D411yescharacteristic
290C2×Q16—22no—
291C2×SD16—22no—
292C4⋊Q8C3×D411yescharacteristic
293C4⋊Q8C3×D411yescharacteristic
294C4⋊Q8C3×D411yescharacteristic
295#127C3×D411yescharacteristic
296#126—22no—
297#126—22no—
298#126—22no—
299#126—22no—
300#125—22no—
301#125—22no—
302#125—21no—
303#124—22no—
304#124—22no—
305#124—21no—
306#123—22no—
307#123—22no—
308#123—22no—
309#123—22no—
310#123—22no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—21no—
316C22⋊Q8C23×C312yes—
317#121—21no—
318#121—21no—
319#121—22no—
320#121—22no—
321#121—21no—
322#121C23×C312yes—
323#120—22no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8C3×D412yes—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—21no—
339C4×D4C3×D412yes—
340#117—22no—
341#117—21no—
342#117—21no—
343#117—21no—
344#117—21no—
345#117—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C3×D411yescharacteristic
350C2×C4⋊C4C23×C311yescharacteristic
351C2×C4⋊C4C23×C311yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4C3×D411yescharacteristic
355C2×C22⋊C4C23×C311yescharacteristic
356C2×C42—22no—
357C2×C42—21no—
358C2×C42C3×D411yescharacteristic
359C2×C42C23×C311yescharacteristic
360#107—22no—
361#106—22no—
362C4⋊C8—21no—
363C4⋊C8—21no—
364#103—22no—
365#103—22no—
366#103—22no—
367#102—22no—
368C22⋊C8C23×C312yes—
369C22⋊C8C23×C312yes—
370#97—21no—
371#97—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95C23×C311yescharacteristic
379#95C23×C311yescharacteristic
380C23×C3—21no—
381C23×C3—22no—
382C23×C3—21no—
383C23×C3—21no—
384C23×C3—41no—
385C23×C3—41no—
386C23×C3C22×D411yescharacteristic
387C3×Q8—22no—
388C3×Q8—22no—
389C3×Q8—42no—
390C3×Q8—41no—
391C3×Q8—41no—
392C3×Q8—42no—
393C3×Q8—81no—
394C3×D4—42no—
395C3×D4—42no—
396C3×D4—42no—
397C3×D4—42no—
398C3×D4—81no—
399C3×D4—82no—
400C3×D4—82no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—22no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—42no—
415C2×C3×C4—42no—
416C2×C3×C4—44no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
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×C4C22×D411yescharacteristic
437C2×C3×C4C22×D411yescharacteristic
438C3×C8—42no—
439C3×C8—42no—
440C24C2×C3×D411yescharacteristic
441#41—42no—
442#41—42no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—42no—
448C2×Q8C2×C3×D412yes—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4C2×C3×D412yes—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4C2×C3×D411yescharacteristic
467C22×C4C2×C3×D411yescharacteristic
468C22×C4C2×C3×D411yescharacteristic
469C22×C4C2×C3×D411yescharacteristic
470C22×C4C2×C3×D411yescharacteristic
471C22×C4C24×C311yescharacteristic
472Q16—44no—
473SD16—44no—
474C2×C8—22no—
475C2×C8—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4C2×C3×D412yes—
497C4⋊C4C2×C3×D412yes—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—22no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—22no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C42—22no—
514C42—22no—
515C42—21no—
516C42—22no—
517C42—21no—
518C42—41no—
519C42—41no—
520C42C2×C3×D411yescharacteristic
521C42C2×C3×D411yescharacteristic
522C42C2×C3×D411yescharacteristic
523C42C2×C3×D411yescharacteristic
524C22×C3—21no—
525C22×C3—21no—
526C22×C3—42no—
527C22×C3—41no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—41no—
531C22×C3—42no—
532C22×C3—41no—
533C22×C3—81no—
534C22×C3C2×#12011yescharacteristic
535C22×C3#57511yescharacteristic
536C22×C3#57511yescharacteristic
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—41no—
544C3×C4—42no—
545C3×C4—41no—
546C3×C4—42no—
547C3×C4—42no—
548C3×C4—41no—
549C3×C4—42no—
550C3×C4—81no—
551C3×C4—81no—
552C3×C4—81no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—21no—
557C23—41no—
558C23—41no—
559C23C22×C3×D411yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—42no—
563Q8—41no—
564Q8—41no—
565Q8—42no—
566Q8—81no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—81no—
572D4—82no—
573D4—82no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4C3×#12011yescharacteristic
606C2×C4C3×#12011yescharacteristic
607C2×C4C3×#12011yescharacteristic
608C2×C4C3×#12011yescharacteristic
609C2×C4C22×C3×D411yescharacteristic
610C2×C4C22×C3×D411yescharacteristic
611C8—42no—
612C8—42no—
613C2×C3—21no—
614C2×C3—21no—
615C2×C3—82no—
616C2×C3—81no—
617C2×C3#300511yescharacteristic
618C2×C3#302211yescharacteristic
619C2×C3#302511yescharacteristic
620C22—21no—
621C22—21no—
622C22—42no—
623C22—41no—
624C22—42no—
625C22—42no—
626C22—41no—
627C22—42no—
628C22—41no—
629C22—81no—
630C22C2×C3×#12011yescharacteristic
631C22C3×#57511yescharacteristic
632C22C3×#57511yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—41no—
640C4—42no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—41no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—81no—
649C3#1815711yescharacteristic, direct product factor, Sylow
650C2—21no—
651C2—21no—
652C2—81no—
653C2—82no—
654C2C3×#300511yescharacteristic
655C2C3×#302211yescharacteristic
656C2C3×#302511yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3455C2
LCSG ▷ C22×C4 ▷ 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×#3113C2semidirect
3. C3×#3112C2semidirect
4. C3×#2599C2semidirect
5. C3×#1612C2semidirect
6. C3×#1609C2semidirect
7. C3×#1606C2semidirect
8. C3×#1603C2semidirect
9. C3×#1515C2semidirect
10. C3×#1514C2semidirect
11. C3×#1464C2semidirect
12. #18157C3direct (N × H)
57. #3113C2×C3semidirect
58. #3112C2×C3semidirect
59. #2599C2×C3semidirect
60. #1612C2×C3semidirect
61. #1609C2×C3semidirect
62. #1606C2×C3semidirect
63. #1603C2×C3semidirect
64. #1515C2×C3semidirect
65. #1514C2×C3semidirect
66. #1464C2×C3semidirect
649. C3#18157direct (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×#31132
C3×#31122
C3×#25992
C3×#16122
C3×#16092
C3×#16062
C3×#16032
C3×#15152
C3×#15142
C3×#14642
#181573