Finite Group Databasebeta

← C3×#78638 #11864046

Subgroups — C3×#78638 #11864046

Order 1152.

1611 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#235C211yesmaximal, characteristic, complemented
2C3×#5791C211yesmaximal, characteristic, complemented
3C2×C3×#909C211yesmaximal, characteristic, complemented
4C2×C3×#908C211yesmaximal, characteristic, complemented
5C32×#407C211yesmaximal, characteristic, complemented
6C3×#5131C211yesmaximal, characteristic, complemented
7C3×#5120C211yesmaximal, characteristic, complemented
8C3×#4734C214yesmaximal, complemented
9C3×#4733C214yesmaximal, complemented
10C3×#3005—31nomaximal
11#78638C311yesmaximal, characteristic, direct product factor
12C23×C32×C4C2211yescharacteristic
13C22×C3×#67—21no—
14C2×C3×#235—22no—
15C2×C3×#235—24no—
16C2×C3×#235—24no—
17C2×C3×#235C2211yescharacteristic
18C2×C3×#235C2214yes—
19C22×C3×D12C2211yescharacteristic
20C22×C3×C4×D3—21no—
21C22×C3×Dic6C2211yescharacteristic
22C2×C32×M16C2211yescharacteristic
23C2×C32×M16C2211yescharacteristic
24C3×#953—21no—
25C3×#930C2212yescomplemented
26C3×#929—21no—
27C3×C4×#67—22no—
28C3×#926C2211yescharacteristic
29C3×#924C2212yescomplemented
30C3×#909—24no—
31C3×#908—24no—
32C2×C3×Dic12C2212yescomplemented
33C2×C3×D24C2212yescomplemented
34C2×C3×#204C2212yescomplemented
35C2×C3×#204C2212yescomplemented
36C32×C22⋊C8C2214yescomplemented
37C3×#821C2214yescomplemented
38C3×#816C2214yescomplemented
39C22×C3×#41—31no—
40C22×#235C2×C311yescharacteristic, complemented
41C2×C3×#137—31no—
42C2×C3×AGL(1,ℤ/8)—31no—
43C3×#525—31no—
44#5791C2×C311yescharacteristic, complemented
45C2×#909C2×C311yescharacteristic, complemented
46C2×#908C2×C311yescharacteristic, complemented
47C3×#452—34no—
48C3×#449—34no—
49C3×#419—31no—
50C3×#418—31no—
51C3×#407—21no—
52C3×#407D311yescharacteristic, complemented
53C3×#407C2×C311yescharacteristic, complemented
54#5131C2×C311yescharacteristic, complemented
55#5120C2×C311yescharacteristic, complemented
56#4734C2×C314yescomplemented
57#4733C2×C314yescomplemented
58C24×C32D411yescharacteristic
59C23×C3×D3—21no—
60C22×C32×C4—22no—
61C22×C32×C4—22no—
62C22×C32×C4D412yes—
63C22×C32×C4D411yescharacteristic
64C22×C32×C4C2312yes—
65C22×C32×C4C2311yescharacteristic
66C2×C3×#67—28no—
67C2×C3×#67—24no—
68C2×C3×#67—22no—
69C2×C3×#67—21no—
70C22×C3×Dic3—21no—
71C3×#235—216no—
72C3×#235—28no—
73C3×#235—22no—
74C3×#235—22no—
75C3×#235—28no—
76C2×C3×D12—21no—
77C2×C3×D12—22no—
78C2×C3×D12D414yescomplemented
79C2×C3×D12C2311yescharacteristic
80C2×C3×D12C2312yes—
81C2×C3×C4×D3—28no—
82C2×C3×C4×D3—24no—
83C2×C3×C4×D3—21no—
84C2×C3×Dic6—21no—
85C2×C3×Dic6—22no—
86C2×C3×Dic6D414yescomplemented
87C2×C3×Dic6C2311yescharacteristic
88C2×C3×Dic6C2312yes—
89C32×M16—22no—
90C32×M16—22no—
91C2×C32×C8C2312yes—
92C2×C32×C8C2312yes—
93C3×#217—22no—
94C3×#217—21no—
95C3×(C2×D3)⋊C4—22no—
96C3×C4⋊Dic3C2312yes—
97C3×Dic3⋊C4—22no—
98C3×C4×Dic3—21no—
99C3×Dic12—24no—
100C3×D24—24no—
101C3×#204—24no—
102C3×#204—24no—
103#3005—31noSylow
104C2×C3×#41—31no—
105C2×C3×#41—34no—
106C2×C3×#41—64no—
107C2×C3×#41—64no—
108C2×C3×#41—62no—
109C22×C3×Q8—31no—
110C22×C3×D4—31no—
111C22×C3×D4—61no—
112C23×C3×C4—21no—
113C23×C3×C4—61no—
114C23×C3×C4C2×D311yescharacteristic
115C23×C3×C4C22×C311yescharacteristic
116C22×#67—21no—
117C2×#235—22no—
118C2×#235—24no—
119C2×#235—24no—
120C2×#235C22×C314yes—
121C2×#235C22×C311yescharacteristic
122C22×D12C22×C311yescharacteristic
123C22×C4×D3—21no—
124C22×Dic6C22×C311yescharacteristic
125C3×#137—64no—
126C3×AGL(1,ℤ/8)—64no—
127C2×C3×Q16—32no—
128C2×C3×SD16—32no—
129C2×C3×SD16—32no—
130C2×C3×D8—32no—
131C2×C3×M16—21no—
132C2×C3×M16—21no—
133C2×C3×M16C2×D311yescharacteristic
134C2×C3×M16C2×D311yescharacteristic
135C2×C3×M16C22×C311yescharacteristic
136C2×C3×M16C22×C311yescharacteristic
137C3×#123—61no—
138C3×C22⋊Q8—32no—
139C3×#121—32no—
140C3×#120—61no—
141C3×C4×D4—62no—
142C3×#117—31no—
143#953—21no—
144#930C22×C312yescomplemented
145#929—21no—
146C4×#67—22no—
147#926C22×C311yescharacteristic
148#924C22×C312yescomplemented
149#909—24no—
150#908—24no—
151C2×Dic12C22×C312yescomplemented
152C2×D24C22×C312yescomplemented
153C2×#204C22×C312yescomplemented
154C2×#204C22×C312yescomplemented
155C3×#103—34no—
156C3×#102—34no—
157C3×C22⋊C8—24no—
158C3×C22⋊C8C2×D314yescomplemented
159C3×C22⋊C8C22×C314yescomplemented
160#821C22×C314yescomplemented
161#816C22×C314yescomplemented
162C23×C32—22no—
163C23×C32—42no—
164C23×C32C2×D412yes—
165C23×C32C2×D411yescharacteristic
166C22×C3×D3—24no—
167C22×C3×D3—22no—
168C22×C3×D3—21no—
169C22×C3×D3—44no—
170C2×C32×C4—22no—
171C2×C32×C4—24no—
172C2×C32×C4—22no—
173C2×C32×C4—22no—
174C2×C32×C4C2×D412yes—
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2×D412yes—
177C2×C32×C4C2×D412yes—
178C2×C32×C4C2411yescharacteristic
179C3×#67—216no—
180C3×#67—44no—
181C3×#67—42no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—21no—
186C3×D12—24no—
187C3×D12—22no—
188C3×D12—21no—
189C3×D12—21no—
190C3×D12C2×D414yes—
191C3×C4×D3—216no—
192C3×C4×D3—42no—
193C3×Dic6—24no—
194C3×Dic6—22no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6C2×D414yes—
198C32×C8—22no—
199C32×C8—22no—
200C22×#41—31no—
201C2×#137—31no—
202C2×AGL(1,ℤ/8)—31no—
203#525—31no—
204#452—34no—
205#449—34no—
206#419—31no—
207#418—31no—
208#407C3×D311yescharacteristic, complemented
209C24×C3—21no—
210C24×C3—61no—
211C24×C3D1211yescharacteristic
212C24×C3C3×D411yescharacteristic
213C23×D3—21no—
214C3×#41—68no—
215C3×#41—68no—
216C3×#41—616no—
217C3×#41—62no—
218C3×#41—62no—
219C2×C3×Q8—34no—
220C2×C3×Q8—32no—
221C2×C3×Q8—31no—
222C2×C3×Q8—61no—
223C2×C3×Q8—62no—
224C2×C3×D4—34no—
225C2×C3×D4—32no—
226C2×C3×D4—31no—
227C2×C3×D4—68no—
228C2×C3×D4—64no—
229C2×C3×D4—62no—
230C2×C3×D4—61no—
231C2×C3×D4—61no—
232C2×C3×D4—62no—
233C22×C3×C4—22no—
234C22×C3×C4—22no—
235C22×C3×C4—22no—
236C22×C3×C4—22no—
237C22×C3×C4—22no—
238C22×C3×C4—21no—
239C22×C3×C4—21no—
240C22×C3×C4—22no—
241C22×C3×C4—42no—
242C22×C3×C4—42no—
243C22×C3×C4—68no—
244C22×C3×C4—61no—
245C22×C3×C4—64no—
246C22×C3×C4—61no—
247C22×C3×C4D1212yes—
248C22×C3×C4D1211yescharacteristic
249C22×C3×C4C3×D412yes—
250C22×C3×C4C3×D411yescharacteristic
251C22×C3×C4C22×D312yes—
252C22×C3×C4C22×D311yescharacteristic
253C22×C3×C4C23×C312yes—
254C22×C3×C4C23×C311yescharacteristic
255C2×#67—28no—
256C2×#67—24no—
257C2×#67—22no—
258C2×#67—21no—
259C22×Dic3—21no—
260#235—216no—
261#235—28no—
262#235—22no—
263#235—22no—
264#235—28no—
265C2×D12—21no—
266C2×D12—22no—
267C2×D12C3×D414yescomplemented
268C2×D12C23×C311yescharacteristic
269C2×D12C23×C312yes—
270C2×C4×D3—28no—
271C2×C4×D3—24no—
272C2×C4×D3—21no—
273C2×Dic6—21no—
274C2×Dic6—22no—
275C2×Dic6C3×D414yescomplemented
276C2×Dic6C23×C311yescharacteristic
277C2×Dic6C23×C312yes—
278C3×Q16—64no—
279C3×SD16—64no—
280C3×SD16—64no—
281C3×D8—64no—
282C3×M16—22no—
283C3×M16—22no—
284C3×M16—22no—
285C3×M16—22no—
286C3×M16—42no—
287C3×M16—42no—
288C2×C3×C8—22no—
289C2×C3×C8—22no—
290C2×C3×C8C22×D312yes—
291C2×C3×C8C22×D312yes—
292C2×C3×C8C23×C312yes—
293C2×C3×C8C23×C312yes—
294C3×C4⋊C4—32no—
295C3×C4⋊C4—62no—
296C3×C22⋊C4—61no—
297C3×C22⋊C4—62no—
298C3×C22⋊C4—62no—
299C3×C42—61no—
300#217—22no—
301#217—21no—
302(C2×D3)⋊C4—22no—
303C4⋊Dic3C23×C312yes—
304Dic3⋊C4—22no—
305C4×Dic3—21no—
306Dic12—24no—
307D24—24no—
308#204—24no—
309#204—24no—
310C22×C32—22no—
311C22×C32—22no—
312C22×C32—22no—
313C22×C32—22no—
314C22×C32—44no—
315C22×C32#12012yes—
316C22×C32C22×D411yescharacteristic
317C2×C3×D3—24no—
318C2×C3×D3—48no—
319C2×C3×D3—44no—
320C2×C3×D3—41no—
321C2×C3×D3—42no—
322C32×C4—22no—
323C32×C4#12012yes—
324C32×C4C22×D411yescharacteristic
325C32×C4C22×D411yescharacteristic
326C3×Dic3—24no—
327C3×Dic3—41no—
328C3×Dic3—42no—
329C2×#41—31no—
330C2×#41—34no—
331C2×#41—64no—
332C2×#41—64no—
333C2×#41—62no—
334C22×Q8—31no—
335C22×D4—31no—
336C22×D4—61no—
337C23×C4—61no—
338C23×C4C2×C3×D311yescharacteristic
339#137—64no—
340AGL(1,ℤ/8)—64no—
341C2×Q16—32no—
342C2×SD16—32no—
343C2×SD16—32no—
344C2×D8—32no—
345C2×M16C2×C3×D311yescharacteristic
346C2×M16C2×C3×D311yescharacteristic
347#123—61no—
348C22⋊Q8—32no—
349#121—32no—
350#120—61no—
351C4×D4—62no—
352#117—31no—
353#103—34no—
354#102—34no—
355C22⋊C8C2×C3×D314yescomplemented
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—21no—
360C23×C3—42no—
361C23×C3—42no—
362C23×C3—42no—
363C23×C3—44no—
364C23×C3—64no—
365C23×C3—62no—
366C23×C3—61no—
367C23×C3—124no—
368C23×C3C2×D1212yes—
369C23×C3C2×D1211yescharacteristic
370C23×C3C2×C3×D412yes—
371C23×C3C2×C3×D411yescharacteristic
372C22×D3—24no—
373C22×D3—22no—
374C22×D3—21no—
375C22×D3—44no—
376C3×Q8—34no—
377C3×Q8—64no—
378C3×Q8—62no—
379C3×Q8—61no—
380C3×Q8—61no—
381C3×D4—34no—
382C3×D4—616no—
383C3×D4—64no—
384C3×D4—62no—
385C3×D4—61no—
386C3×D4—61no—
387C3×D4—124no—
388C3×D4—122no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—24no—
392C2×C3×C4—24no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—44no—
403C2×C3×C4—42no—
404C2×C3×C4—42no—
405C2×C3×C4—42no—
406C2×C3×C4—616no—
407C2×C3×C4—64no—
408C2×C3×C4—62no—
409C2×C3×C4—62no—
410C2×C3×C4—61no—
411C2×C3×C4—122no—
412C2×C3×C4C2×D1212yes—
413C2×C3×C4C2×D1211yescharacteristic
414C2×C3×C4D3×D412yes—
415C2×C3×C4D3×D412yes—
416C2×C3×C4C2×C3×D412yes—
417C2×C3×C4C2×C3×D411yescharacteristic
418C2×C3×C4C2×C3×D412yes—
419C2×C3×C4C2×C3×D412yes—
420C2×C3×C4C23×D311yescharacteristic
421C2×C3×C4C24×C311yescharacteristic
422#67—216no—
423#67—44no—
424#67—42no—
425C2×Dic3—24no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—21no—
429D12—24no—
430D12—22no—
431D12—21no—
432D12—21no—
433D12C2×C3×D414yes—
434C4×D3—216no—
435C4×D3—42no—
436Dic6—24no—
437Dic6—22no—
438Dic6—21no—
439Dic6—21no—
440Dic6C2×C3×D414yes—
441C3×C8—22no—
442C3×C8—22no—
443C3×C8—22no—
444C3×C8—22no—
445C3×C8—42no—
446C3×C8—42no—
447C2×C32—22no—
448C2×C32—42no—
449C2×C32C2×#12011yescharacteristic
450C2×C32#57512yes—
451C3×D3—44no—
452C3×D3—81no—
453C24—61no—
454C24C3×D1211yescharacteristic
455#41—68no—
456#41—68no—
457#41—616no—
458#41—62no—
459#41—62no—
460C2×Q8—32no—
461C2×Q8—31no—
462C2×Q8—34no—
463C2×Q8—61no—
464C2×Q8—62no—
465C2×D4—32no—
466C2×D4—31no—
467C2×D4—34no—
468C2×D4—68no—
469C2×D4—64no—
470C2×D4—62no—
471C2×D4—61no—
472C2×D4—61no—
473C2×D4—62no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—68no—
477C22×C4—61no—
478C22×C4—64no—
479C22×C4—61no—
480C22×C4C3×D1212yes—
481C22×C4C3×D1211yescharacteristic
482C22×C4C22×C3×D312yes—
483C22×C4C22×C3×D311yescharacteristic
484Q16—64no—
485SD16—64no—
486SD16—64no—
487D8—64no—
488M16—22no—
489M16—22no—
490C2×C8C22×C3×D312yes—
491C2×C8C22×C3×D312yes—
492C4⋊C4—32no—
493C4⋊C4—62no—
494C22⋊C4—61no—
495C22⋊C4—62no—
496C22⋊C4—62no—
497C42—61no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—24no—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—21no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—44no—
513C22×C3—44no—
514C22×C3—48no—
515C22×C3—64no—
516C22×C3—128no—
517C22×C3—124no—
518C22×C3—121no—
519C22×C3—122no—
520C22×C3#88212yes—
521C22×C3C3×#12012yes—
522C22×C3C22×D1211yescharacteristic
523C22×C3C22×C3×D411yescharacteristic
524C2×D3—24no—
525C2×D3—48no—
526C2×D3—44no—
527C2×D3—42no—
528C2×D3—41no—
529C3×C4—22no—
530C3×C4—22no—
531C3×C4—22no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—42no—
535C3×C4—64no—
536C3×C4—121no—
537C3×C4—122no—
538C3×C4#88212yes—
539C3×C4C3×#12012yes—
540C3×C4C2×D3×D411yescharacteristic
541C3×C4C2×D3×D411yescharacteristic
542C3×C4C22×C3×D411yescharacteristic
543C3×C4C22×C3×D411yescharacteristic
544Dic3—24no—
545Dic3—42no—
546Dic3—41no—
547C32#300511yescharacteristic, complemented, Sylow
548C23—22no—
549C23—42no—
550C23—64no—
551C23—62no—
552C23—61no—
553C23—124no—
554C23C2×C3×D1212yes—
555C23C2×C3×D1211yescharacteristic
556Q8—34no—
557Q8—64no—
558Q8—62no—
559Q8—61no—
560Q8—61no—
561D4—34no—
562D4—616no—
563D4—64no—
564D4—62no—
565D4—61no—
566D4—61no—
567D4—124no—
568D4—122no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—616no—
574C2×C4—64no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—61no—
578C2×C4—122no—
579C2×C4C2×C3×D1212yes—
580C2×C4C2×C3×D1211yescharacteristic
581C2×C4C3×D3×D412yes—
582C2×C4C3×D3×D412yes—
583C2×C4C23×C3×D311yescharacteristic
584C8—22no—
585C8—22no—
586C2×C3—22no—
587C2×C3—22no—
588C2×C3—21no—
589C2×C3—22no—
590C2×C3—42no—
591C2×C3—44no—
592C2×C3—42no—
593C2×C3—42no—
594C2×C3—124no—
595C2×C3—241no—
596C2×C3C2×#88211yescharacteristic
597C2×C3#574812yes—
598C2×C3C2×C3×#12011yescharacteristic
599C2×C3C3×#57512yes—
600D3—44no—
601D3—81no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—44no—
607C22—64no—
608C22—128no—
609C22—124no—
610C22—121no—
611C22—122no—
612C22C3×#88212yes—
613C22C22×C3×D1211yescharacteristic
614C4—22no—
615C4—64no—
616C4—121no—
617C4—122no—
618C4C3×#88212yes—
619C4C2×C3×D3×D411yescharacteristic
620C4C2×C3×D3×D411yescharacteristic
621C3—21no—
622C3#7863811yescharacteristic, direct product factor
623C3C3×#300511yescharacteristic, complemented
624C2—22no—
625C2—42no—
626C2—124no—
627C2—241no—
628C2C2×C3×#88211yescharacteristic
629C2C3×#574812yes—
subgroupG/H
centerC22×C3C22×D12
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#407C2
socleC22×C32C22×D4
hypercenterC3×#407D3
hypocenterC3C3×#3005
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C3×#407

Complements

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

NH ≅ G/Ntype
1. C22×C3×#235C2semidirect
2. C3×#5791C2semidirect
3. C2×C3×#909C2semidirect
4. C2×C3×#908C2semidirect
5. C32×#407C2semidirect
6. C3×#5131C2semidirect
7. C3×#5120C2semidirect
8. C3×#4734C2semidirect
9. C3×#4733C2semidirect
11. #78638C3direct (N × H)
25. C3×#930C22semidirect
29. C3×#924C22semidirect
32. C2×C3×Dic12C22semidirect
33. C2×C3×D24C22semidirect
34. C2×C3×#204C22semidirect
35. C2×C3×#204C22semidirect
36. C32×C22⋊C8C22semidirect
37. C3×#821C22semidirect
38. C3×#816C22semidirect
40. C22×#235C2×C3semidirect
44. #5791C2×C3semidirect
45. C2×#909C2×C3semidirect
46. C2×#908C2×C3semidirect
52. C3×#407D3semidirect
53. C3×#407C2×C3semidirect
54. #5131C2×C3semidirect
55. #5120C2×C3semidirect
56. #4734C2×C3semidirect
57. #4733C2×C3semidirect
78. C2×C3×D12D4semidirect
86. C2×C3×Dic6D4semidirect
144. #930C22×C3semidirect
148. #924C22×C3semidirect
151. C2×Dic12C22×C3semidirect
152. C2×D24C22×C3semidirect
153. C2×#204C22×C3semidirect
154. C2×#204C22×C3semidirect
158. C3×C22⋊C8C2×D3semidirect
159. C3×C22⋊C8C22×C3semidirect
160. #821C22×C3semidirect
161. #816C22×C3semidirect
208. #407C3×D3semidirect
267. C2×D12C3×D4semidirect
275. C2×Dic6C3×D4semidirect
355. C22⋊C8C2×C3×D3semidirect
547. C32#3005semidirect
622. C3#78638direct (N × H)
623. C3C3×#3005semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2352
C3×#57912
C2×C3×#9092
C2×C3×#9082
C32×#4072
C3×#51312
C3×#51202
C3×#47342
C3×#47332
C3×#30053
#786383