Finite Group Databasebeta

← C3×#85493 #11922779

Subgroups — C3×#85493 #11922779

Order 1152.

1460 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 1460 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×S4C211yesmaximal, characteristic, complemented
2C2×C3×D4×A4C211yesmaximal, characteristic, complemented
3C2×C3×GL(2,ℤ/4)C212yesmaximal, complemented
4C3×#5432C211yesmaximal, characteristic, complemented
5C3×(C2×S4)⋊C4C212yesmaximal, complemented
6#85493C311yesmaximal, characteristic, direct product factor
7C3×#2403—31nomaximal
8C23×C3×A4C2211yescharacteristic
9C23×C3×A4C2211yescharacteristic, complemented
10C22×C3×S4—22no—
11C22×C3×S4—24no—
12C22×C3×S4C2212yes—
13C3×D4×A4—22no—
14C2×C3×C4×A4C2211yescharacteristic, complemented
15C3×GL(2,ℤ/4)—24no—
16C2×C3×SL(2,ℤ/4)C2212yescomplemented
17C3×#937—41nomaximal
18C23×S4C2×C311yescharacteristic, complemented
19C23×C3×D4—31no—
20C23×C3×D4D311yescharacteristic, complemented
21C2×D4×A4—21no—
22C2×D4×A4C2×C311yescharacteristic, complemented
23C2×GL(2,ℤ/4)C2×C312yescomplemented
24C2×C3×#120—31no—
25C2×C3×#120—31no—
26C2×C3×#120—32no—
27#5432C2×C311yescharacteristic, complemented
28(C2×S4)⋊C4C2×C312yescomplemented
29C3×#392—32no—
30C3×#392—32no—
31C3×#386—31no—
32C3×#386—31no—
33C3×#386—32no—
34C3×#379—31no—
35C23×C3×D3—41no—
36C22×C3×A4—22no—
37C22×C3×A4—21no—
38C22×C3×A4—22no—
39C22×C3×A4D412yescomplemented
40C22×C3×A4C2311yescharacteristic
41C2×C3×S4—28no—
42C2×C3×S4—22no—
43C2×C3×S4—22no—
44C2×C3×S4D414yescomplemented
45C2×C32×D4—41no—
46C2×C3×#67—42no—
47C3×C4×A4—21no—
48C3×SL(2,ℤ/4)—22no—
49C3×#217—41no—
50C3×(C2×D3)⋊C4—42no—
51#2403—31noSylow
52C25×C3—31no—
53C25×C3C2×D311yescharacteristic
54C25×C3C2×D311yescharacteristic, complemented
55C23×A4—21no—
56C23×A4—21no—
57C23×A4C22×C311yescharacteristic
58C23×A4C22×C311yescharacteristic, complemented
59C22×S4—22no—
60C22×S4—24no—
61C22×S4C22×C312yes—
62C22×C3×D4—22no—
63C22×C3×D4—31no—
64C22×C3×D4—32no—
65C22×C3×D4—31no—
66C22×C3×D4—31no—
67C22×C3×D4—31no—
68C22×C3×D4—32no—
69C22×C3×D4—32no—
70C22×C3×D4—32no—
71C22×C3×D4—62no—
72C22×C3×D4—64no—
73C22×C3×D4—64no—
74C22×C3×D4—62no—
75C22×C3×D4—62no—
76C23×C3×C4—31no—
77C23×C3×C4C2×D311yescharacteristic, complemented
78D4×A4—24no—
79D4×A4—22no—
80C2×C4×A4—21no—
81C2×C4×A4C22×C311yescharacteristic, complemented
82GL(2,ℤ/4)—24no—
83C2×SL(2,ℤ/4)C22×C312yescomplemented
84C3×#120—64no—
85C3×#120—64no—
86C3×#120—64no—
87C3×#120—61no—
88C3×#120—61no—
89C3×#120—61no—
90C3×#120—61no—
91C2×C3×C22⋊C4—31no—
92C2×C3×C22⋊C4—31no—
93C2×C3×C22⋊C4—32no—
94C2×C3×C22⋊C4—32no—
95C2×C3×C22⋊C4—32no—
96C2×C3×C22⋊C4—32no—
97C2×C3×C22⋊C4—32no—
98C2×C3×C22⋊C4—32no—
99#937—41no—
100C3×#95—32no—
101C3×#95—32no—
102C23×C32—41no—
103C23×C32—41no—
104C22×C3×D3—42no—
105C22×C3×D3—84no—
106C22×C3×D3—82no—
107C2×C3×A4—22no—
108C2×C3×A4—41no—
109C2×C3×A4C2×D412yes—
110C2×C3×A4C2×D411yescharacteristic
111C3×S4—24no—
112C32×D4—82no—
113C2×C32×C4—41no—
114C3×#67—84no—
115C2×C3×Dic3—42no—
116C23×D4—31no—
117C23×D4C3×D311yescharacteristic, complemented
118C2×#120—31no—
119C2×#120—31no—
120C2×#120—32no—
121#392—32no—
122#392—32no—
123#386—31no—
124#386—31no—
125#386—32no—
126#379—31no—
127C24×C3—22no—
128C24×C3—22no—
129C24×C3—21no—
130C24×C3—31no—
131C24×C3—31no—
132C24×C3—31no—
133C24×C3—32no—
134C24×C3—32no—
135C24×C3—31no—
136C24×C3—61no—
137C24×C3—62no—
138C24×C3—62no—
139C24×C3—64no—
140C24×C3—62no—
141C24×C3—62no—
142C24×C3—61no—
143C24×C3—61no—
144C24×C3—122no—
145C24×C3#6712yescomplemented
146C24×C3C22×D311yescharacteristic
147C23×D3—41no—
148C22×A4—22no—
149C22×A4—22no—
150C22×A4—22no—
151C22×A4—21no—
152C22×A4—22no—
153C22×A4—21no—
154C22×A4—42no—
155C22×A4—42no—
156C22×A4C3×D412yescomplemented
157C22×A4C23×C311yescharacteristic
158C2×S4—28no—
159C2×S4—22no—
160C2×S4—22no—
161C2×S4C3×D414yescomplemented
162C2×C3×D4—34no—
163C2×C3×D4—34no—
164C2×C3×D4—34no—
165C2×C3×D4—31no—
166C2×C3×D4—31no—
167C2×C3×D4—31no—
168C2×C3×D4—41no—
169C2×C3×D4—61no—
170C2×C3×D4—62no—
171C2×C3×D4—64no—
172C2×C3×D4—62no—
173C2×C3×D4—62no—
174C2×C3×D4—68no—
175C2×C3×D4—62no—
176C2×C3×D4—62no—
177C2×C3×D4—62no—
178C2×C3×D4—64no—
179C2×C3×D4—64no—
180C2×C3×D4—68no—
181C2×C3×D4—616no—
182C2×C3×D4—62no—
183C2×C3×D4—62no—
184C2×C3×D4—62no—
185C2×C3×D4—62no—
186C2×C3×D4—62no—
187C2×C3×D4—64no—
188C2×C3×D4—64no—
189C2×C3×D4—64no—
190C2×C3×D4—62no—
191C2×C3×D4—62no—
192C2×C3×D4—81no—
193C2×C3×D4—122no—
194C2×C3×D4—122no—
195C2×C3×D4S411yescharacteristic, complemented
196C22×C3×C4—21no—
197C22×C3×C4—31no—
198C22×C3×C4—32no—
199C22×C3×C4—31no—
200C22×C3×C4—32no—
201C22×C3×C4—32no—
202C22×C3×C4—61no—
203C22×C3×C4—64no—
204C22×C3×C4—62no—
205C2×#67—42no—
206C4×A4—22no—
207C4×A4—21no—
208SL(2,ℤ/4)—22no—
209C3×C22⋊C4—64no—
210C3×C22⋊C4—64no—
211C3×C22⋊C4—64no—
212C3×C22⋊C4—62no—
213C3×C22⋊C4—62no—
214C3×C22⋊C4—62no—
215C3×C22⋊C4—61no—
216C3×C22⋊C4—62no—
217C3×C22⋊C4—61no—
218C3×C22⋊C4—62no—
219C3×C22⋊C4—122no—
220#217—41no—
221(C2×D3)⋊C4—42no—
222C22×C32—42no—
223C22×C32—41no—
224C22×C32—81no—
225C22×C32—82no—
226C22×C32—82no—
227C2×C3×D3—44no—
228C2×C3×D3—82no—
229C2×C3×D3—82no—
230C2×C3×D3—88no—
231C3×A4#12011yescharacteristic, complemented
232C32×C4—81no—
233C3×Dic3—82no—
234C25—31no—
235C25C2×C3×D311yescharacteristic
236C25C2×C3×D311yescharacteristic, complemented
237C22×D4—22no—
238C22×D4—31no—
239C22×D4—32no—
240C22×D4—31no—
241C22×D4—31no—
242C22×D4—31no—
243C22×D4—32no—
244C22×D4—32no—
245C22×D4—32no—
246C22×D4—62no—
247C22×D4—64no—
248C22×D4—64no—
249C22×D4—62no—
250C22×D4—62no—
251C23×C4—31no—
252C23×C4C2×C3×D311yescharacteristic, complemented
253#120—64no—
254#120—64no—
255#120—64no—
256#120—61no—
257#120—61no—
258#120—61no—
259#120—61no—
260C2×C22⋊C4—31no—
261C2×C22⋊C4—31no—
262C2×C22⋊C4—32no—
263C2×C22⋊C4—32no—
264C2×C22⋊C4—32no—
265C2×C22⋊C4—32no—
266C2×C22⋊C4—32no—
267C2×C22⋊C4—32no—
268#95—32no—
269#95—32no—
270C23×C3—22no—
271C23×C3—31no—
272C23×C3—32no—
273C23×C3—31no—
274C23×C3—32no—
275C23×C3—32no—
276C23×C3—32no—
277C23×C3—31no—
278C23×C3—34no—
279C23×C3—32no—
280C23×C3—31no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—62no—
285C23×C3—62no—
286C23×C3—62no—
287C23×C3—61no—
288C23×C3—61no—
289C23×C3—62no—
290C23×C3—61no—
291C23×C3—62no—
292C23×C3—62no—
293C23×C3—62no—
294C23×C3—64no—
295C23×C3—62no—
296C23×C3—62no—
297C23×C3—62no—
298C23×C3—62no—
299C23×C3—62no—
300C23×C3—62no—
301C23×C3—62no—
302C23×C3—61no—
303C23×C3—68no—
304C23×C3—62no—
305C23×C3—81no—
306C23×C3—81no—
307C23×C3—124no—
308C23×C3—122no—
309C23×C3—121no—
310C23×C3—122no—
311C23×C3—122no—
312C23×C3—122no—
313C23×C3—128no—
314C23×C3—124no—
315C23×C3—124no—
316C23×C3—122no—
317C23×C3D3×D412yes—
318C23×C3C2×#6711yescharacteristic
319C23×C3C2×S411yescharacteristic
320C23×C3C2×S411yescharacteristic, complemented
321C22×D3—42no—
322C22×D3—84no—
323C22×D3—82no—
324C2×A4—22no—
325C2×A4—22no—
326C2×A4—21no—
327C2×A4—42no—
328C2×A4—42no—
329C2×A4—41no—
330C2×A4C2×C3×D412yes—
331C2×A4C2×C3×D411yescharacteristic
332S4—24no—
333C3×D4—22no—
334C3×D4—62no—
335C3×D4—62no—
336C3×D4—64no—
337C3×D4—64no—
338C3×D4—64no—
339C3×D4—64no—
340C3×D4—616no—
341C3×D4—64no—
342C3×D4—62no—
343C3×D4—82no—
344C3×D4—84no—
345C3×D4—124no—
346C3×D4—128no—
347C3×D4—124no—
348C2×C3×C4—34no—
349C2×C3×C4—32no—
350C2×C3×C4—31no—
351C2×C3×C4—41no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—68no—
357C2×C3×C4—61no—
358C2×C3×C4—62no—
359C2×C3×C4—62no—
360C2×C3×C4—62no—
361C2×C3×C4—62no—
362C2×C3×C4—62no—
363C2×C3×C4—62no—
364C2×C3×C4—81no—
365C2×C3×C4C2×S411yescharacteristic, complemented
366#67—84no—
367C2×Dic3—42no—
368C2×C32—42no—
369C2×C32—41no—
370C2×C32—82no—
371C2×C32—161no—
372C3×D3—84no—
373C24—22no—
374C24—22no—
375C24—21no—
376C24—31no—
377C24—31no—
378C24—31no—
379C24—32no—
380C24—32no—
381C24—31no—
382C24—61no—
383C24—62no—
384C24—62no—
385C24—64no—
386C24—62no—
387C24—62no—
388C24—61no—
389C24—61no—
390C24—122no—
391C24C3×#6712yescomplemented
392C24C22×C3×D311yescharacteristic
393C2×D4—34no—
394C2×D4—34no—
395C2×D4—31no—
396C2×D4—31no—
397C2×D4—34no—
398C2×D4—31no—
399C2×D4—61no—
400C2×D4—62no—
401C2×D4—64no—
402C2×D4—62no—
403C2×D4—62no—
404C2×D4—68no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—64no—
409C2×D4—64no—
410C2×D4—68no—
411C2×D4—616no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—62no—
417C2×D4—64no—
418C2×D4—64no—
419C2×D4—64no—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4—122no—
423C2×D4—122no—
424C2×D4C3×S411yescharacteristic, complemented
425C22×C4—21no—
426C22×C4—31no—
427C22×C4—32no—
428C22×C4—31no—
429C22×C4—32no—
430C22×C4—32no—
431C22×C4—61no—
432C22×C4—64no—
433C22×C4—62no—
434C22⋊C4—64no—
435C22⋊C4—64no—
436C22⋊C4—64no—
437C22⋊C4—62no—
438C22⋊C4—62no—
439C22⋊C4—62no—
440C22⋊C4—61no—
441C22⋊C4—61no—
442C22⋊C4—62no—
443C22⋊C4—62no—
444C22⋊C4—122no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—21no—
448C22×C3—32no—
449C22×C3—31no—
450C22×C3—32no—
451C22×C3—32no—
452C22×C3—31no—
453C22×C3—31no—
454C22×C3—32no—
455C22×C3—42no—
456C22×C3—41no—
457C22×C3—62no—
458C22×C3—64no—
459C22×C3—62no—
460C22×C3—61no—
461C22×C3—62no—
462C22×C3—62no—
463C22×C3—64no—
464C22×C3—64no—
465C22×C3—64no—
466C22×C3—62no—
467C22×C3—62no—
468C22×C3—64no—
469C22×C3—61no—
470C22×C3—82no—
471C22×C3—82no—
472C22×C3—81no—
473C22×C3—82no—
474C22×C3—82no—
475C22×C3—81no—
476C22×C3—128no—
477C22×C3—121no—
478C22×C3—122no—
479C22×C3—121no—
480C22×C3—122no—
481C22×C3—121no—
482C22×C3—121no—
483C22×C3—122no—
484C22×C3—122no—
485C22×C3—128no—
486C22×C3—122no—
487C22×C3—122no—
488C22×C3—162no—
489C22×C3—162no—
490C22×C3#93711yescharacteristic, complemented
491C22×C3GL(2,ℤ/4)12yescomplemented
492C22×C3C22×S411yescharacteristic
493C2×D3—44no—
494C2×D3—82no—
495C2×D3—82no—
496C2×D3—88no—
497A4—21no—
498A4C3×#12011yescharacteristic, complemented
499C3×C4—21no—
500C3×C4—64no—
501C3×C4—61no—
502C3×C4—81no—
503C3×C4—82no—
504C3×C4—122no—
505C3×C4—122no—
506Dic3—82no—
507C32—41noSylow
508C23—22no—
509C23—31no—
510C23—32no—
511C23—31no—
512C23—32no—
513C23—32no—
514C23—32no—
515C23—31no—
516C23—34no—
517C23—32no—
518C23—31no—
519C23—41no—
520C23—62no—
521C23—62no—
522C23—62no—
523C23—61no—
524C23—61no—
525C23—62no—
526C23—61no—
527C23—62no—
528C23—62no—
529C23—62no—
530C23—64no—
531C23—62no—
532C23—62no—
533C23—62no—
534C23—62no—
535C23—62no—
536C23—62no—
537C23—62no—
538C23—61no—
539C23—68no—
540C23—62no—
541C23—124no—
542C23—122no—
543C23—121no—
544C23—122no—
545C23—122no—
546C23—122no—
547C23—128no—
548C23—124no—
549C23—124no—
550C23—122no—
551C23C3×D3×D412yes—
552C23C2×C3×#6711yescharacteristic
553C23C2×C3×S411yescharacteristic
554C23C2×C3×S411yescharacteristic, complemented
555D4—22no—
556D4—62no—
557D4—62no—
558D4—64no—
559D4—64no—
560D4—64no—
561D4—64no—
562D4—616no—
563D4—62no—
564D4—64no—
565D4—124no—
566D4—128no—
567D4—124no—
568C2×C4—34no—
569C2×C4—32no—
570C2×C4—31no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—68no—
576C2×C4—61no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4C2×C3×S411yescharacteristic, complemented
584C2×C3—22no—
585C2×C3—31no—
586C2×C3—32no—
587C2×C3—31no—
588C2×C3—41no—
589C2×C3—42no—
590C2×C3—41no—
591C2×C3—62no—
592C2×C3—82no—
593C2×C3—82no—
594C2×C3—81no—
595C2×C3—121no—
596C2×C3—124no—
597C2×C3—162no—
598C2×C3—161no—
599C2×C3—162no—
600C2×C3D4×S412yes—
601C2×C3C2×GL(2,ℤ/4)11yescharacteristic
602D3—84no—
603C22—22no—
604C22—22no—
605C22—21no—
606C22—32no—
607C22—31no—
608C22—32no—
609C22—32no—
610C22—31no—
611C22—31no—
612C22—32no—
613C22—62no—
614C22—64no—
615C22—62no—
616C22—61no—
617C22—62no—
618C22—62no—
619C22—64no—
620C22—64no—
621C22—64no—
622C22—62no—
623C22—62no—
624C22—64no—
625C22—61no—
626C22—128no—
627C22—121no—
628C22—122no—
629C22—121no—
630C22—122no—
631C22—121no—
632C22—121no—
633C22—122no—
634C22—122no—
635C22—128no—
636C22—122no—
637C22—122no—
638C22C3×#93711yescharacteristic, complemented
639C22C3×GL(2,ℤ/4)12yescomplemented
640C22C22×C3×S411yescharacteristic
641C4—21no—
642C4—64no—
643C4—61no—
644C4—122no—
645C4—122no—
646C3—41no—
647C3—81no—
648C3#8549311yescharacteristic, direct product factor
649C2—22no—
650C2—31no—
651C2—32no—
652C2—31no—
653C2—41no—
654C2—62no—
655C2—121no—
656C2—124no—
657C2C3×D4×S412yes—
658C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutator, cosocleC22×A4C23×C3
FrattiniC22C22×C3×S4
FittingC23×C3×D4D3
socleC24×C3C22×D3
hypercenterC2×C3×D4S4
hypocenterA4C3×#120
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C23×C3×S4C2semidirect
2. C2×C3×D4×A4C2semidirect
3. C2×C3×GL(2,ℤ/4)C2semidirect
4. C3×#5432C2semidirect
5. C3×(C2×S4)⋊C4C2semidirect
6. #85493C3direct (N × H)
9. C23×C3×A4C22semidirect
14. C2×C3×C4×A4C22semidirect
16. C2×C3×SL(2,ℤ/4)C22semidirect
18. C23×S4C2×C3semidirect
20. C23×C3×D4D3semidirect
22. C2×D4×A4C2×C3semidirect
23. C2×GL(2,ℤ/4)C2×C3semidirect
27. #5432C2×C3semidirect
28. (C2×S4)⋊C4C2×C3semidirect
39. C22×C3×A4D4semidirect
44. C2×C3×S4D4semidirect
54. C25×C3C2×D3semidirect
58. C23×A4C22×C3semidirect
77. C23×C3×C4C2×D3semidirect
81. C2×C4×A4C22×C3semidirect
83. C2×SL(2,ℤ/4)C22×C3semidirect
117. C23×D4C3×D3semidirect
145. C24×C3#67semidirect
156. C22×A4C3×D4semidirect
161. C2×S4C3×D4semidirect
195. C2×C3×D4S4semidirect
231. C3×A4#120semidirect
236. C25C2×C3×D3semidirect
252. C23×C4C2×C3×D3semidirect
320. C23×C3C2×S4semidirect
365. C2×C3×C4C2×S4semidirect
391. C24C3×#67semidirect
424. C2×D4C3×S4semidirect
490. C22×C3#937semidirect
491. C22×C3GL(2,ℤ/4)semidirect
498. A4C3×#120semidirect
554. C23C2×C3×S4semidirect
583. C2×C4C2×C3×S4semidirect
638. C22C3×#937semidirect
639. C22C3×GL(2,ℤ/4)semidirect
648. C3#85493direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×S42
C2×C3×D4×A42
C2×C3×GL(2,ℤ/4)2
C3×#54322
C3×(C2×S4)⋊C42
#854933
C3×#24033
C3×#9374