Finite Group Databasebeta

← C4×D32×A4 #13180886

Subgroups — C4×D32×A4 #13180886

Order 1728.

1410 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1410 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×D32×A4C211yesmaximal, characteristic
2C4×A4×C3⋊D3C211yesmaximal, characteristic, complemented
3C3×C4×D3×A4C212yesmaximal, complemented
4A4×#624C211yesmaximal, characteristic, complemented
5D3×Dic3×A4C212yesmaximal, complemented
6C22×C4×D32C311yesmaximal, characteristic, complemented
7C2×C4×D3×A4—32nomaximal
8C2×A4×C3⋊D3C2211yescharacteristic
9C2×C3×D3×A4C2212yes—
10D32×A4C414yesdirect product factor
11C32×C4×A4C2211yescharacteristic, complemented
12C3×C4×D32—41nomaximal
13A4×C3⋊Dic3C2211yescharacteristic, complemented
14C3×Dic3×A4C2212yescomplemented
15C23×D32C2×C311yescharacteristic
16C22×D3×A4—32no—
17C22×C4×C3⋊D3C2×C311yescharacteristic, complemented
18C22×C3×C4×D3C2×C312yescomplemented
19C2×C3×C4×A4—32no—
20C22×#624C2×C311yescharacteristic, complemented
21C22×D3×Dic3C2×C312yescomplemented
22C2×C4×D32—31no—
23C2×C4×D32—31no—
24C2×C4×D32—31no—
25C2×C4×D32—32no—
26C2×C4×D32—31no—
27C2×C4×D32—32no—
28C2×Dic3×A4—32no—
29C4×D3×A4—22no—
30C4×D3×A4—32no—
31C4×D3×A4—32no—
32C4×D3×A4—32no—
33C4×D3×A4—61no—
34C4×D3×A4D312yesdirect product factor
35C2×C32×A4C2311yescharacteristic
36C2×C3×D32—41no—
37A4×C3⋊D3C2×C412yescomplemented
38C3×D3×A4C2×C414yescomplemented
39C3×C4×C3⋊D3—41no—
40C32×C4×D3—42no—
41C3×#624—41no—
42C3×D3×Dic3—42no—
43C23×C4×D3—32no—
44C22×C4×A4—91no—
45C23×C3⋊D3C22×C311yescharacteristic
46C23×C3×D3C22×C312yes—
47C22×C3×A4—32no—
48C22×D32—31no—
49C22×D32—31no—
50C22×D32—32no—
51C22×D32—34no—
52C22×D32C3×C414yescomplemented
53C2×D3×A4—22no—
54C2×D3×A4—32no—
55C2×D3×A4—38no—
56C2×D3×A4—61no—
57C2×D3×A4C2×D312yes—
58C22×C32×C4C22×C311yescharacteristic, complemented
59C22×C3⋊Dic3C22×C311yescharacteristic, complemented
60C2×C4×C3⋊D3—31no—
61C2×C4×C3⋊D3—31no—
62C2×C4×C3⋊D3—31no—
63C2×C4×C3⋊D3—31no—
64C22×C3×Dic3C22×C312yescomplemented
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—32no—
67C2×C3×C4×D3—32no—
68C2×C3×C4×D3—32no—
69C2×C3×C4×D3—122no—
70C3×C4×A4—22no—
71C3×C4×A4—21no—
72C3×C4×A4—32no—
73C3×C4×A4—41no—
74C3×C4×A4C2×D312yescomplemented
75C2×#624—31no—
76C2×#624—31no—
77C2×#624—32no—
78C2×D3×Dic3—32no—
79C2×D3×Dic3—32no—
80C2×D3×Dic3—32no—
81C2×D3×Dic3—32no—
82C4×D32—32no—
83C4×D32—32no—
84C4×D32—34no—
85C4×D32—34no—
86C4×D32—32no—
87C4×D32—31no—
88C4×D32—31no—
89C4×D32—32no—
90C4×D32—31no—
91C4×D32—32no—
92C4×D32A411yescharacteristic, direct product factor
93Dic3×A4—22no—
94Dic3×A4—32no—
95Dic3×A4—61no—
96Dic3×A4C2×D312yescomplemented
97C2×C3×C3⋊D3—41no—
98C2×C32×D3—42no—
99C32×A4C22×C411yescharacteristic, complemented
100C3×D32—44no—
101C33×C4—41no—
102C3×C3⋊Dic3—41no—
103C32×Dic3—42no—
104C24×D3—32no—
105C23×A4—91no—
106C23×C3×C4—32no—
107C23×Dic3—32no—
108C22×C4×D3—32no—
109C22×C4×D3—32no—
110C22×C4×D3—32no—
111C22×C4×D3—61no—
112C22×C4×D3—92no—
113C22×C4×D3—92no—
114C22×C4×D3—92no—
115C22×C4×D3—92no—
116C22×C4×D3—92no—
117C22×C4×D3—92no—
118C22×C4×D3—92no—
119C22×C4×D3—92no—
120C22×C4×D3C3×D312yescomplemented
121C2×C4×A4—32no—
122C2×C4×A4—62no—
123C2×C4×A4—91no—
124C2×C4×A4—92no—
125C2×C4×A4—91no—
126C23×C32C23×C311yescharacteristic
127C22×C3⋊D3—31no—
128C22×C3⋊D3—32no—
129C22×C3⋊D3—31no—
130C22×C3⋊D3C2×C3×C412yescomplemented
131C22×C3×D3—32no—
132C22×C3×D3—34no—
133C22×C3×D3—32no—
134C22×C3×D3—122no—
135C22×C3×D3C2×C3×C414yescomplemented
136C2×C3×A4—22no—
137C2×C3×A4—21no—
138C2×C3×A4—34no—
139C2×C3×A4—41no—
140C2×C3×A4C22×D312yes—
141C2×D32—34no—
142C2×D32—38no—
143C2×D32—32no—
144C2×D32—31no—
145C2×D32—32no—
146C2×D32—34no—
147C2×D32—34no—
148C2×D32—34no—
149C2×D32—38no—
150C2×D32C2×A411yescharacteristic
151D3×A4—24no—
152D3×A4—34no—
153D3×A4—62no—
154D3×A4C4×D314yesdirect product factor
155C2×C32×C4—31no—
156C2×C32×C4—31no—
157C2×C32×C4—122no—
158C2×C3⋊Dic3—31no—
159C2×C3⋊Dic3—31no—
160C4×C3⋊D3—32no—
161C4×C3⋊D3—31no—
162C4×C3⋊D3—31no—
163C4×C3⋊D3—31no—
164C4×C3⋊D3C2×A411yescharacteristic, complemented
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—122no—
168C3×C4×D3—34no—
169C3×C4×D3—32no—
170C3×C4×D3—32no—
171C3×C4×D3—32no—
172C3×C4×D3—42no—
173C3×C4×D3—82no—
174C3×C4×D3—122no—
175C3×C4×D3—122no—
176C3×C4×D3—122no—
177C3×C4×D3—241no—
178C3×C4×D3C2×A412yescomplemented
179#624—32no—
180#624—31no—
181#624—32no—
182#624C2×A411yescharacteristic, complemented
183D3×Dic3—34no—
184D3×Dic3—32no—
185D3×Dic3—32no—
186D3×Dic3—32no—
187D3×Dic3C2×A412yescomplemented
188C24×C4—91noSylow
189C2×C33—41no—
190C3×C3⋊D3—42no—
191C32×D3—44no—
192C24×C3—32no—
193C23×D3—32no—
194C23×D3—38no—
195C23×D3—61no—
196C23×D3—92no—
197C23×D3—92no—
198C23×D3—92no—
199C23×D3—92no—
200C23×D3—98no—
201C23×D3C2×C3×D312yes—
202C22×A4—32no—
203C22×A4—62no—
204C22×A4—94no—
205C22×A4—91no—
206C22×C3×C4—21no—
207C22×C3×C4—32no—
208C22×C3×C4—92no—
209C22×C3×C4—92no—
210C22×C3×C4—92no—
211C22×C3×C4—92no—
212C22×C3×C4—361no—
213C22×C3×C4C2×C3×D312yescomplemented
214C22×Dic3—32no—
215C22×Dic3—61no—
216C22×Dic3—92no—
217C22×Dic3—92no—
218C22×Dic3—92no—
219C22×Dic3—92no—
220C22×Dic3C2×C3×D312yescomplemented
221C2×C4×D3—32no—
222C2×C4×D3—32no—
223C2×C4×D3—32no—
224C2×C4×D3—32no—
225C2×C4×D3—32no—
226C2×C4×D3—92no—
227C2×C4×D3—92no—
228C2×C4×D3—92no—
229C2×C4×D3—92no—
230C2×C4×D3—92no—
231C2×C4×D3—92no—
232C2×C4×D3—92no—
233C2×C4×D3—92no—
234C2×C4×D3—92no—
235C2×C4×D3—94no—
236C2×C4×D3—94no—
237C2×C4×D3—94no—
238C2×C4×D3—94no—
239C2×C4×D3—94no—
240C2×C4×D3—94no—
241C2×C4×D3—94no—
242C2×C4×D3—92no—
243C2×C4×D3—92no—
244C2×C4×D3—92no—
245C2×C4×D3—92no—
246C2×C4×D3—92no—
247C2×C4×D3—92no—
248C2×C4×D3—92no—
249C2×C4×D3—92no—
250C2×C4×D3—92no—
251C2×C4×D3—92no—
252C2×C4×D3—181no—
253C2×C4×D3—181no—
254C2×C4×D3—181no—
255C2×C4×D3—181no—
256C4×A4—22no—
257C4×A4—32no—
258C4×A4—41no—
259C4×A4—62no—
260C4×A4—91no—
261C4×A4D3211yescharacteristic, direct product factor
262C22×C32—31no—
263C22×C32—31no—
264C22×C32—122no—
265C22×C32C22×C3×C411yescharacteristic, complemented
266C2×C3⋊D3—32no—
267C2×C3⋊D3—31no—
268C2×C3⋊D3—32no—
269C2×C3⋊D3—32no—
270C2×C3⋊D3—32no—
271C2×C3⋊D3C22×A411yescharacteristic
272C2×C3×D3—34no—
273C2×C3×D3—32no—
274C2×C3×D3—34no—
275C2×C3×D3—34no—
276C2×C3×D3—34no—
277C2×C3×D3—42no—
278C2×C3×D3—82no—
279C2×C3×D3—128no—
280C2×C3×D3—122no—
281C2×C3×D3—241no—
282C2×C3×D3C22×A412yes—
283C3×A4—22no—
284C3×A4—21no—
285C3×A4—41no—
286C3×A4C2×C4×D312yescomplemented
287D32—38no—
288D32—34no—
289D32—38no—
290D32C4×A414yesdirect product factor
291C32×C4—31no—
292C32×C4—42no—
293C32×C4—81no—
294C32×C4—82no—
295C32×C4—122no—
296C32×C4—161no—
297C32×C4C22×A411yescharacteristic, complemented
298C3⋊Dic3—31no—
299C3⋊Dic3C22×A411yescharacteristic, complemented
300C3×Dic3—32no—
301C3×Dic3—42no—
302C3×Dic3—82no—
303C3×Dic3—122no—
304C3×Dic3—241no—
305C3×Dic3C22×A412yescomplemented
306C25—91no—
307C23×C4—32no—
308C23×C4—91no—
309C23×C4—92no—
310C23×C4—91no—
311C23×C4—271no—
312C23×C4—272no—
313C23×C4—271no—
314C23×C4—271no—
315C23×C4—272no—
316C23×C4—271no—
317C33—41noSylow
318C23×C3—21no—
319C23×C3—34no—
320C23×C3—92no—
321C23×C3—92no—
322C23×C3—94no—
323C23×C3—361no—
324C23×C3C22×C3×D312yes—
325C22×D3—32no—
326C22×D3—32no—
327C22×D3—32no—
328C22×D3—34no—
329C22×D3—34no—
330C22×D3—62no—
331C22×D3—92no—
332C22×D3—98no—
333C22×D3—98no—
334C22×D3—92no—
335C22×D3—92no—
336C22×D3—94no—
337C22×D3—98no—
338C22×D3—98no—
339C22×D3—98no—
340C22×D3—98no—
341C22×D3—98no—
342C22×D3—98no—
343C22×D3—92no—
344C22×D3—92no—
345C22×D3—94no—
346C22×D3—181no—
347C22×D3—181no—
348C22×D3—182no—
349C22×D3C3×C4×D314yescomplemented
350C2×A4—22no—
351C2×A4—34no—
352C2×A4—41no—
353C2×A4—64no—
354C2×A4—92no—
355C2×A4C2×D3211yescharacteristic
356C2×C3×C4—32no—
357C2×C3×C4—32no—
358C2×C3×C4—32no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—92no—
362C2×C3×C4—92no—
363C2×C3×C4—92no—
364C2×C3×C4—94no—
365C2×C3×C4—92no—
366C2×C3×C4—92no—
367C2×C3×C4—122no—
368C2×C3×C4—242no—
369C2×C3×C4—361no—
370C2×C3×C4—361no—
371C2×C3×C4—362no—
372C2×Dic3—32no—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—92no—
376C2×Dic3—92no—
377C2×Dic3—92no—
378C2×Dic3—94no—
379C2×Dic3—92no—
380C2×Dic3—92no—
381C2×Dic3—181no—
382C2×Dic3—181no—
383C4×D3—34no—
384C4×D3—32no—
385C4×D3—32no—
386C4×D3—32no—
387C4×D3—32no—
388C4×D3—32no—
389C4×D3—32no—
390C4×D3—61no—
391C4×D3—92no—
392C4×D3—92no—
393C4×D3—92no—
394C4×D3—92no—
395C4×D3—94no—
396C4×D3—92no—
397C4×D3—92no—
398C4×D3—94no—
399C4×D3—94no—
400C4×D3—92no—
401C4×D3—92no—
402C4×D3—92no—
403C4×D3—181no—
404C4×D3—181no—
405C4×D3—181no—
406C4×D3—182no—
407C4×D3D3×A412yesdirect product factor
408C2×C32—31no—
409C2×C32—31no—
410C2×C32—42no—
411C2×C32—81no—
412C2×C32—82no—
413C2×C32—124no—
414C2×C32—161no—
415C2×C32C23×A411yescharacteristic
416C3⋊D3—32no—
417C3⋊D3C2×C4×A412yescomplemented
418C3×D3—34no—
419C3×D3—44no—
420C3×D3—84no—
421C3×D3—124no—
422C3×D3—242no—
423C3×D3C2×C4×A414yescomplemented
424C24—32no—
425C24—94no—
426C24—91no—
427C24—271no—
428C24—272no—
429C24—271no—
430C24—274no—
431C22×C4—32no—
432C22×C4—92no—
433C22×C4—91no—
434C22×C4—92no—
435C22×C4—92no—
436C22×C4—92no—
437C22×C4—91no—
438C22×C4—271no—
439C22×C4—272no—
440C22×C4—271no—
441C22×C4—271no—
442C22×C4—271no—
443C22×C4—271no—
444C22×C4—271no—
445C22×C4—272no—
446C22×C4—271no—
447C22×C4—272no—
448C22×C4—271no—
449C22×C4—274no—
450C22×C4—274no—
451C22×C4—272no—
452C22×C4—272no—
453C22×C4—272no—
454C22×C4—272no—
455C22×C4—271no—
456C22×C4—272no—
457C22×C4—272no—
458C22×C4—272no—
459C22×C4C3×D3211yescharacteristic, complemented
460C22×C3—21no—
461C22×C3—32no—
462C22×C3—32no—
463C22×C3—32no—
464C22×C3—61no—
465C22×C3—61no—
466C22×C3—94no—
467C22×C3—94no—
468C22×C3—94no—
469C22×C3—94no—
470C22×C3—92no—
471C22×C3—122no—
472C22×C3—242no—
473C22×C3—361no—
474C22×C3—364no—
475C22×C3C2×C3×C4×D312yescomplemented
476C2×D3—34no—
477C2×D3—32no—
478C2×D3—32no—
479C2×D3—34no—
480C2×D3—34no—
481C2×D3—38no—
482C2×D3—34no—
483C2×D3—61no—
484C2×D3—98no—
485C2×D3—94no—
486C2×D3—98no—
487C2×D3—94no—
488C2×D3—916no—
489C2×D3—94no—
490C2×D3—92no—
491C2×D3—98no—
492C2×D3—94no—
493C2×D3—182no—
494C2×D3—182no—
495C2×D3—181no—
496C2×D3—182no—
497C2×D3—182no—
498C2×D3C2×D3×A412yes—
499A4—22no—
500A4—41no—
501A4C4×D3211yescharacteristic, direct product factor
502C3×C4—21no—
503C3×C4—32no—
504C3×C4—32no—
505C3×C4—41no—
506C3×C4—61no—
507C3×C4—82no—
508C3×C4—92no—
509C3×C4—122no—
510C3×C4—161no—
511C3×C4—242no—
512C3×C4—361no—
513C3×C4C2×D3×A412yescomplemented
514Dic3—32no—
515Dic3—32no—
516Dic3—61no—
517Dic3—92no—
518Dic3—181no—
519Dic3C2×D3×A412yescomplemented
520C32—42no—
521C32—81no—
522C32—82no—
523C32—161no—
524C32C22×C4×A411yescharacteristic, complemented
525C23—34no—
526C23—92no—
527C23—91no—
528C23—92no—
529C23—94no—
530C23—92no—
531C23—271no—
532C23—272no—
533C23—271no—
534C23—272no—
535C23—271no—
536C23—274no—
537C23—274no—
538C23—278no—
539C23—274no—
540C23—272no—
541C23—278no—
542C23—274no—
543C23C2×C3×D3211yescharacteristic
544C2×C4—31no—
545C2×C4—32no—
546C2×C4—31no—
547C2×C4—92no—
548C2×C4—91no—
549C2×C4—92no—
550C2×C4—92no—
551C2×C4—92no—
552C2×C4—94no—
553C2×C4—92no—
554C2×C4—91no—
555C2×C4—92no—
556C2×C4—272no—
557C2×C4—271no—
558C2×C4—271no—
559C2×C4—272no—
560C2×C4—272no—
561C2×C4—271no—
562C2×C4—272no—
563C2×C4—272no—
564C2×C4—271no—
565C2×C4—272no—
566C2×C4—274no—
567C2×C4—271no—
568C2×C4—271no—
569C2×C3—21no—
570C2×C3—32no—
571C2×C3—32no—
572C2×C3—34no—
573C2×C3—41no—
574C2×C3—61no—
575C2×C3—61no—
576C2×C3—82no—
577C2×C3—94no—
578C2×C3—124no—
579C2×C3—161no—
580C2×C3—244no—
581C2×C3—362no—
582C2×C3C22×D3×A412yes—
583D3—34no—
584D3—34no—
585D3—62no—
586D3—94no—
587D3—182no—
588D3C4×D3×A414yesdirect product factor
589C22—31no—
590C22—31no—
591C22—32no—
592C22—94no—
593C22—94no—
594C22—94no—
595C22—94no—
596C22—94no—
597C22—92no—
598C22—91no—
599C22—272no—
600C22—278no—
601C22—272no—
602C22—272no—
603C22—272no—
604C22—274no—
605C22—278no—
606C22—271no—
607C22C3×C4×D3211yescharacteristic, complemented
608C4—31no—
609C4—32no—
610C4—92no—
611C4—91no—
612C4—271no—
613C4D32×A411yescharacteristic, direct product factor
614C3—21no—
615C3—41no—
616C3—82no—
617C3—161no—
618C3C2×C4×D3×A412yescomplemented
619C2—31no—
620C2—31no—
621C2—34no—
622C2—94no—
623C2—92no—
624C2—272no—
625C2C2×D32×A411yescharacteristic
subgroupG/H
center, hypercenterC4D32×A4
commutator, hypocenterC22×C32C22×C3×C4
FrattiniC2C2×D32×A4
FittingC22×C32×C4C22×C3
socle, cosocleC23×C32C23×C3
prime radicalC2×D32×A4C2
LCSG ▷ C22×C32
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C4×A4×C3⋊D3C2semidirect
3. C3×C4×D3×A4C2semidirect
4. A4×#624C2semidirect
5. D3×Dic3×A4C2semidirect
6. C22×C4×D32C3semidirect
10. D32×A4C4direct (N × H)
11. C32×C4×A4C22semidirect
13. A4×C3⋊Dic3C22semidirect
14. C3×Dic3×A4C22semidirect
17. C22×C4×C3⋊D3C2×C3semidirect
18. C22×C3×C4×D3C2×C3semidirect
20. C22×#624C2×C3semidirect
21. C22×D3×Dic3C2×C3semidirect
34. C4×D3×A4D3direct (N × H)
37. A4×C3⋊D3C2×C4semidirect
38. C3×D3×A4C2×C4semidirect
52. C22×D32C3×C4semidirect
58. C22×C32×C4C22×C3semidirect
59. C22×C3⋊Dic3C22×C3semidirect
64. C22×C3×Dic3C22×C3semidirect
74. C3×C4×A4C2×D3semidirect
92. C4×D32A4direct (N × H)
96. Dic3×A4C2×D3semidirect
99. C32×A4C22×C4semidirect
120. C22×C4×D3C3×D3semidirect
130. C22×C3⋊D3C2×C3×C4semidirect
135. C22×C3×D3C2×C3×C4semidirect
154. D3×A4C4×D3direct (N × H)
164. C4×C3⋊D3C2×A4semidirect
178. C3×C4×D3C2×A4semidirect
182. #624C2×A4semidirect
187. D3×Dic3C2×A4semidirect
213. C22×C3×C4C2×C3×D3semidirect
220. C22×Dic3C2×C3×D3semidirect
261. C4×A4D32direct (N × H)
265. C22×C32C22×C3×C4semidirect
286. C3×A4C2×C4×D3semidirect
290. D32C4×A4direct (N × H)
297. C32×C4C22×A4semidirect
299. C3⋊Dic3C22×A4semidirect
305. C3×Dic3C22×A4semidirect
349. C22×D3C3×C4×D3semidirect
407. C4×D3D3×A4direct (N × H)
417. C3⋊D3C2×C4×A4semidirect
423. C3×D3C2×C4×A4semidirect
459. C22×C4C3×D32semidirect
475. C22×C3C2×C3×C4×D3semidirect
501. A4C4×D32direct (N × H)
513. C3×C4C2×D3×A4semidirect
519. Dic3C2×D3×A4semidirect
524. C32C22×C4×A4semidirect
588. D3C4×D3×A4direct (N × H)
607. C22C3×C4×D32semidirect
613. C4D32×A4direct (N × H)
618. C3C2×C4×D3×A4semidirect

Maximal subgroups

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

subgroupindex
C2×D32×A42
C4×A4×C3⋊D32
C3×C4×D3×A42
A4×#6242
D3×Dic3×A42
C22×C4×D323
C2×C4×D3×A43
C3×C4×D324