Finite Group Databasebeta

← A4×#898 #11923171

Subgroups — A4×#898 #11923171

Order 1152.

666 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
2A4×(C2×D3)⋊C4C211yesmaximal, characteristic
3A4×(C2×D3)⋊C4C211yesmaximal, characteristic
4A4×(C2×D3)⋊C4C211yesmaximal, characteristic
5A4×C4⋊Dic3C211yesmaximal, characteristic, complemented
6A4×Dic3⋊C4C211yesmaximal, characteristic, complemented
7C4×Dic3×A4C211yesmaximal, characteristic, complemented
8C22×#898C311yesmaximal, characteristic, complemented
9A4×#126—31nomaximal
10C22×D3×A4C2211yescharacteristic
11C2×C3×C4×A4C2211yescharacteristic
12C2×C3×C4×A4C2211yescharacteristic
13C2×C3×C4×A4C2211yescharacteristic
14C2×Dic3×A4C2211yescharacteristic
15C2×Dic3×A4C2211yescharacteristic
16C2×Dic3×A4C2211yescharacteristic
17C3×#898—41nomaximal
18C22×C3×C4⋊C4C2×C311yescharacteristic, complemented
19C22×(C2×D3)⋊C4C2×C311yescharacteristic
20C22×(C2×D3)⋊C4C2×C311yescharacteristic
21C22×(C2×D3)⋊C4C2×C311yescharacteristic
22C22×C4⋊Dic3C2×C311yescharacteristic, complemented
23C22×Dic3⋊C4C2×C311yescharacteristic, complemented
24C22×C4×Dic3C2×C311yescharacteristic, complemented
25C2×#898—31no—
26C2×#898—31no—
27C2×#898—31no—
28C2×#898—31no—
29C2×#898—31no—
30C2×#898—31no—
31C2×#898—31no—
32C2×#898—31no—
33A4×C4⋊C4—21no—
34A4×C4⋊C4—31no—
35A4×C4⋊C4—31no—
36A4×C4⋊C4D311yescharacteristic, complemented
37A4×C22⋊C4—31no—
38A4×C22⋊C4—31no—
39A4×C22⋊C4—31no—
40C42×A4—31no—
41C22×C3×A4C2311yescharacteristic
42C2×D3×A4—21no—
43C2×D3×A4—21no—
44C2×D3×A4—21no—
45C3×C4×A4—21no—
46C3×C4×A4—21no—
47C3×C4×A4—21no—
48Dic3×A4—21no—
49Dic3×A4—21no—
50Dic3×A4—21no—
51C32×C4⋊C4—41no—
52C3×(C2×D3)⋊C4—41no—
53C3×(C2×D3)⋊C4—41no—
54C3×(C2×D3)⋊C4—41no—
55C3×C4⋊Dic3—41no—
56C3×Dic3⋊C4—41no—
57C3×C4×Dic3—41no—
58C22×#126—31noSylow
59C24×D3C22×C311yescharacteristic
60C23×A4—31no—
61C23×C3×C4C22×C311yescharacteristic
62C23×C3×C4C22×C311yescharacteristic
63C23×C3×C4C22×C311yescharacteristic
64C23×Dic3C22×C311yescharacteristic
65C23×Dic3C22×C311yescharacteristic
66C23×Dic3C22×C311yescharacteristic
67C2×C4×A4—21no—
68C2×C4×A4—21no—
69C2×C4×A4—21no—
70C2×C4×A4—31no—
71C2×C4×A4—31no—
72C2×C4×A4—31no—
73C2×C4×A4C2×D311yescharacteristic
74C2×C4×A4C2×D311yescharacteristic
75C2×C4×A4C2×D311yescharacteristic
76C3×#126—121no—
77C2×C3×C4⋊C4—31no—
78C2×C3×C4⋊C4—31no—
79C2×C3×C4⋊C4—31no—
80C2×C3×C4⋊C4—31no—
81C2×(C2×D3)⋊C4—31no—
82C2×(C2×D3)⋊C4—31no—
83C2×(C2×D3)⋊C4—31no—
84C2×(C2×D3)⋊C4—31no—
85C2×(C2×D3)⋊C4—31no—
86C2×(C2×D3)⋊C4—31no—
87C2×(C2×D3)⋊C4—31no—
88C2×(C2×D3)⋊C4—31no—
89C2×(C2×D3)⋊C4—31no—
90C2×(C2×D3)⋊C4—31no—
91C2×(C2×D3)⋊C4—31no—
92C2×(C2×D3)⋊C4—31no—
93C2×C4⋊Dic3—31no—
94C2×C4⋊Dic3—31no—
95C2×C4⋊Dic3—31no—
96C2×C4⋊Dic3—31no—
97C2×Dic3⋊C4—31no—
98C2×Dic3⋊C4—31no—
99C2×Dic3⋊C4—31no—
100C2×Dic3⋊C4—31no—
101C2×C4×Dic3—31no—
102C2×C4×Dic3—31no—
103C2×C4×Dic3—31no—
104C2×C4×Dic3—31no—
105#898—32no—
106#898—32no—
107#898—32no—
108#898—32no—
109#898—32no—
110#898—32no—
111#898—32no—
112#898—31no—
113#898—31no—
114#898—31no—
115#898—31no—
116#898—31no—
117#898—31no—
118#898—31no—
119#898A411yescharacteristic, direct product factor
120C22×C3×D3—41no—
121C2×C3×A4#4111yescharacteristic
122C2×C3×A4#4111yescharacteristic
123C2×C3×A4#4111yescharacteristic
124D3×A4—41no—
125C2×C32×C4—41no—
126C2×C32×C4—41no—
127C2×C32×C4—41no—
128C2×C3×Dic3—41no—
129C2×C3×Dic3—41no—
130C2×C3×Dic3—41no—
131C2×#126—91no—
132C2×#126—91no—
133C2×#126—91no—
134C2×#126—91no—
135C2×#126—91no—
136C2×#126—91no—
137C2×#126—91no—
138C2×#126—91no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4—31no—
141C22×C4⋊C4C3×D311yescharacteristic, complemented
142C22×C22⋊C4—31no—
143C22×C22⋊C4—31no—
144C22×C22⋊C4—31no—
145C22×C42—31no—
146C24×C3C23×C311yescharacteristic
147C23×D3—21no—
148C23×D3—21no—
149C23×D3—21no—
150C23×D3—31no—
151C23×D3—31no—
152C23×D3—61no—
153C23×D3—61no—
154C23×D3—61no—
155C22×A4—21no—
156C22×A4—61no—
157C22×A4—61no—
158C22×A4—61no—
159C22×A4C22×D311yescharacteristic
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—31no—
164C22×C3×C4—31no—
165C22×C3×C4—31no—
166C22×C3×C4—31no—
167C22×C3×C4—31no—
168C22×C3×C4—31no—
169C22×C3×C4—61no—
170C22×C3×C4—61no—
171C22×C3×C4—61no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—31no—
176C22×Dic3—31no—
177C22×Dic3—31no—
178C22×Dic3—31no—
179C22×Dic3—31no—
180C22×Dic3—31no—
181C22×Dic3—61no—
182C22×Dic3—61no—
183C22×Dic3—61no—
184C4×A4—22no—
185C4×A4—21no—
186C4×A4—21no—
187C4×A4—21no—
188C4×A4—41no—
189C4×A4—41no—
190C4×A4—61no—
191C4×A4—61no—
192C4×A4—61no—
193C3×C4⋊C4—32no—
194C3×C4⋊C4—31no—
195C3×C4⋊C4—31no—
196C3×C4⋊C4—31no—
197C3×C4⋊C4—41no—
198C3×C4⋊C4—81no—
199C3×C4⋊C4—121no—
200C3×C4⋊C4—121no—
201C3×C4⋊C4C2×A411yescharacteristic, complemented
202C3×C22⋊C4—121no—
203C3×C22⋊C4—121no—
204C3×C22⋊C4—121no—
205C3×C42—121no—
206(C2×D3)⋊C4—32no—
207(C2×D3)⋊C4—32no—
208(C2×D3)⋊C4—32no—
209(C2×D3)⋊C4—31no—
210(C2×D3)⋊C4—31no—
211(C2×D3)⋊C4—31no—
212(C2×D3)⋊C4—31no—
213(C2×D3)⋊C4—31no—
214(C2×D3)⋊C4—31no—
215(C2×D3)⋊C4—31no—
216(C2×D3)⋊C4—31no—
217(C2×D3)⋊C4—31no—
218(C2×D3)⋊C4C2×A411yescharacteristic
219(C2×D3)⋊C4C2×A411yescharacteristic
220(C2×D3)⋊C4C2×A411yescharacteristic
221C4⋊Dic3—32no—
222C4⋊Dic3—31no—
223C4⋊Dic3—31no—
224C4⋊Dic3—31no—
225C4⋊Dic3C2×A411yescharacteristic, complemented
226Dic3⋊C4—32no—
227Dic3⋊C4—31no—
228Dic3⋊C4—31no—
229Dic3⋊C4—31no—
230Dic3⋊C4C2×A411yescharacteristic, complemented
231C4×Dic3—32no—
232C4×Dic3—31no—
233C4×Dic3—31no—
234C4×Dic3—31no—
235C4×Dic3C2×A411yescharacteristic, complemented
236C22×C32—41no—
237C2×C3×D3—81no—
238C2×C3×D3—81no—
239C2×C3×D3—81no—
240C3×A4#12611yescharacteristic, complemented
241C32×C4—81no—
242C32×C4—81no—
243C32×C4—81no—
244C3×Dic3—81no—
245C3×Dic3—81no—
246C3×Dic3—81no—
247C25—31no—
248C23×C4—31no—
249C23×C4—31no—
250C23×C4—31no—
251C23×C4C2×C3×D311yescharacteristic
252C23×C4C2×C3×D311yescharacteristic
253C23×C4C2×C3×D311yescharacteristic
254#126—31no—
255#126—91no—
256#126—91no—
257#126—91no—
258#126—91no—
259#126—91no—
260#126—91no—
261#126—92no—
262#126—92no—
263#126—92no—
264#126—92no—
265#126—92no—
266#126—92no—
267#126—92no—
268#126—91no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—91no—
274C2×C4⋊C4—91no—
275C2×C4⋊C4—91no—
276C2×C4⋊C4—91no—
277C2×C4⋊C4—91no—
278C2×C4⋊C4—91no—
279C2×C4⋊C4—91no—
280C2×C4⋊C4—91no—
281C2×C22⋊C4—91no—
282C2×C22⋊C4—91no—
283C2×C22⋊C4—91no—
284C2×C22⋊C4—91no—
285C2×C22⋊C4—91no—
286C2×C22⋊C4—91no—
287C2×C22⋊C4—91no—
288C2×C22⋊C4—91no—
289C2×C22⋊C4—91no—
290C2×C22⋊C4—91no—
291C2×C22⋊C4—91no—
292C2×C22⋊C4—91no—
293C2×C42—91no—
294C2×C42—91no—
295C2×C42—91no—
296C2×C42—91no—
297C23×C3—31no—
298C23×C3—31no—
299C23×C3—31no—
300C23×C3—31no—
301C23×C3—121no—
302C23×C3C3×#4111yescharacteristic
303C23×C3C3×#4111yescharacteristic
304C23×C3C3×#4111yescharacteristic
305C22×D3—31no—
306C22×D3—41no—
307C22×D3—61no—
308C22×D3—61no—
309C22×D3—61no—
310C22×D3—61no—
311C22×D3—61no—
312C22×D3—61no—
313C22×D3—61no—
314C22×D3—61no—
315C22×D3—61no—
316C22×D3—61no—
317C22×D3—61no—
318C22×D3—61no—
319C22×D3—122no—
320C22×D3—121no—
321C22×D3—121no—
322C22×D3—121no—
323C22×D3C22×A411yescharacteristic
324C2×A4—21no—
325C2×A4—21no—
326C2×A4—21no—
327C2×A4—121no—
328C2×A4#23511yescharacteristic
329C2×A4#23711yescharacteristic
330C2×A4#23911yescharacteristic
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—81no—
350C2×C3×C4—81no—
351C2×C3×C4—81no—
352C2×C3×C4—121no—
353C2×C3×C4—121no—
354C2×C3×C4—121no—
355C2×C3×C4C22×A411yescharacteristic
356C2×C3×C4C22×A411yescharacteristic
357C2×C3×C4C22×A411yescharacteristic
358C2×Dic3—31no—
359C2×Dic3—31no—
360C2×Dic3—31no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—61no—
371C2×Dic3—61no—
372C2×Dic3—61no—
373C2×Dic3C22×A411yescharacteristic
374C2×Dic3C22×A411yescharacteristic
375C2×Dic3C22×A411yescharacteristic
376C2×C32—41no—
377C2×C32—41no—
378C2×C32—41no—
379C3×D3—161no—
380C24—61no—
381C24—61no—
382C24—61no—
383C24—91no—
384C24—91no—
385C24—181no—
386C24—181no—
387C24—181no—
388C24C22×C3×D311yescharacteristic
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—91no—
405C22×C4—91no—
406C22×C4—91no—
407C22×C4—91no—
408C22×C4—91no—
409C22×C4—91no—
410C22×C4—181no—
411C22×C4—181no—
412C22×C4—181no—
413C4⋊C4—32no—
414C4⋊C4—31no—
415C4⋊C4—31no—
416C4⋊C4—31no—
417C4⋊C4—31no—
418C4⋊C4—31no—
419C4⋊C4—91no—
420C4⋊C4—91no—
421C4⋊C4—91no—
422C4⋊C4—91no—
423C4⋊C4—92no—
424C4⋊C4—92no—
425C4⋊C4—91no—
426C4⋊C4—91no—
427C4⋊C4D3×A411yescharacteristic, complemented
428C22⋊C4—31no—
429C22⋊C4—31no—
430C22⋊C4—31no—
431C22⋊C4—91no—
432C22⋊C4—91no—
433C22⋊C4—91no—
434C22⋊C4—91no—
435C22⋊C4—91no—
436C22⋊C4—91no—
437C22⋊C4—92no—
438C22⋊C4—92no—
439C22⋊C4—92no—
440C22⋊C4—91no—
441C22⋊C4—91no—
442C22⋊C4—91no—
443C42—31no—
444C42—91no—
445C42—91no—
446C42—92no—
447C42—91no—
448C22×C3—31no—
449C22×C3—31no—
450C22×C3—31no—
451C22×C3—32no—
452C22×C3—31no—
453C22×C3—31no—
454C22×C3—31no—
455C22×C3—31no—
456C22×C3—31no—
457C22×C3—31no—
458C22×C3—41no—
459C22×C3—81no—
460C22×C3—241no—
461C22×C3—241no—
462C22×C3—241no—
463C22×C3C3×#12611yescharacteristic, complemented
464C22×C3C23×A411yescharacteristic
465C2×D3—21no—
466C2×D3—21no—
467C2×D3—21no—
468C2×D3—61no—
469C2×D3—61no—
470C2×D3—61no—
471C2×D3—121no—
472C2×D3—121no—
473C2×D3—121no—
474C2×D3—121no—
475C2×D3—121no—
476C2×D3—121no—
477C2×D3—121no—
478C2×D3—121no—
479A4—21no—
480A4#89811yescharacteristic, direct product factor
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—61no—
485C3×C4—61no—
486C3×C4—61no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4—82no—
491C3×C4—161no—
492C3×C4—161no—
493C3×C4—241no—
494C3×C4—241no—
495C3×C4—241no—
496Dic3—21no—
497Dic3—21no—
498Dic3—21no—
499Dic3—61no—
500Dic3—61no—
501Dic3—61no—
502C32—41noSylow
503C23—31no—
504C23—31no—
505C23—31no—
506C23—31no—
507C23—31no—
508C23—91no—
509C23—121no—
510C23—181no—
511C23—181no—
512C23—181no—
513C23—181no—
514C23—181no—
515C23—181no—
516C23—181no—
517C23—181no—
518C23—181no—
519C23—181no—
520C23—181no—
521C23—181no—
522C23—361no—
523C23—361no—
524C23—361no—
525C23—362no—
526C23C3×#23511yescharacteristic
527C23C3×#23711yescharacteristic
528C23C3×#23911yescharacteristic
529C2×C4—31no—
530C2×C4—31no—
531C2×C4—31no—
532C2×C4—31no—
533C2×C4—31no—
534C2×C4—31no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—91no—
548C2×C4—91no—
549C2×C4—91no—
550C2×C4—181no—
551C2×C4—181no—
552C2×C4—181no—
553C2×C4—181no—
554C2×C4—181no—
555C2×C4—181no—
556C2×C4—181no—
557C2×C4—181no—
558C2×C4—181no—
559C2×C4—181no—
560C2×C4—181no—
561C2×C4—181no—
562C2×C4C2×D3×A411yescharacteristic
563C2×C4C2×D3×A411yescharacteristic
564C2×C4C2×D3×A411yescharacteristic
565C2×C3—31no—
566C2×C3—31no—
567C2×C3—31no—
568C2×C3—31no—
569C2×C3—41no—
570C2×C3—41no—
571C2×C3—41no—
572C2×C3—81no—
573C2×C3—81no—
574C2×C3—81no—
575C2×C3—481no—
576C2×C3A4×#4111yescharacteristic
577C2×C3A4×#4111yescharacteristic
578C2×C3A4×#4111yescharacteristic
579D3—41no—
580D3—121no—
581C22—31no—
582C22—31no—
583C22—31no—
584C22—32no—
585C22—31no—
586C22—31no—
587C22—31no—
588C22—31no—
589C22—31no—
590C22—31no—
591C22—61no—
592C22—61no—
593C22—61no—
594C22—181no—
595C22—181no—
596C22—181no—
597C22—361no—
598C22—361no—
599C22—361no—
600C22—361no—
601C22—361no—
602C22—361no—
603C22—361no—
604C22—361no—
605C22C3×#89811yescharacteristic, complemented
606C22C22×D3×A411yescharacteristic
607C4—21no—
608C4—21no—
609C4—21no—
610C4—61no—
611C4—61no—
612C4—61no—
613C4—61no—
614C4—61no—
615C4—61no—
616C4—181no—
617C4—181no—
618C4—181no—
619C3—41no—
620C3—81no—
621C3A4×#12611yescharacteristic, complemented
622C2—31no—
623C2—31no—
624C2—31no—
625C2—31no—
626C2—121no—
627C2—361no—
628C2A4×#23511yescharacteristic
629C2A4×#23711yescharacteristic
630C2A4×#23911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, socle, cosocleC24×C3C23×C3
FittingC22×C3×C4⋊C4C2×C3
prime radicalC22×D3×A4C22
hypercenterC4⋊C4D3×A4
hypocenterC22×C3C3×#126
LCSG ▷ C24×C3 ▷ C22×C3
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×A4×C4⋊C4C2semidirect
5. A4×C4⋊Dic3C2semidirect
6. A4×Dic3⋊C4C2semidirect
7. C4×Dic3×A4C2semidirect
8. C22×#898C3semidirect
18. C22×C3×C4⋊C4C2×C3semidirect
22. C22×C4⋊Dic3C2×C3semidirect
23. C22×Dic3⋊C4C2×C3semidirect
24. C22×C4×Dic3C2×C3semidirect
36. A4×C4⋊C4D3semidirect
119. #898A4direct (N × H)
141. C22×C4⋊C4C3×D3semidirect
201. C3×C4⋊C4C2×A4semidirect
225. C4⋊Dic3C2×A4semidirect
230. Dic3⋊C4C2×A4semidirect
235. C4×Dic3C2×A4semidirect
240. C3×A4#126semidirect
427. C4⋊C4D3×A4semidirect
463. C22×C3C3×#126semidirect
480. A4#898direct (N × H)
605. C22C3×#898semidirect
621. C3A4×#126semidirect

Maximal subgroups

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

subgroupindex
C3×A4×C4⋊C42
A4×(C2×D3)⋊C42
A4×C4⋊Dic32
A4×Dic3⋊C42
C4×Dic3×A42
C22×#8983
A4×#1263
C3×#8984