Finite Group Databasebeta

← A4×#924 #11923344

Subgroups — A4×#924 #11923344

Order 1152.

914 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C22×C3×C4×A4C211yesmaximal, characteristic
2C2×A4×Dic6C211yesmaximal, characteristic, complemented
3A4×#217C212yesmaximal
4A4×C4⋊Dic3C211yesmaximal, characteristic, complemented
5A4×Dic3⋊C4C212yesmaximal, complemented
6C22×#924C311yesmaximal, characteristic, complemented
7A4×C22⋊Q8—31nomaximal
8C23×C3×A4C2211yescharacteristic
9C2×C3×C4×A4—22no—
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3×C4×A4C2211yescharacteristic
12C2×Dic3×A4C2212yes—
13C2×Dic3×A4C2212yes—
14A4×Dic6—22no—
15C3×#924—41nomaximal
16C24×C3×C4C2×C311yescharacteristic
17C23×Dic6C2×C311yescharacteristic, complemented
18C2×Q8×A4—31no—
19C22×C4×A4—21no—
20C22×C4×A4D311yescharacteristic
21C22×#217C2×C312yes—
22C22×C4⋊Dic3C2×C311yescharacteristic, complemented
23C2×#924—31no—
24C2×#924—32no—
25C2×#924—31no—
26C2×#924—31no—
27C2×#924—32no—
28C2×#924—31no—
29C22×Dic3⋊C4C2×C312yescomplemented
30A4×C4⋊C4—32no—
31A4×C4⋊C4—31no—
32A4×C22⋊C4—32no—
33C22×C3×A4—21no—
34C22×C3×A4—21no—
35C22×C3×A4Q812yescomplemented
36C22×C3×A4C2311yescharacteristic
37C22×C32×C4—41no—
38C2×C3×Dic6—41no—
39C3×C4×A4—21no—
40C3×C4×A4D412yes—
41Dic3×A4—22no—
42Dic3×A4—22no—
43C3×#217—42no—
44C3×C4⋊Dic3—41no—
45C3×Dic3⋊C4—42no—
46C22×C22⋊Q8—31noSylow
47C25×C3C22×C311yescharacteristic
48C23×A4—21no—
49C23×A4C2×D311yescharacteristic
50C23×C3×C4—22no—
51C23×C3×C4—31no—
52C23×C3×C4—31no—
53C23×C3×C4—31no—
54C23×C3×C4—31no—
55C23×C3×C4—62no—
56C23×C3×C4C22×C311yescharacteristic
57C23×C3×C4C22×C311yescharacteristic
58C23×Dic3C22×C312yes—
59C23×Dic3C22×C312yes—
60C22×Dic6—22no—
61C22×Dic6—31no—
62C22×Dic6—31no—
63C22×Dic6—32no—
64C22×Dic6—62no—
65Q8×A4—62no—
66C2×C4×A4—24no—
67C2×C4×A4—22no—
68C2×C4×A4—21no—
69C2×C4×A4—21no—
70C2×C4×A4—32no—
71C2×C4×A4—32no—
72C2×C4×A4C2×D311yescharacteristic
73C2×C4×A4C2×D311yescharacteristic
74C3×C22⋊Q8—121no—
75C2×#217—32no—
76C2×#217—32no—
77C2×#217—32no—
78C2×#217—32no—
79C2×C4⋊Dic3—31no—
80C2×C4⋊Dic3—31no—
81C2×C4⋊Dic3—32no—
82#924—32no—
83#924—32no—
84#924—34no—
85#924—34no—
86#924—32no—
87#924—31no—
88#924—31no—
89#924—32no—
90#924—31no—
91#924—32no—
92#924A411yescharacteristic, direct product factor
93C2×Dic3⋊C4—32no—
94C2×Dic3⋊C4—32no—
95C2×Dic3⋊C4—32no—
96C2×Dic3⋊C4—32no—
97C23×C32—41no—
98C2×C3×A4—22no—
99C2×C3×A4C2×D411yescharacteristic
100C2×C3×A4C2×Q811yescharacteristic
101C2×C3×A4#4111yescharacteristic
102C2×C32×C4—41no—
103C2×C32×C4—41no—
104C2×C32×C4—82no—
105C2×C3×Dic3—42no—
106C2×C3×Dic3—42no—
107C3×Dic6—82no—
108C23×Q8—31no—
109C24×C4C3×D311yescharacteristic
110C2×C22⋊Q8—91no—
111C2×C22⋊Q8—92no—
112C2×C22⋊Q8—92no—
113C2×C22⋊Q8—91no—
114C2×C22⋊Q8—91no—
115C2×C22⋊Q8—91no—
116C22×C4⋊C4—32no—
117C22×C4⋊C4—31no—
118C22×C22⋊C4—32no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3—32no—
122C24×C3—31no—
123C24×C3—31no—
124C24×C3—61no—
125C24×C3—61no—
126C24×C3C3×Q812yescomplemented
127C24×C3C23×C311yescharacteristic
128C22×A4—22no—
129C22×A4—22no—
130C22×A4—21no—
131C22×A4—21no—
132C22×A4—22no—
133C22×A4—21no—
134C22×A4Dic612yescomplemented
135C22×A4C22×D311yescharacteristic
136C2×C3×Q8—121no—
137C22×C3×C4—21no—
138C22×C3×C4—31no—
139C22×C3×C4—31no—
140C22×C3×C4—32no—
141C22×C3×C4—32no—
142C22×C3×C4—31no—
143C22×C3×C4—31no—
144C22×C3×C4—31no—
145C22×C3×C4—31no—
146C22×C3×C4—31no—
147C22×C3×C4—41no—
148C22×C3×C4—62no—
149C22×C3×C4—62no—
150C22×C3×C4—62no—
151C22×C3×C4—62no—
152C22×C3×C4—62no—
153C22×C3×C4—62no—
154C22×C3×C4—62no—
155C22×C3×C4—62no—
156C22×C3×C4—61no—
157C22×C3×C4—81no—
158C22×C3×C4C3×D412yes—
159C22×C3×C4C2×A411yescharacteristic
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—32no—
163C22×Dic3—32no—
164C22×Dic3—32no—
165C22×Dic3—32no—
166C22×Dic3—62no—
167C22×Dic3—62no—
168C2×Dic6—32no—
169C2×Dic6—31no—
170C2×Dic6—32no—
171C2×Dic6—62no—
172C2×Dic6—62no—
173C2×Dic6—64no—
174C2×Dic6—64no—
175C2×Dic6—62no—
176C2×Dic6—62no—
177C2×Dic6C2×A411yescharacteristic, complemented
178C4×A4—22no—
179C4×A4—21no—
180C4×A4—22no—
181C4×A4—62no—
182C4×A4—62no—
183C4×A4#6712yes—
184C3×C4⋊C4—122no—
185C3×C4⋊C4—121no—
186C3×C22⋊C4—122no—
187#217—34no—
188#217—32no—
189#217—32no—
190#217—32no—
191#217C2×A412yes—
192C4⋊Dic3—32no—
193C4⋊Dic3—31no—
194C4⋊Dic3—32no—
195C4⋊Dic3C2×A411yescharacteristic, complemented
196Dic3⋊C4—34no—
197Dic3⋊C4—32no—
198Dic3⋊C4—32no—
199Dic3⋊C4—32no—
200Dic3⋊C4C2×A412yescomplemented
201C22×C32—42no—
202C22×C32—41no—
203C22×C32—81no—
204C22×C32—81no—
205C3×A4C22⋊Q811yescharacteristic, complemented
206C32×C4—42no—
207C32×C4—81no—
208C3×Dic3—82no—
209C3×Dic3—82no—
210C25C2×C3×D311yescharacteristic
211C22×Q8—62no—
212C22×Q8—91no—
213C22×Q8—91no—
214C22×Q8—92no—
215C22×Q8—182no—
216C23×C4—22no—
217C23×C4—31no—
218C23×C4—31no—
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—32no—
222C23×C4—32no—
223C23×C4—62no—
224C23×C4C2×C3×D311yescharacteristic
225C23×C4C2×C3×D311yescharacteristic
226C22⋊Q8—31no—
227C22⋊Q8—92no—
228C22⋊Q8—92no—
229C22⋊Q8—91no—
230C22⋊Q8—91no—
231C22⋊Q8—94no—
232C22⋊Q8—94no—
233C22⋊Q8—92no—
234C22⋊Q8—92no—
235C22⋊Q8—92no—
236C22⋊Q8—91no—
237C2×C4⋊C4—92no—
238C2×C4⋊C4—91no—
239C2×C4⋊C4—92no—
240C2×C4⋊C4—92no—
241C2×C4⋊C4—92no—
242C2×C4⋊C4—92no—
243C2×C4⋊C4—91no—
244C2×C22⋊C4—92no—
245C2×C22⋊C4—92no—
246C2×C22⋊C4—92no—
247C2×C22⋊C4—92no—
248C23×C3—22no—
249C23×C3—31no—
250C23×C3—31no—
251C23×C3—31no—
252C23×C3—31no—
253C23×C3—32no—
254C23×C3—32no—
255C23×C3—32no—
256C23×C3—32no—
257C23×C3—31no—
258C23×C3—41no—
259C23×C3—61no—
260C23×C3—61no—
261C23×C3—61no—
262C23×C3—61no—
263C23×C3—62no—
264C23×C3—61no—
265C23×C3—61no—
266C23×C3—62no—
267C23×C3—62no—
268C23×C3—61no—
269C23×C3—61no—
270C23×C3—64no—
271C23×C3—81no—
272C23×C3C2×C3×D411yescharacteristic
273C23×C3C2×C3×Q811yescharacteristic
274C23×C3C3×#4111yescharacteristic
275C23×C3C22×A411yescharacteristic
276C2×A4—24no—
277C2×A4—22no—
278C2×A4—21no—
279C2×A4—21no—
280C2×A4—21no—
281C2×A4C2×Dic611yescharacteristic
282C2×A4#23511yescharacteristic
283C2×A4C2×#6711yescharacteristic
284C3×Q8—242no—
285C2×C3×C4—22no—
286C2×C3×C4—32no—
287C2×C3×C4—31no—
288C2×C3×C4—32no—
289C2×C3×C4—32no—
290C2×C3×C4—32no—
291C2×C3×C4—31no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—62no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—62no—
298C2×C3×C4—64no—
299C2×C3×C4—62no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—82no—
303C2×C3×C4—84no—
304C2×C3×C4—81no—
305C2×C3×C4—81no—
306C2×C3×C4—122no—
307C2×C3×C4—122no—
308C2×C3×C4C22×A411yescharacteristic
309C2×C3×C4C22×A411yescharacteristic
310C2×Dic3—32no—
311C2×Dic3—32no—
312C2×Dic3—62no—
313C2×Dic3—62no—
314C2×Dic3—62no—
315C2×Dic3—62no—
316C2×Dic3—62no—
317C2×Dic3—62no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C2×Dic3C22×A412yes—
321C2×Dic3C22×A412yes—
322Dic6—22no—
323Dic6—64no—
324Dic6—64no—
325Dic6—62no—
326C2×C32—41no—
327C2×C32—41no—
328C2×C32—41no—
329C2×C32—82no—
330C24—21no—
331C24—21no—
332C24—31no—
333C24—32no—
334C24—31no—
335C24—61no—
336C24—61no—
337C24C3×Dic612yescomplemented
338C24C22×C3×D311yescharacteristic
339C2×Q8—31no—
340C2×Q8—92no—
341C2×Q8—92no—
342C2×Q8—91no—
343C2×Q8—182no—
344C2×Q8—182no—
345C2×Q8—182no—
346C2×Q8—184no—
347C2×Q8—182no—
348C2×Q8—184no—
349C22×C4—21no—
350C22×C4—31no—
351C22×C4—31no—
352C22×C4—32no—
353C22×C4—32no—
354C22×C4—31no—
355C22×C4—31no—
356C22×C4—31no—
357C22×C4—31no—
358C22×C4—31no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—61no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—92no—
371C22×C4—92no—
372C22×C4—92no—
373C22×C4—92no—
374C22×C4—182no—
375C22×C4—182no—
376C22×C4C3×#6712yes—
377C22×C4D3×A411yescharacteristic
378C4⋊C4—32no—
379C4⋊C4—31no—
380C4⋊C4—92no—
381C4⋊C4—92no—
382C4⋊C4—92no—
383C4⋊C4—94no—
384C4⋊C4—92no—
385C4⋊C4—92no—
386C4⋊C4—91no—
387C22⋊C4—32no—
388C22⋊C4—92no—
389C22⋊C4—92no—
390C22⋊C4—94no—
391C22⋊C4—92no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—31no—
395C22×C3—31no—
396C22×C3—31no—
397C22×C3—32no—
398C22×C3—31no—
399C22×C3—31no—
400C22×C3—31no—
401C22×C3—31no—
402C22×C3—31no—
403C22×C3—31no—
404C22×C3—32no—
405C22×C3—42no—
406C22×C3—41no—
407C22×C3—62no—
408C22×C3—62no—
409C22×C3—62no—
410C22×C3—62no—
411C22×C3—62no—
412C22×C3—62no—
413C22×C3—62no—
414C22×C3—62no—
415C22×C3—61no—
416C22×C3—61no—
417C22×C3—81no—
418C22×C3—81no—
419C22×C3—82no—
420C22×C3—82no—
421C22×C3—82no—
422C22×C3—81no—
423C22×C3C3×C22⋊Q811yescharacteristic, complemented
424C22×C3Q8×A412yescomplemented
425C22×C3C23×A411yescharacteristic
426A4—21no—
427A4#92411yescharacteristic, direct product factor
428C3×C4—21no—
429C3×C4—32no—
430C3×C4—42no—
431C3×C4—61no—
432C3×C4—81no—
433C3×C4—82no—
434C3×C4—82no—
435C3×C4—242no—
436C3×C4—242no—
437C3×C4D4×A412yes—
438Dic3—22no—
439Dic3—22no—
440Dic3—62no—
441Dic3—62no—
442C32—41noSylow
443C23—22no—
444C23—31no—
445C23—31no—
446C23—31no—
447C23—31no—
448C23—32no—
449C23—32no—
450C23—32no—
451C23—32no—
452C23—31no—
453C23—61no—
454C23—61no—
455C23—61no—
456C23—61no—
457C23—62no—
458C23—61no—
459C23—61no—
460C23—62no—
461C23—62no—
462C23—61no—
463C23—61no—
464C23—64no—
465C23C2×C3×Dic611yescharacteristic
466C23C3×#23511yescharacteristic
467C23C2×C3×#6711yescharacteristic
468C23C2×D3×A411yescharacteristic
469Q8—62no—
470Q8—184no—
471Q8—184no—
472Q8—182no—
473C2×C4—22no—
474C2×C4—32no—
475C2×C4—31no—
476C2×C4—32no—
477C2×C4—32no—
478C2×C4—32no—
479C2×C4—32no—
480C2×C4—32no—
481C2×C4—31no—
482C2×C4—62no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—62no—
486C2×C4—64no—
487C2×C4—62no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—92no—
491C2×C4—92no—
492C2×C4—182no—
493C2×C4—182no—
494C2×C4—182no—
495C2×C4—182no—
496C2×C4—182no—
497C2×C4—182no—
498C2×C4—182no—
499C2×C4—182no—
500C2×C4C2×D3×A411yescharacteristic
501C2×C4C2×D3×A411yescharacteristic
502C2×C3—22no—
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—31no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3—41no—
510C2×C3—62no—
511C2×C3—82no—
512C2×C3—84no—
513C2×C3—81no—
514C2×C3—81no—
515C2×C3—81no—
516C2×C3C2×D4×A411yescharacteristic
517C2×C3C2×Q8×A411yescharacteristic
518C2×C3A4×#4111yescharacteristic
519C22—21no—
520C22—21no—
521C22—31no—
522C22—31no—
523C22—31no—
524C22—32no—
525C22—31no—
526C22—31no—
527C22—31no—
528C22—31no—
529C22—31no—
530C22—31no—
531C22—32no—
532C22—62no—
533C22—62no—
534C22—62no—
535C22—62no—
536C22—62no—
537C22—62no—
538C22—62no—
539C22—62no—
540C22—61no—
541C22—61no—
542C22C3×#92411yescharacteristic, complemented
543C22A4×Dic612yescomplemented
544C22C22×D3×A411yescharacteristic
545C4—21no—
546C4—32no—
547C4—62no—
548C4—61no—
549C4—62no—
550C4—182no—
551C4—182no—
552C4A4×#6712yes—
553C3—41no—
554C3—81no—
555C3A4×C22⋊Q811yescharacteristic, complemented
556C2—22no—
557C2—31no—
558C2—31no—
559C2—31no—
560C2—31no—
561C2—62no—
562C2C2×A4×Dic611yescharacteristic
563C2A4×#23511yescharacteristic
564C2C2×A4×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, socle, cosocleC24×C3C23×C3
FittingC24×C3×C4C2×C3
prime radicalC23×C3×A4C22
hypercenterC22×C4D3×A4
hypocenterC22×C3C3×C22⋊Q8
LCSG ▷ C24×C3 ▷ C22×C3
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×A4×Dic6C2semidirect
4. A4×C4⋊Dic3C2semidirect
5. A4×Dic3⋊C4C2semidirect
6. C22×#924C3semidirect
17. C23×Dic6C2×C3semidirect
22. C22×C4⋊Dic3C2×C3semidirect
29. C22×Dic3⋊C4C2×C3semidirect
35. C22×C3×A4Q8semidirect
92. #924A4direct (N × H)
126. C24×C3C3×Q8semidirect
134. C22×A4Dic6semidirect
177. C2×Dic6C2×A4semidirect
195. C4⋊Dic3C2×A4semidirect
200. Dic3⋊C4C2×A4semidirect
205. C3×A4C22⋊Q8semidirect
337. C24C3×Dic6semidirect
423. C22×C3C3×C22⋊Q8semidirect
424. C22×C3Q8×A4semidirect
427. A4#924direct (N × H)
542. C22C3×#924semidirect
543. C22A4×Dic6semidirect
555. C3A4×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×A42
C2×A4×Dic62
A4×#2172
A4×C4⋊Dic32
A4×Dic3⋊C42
C22×#9243
A4×C22⋊Q83
C3×#9244