Finite Group Databasebeta

← A4×#897 #11923170

Subgroups — A4×#897 #11923170

Order 1152.

886 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 886 conjugacy classes of subgroups — too many to draw. Only its 105 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×C4×D3×A4C211yesmaximal, characteristic
2C2×A4×Dic6C211yesmaximal, characteristic, complemented
3C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4A4×(C2×D3)⋊C4C212yesmaximal
5A4×C4⋊Dic3C212yesmaximal, complemented
6C22×#897C311yesmaximal, characteristic, complemented
7A4×C22⋊Q8—31nomaximal
8C22×D3×A4C2211yescharacteristic
9C2×C3×C4×A4C2212yes—
10C2×C3×C4×A4C2211yescharacteristic
11C2×Dic3×A4C2211yescharacteristic
12C2×Dic3×A4C2212yes—
13C4×D3×A4—22no—
14A4×Dic6—22no—
15C3×#897—41nomaximal
16C23×C4×D3C2×C311yescharacteristic
17C23×Dic6C2×C311yescharacteristic, complemented
18C2×Q8×A4—31no—
19C22×C4×A4—31no—
20C22×C3×C4⋊C4C2×C311yescharacteristic, complemented
21C22×(C2×D3)⋊C4C2×C312yes—
22C22×C4⋊Dic3C2×C312yescomplemented
23C2×#897—31no—
24C2×#897—31no—
25C2×#897—31no—
26C2×#897—31no—
27C2×#897—32no—
28C2×#897—32no—
29A4×C4⋊C4—21no—
30A4×C4⋊C4—32no—
31A4×C4⋊C4D311yescharacteristic, complemented
32A4×C22⋊C4—32no—
33C22×C3×A4C2311yescharacteristic
34C2×D3×A4—21no—
35C2×D3×A4—21no—
36C2×D3×A4Q812yescomplemented
37C2×C3×C4×D3—41no—
38C2×C3×Dic6—41no—
39C3×C4×A4—22no—
40C3×C4×A4D412yes—
41Dic3×A4—22no—
42Dic3×A4—21no—
43C32×C4⋊C4—41no—
44C3×(C2×D3)⋊C4—42no—
45C3×C4⋊Dic3—42no—
46C22×C22⋊Q8—31noSylow
47C24×D3C22×C311yescharacteristic
48C23×A4—31no—
49C23×C3×C4C22×C312yes—
50C23×C3×C4C22×C311yescharacteristic
51C23×Dic3C22×C312yes—
52C23×Dic3C22×C311yescharacteristic
53C22×C4×D3—22no—
54C22×C4×D3—31no—
55C22×C4×D3—31no—
56C22×C4×D3—31no—
57C22×C4×D3—31no—
58C22×C4×D3—62no—
59C22×Dic6—22no—
60C22×Dic6—31no—
61C22×Dic6—31no—
62C22×Dic6—32no—
63C22×Dic6—62no—
64Q8×A4—62no—
65C2×C4×A4—21no—
66C2×C4×A4—22no—
67C2×C4×A4—31no—
68C2×C4×A4—32no—
69C2×C4×A4—62no—
70C2×C4×A4C2×D312yes—
71C2×C4×A4C2×D311yescharacteristic
72C3×C22⋊Q8—121no—
73C2×C3×C4⋊C4—31no—
74C2×C3×C4⋊C4—31no—
75C2×C3×C4⋊C4—32no—
76C2×(C2×D3)⋊C4—32no—
77C2×(C2×D3)⋊C4—32no—
78C2×(C2×D3)⋊C4—32no—
79C2×(C2×D3)⋊C4—32no—
80C2×C4⋊Dic3—32no—
81C2×C4⋊Dic3—32no—
82C2×C4⋊Dic3—32no—
83C2×C4⋊Dic3—32no—
84#897—32no—
85#897—32no—
86#897—34no—
87#897—34no—
88#897—32no—
89#897—31no—
90#897—31no—
91#897—31no—
92#897—32no—
93#897—32no—
94#897A411yescharacteristic, direct product factor
95C22×C3×D3—41no—
96C2×C3×A4C2×D411yescharacteristic
97C2×C3×A4C2×Q811yescharacteristic
98C2×C3×A4#4111yescharacteristic
99D3×A4—22no—
100C2×C32×C4—41no—
101C2×C32×C4—42no—
102C2×C3×Dic3—41no—
103C2×C3×Dic3—42no—
104C3×C4×D3—82no—
105C3×Dic6—82no—
106C23×Q8—31no—
107C24×C4—31no—
108C2×C22⋊Q8—91no—
109C2×C22⋊Q8—92no—
110C2×C22⋊Q8—91no—
111C2×C22⋊Q8—92no—
112C2×C22⋊Q8—91no—
113C2×C22⋊Q8—91no—
114C22×C4⋊C4—32no—
115C22×C4⋊C4C3×D311yescharacteristic, complemented
116C22×C22⋊C4—32no—
117C24×C3C23×C311yescharacteristic
118C23×D3—21no—
119C23×D3—21no—
120C23×D3—31no—
121C23×D3—31no—
122C23×D3—32no—
123C23×D3—61no—
124C23×D3—61no—
125C23×D3C3×Q812yescomplemented
126C22×A4—21no—
127C22×A4—32no—
128C22×A4—61no—
129C22×A4—61no—
130C22×A4C22×D311yescharacteristic
131C2×C3×Q8—121no—
132C22×C3×C4—22no—
133C22×C3×C4—31no—
134C22×C3×C4—31no—
135C22×C3×C4—32no—
136C22×C3×C4—32no—
137C22×C3×C4—32no—
138C22×C3×C4—62no—
139C22×C3×C4—121no—
140C22×C3×C4C3×D412yes—
141C22×Dic3—22no—
142C22×Dic3—21no—
143C22×Dic3—31no—
144C22×Dic3—32no—
145C22×Dic3—31no—
146C22×Dic3—32no—
147C22×Dic3—62no—
148C22×Dic3—61no—
149C2×C4×D3—32no—
150C2×C4×D3—31no—
151C2×C4×D3—31no—
152C2×C4×D3—31no—
153C2×C4×D3—62no—
154C2×C4×D3—62no—
155C2×C4×D3—62no—
156C2×C4×D3—62no—
157C2×C4×D3—62no—
158C2×C4×D3—62no—
159C2×C4×D3—62no—
160C2×C4×D3—62no—
161C2×C4×D3C2×A411yescharacteristic
162C2×Dic6—32no—
163C2×Dic6—31no—
164C2×Dic6—32no—
165C2×Dic6—62no—
166C2×Dic6—62no—
167C2×Dic6—64no—
168C2×Dic6—64no—
169C2×Dic6—62no—
170C2×Dic6—62no—
171C2×Dic6C2×A411yescharacteristic, complemented
172C4×A4—22no—
173C4×A4—22no—
174C4×A4—42no—
175C4×A4—61no—
176C4×A4—62no—
177C4×A4D1212yes—
178C3×C4⋊C4—32no—
179C3×C4⋊C4—32no—
180C3×C4⋊C4—31no—
181C3×C4⋊C4—41no—
182C3×C4⋊C4—81no—
183C3×C4⋊C4—122no—
184C3×C4⋊C4C2×A411yescharacteristic, complemented
185C3×C22⋊C4—122no—
186(C2×D3)⋊C4—34no—
187(C2×D3)⋊C4—32no—
188(C2×D3)⋊C4—32no—
189(C2×D3)⋊C4—32no—
190(C2×D3)⋊C4C2×A412yes—
191C4⋊Dic3—34no—
192C4⋊Dic3—32no—
193C4⋊Dic3—32no—
194C4⋊Dic3—32no—
195C4⋊Dic3C2×A412yescomplemented
196C22×C32—41no—
197C2×C3×D3—42no—
198C2×C3×D3—81no—
199C2×C3×D3—81no—
200C3×A4C22⋊Q811yescharacteristic, complemented
201C32×C4—42no—
202C32×C4—82no—
203C3×Dic3—82no—
204C3×Dic3—81no—
205C25—31no—
206C22×Q8—62no—
207C22×Q8—91no—
208C22×Q8—91no—
209C22×Q8—92no—
210C22×Q8—182no—
211C23×C4—32no—
212C23×C4—31no—
213C23×C4—62no—
214C23×C4—91no—
215C23×C4—91no—
216C23×C4—91no—
217C23×C4—91no—
218C23×C4—182no—
219C23×C4C2×C3×D312yes—
220C23×C4C2×C3×D311yescharacteristic
221C22⋊Q8—31no—
222C22⋊Q8—92no—
223C22⋊Q8—92no—
224C22⋊Q8—91no—
225C22⋊Q8—91no—
226C22⋊Q8—94no—
227C22⋊Q8—92no—
228C22⋊Q8—92no—
229C22⋊Q8—92no—
230C22⋊Q8—94no—
231C22⋊Q8—91no—
232C2×C4⋊C4—31no—
233C2×C4⋊C4—31no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—92no—
236C2×C4⋊C4—92no—
237C2×C4⋊C4—92no—
238C2×C4⋊C4—92no—
239C2×C22⋊C4—92no—
240C2×C22⋊C4—92no—
241C2×C22⋊C4—92no—
242C2×C22⋊C4—92no—
243C23×C3—31no—
244C23×C3—31no—
245C23×C3—31no—
246C23×C3—31no—
247C23×C3—121no—
248C23×C3C2×C3×D411yescharacteristic
249C23×C3C2×C3×Q811yescharacteristic
250C23×C3C3×#4111yescharacteristic
251C22×D3—22no—
252C22×D3—32no—
253C22×D3—31no—
254C22×D3—32no—
255C22×D3—32no—
256C22×D3—32no—
257C22×D3—61no—
258C22×D3—61no—
259C22×D3—61no—
260C22×D3—61no—
261C22×D3—64no—
262C22×D3—61no—
263C22×D3—61no—
264C22×D3—62no—
265C22×D3—62no—
266C22×D3—61no—
267C22×D3—61no—
268C22×D3—62no—
269C22×D3C22×A411yescharacteristic
270C2×A4—21no—
271C2×A4—21no—
272C2×A4—21no—
273C2×A4—62no—
274C2×A4C2×D1211yescharacteristic
275C2×A4#23711yescharacteristic
276C2×A4D3×Q811yescharacteristic
277C3×Q8—242no—
278C2×C3×C4—32no—
279C2×C3×C4—31no—
280C2×C3×C4—32no—
281C2×C3×C4—32no—
282C2×C3×C4—32no—
283C2×C3×C4—32no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—62no—
287C2×C3×C4—62no—
288C2×C3×C4—62no—
289C2×C3×C4—62no—
290C2×C3×C4—81no—
291C2×C3×C4—82no—
292C2×C3×C4—122no—
293C2×C3×C4—121no—
294C2×C3×C4—242no—
295C2×C3×C4C22×A412yes—
296C2×C3×C4C22×A411yescharacteristic
297C2×Dic3—31no—
298C2×Dic3—32no—
299C2×Dic3—62no—
300C2×Dic3—62no—
301C2×Dic3—61no—
302C2×Dic3—61no—
303C2×Dic3—62no—
304C2×Dic3—62no—
305C2×Dic3—61no—
306C2×Dic3—61no—
307C2×Dic3C22×A411yescharacteristic
308C2×Dic3C22×A412yes—
309C4×D3—22no—
310C4×D3—62no—
311C4×D3—62no—
312C4×D3—64no—
313C4×D3—62no—
314Dic6—22no—
315Dic6—64no—
316Dic6—64no—
317Dic6—62no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—41no—
321C3×D3—82no—
322C24—32no—
323C24—61no—
324C24—61no—
325C24—91no—
326C24—91no—
327C24—92no—
328C24—181no—
329C24—181no—
330C24C22×C3×D311yescharacteristic
331C2×Q8—31no—
332C2×Q8—92no—
333C2×Q8—92no—
334C2×Q8—91no—
335C2×Q8—182no—
336C2×Q8—182no—
337C2×Q8—182no—
338C2×Q8—184no—
339C2×Q8—182no—
340C2×Q8—184no—
341C22×C4—22no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—32no—
345C22×C4—32no—
346C22×C4—31no—
347C22×C4—32no—
348C22×C4—62no—
349C22×C4—61no—
350C22×C4—62no—
351C22×C4—91no—
352C22×C4—92no—
353C22×C4—91no—
354C22×C4—91no—
355C22×C4—92no—
356C22×C4—91no—
357C22×C4—92no—
358C22×C4—91no—
359C22×C4—182no—
360C22×C4—182no—
361C22×C4—182no—
362C22×C4—182no—
363C22×C4—182no—
364C22×C4—182no—
365C22×C4—182no—
366C22×C4—182no—
367C22×C4—181no—
368C22×C4—182no—
369C22×C4C3×D1212yes—
370C4⋊C4—32no—
371C4⋊C4—32no—
372C4⋊C4—31no—
373C4⋊C4—32no—
374C4⋊C4—92no—
375C4⋊C4—92no—
376C4⋊C4—94no—
377C4⋊C4—92no—
378C4⋊C4D3×A411yescharacteristic, complemented
379C22⋊C4—32no—
380C22⋊C4—92no—
381C22⋊C4—92no—
382C22⋊C4—94no—
383C22⋊C4—92no—
384C22×C3—31no—
385C22×C3—31no—
386C22×C3—31no—
387C22×C3—32no—
388C22×C3—31no—
389C22×C3—31no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—41no—
395C22×C3—81no—
396C22×C3—122no—
397C22×C3—241no—
398C22×C3—241no—
399C22×C3C3×C22⋊Q811yescharacteristic, complemented
400C22×C3C23×A411yescharacteristic
401C2×D3—21no—
402C2×D3—21no—
403C2×D3—32no—
404C2×D3—62no—
405C2×D3—62no—
406C2×D3—62no—
407C2×D3—62no—
408C2×D3—62no—
409C2×D3—62no—
410C2×D3—61no—
411C2×D3—61no—
412C2×D3—62no—
413C2×D3—62no—
414C2×D3Q8×A412yescomplemented
415A4—21no—
416A4#89711yescharacteristic, direct product factor
417C3×C4—22no—
418C3×C4—32no—
419C3×C4—42no—
420C3×C4—62no—
421C3×C4—82no—
422C3×C4—82no—
423C3×C4—162no—
424C3×C4—241no—
425C3×C4—242no—
426C3×C4D4×A412yes—
427Dic3—22no—
428Dic3—21no—
429Dic3—62no—
430Dic3—61no—
431C32—41noSylow
432C23—31no—
433C23—31no—
434C23—31no—
435C23—31no—
436C23—31no—
437C23—62no—
438C23—92no—
439C23—92no—
440C23—92no—
441C23—92no—
442C23—91no—
443C23—181no—
444C23—181no—
445C23—181no—
446C23—181no—
447C23—182no—
448C23—181no—
449C23—181no—
450C23—182no—
451C23—182no—
452C23—181no—
453C23—181no—
454C23—184no—
455C23C2×C3×D1211yescharacteristic
456C23C3×#23711yescharacteristic
457C23C3×D3×Q811yescharacteristic
458Q8—62no—
459Q8—184no—
460Q8—184no—
461Q8—182no—
462C2×C4—32no—
463C2×C4—31no—
464C2×C4—32no—
465C2×C4—32no—
466C2×C4—32no—
467C2×C4—32no—
468C2×C4—32no—
469C2×C4—31no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—92no—
476C2×C4—91no—
477C2×C4—182no—
478C2×C4—182no—
479C2×C4—184no—
480C2×C4—181no—
481C2×C4—182no—
482C2×C4—181no—
483C2×C4—182no—
484C2×C4—182no—
485C2×C4—181no—
486C2×C4—182no—
487C2×C4—181no—
488C2×C4—182no—
489C2×C4C2×D3×A412yes—
490C2×C4C2×D3×A411yescharacteristic
491C2×C3—31no—
492C2×C3—31no—
493C2×C3—31no—
494C2×C3—31no—
495C2×C3—41no—
496C2×C3—41no—
497C2×C3—41no—
498C2×C3—81no—
499C2×C3—81no—
500C2×C3—81no—
501C2×C3—242no—
502C2×C3C2×D4×A411yescharacteristic
503C2×C3C2×Q8×A411yescharacteristic
504C2×C3A4×#4111yescharacteristic
505D3—22no—
506D3—62no—
507C22—31no—
508C22—31no—
509C22—31no—
510C22—32no—
511C22—31no—
512C22—31no—
513C22—31no—
514C22—31no—
515C22—31no—
516C22—31no—
517C22—32no—
518C22—61no—
519C22—61no—
520C22—92no—
521C22—182no—
522C22—182no—
523C22—182no—
524C22—182no—
525C22—182no—
526C22—182no—
527C22—182no—
528C22—182no—
529C22—181no—
530C22—181no—
531C22C3×#89711yescharacteristic, complemented
532C22C22×D3×A411yescharacteristic
533C4—22no—
534C4—32no—
535C4—61no—
536C4—62no—
537C4—62no—
538C4—181no—
539C4—182no—
540C4A4×D1212yes—
541C3—41no—
542C3—81no—
543C3A4×C22⋊Q811yescharacteristic, complemented
544C2—31no—
545C2—31no—
546C2—31no—
547C2—31no—
548C2—62no—
549C2—182no—
550C2C2×A4×D1211yescharacteristic
551C2A4×#23711yescharacteristic
552C2D3×Q8×A411yescharacteristic
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×C22⋊Q8
LCSG ▷ C24×C3 ▷ C22×C3
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×A4×Dic6C2semidirect
3. C3×A4×C4⋊C4C2semidirect
5. A4×C4⋊Dic3C2semidirect
6. C22×#897C3semidirect
17. C23×Dic6C2×C3semidirect
20. C22×C3×C4⋊C4C2×C3semidirect
22. C22×C4⋊Dic3C2×C3semidirect
31. A4×C4⋊C4D3semidirect
36. C2×D3×A4Q8semidirect
94. #897A4direct (N × H)
115. C22×C4⋊C4C3×D3semidirect
125. C23×D3C3×Q8semidirect
171. C2×Dic6C2×A4semidirect
184. C3×C4⋊C4C2×A4semidirect
195. C4⋊Dic3C2×A4semidirect
200. C3×A4C22⋊Q8semidirect
378. C4⋊C4D3×A4semidirect
399. C22×C3C3×C22⋊Q8semidirect
414. C2×D3Q8×A4semidirect
416. A4#897direct (N × H)
531. C22C3×#897semidirect
543. C3A4×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×A42
C2×A4×Dic62
C3×A4×C4⋊C42
A4×(C2×D3)⋊C42
A4×C4⋊Dic32
C22×#8973
A4×C22⋊Q83
C3×#8974