Finite Group Databasebeta

← A4×#877 #11923150

Subgroups — A4×#877 #11923150

Order 1152.

914 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 123 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×Dic3×A4C211yesmaximal, characteristic
2C3×A4×C22⋊C4C211yesmaximal, characteristic
3A4×#217C211yesmaximal, characteristic
4A4×Dic3⋊C4C212yesmaximal, complemented
5C4×Dic3×A4C212yesmaximal, complemented
6C22×#877C311yesmaximal, characteristic, complemented
7A4×#117—31nomaximal
8C23×C3×A4C2211yescharacteristic
9C2×C3×C4×A4C2212yes—
10C2×Dic3×A4C414yescomplemented
11C2×Dic3×A4C2212yes—
12C2×Dic3×A4C2211yescharacteristic
13C2×Dic3×A4C2211yescharacteristic
14C3×#877—41nomaximal
15C24×Dic3C2×C311yescharacteristic
16C22×C4×A4—31no—
17C22×C3×C22⋊C4C2×C311yescharacteristic
18C22×#217C2×C311yescharacteristic
19C22×Dic3⋊C4C2×C312yescomplemented
20C22×C4×Dic3C2×C312yescomplemented
21C2×#877—31no—
22C2×#877—31no—
23C2×#877—31no—
24C2×#877—31no—
25C2×#877—32no—
26C2×#877—32no—
27A4×C4⋊C4—32no—
28A4×C22⋊C4—21no—
29A4×C22⋊C4—31no—
30A4×C22⋊C4D311yescharacteristic
31C42×A4—32no—
32C22×C3×A4—22no—
33C22×C3×A4C2×C412yes—
34C22×C3×A4C2311yescharacteristic
35C22×C3×Dic3—41no—
36C3×C4×A4—22no—
37Dic3×A4—22no—
38Dic3×A4C2×C412yes—
39Dic3×A4C2×C412yes—
40C32×C22⋊C4—41no—
41C3×#217—41no—
42C3×Dic3⋊C4—42no—
43C3×C4×Dic3—42no—
44C22×#117—31noSylow
45C25×C3C22×C311yescharacteristic
46C23×A4—21no—
47C23×A4C2×D311yescharacteristic
48C23×C3×C4C22×C312yes—
49C23×Dic3—31no—
50C23×Dic3—34no—
51C23×Dic3—31no—
52C23×Dic3—31no—
53C23×Dic3—31no—
54C23×Dic3C3×C414yescomplemented
55C23×Dic3C22×C312yes—
56C23×Dic3C22×C311yescharacteristic
57C23×Dic3C22×C311yescharacteristic
58C2×C4×A4—22no—
59C2×C4×A4—34no—
60C2×C4×A4—31no—
61C2×C4×A4—31no—
62C2×C4×A4—32no—
63C2×C4×A4C2×D312yes—
64C3×#117—121no—
65C2×C3×C22⋊C4—31no—
66C2×C3×C22⋊C4—31no—
67C2×C3×C22⋊C4—32no—
68C2×#217—31no—
69C2×#217—31no—
70C2×#217—32no—
71C2×Dic3⋊C4—32no—
72C2×Dic3⋊C4—32no—
73C2×Dic3⋊C4—32no—
74C2×Dic3⋊C4—32no—
75C2×C4×Dic3—32no—
76C2×C4×Dic3—32no—
77C2×C4×Dic3—32no—
78C2×C4×Dic3—32no—
79#877—32no—
80#877—32no—
81#877—34no—
82#877—34no—
83#877—32no—
84#877—31no—
85#877—31no—
86#877—31no—
87#877—32no—
88#877—32no—
89#877A411yescharacteristic, direct product factor
90C23×C32—41no—
91C2×C3×A4—22no—
92C2×C3×A4C22×C411yescharacteristic
93C2×C3×A4#4112yes—
94C2×C32×C4—42no—
95C2×C3×Dic3—44no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3—42no—
99C24×C4—31no—
100C2×#117—91no—
101C2×#117—92no—
102C2×#117—91no—
103C2×#117—91no—
104C2×#117—92no—
105C2×#117—91no—
106C22×C4⋊C4—32no—
107C22×C22⋊C4—31no—
108C22×C22⋊C4C3×D311yescharacteristic
109C22×C42—32no—
110C24×C3—22no—
111C24×C3—31no—
112C24×C3—32no—
113C24×C3—31no—
114C24×C3—62no—
115C24×C3C2×C3×C412yes—
116C24×C3C23×C311yescharacteristic
117C22×A4—22no—
118C22×A4—22no—
119C22×A4—21no—
120C22×A4—42no—
121C22×A4C4×D312yes—
122C22×A4C22×D311yescharacteristic
123C22×C3×C4—22no—
124C22×C3×C4—32no—
125C22×C3×C4—32no—
126C22×C3×C4—62no—
127C22×C3×C4—121no—
128C22×Dic3—22no—
129C22×Dic3—32no—
130C22×Dic3—34no—
131C22×Dic3—34no—
132C22×Dic3—34no—
133C22×Dic3—31no—
134C22×Dic3—34no—
135C22×Dic3—34no—
136C22×Dic3—31no—
137C22×Dic3—31no—
138C22×Dic3—31no—
139C22×Dic3—31no—
140C22×Dic3—31no—
141C22×Dic3—31no—
142C22×Dic3—34no—
143C22×Dic3—34no—
144C22×Dic3—34no—
145C22×Dic3—32no—
146C22×Dic3—32no—
147C22×Dic3—32no—
148C22×Dic3—32no—
149C22×Dic3—62no—
150C22×Dic3C2×C3×C412yes—
151C22×Dic3C2×C3×C412yes—
152C22×Dic3C2×A411yescharacteristic
153C4×A4—22no—
154C4×A4—32no—
155C4×A4—32no—
156C4×A4—42no—
157C4×A4—62no—
158C3×C4⋊C4—122no—
159C3×C22⋊C4—32no—
160C3×C22⋊C4—31no—
161C3×C22⋊C4—32no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—81no—
164C3×C22⋊C4—121no—
165C3×C22⋊C4C2×A411yescharacteristic
166C3×C42—122no—
167#217—32no—
168#217—31no—
169#217—32no—
170#217C2×A411yescharacteristic
171Dic3⋊C4—34no—
172Dic3⋊C4—32no—
173Dic3⋊C4—32no—
174Dic3⋊C4—32no—
175Dic3⋊C4C2×A412yescomplemented
176C4×Dic3—34no—
177C4×Dic3—32no—
178C4×Dic3—32no—
179C4×Dic3—32no—
180C4×Dic3C2×A412yescomplemented
181C22×C32—42no—
182C22×C32—41no—
183C22×C32—82no—
184C3×A4#11711yescharacteristic, complemented
185C32×C4—82no—
186C3×Dic3—42no—
187C3×Dic3—42no—
188C3×Dic3—82no—
189C25C2×C3×D311yescharacteristic
190C23×C4—34no—
191C23×C4—31no—
192C23×C4—31no—
193C23×C4—32no—
194C23×C4—91no—
195C23×C4—91no—
196C23×C4—91no—
197C23×C4—91no—
198C23×C4—94no—
199C23×C4C2×C3×D312yes—
200#117—31no—
201#117—92no—
202#117—91no—
203#117—92no—
204#117—91no—
205#117—92no—
206#117—94no—
207#117—92no—
208#117—92no—
209#117—94no—
210#117—91no—
211C2×C4⋊C4—92no—
212C2×C4⋊C4—92no—
213C2×C4⋊C4—92no—
214C2×C4⋊C4—92no—
215C2×C22⋊C4—31no—
216C2×C22⋊C4—31no—
217C2×C22⋊C4—32no—
218C2×C22⋊C4—91no—
219C2×C22⋊C4—92no—
220C2×C22⋊C4—91no—
221C2×C42—92no—
222C2×C42—92no—
223C2×C42—92no—
224C2×C42—92no—
225C23×C3—22no—
226C23×C3—31no—
227C23×C3—32no—
228C23×C3—31no—
229C23×C3—32no—
230C23×C3—32no—
231C23×C3—32no—
232C23×C3—32no—
233C23×C3—31no—
234C23×C3—41no—
235C23×C3—62no—
236C23×C3—62no—
237C23×C3—62no—
238C23×C3—64no—
239C23×C3—62no—
240C23×C3—62no—
241C23×C3—64no—
242C23×C3—81no—
243C23×C3C22×C3×C411yescharacteristic
244C23×C3C3×#4112yes—
245C23×C3C22×A411yescharacteristic
246C2×A4—22no—
247C2×A4—21no—
248C2×A4—22no—
249C2×A4—42no—
250C2×A4C2×C4×D311yescharacteristic
251C2×A4#23712yes—
252C2×C3×C4—32no—
253C2×C3×C4—42no—
254C2×C3×C4—62no—
255C2×C3×C4—62no—
256C2×C3×C4—62no—
257C2×C3×C4—62no—
258C2×C3×C4—82no—
259C2×C3×C4—121no—
260C2×C3×C4—121no—
261C2×C3×C4—124no—
262C2×C3×C4—122no—
263C2×C3×C4C22×A412yes—
264C2×Dic3—32no—
265C2×Dic3—32no—
266C2×Dic3—38no—
267C2×Dic3—34no—
268C2×Dic3—31no—
269C2×Dic3—31no—
270C2×Dic3—32no—
271C2×Dic3—32no—
272C2×Dic3—34no—
273C2×Dic3—34no—
274C2×Dic3—32no—
275C2×Dic3—32no—
276C2×Dic3—32no—
277C2×Dic3—32no—
278C2×Dic3—32no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×Dic3—62no—
282C2×Dic3—62no—
283C2×Dic3C4×A414yescomplemented
284C2×Dic3C22×A412yes—
285C2×Dic3C22×A411yescharacteristic
286C2×Dic3C22×A411yescharacteristic
287C2×C32—41no—
288C2×C32—42no—
289C2×C32—82no—
290C24—22no—
291C24—31no—
292C24—32no—
293C24—31no—
294C24—62no—
295C24C3×C4×D312yes—
296C24C22×C3×D311yescharacteristic
297C22×C4—22no—
298C22×C4—31no—
299C22×C4—32no—
300C22×C4—32no—
301C22×C4—32no—
302C22×C4—32no—
303C22×C4—62no—
304C22×C4—62no—
305C22×C4—91no—
306C22×C4—91no—
307C22×C4—91no—
308C22×C4—91no—
309C22×C4—94no—
310C22×C4—94no—
311C22×C4—92no—
312C22×C4—92no—
313C22×C4—94no—
314C22×C4—94no—
315C22×C4—94no—
316C22×C4—91no—
317C22×C4—91no—
318C22×C4—91no—
319C22×C4—92no—
320C22×C4—92no—
321C22×C4—94no—
322C22×C4—94no—
323C22×C4—94no—
324C22×C4—92no—
325C22×C4—182no—
326C4⋊C4—32no—
327C4⋊C4—92no—
328C4⋊C4—92no—
329C4⋊C4—94no—
330C4⋊C4—92no—
331C22⋊C4—32no—
332C22⋊C4—31no—
333C22⋊C4—31no—
334C22⋊C4—32no—
335C22⋊C4—92no—
336C22⋊C4—92no—
337C22⋊C4—91no—
338C22⋊C4D3×A411yescharacteristic
339C42—32no—
340C42—92no—
341C42—92no—
342C42—94no—
343C42—92no—
344C22×C3—22no—
345C22×C3—31no—
346C22×C3—32no—
347C22×C3—32no—
348C22×C3—31no—
349C22×C3—31no—
350C22×C3—32no—
351C22×C3—32no—
352C22×C3—32no—
353C22×C3—42no—
354C22×C3—41no—
355C22×C3—62no—
356C22×C3—64no—
357C22×C3—62no—
358C22×C3—62no—
359C22×C3—62no—
360C22×C3—64no—
361C22×C3—62no—
362C22×C3—82no—
363C22×C3—82no—
364C22×C3—81no—
365C22×C3—162no—
366C22×C3C3×#11711yescharacteristic, complemented
367C22×C3C2×C4×A412yes—
368C22×C3C23×A411yescharacteristic
369A4—21no—
370A4#87711yescharacteristic, direct product factor
371C3×C4—22no—
372C3×C4—62no—
373C3×C4—82no—
374C3×C4—122no—
375C3×C4—122no—
376C3×C4—162no—
377C3×C4—242no—
378Dic3—22no—
379Dic3—32no—
380Dic3—32no—
381Dic3—62no—
382Dic3C2×C4×A412yes—
383Dic3C2×C4×A412yes—
384C32—41noSylow
385C23—22no—
386C23—32no—
387C23—31no—
388C23—31no—
389C23—32no—
390C23—32no—
391C23—32no—
392C23—32no—
393C23—31no—
394C23—62no—
395C23—62no—
396C23—62no—
397C23—64no—
398C23—62no—
399C23—62no—
400C23—64no—
401C23C2×C3×C4×D311yescharacteristic
402C23C3×#23712yes—
403C23C2×D3×A411yescharacteristic
404C2×C4—34no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—32no—
408C2×C4—32no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—92no—
414C2×C4—92no—
415C2×C4—92no—
416C2×C4—92no—
417C2×C4—94no—
418C2×C4—94no—
419C2×C4—92no—
420C2×C4—92no—
421C2×C4—98no—
422C2×C4—94no—
423C2×C4—91no—
424C2×C4—91no—
425C2×C4—92no—
426C2×C4—92no—
427C2×C4—92no—
428C2×C4—182no—
429C2×C4—182no—
430C2×C4—182no—
431C2×C4—182no—
432C2×C4C2×D3×A412yes—
433C2×C3—22no—
434C2×C3—31no—
435C2×C3—32no—
436C2×C3—31no—
437C2×C3—41no—
438C2×C3—42no—
439C2×C3—62no—
440C2×C3—82no—
441C2×C3—81no—
442C2×C3—82no—
443C2×C3—162no—
444C2×C3C22×C4×A411yescharacteristic
445C2×C3A4×#4112yes—
446C22—22no—
447C22—31no—
448C22—32no—
449C22—32no—
450C22—31no—
451C22—31no—
452C22—32no—
453C22—32no—
454C22—32no—
455C22—62no—
456C22—62no—
457C22—64no—
458C22—62no—
459C22—62no—
460C22—64no—
461C22—62no—
462C22C3×#87711yescharacteristic, complemented
463C22C4×D3×A412yes—
464C22C22×D3×A411yescharacteristic
465C4—22no—
466C4—32no—
467C4—32no—
468C4—62no—
469C4—62no—
470C4—92no—
471C4—92no—
472C4—182no—
473C3—41no—
474C3—81no—
475C3A4×#11711yescharacteristic, complemented
476C2—22no—
477C2—31no—
478C2—31no—
479C2—32no—
480C2—62no—
481C2C2×C4×D3×A411yescharacteristic
482C2A4×#23712yes—
subgroupG/H
center, FrattiniC22C22×D3×A4
commutatorC23×C3C22×C3×C4
FittingC22×C3×C22⋊C4C2×C3
socle, cosocleC24×C3C23×C3
prime radicalC23×C3×A4C22
hypercenterC22⋊C4D3×A4
hypocenterC22×C3C3×#117
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
4. A4×Dic3⋊C4C2semidirect
5. C4×Dic3×A4C2semidirect
6. C22×#877C3semidirect
10. C2×Dic3×A4C4semidirect
19. C22×Dic3⋊C4C2×C3semidirect
20. C22×C4×Dic3C2×C3semidirect
54. C23×Dic3C3×C4semidirect
89. #877A4direct (N × H)
175. Dic3⋊C4C2×A4semidirect
180. C4×Dic3C2×A4semidirect
184. C3×A4#117semidirect
283. C2×Dic3C4×A4semidirect
366. C22×C3C3×#117semidirect
370. A4#877direct (N × H)
462. C22C3×#877semidirect
475. C3A4×#117semidirect

Maximal subgroups

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

subgroupindex
C22×Dic3×A42
C3×A4×C22⋊C42
A4×#2172
A4×Dic3⋊C42
C4×Dic3×A42
C22×#8773
A4×#1173
C3×#8774