Finite Group Databasebeta

← A4×#887 #11923160

Subgroups — A4×#887 #11923160

Order 1152.

814 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 129 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×A4×Dic6C211yesmaximal, characteristic
2C3×A4×C4⋊C4C211yesmaximal, characteristic
3A4×Dic3⋊C4C212yesmaximal
4C4×Dic3×A4C211yesmaximal, characteristic
5C4×Dic3×A4C212yesmaximal
6C22×#887C311yesmaximal, characteristic, complemented
7C4×Q8×A4—31nomaximal
8C2×C3×C4×A4C2212yes—
9C2×C3×C4×A4C2211yescharacteristic
10C2×Dic3×A4C2211yescharacteristic
11C2×Dic3×A4C2212yes—
12C2×Dic3×A4C2211yescharacteristic
13A4×Dic6C414yescomplemented
14C3×#887—41nomaximal
15C23×Dic6C2×C311yescharacteristic
16C2×Q8×A4—31no—
17C22×C3×C4⋊C4C2×C311yescharacteristic
18C22×Dic3⋊C4C2×C312yes—
19C22×C4×Dic3C2×C311yescharacteristic
20C22×C4×Dic3C2×C312yes—
21C2×#887—31no—
22C2×#887—31no—
23C2×#887—31no—
24C2×#887—32no—
25C2×#887—31no—
26C2×#887—32no—
27A4×C4⋊C4—21no—
28A4×C4⋊C4—32no—
29A4×C4⋊C4D311yescharacteristic
30C42×A4—32no—
31C42×A4—31no—
32C22×C3×A4C2311yescharacteristic
33C2×C3×Dic6—41no—
34C3×C4×A4—22no—
35C3×C4×A4C2×C412yes—
36Dic3×A4—21no—
37Dic3×A4C2×C414yes—
38Dic3×A4Q812yescomplemented
39C32×C4⋊C4—41no—
40C3×Dic3⋊C4—42no—
41C3×C4×Dic3—42no—
42C3×C4×Dic3—41no—
43C22×C4×Q8—31noSylow
44C23×C3×C4C22×C312yes—
45C23×C3×C4C22×C311yescharacteristic
46C23×Dic3C22×C311yescharacteristic
47C23×Dic3C22×C312yes—
48C23×Dic3C22×C311yescharacteristic
49C22×Dic6—31no—
50C22×Dic6—34no—
51C22×Dic6—31no—
52C22×Dic6—32no—
53C22×Dic6C3×C414yescomplemented
54Q8×A4—34no—
55C2×C4×A4—21no—
56C2×C4×A4—22no—
57C2×C4×A4—32no—
58C2×C4×A4—31no—
59C2×C4×A4—31no—
60C2×C4×A4C2×D312yes—
61C2×C4×A4C2×D311yescharacteristic
62C3×C4×Q8—121no—
63C2×C3×C4⋊C4—31no—
64C2×C3×C4⋊C4—31no—
65C2×C3×C4⋊C4—32no—
66C2×Dic3⋊C4—32no—
67C2×Dic3⋊C4—32no—
68C2×Dic3⋊C4—32no—
69C2×Dic3⋊C4—32no—
70C2×C4×Dic3—32no—
71C2×C4×Dic3—31no—
72C2×C4×Dic3—32no—
73C2×C4×Dic3—31no—
74C2×C4×Dic3—32no—
75C2×C4×Dic3—31no—
76C2×C4×Dic3—32no—
77C2×C4×Dic3—31no—
78#887—32no—
79#887—34no—
80#887—34no—
81#887—32no—
82#887—32no—
83#887—31no—
84#887—31no—
85#887—32no—
86#887—32no—
87#887—31no—
88#887A411yescharacteristic, direct product factor
89C2×C3×A4C22×C411yescharacteristic
90C2×C3×A4C2×Q811yescharacteristic
91C2×C3×A4#4111yescharacteristic
92C2×C32×C4—41no—
93C2×C32×C4—42no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—42no—
96C2×C3×Dic3—41no—
97C3×Dic6—44no—
98C23×Q8—31no—
99C2×C4×Q8—91no—
100C2×C4×Q8—91no—
101C2×C4×Q8—92no—
102C2×C4×Q8—91no—
103C2×C4×Q8—92no—
104C2×C4×Q8—91no—
105C22×C4⋊C4—32no—
106C22×C4⋊C4C3×D311yescharacteristic
107C22×C42—32no—
108C22×C42—31no—
109C24×C3C23×C311yescharacteristic
110C22×A4—21no—
111C22×A4C22×D311yescharacteristic
112C2×C3×Q8—121no—
113C22×C3×C4—22no—
114C22×C3×C4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—32no—
117C22×C3×C4—32no—
118C22×C3×C4—32no—
119C22×C3×C4—62no—
120C22×C3×C4C2×C3×C412yes—
121C22×Dic3—21no—
122C22×Dic3—31no—
123C22×Dic3—32no—
124C22×Dic3—31no—
125C22×Dic3—32no—
126C22×Dic3—34no—
127C22×Dic3—32no—
128C22×Dic3—31no—
129C22×Dic3—31no—
130C22×Dic3—61no—
131C22×Dic3C2×C3×C414yes—
132C22×Dic3C3×Q812yescomplemented
133C2×Dic6—32no—
134C2×Dic6—34no—
135C2×Dic6—38no—
136C2×Dic6—31no—
137C2×Dic6—32no—
138C2×Dic6—34no—
139C2×Dic6—34no—
140C2×Dic6—34no—
141C2×Dic6—38no—
142C2×Dic6C2×A411yescharacteristic
143C4×A4—22no—
144C4×A4—22no—
145C4×A4—34no—
146C4×A4—32no—
147C4×A4—42no—
148C4×A4—61no—
149C4×A4C4×D312yes—
150C3×C4⋊C4—32no—
151C3×C4⋊C4—32no—
152C3×C4⋊C4—31no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—81no—
155C3×C4⋊C4—122no—
156C3×C4⋊C4C2×A411yescharacteristic
157C3×C42—121no—
158C3×C42—122no—
159Dic3⋊C4—34no—
160Dic3⋊C4—32no—
161Dic3⋊C4—32no—
162Dic3⋊C4—32no—
163Dic3⋊C4C2×A412yes—
164C4×Dic3—34no—
165C4×Dic3—32no—
166C4×Dic3—32no—
167C4×Dic3—31no—
168C4×Dic3—31no—
169C4×Dic3—32no—
170C4×Dic3—32no—
171C4×Dic3—31no—
172C4×Dic3C2×A411yescharacteristic
173C4×Dic3C2×A412yes—
174C22×C32—41no—
175C3×A4C4×Q811yescharacteristic, complemented
176C32×C4—42no—
177C32×C4—82no—
178C3×Dic3—44no—
179C3×Dic3—42no—
180C3×Dic3—81no—
181C22×Q8—34no—
182C22×Q8—91no—
183C22×Q8—91no—
184C22×Q8—92no—
185C22×Q8—94no—
186C23×C4—32no—
187C23×C4—31no—
188C23×C4—31no—
189C23×C4C2×C3×D312yes—
190C23×C4C2×C3×D311yescharacteristic
191C4×Q8—31no—
192C4×Q8—91no—
193C4×Q8—92no—
194C4×Q8—92no—
195C4×Q8—91no—
196C4×Q8—92no—
197C4×Q8—92no—
198C4×Q8—94no—
199C4×Q8—92no—
200C4×Q8—94no—
201C4×Q8—91no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—32no—
205C2×C4⋊C4—92no—
206C2×C4⋊C4—92no—
207C2×C4⋊C4—92no—
208C2×C4⋊C4—92no—
209C2×C42—91no—
210C2×C42—92no—
211C2×C42—91no—
212C2×C42—91no—
213C2×C42—92no—
214C2×C42—92no—
215C2×C42—91no—
216C2×C42—92no—
217C23×C3—31no—
218C23×C3—31no—
219C23×C3—31no—
220C23×C3—31no—
221C23×C3C22×C3×C411yescharacteristic
222C23×C3C2×C3×Q811yescharacteristic
223C23×C3C3×#4111yescharacteristic
224C2×A4—21no—
225C2×A4—21no—
226C2×A4—21no—
227C2×A4C2×C4×D311yescharacteristic
228C2×A4#23711yescharacteristic
229C2×A4D3×Q811yescharacteristic
230C3×Q8—124no—
231C2×C3×C4—32no—
232C2×C3×C4—31no—
233C2×C3×C4—32no—
234C2×C3×C4—32no—
235C2×C3×C4—32no—
236C2×C3×C4—32no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4—81no—
244C2×C3×C4—82no—
245C2×C3×C4—121no—
246C2×C3×C4—122no—
247C2×C3×C4—121no—
248C2×C3×C4C22×A412yes—
249C2×C3×C4C22×A411yescharacteristic
250C2×Dic3—34no—
251C2×Dic3—32no—
252C2×Dic3—31no—
253C2×Dic3—32no—
254C2×Dic3—34no—
255C2×Dic3—34no—
256C2×Dic3—32no—
257C2×Dic3—32no—
258C2×Dic3—34no—
259C2×Dic3—32no—
260C2×Dic3—31no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3C22×A411yescharacteristic
266C2×Dic3C22×A412yes—
267C2×Dic3C22×A411yescharacteristic
268Dic6—38no—
269Dic6—34no—
270Dic6—38no—
271Dic6C4×A414yescomplemented
272C2×C32—41no—
273C2×C32—41no—
274C2×C32—41no—
275C24C22×C3×D311yescharacteristic
276C2×Q8—31no—
277C2×Q8—92no—
278C2×Q8—94no—
279C2×Q8—94no—
280C2×Q8—94no—
281C2×Q8—98no—
282C2×Q8—92no—
283C2×Q8—98no—
284C2×Q8—94no—
285C2×Q8—91no—
286C22×C4—22no—
287C22×C4—31no—
288C22×C4—31no—
289C22×C4—32no—
290C22×C4—32no—
291C22×C4—32no—
292C22×C4—34no—
293C22×C4—32no—
294C22×C4—62no—
295C22×C4—61no—
296C22×C4—91no—
297C22×C4—92no—
298C22×C4—91no—
299C22×C4—91no—
300C22×C4—92no—
301C22×C4—94no—
302C22×C4—92no—
303C22×C4—91no—
304C22×C4—181no—
305C22×C4C3×C4×D312yes—
306C4⋊C4—32no—
307C4⋊C4—32no—
308C4⋊C4—31no—
309C4⋊C4—32no—
310C4⋊C4—92no—
311C4⋊C4—92no—
312C4⋊C4—94no—
313C4⋊C4—92no—
314C4⋊C4D3×A411yescharacteristic
315C42—32no—
316C42—31no—
317C42—91no—
318C42—92no—
319C42—92no—
320C42—91no—
321C42—92no—
322C42—94no—
323C42—92no—
324C42—91no—
325C22×C3—31no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—32no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—41no—
336C22×C3—81no—
337C22×C3C3×C4×Q811yescharacteristic, complemented
338C22×C3C23×A411yescharacteristic
339A4—21no—
340A4#88711yescharacteristic, direct product factor
341C3×C4—22no—
342C3×C4—32no—
343C3×C4—42no—
344C3×C4—62no—
345C3×C4—82no—
346C3×C4—82no—
347C3×C4—124no—
348C3×C4—122no—
349C3×C4—162no—
350C3×C4—241no—
351C3×C4C2×C4×A412yes—
352Dic3—21no—
353Dic3—34no—
354Dic3—32no—
355Dic3—61no—
356Dic3C2×C4×A414yes—
357Dic3Q8×A412yescomplemented
358C32—41noSylow
359C23—31no—
360C23—31no—
361C23—31no—
362C23—31no—
363C23C2×C3×C4×D311yescharacteristic
364C23C3×#23711yescharacteristic
365C23C3×D3×Q811yescharacteristic
366Q8—34no—
367Q8—98no—
368Q8—98no—
369Q8—94no—
370C2×C4—32no—
371C2×C4—31no—
372C2×C4—32no—
373C2×C4—31no—
374C2×C4—32no—
375C2×C4—32no—
376C2×C4—32no—
377C2×C4—31no—
378C2×C4—32no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—94no—
384C2×C4—92no—
385C2×C4—94no—
386C2×C4—92no—
387C2×C4—94no—
388C2×C4—92no—
389C2×C4—91no—
390C2×C4—92no—
391C2×C4—94no—
392C2×C4—92no—
393C2×C4—91no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4—181no—
397C2×C4—181no—
398C2×C4C2×D3×A412yes—
399C2×C4C2×D3×A411yescharacteristic
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—41no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—81no—
408C2×C3—81no—
409C2×C3—81no—
410C2×C3C22×C4×A411yescharacteristic
411C2×C3C2×Q8×A411yescharacteristic
412C2×C3A4×#4111yescharacteristic
413C22—31no—
414C22—31no—
415C22—31no—
416C22—32no—
417C22—31no—
418C22—31no—
419C22—31no—
420C22—31no—
421C22—31no—
422C22—31no—
423C22C3×#88711yescharacteristic, complemented
424C22C22×D3×A411yescharacteristic
425C4—22no—
426C4—32no—
427C4—34no—
428C4—32no—
429C4—62no—
430C4—61no—
431C4—94no—
432C4—92no—
433C4—181no—
434C4C4×D3×A412yes—
435C3—41no—
436C3—81no—
437C3C4×Q8×A411yescharacteristic, complemented
438C2—31no—
439C2—31no—
440C2—31no—
441C2—31no—
442C2C2×C4×D3×A411yescharacteristic
443C2A4×#23711yescharacteristic
444C2D3×Q8×A411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutatorC23×C3C22×C3×C4
FittingC22×C3×C4⋊C4C2×C3
socle, cosocleC24×C3C23×C3
prime radicalC22×C3×A4C23
hypercenterC4⋊C4D3×A4
hypocenterC22×C3C3×C4×Q8
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
6. C22×#887C3semidirect
13. A4×Dic6C4semidirect
38. Dic3×A4Q8semidirect
53. C22×Dic6C3×C4semidirect
88. #887A4direct (N × H)
132. C22×Dic3C3×Q8semidirect
175. C3×A4C4×Q8semidirect
271. Dic6C4×A4semidirect
337. C22×C3C3×C4×Q8semidirect
340. A4#887direct (N × H)
357. Dic3Q8×A4semidirect
423. C22C3×#887semidirect
437. C3C4×Q8×A4semidirect

Maximal subgroups

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

subgroupindex
C2×A4×Dic62
C3×A4×C4⋊C42
A4×Dic3⋊C42
C4×Dic3×A42
C22×#8873
C4×Q8×A43
C3×#8874