Finite Group Databasebeta

← A4×#890 #11923164

Subgroups — A4×#890 #11923164

Order 1152.

610 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 99 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
2A4×C4⋊Dic3C212yesmaximal
3A4×C4⋊Dic3C211yesmaximal, characteristic
4A4×Dic3⋊C4C212yesmaximal
5C4×Dic3×A4C211yesmaximal, characteristic
6C22×#890C311yesmaximal, characteristic, complemented
7A4×#125—31nomaximal
8C2×C3×C4×A4C2212yes—
9C2×C3×C4×A4C2211yescharacteristic
10C2×Dic3×A4C2211yescharacteristic
11C2×Dic3×A4C2211yescharacteristic
12C2×Dic3×A4C2212yes—
13C3×#890—41nomaximal
14C22×C3×C4⋊C4C2×C311yescharacteristic
15C22×C4⋊Dic3C2×C312yes—
16C22×C4⋊Dic3C2×C311yescharacteristic
17C22×Dic3⋊C4C2×C312yes—
18C22×C4×Dic3C2×C311yescharacteristic
19C2×#890—31no—
20C2×#890—31no—
21C2×#890—32no—
22C2×#890—31no—
23C2×#890—32no—
24C2×#890—31no—
25A4×C4⋊C4—21no—
26A4×C4⋊C4—32no—
27A4×C4⋊C4—32no—
28A4×C4⋊C4—31no—
29A4×C4⋊C4D311yescharacteristic
30C42×A4—31no—
31C22×C3×A4C2311yescharacteristic
32C3×C4×A4—22no—
33C3×C4×A4Q812yes—
34Dic3×A4—21no—
35Dic3×A4—21no—
36Dic3×A4—22no—
37C32×C4⋊C4—41no—
38C3×C4⋊Dic3—42no—
39C3×C4⋊Dic3—41no—
40C3×Dic3⋊C4—42no—
41C3×C4×Dic3—41no—
42C22×#125—31noSylow
43C23×C3×C4C22×C312yes—
44C23×C3×C4C22×C311yescharacteristic
45C23×Dic3C22×C311yescharacteristic
46C23×Dic3C22×C311yescharacteristic
47C23×Dic3C22×C312yes—
48C2×C4×A4—21no—
49C2×C4×A4—22no—
50C2×C4×A4—31no—
51C2×C4×A4—31no—
52C2×C4×A4—32no—
53C2×C4×A4C2×D312yes—
54C2×C4×A4C2×D311yescharacteristic
55C3×#125—121no—
56C2×C3×C4⋊C4—31no—
57C2×C3×C4⋊C4—31no—
58C2×C3×C4⋊C4—32no—
59C2×C4⋊Dic3—32no—
60C2×C4⋊Dic3—31no—
61C2×C4⋊Dic3—32no—
62C2×C4⋊Dic3—31no—
63C2×C4⋊Dic3—32no—
64C2×C4⋊Dic3—32no—
65C2×C4⋊Dic3—32no—
66C2×Dic3⋊C4—32no—
67C2×Dic3⋊C4—32no—
68C2×Dic3⋊C4—32no—
69C2×Dic3⋊C4—32no—
70C2×C4×Dic3—31no—
71C2×C4×Dic3—31no—
72C2×C4×Dic3—31no—
73C2×C4×Dic3—31no—
74#890—34no—
75#890—34no—
76#890—32no—
77#890—32no—
78#890—32no—
79#890—31no—
80#890—31no—
81#890—32no—
82#890—32no—
83#890—31no—
84#890A411yescharacteristic, direct product factor
85C2×C3×A4C2×Q811yescharacteristic
86C2×C3×A4#4111yescharacteristic
87C2×C3×A4#4111yescharacteristic
88C2×C32×C4—41no—
89C2×C32×C4—42no—
90C2×C3×Dic3—41no—
91C2×C3×Dic3—41no—
92C2×C3×Dic3—42no—
93C2×#125—91no—
94C2×#125—91no—
95C2×#125—91no—
96C2×#125—92no—
97C2×#125—92no—
98C2×#125—91no—
99C22×C4⋊C4—32no—
100C22×C4⋊C4—31no—
101C22×C4⋊C4—32no—
102C22×C4⋊C4C3×D311yescharacteristic
103C22×C42—31no—
104C24×C3C23×C311yescharacteristic
105C22×A4—21no—
106C22×A4C22×D311yescharacteristic
107C22×C3×C4—22no—
108C22×C3×C4—31no—
109C22×C3×C4—31no—
110C22×C3×C4—32no—
111C22×C3×C4—32no—
112C22×C3×C4—32no—
113C22×C3×C4—62no—
114C22×C3×C4C3×Q812yes—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—22no—
118C22×Dic3—31no—
119C22×Dic3—31no—
120C22×Dic3—32no—
121C22×Dic3—31no—
122C22×Dic3—31no—
123C22×Dic3—32no—
124C22×Dic3—61no—
125C22×Dic3—62no—
126C22×Dic3—61no—
127C4×A4—22no—
128C4×A4—22no—
129C4×A4—42no—
130C4×A4—61no—
131C4×A4—61no—
132C4×A4—62no—
133C4×A4Dic612yes—
134C3×C4⋊C4—32no—
135C3×C4⋊C4—32no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—81no—
139C3×C4⋊C4—122no—
140C3×C4⋊C4—121no—
141C3×C4⋊C4—122no—
142C3×C4⋊C4C2×A411yescharacteristic
143C3×C42—121no—
144C4⋊Dic3—34no—
145C4⋊Dic3—32no—
146C4⋊Dic3—32no—
147C4⋊Dic3—31no—
148C4⋊Dic3—32no—
149C4⋊Dic3—32no—
150C4⋊Dic3—32no—
151C4⋊Dic3C2×A412yes—
152C4⋊Dic3C2×A411yescharacteristic
153Dic3⋊C4—34no—
154Dic3⋊C4—32no—
155Dic3⋊C4—32no—
156Dic3⋊C4—32no—
157Dic3⋊C4C2×A412yes—
158C4×Dic3—32no—
159C4×Dic3—31no—
160C4×Dic3—31no—
161C4×Dic3—31no—
162C4×Dic3C2×A411yescharacteristic
163C22×C32—41no—
164C3×A4#12511yescharacteristic, complemented
165C32×C4—42no—
166C32×C4—82no—
167C3×Dic3—81no—
168C3×Dic3—82no—
169C3×Dic3—81no—
170C23×C4—32no—
171C23×C4—31no—
172C23×C4—31no—
173C23×C4C2×C3×D312yes—
174C23×C4C2×C3×D311yescharacteristic
175#125—31no—
176#125—91no—
177#125—92no—
178#125—92no—
179#125—91no—
180#125—92no—
181#125—92no—
182#125—94no—
183#125—94no—
184#125—92no—
185#125—91no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—92no—
190C2×C4⋊C4—91no—
191C2×C4⋊C4—92no—
192C2×C4⋊C4—92no—
193C2×C4⋊C4—92no—
194C2×C4⋊C4—92no—
195C2×C4⋊C4—92no—
196C2×C4⋊C4—92no—
197C2×C4⋊C4—92no—
198C2×C4⋊C4—91no—
199C2×C4⋊C4—92no—
200C2×C42—91no—
201C2×C42—91no—
202C2×C42—91no—
203C2×C42—91no—
204C23×C3—31no—
205C23×C3—31no—
206C23×C3—31no—
207C23×C3—31no—
208C23×C3C2×C3×Q811yescharacteristic
209C23×C3C3×#4111yescharacteristic
210C23×C3C3×#4111yescharacteristic
211C2×A4—21no—
212C2×A4—21no—
213C2×A4—21no—
214C2×A4C2×Dic611yescharacteristic
215C2×A4#23711yescharacteristic
216C2×A4#23911yescharacteristic
217C2×C3×C4—32no—
218C2×C3×C4—31no—
219C2×C3×C4—32no—
220C2×C3×C4—32no—
221C2×C3×C4—32no—
222C2×C3×C4—32no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—62no—
229C2×C3×C4—81no—
230C2×C3×C4—82no—
231C2×C3×C4—121no—
232C2×C3×C4—121no—
233C2×C3×C4—122no—
234C2×C3×C4C22×A412yes—
235C2×C3×C4C22×A411yescharacteristic
236C2×Dic3—31no—
237C2×Dic3—31no—
238C2×Dic3—32no—
239C2×Dic3—61no—
240C2×Dic3—62no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—62no—
244C2×Dic3—61no—
245C2×Dic3—61no—
246C2×Dic3—62no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—62no—
250C2×Dic3—61no—
251C2×Dic3C22×A411yescharacteristic
252C2×Dic3C22×A411yescharacteristic
253C2×Dic3C22×A412yes—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—41no—
257C24C22×C3×D311yescharacteristic
258C22×C4—22no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—32no—
262C22×C4—32no—
263C22×C4—32no—
264C22×C4—62no—
265C22×C4—61no—
266C22×C4—62no—
267C22×C4—61no—
268C22×C4—91no—
269C22×C4—91no—
270C22×C4—92no—
271C22×C4—91no—
272C22×C4—91no—
273C22×C4—92no—
274C22×C4—181no—
275C22×C4—182no—
276C22×C4—181no—
277C22×C4C3×Dic612yes—
278C4⋊C4—32no—
279C4⋊C4—32no—
280C4⋊C4—31no—
281C4⋊C4—32no—
282C4⋊C4—32no—
283C4⋊C4—31no—
284C4⋊C4—92no—
285C4⋊C4—92no—
286C4⋊C4—92no—
287C4⋊C4—92no—
288C4⋊C4—92no—
289C4⋊C4—94no—
290C4⋊C4—92no—
291C4⋊C4—94no—
292C4⋊C4—92no—
293C4⋊C4—92no—
294C4⋊C4—91no—
295C4⋊C4D3×A411yescharacteristic
296C42—31no—
297C42—91no—
298C42—91no—
299C42—92no—
300C42—91no—
301C22×C3—31no—
302C22×C3—31no—
303C22×C3—31no—
304C22×C3—32no—
305C22×C3—31no—
306C22×C3—31no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—41no—
312C22×C3—81no—
313C22×C3C3×#12511yescharacteristic, complemented
314C22×C3C23×A411yescharacteristic
315A4—21no—
316A4#89011yescharacteristic, direct product factor
317C3×C4—22no—
318C3×C4—32no—
319C3×C4—42no—
320C3×C4—62no—
321C3×C4—82no—
322C3×C4—82no—
323C3×C4—162no—
324C3×C4—241no—
325C3×C4—241no—
326C3×C4—242no—
327C3×C4Q8×A412yes—
328Dic3—21no—
329Dic3—21no—
330Dic3—22no—
331Dic3—61no—
332Dic3—61no—
333Dic3—62no—
334C32—41noSylow
335C23—31no—
336C23—31no—
337C23—31no—
338C23—31no—
339C23C2×C3×Dic611yescharacteristic
340C23C3×#23711yescharacteristic
341C23C3×#23911yescharacteristic
342C2×C4—32no—
343C2×C4—31no—
344C2×C4—32no—
345C2×C4—32no—
346C2×C4—32no—
347C2×C4—31no—
348C2×C4—31no—
349C2×C4—32no—
350C2×C4—32no—
351C2×C4—62no—
352C2×C4—62no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—91no—
356C2×C4—91no—
357C2×C4—92no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4—182no—
361C2×C4—181no—
362C2×C4—181no—
363C2×C4—182no—
364C2×C4—181no—
365C2×C4—181no—
366C2×C4—182no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—182no—
370C2×C4C2×D3×A412yes—
371C2×C4C2×D3×A411yescharacteristic
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—81no—
380C2×C3—81no—
381C2×C3—81no—
382C2×C3C2×Q8×A411yescharacteristic
383C2×C3A4×#4111yescharacteristic
384C2×C3A4×#4111yescharacteristic
385C22—31no—
386C22—31no—
387C22—31no—
388C22—32no—
389C22—31no—
390C22—31no—
391C22—31no—
392C22—31no—
393C22—31no—
394C22—31no—
395C22C3×#89011yescharacteristic, complemented
396C22C22×D3×A411yescharacteristic
397C4—22no—
398C4—32no—
399C4—62no—
400C4—61no—
401C4—61no—
402C4—62no—
403C4—181no—
404C4—181no—
405C4—182no—
406C4A4×Dic612yes—
407C3—41no—
408C3—81no—
409C3A4×#12511yescharacteristic, complemented
410C2—31no—
411C2—31no—
412C2—31no—
413C2—31no—
414C2C2×A4×Dic611yescharacteristic
415C2A4×#23711yescharacteristic
416C2A4×#23911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, socle, cosocleC24×C3C23×C3
FittingC22×C3×C4⋊C4C2×C3
prime radicalC22×C3×A4C23
hypercenterC4⋊C4D3×A4
hypocenterC22×C3C3×#125
LCSG ▷ C24×C3 ▷ C22×C3
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
6. C22×#890C3semidirect
84. #890A4direct (N × H)
164. C3×A4#125semidirect
313. C22×C3C3×#125semidirect
316. A4#890direct (N × H)
395. C22C3×#890semidirect
409. C3A4×#125semidirect

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×C4⋊Dic32
A4×Dic3⋊C42
C4×Dic3×A42
C22×#8903
A4×#1253
C3×#8904