Finite Group Databasebeta

← A4×#3828 #13181459

Subgroups — A4×#3828 #13181459

Order 1728.

1418 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 1418 conjugacy classes of subgroups — too many to draw. Only its 141 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×C32×C4×A4C211yesmaximal, characteristic, complemented
2A4×#638C212yesmaximal, complemented
3A4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C4×A4×C3⋊D3C212yesmaximal, complemented
5A4×#634C211yesmaximal, characteristic, complemented
6C22×#3828C311yesmaximal, characteristic, complemented
7A4×#235—34nomaximal
8C22×C32×A4C2211yescharacteristic
9C2×A4×C3⋊D3C2212yes—
10C3×#3828—41nomaximal
11C32×C4×A4C2211yescharacteristic
12C32×C4×A4C2211yescharacteristic
13A4×C3⋊Dic3C2212yes—
14C23×C32×C4C2×C311yescharacteristic, complemented
15C22×#638C2×C312yescomplemented
16C2×#3828—31no—
17C2×#3828—31no—
18C2×#3828—32no—
19C2×#3828—31no—
20C2×#3828—31no—
21C2×#3828—32no—
22C22×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
23C22×C4×C3⋊D3C2×C312yescomplemented
24C22×#634C2×C311yescharacteristic, complemented
25C2×C3×C4×A4—24no—
26C2×C3×C4×A4D314yescomplemented
27A4×#67—38no—
28A4×D12—34no—
29C4×D3×A4—38no—
30A4×Dic6—34no—
31C2×C32×A4—21no—
32C2×C32×A4C2311yescharacteristic
33A4×C3⋊D3—22no—
34C2×C33×C4—41no—
35C3×#638—42no—
36C3×C4⋊(C3⋊D3)—41no—
37C3×C4×C3⋊D3—42no—
38C3×#634—41no—
39C22×#235—34no—
40A4×#41—91no—
41C24×C32C22×C311yescharacteristic
42C23×C3⋊D3C22×C312yes—
43C22×C3×A4—24no—
44C22×C3×A4C2×D314yes—
45C2×D3×A4—38no—
46C22×C32×C4—31no—
47C22×C32×C4—31no—
48C22×C32×C4—31no—
49C22×C32×C4—31no—
50C22×C32×C4C22×C311yescharacteristic
51C22×C32×C4C22×C311yescharacteristic
52C2×#638—32no—
53C2×#638—32no—
54C2×#638—32no—
55C2×#638—32no—
56C22×C3⋊Dic3C22×C312yes—
57#3828—32no—
58#3828—34no—
59#3828—32no—
60#3828—32no—
61#3828—34no—
62#3828—31no—
63#3828—32no—
64#3828—31no—
65#3828—32no—
66#3828—31no—
67#3828A411yescharacteristic, direct product factor
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4⋊(C3⋊D3)—32no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4×C3⋊D3—32no—
72C2×C4×C3⋊D3—32no—
73C2×C4×C3⋊D3—32no—
74C2×C4×C3⋊D3—32no—
75C2×#634—31no—
76C2×#634—31no—
77C2×#634—32no—
78C3×#235—124no—
79C3×C4×A4—24no—
80C3×C4×A4—24no—
81C3×C4×A4C2×D314yes—
82C3×C4×A4C2×D314yes—
83Dic3×A4—38no—
84C22×C33—41no—
85C2×C3×C3⋊D3—42no—
86C32×A4#4111yescharacteristic, complemented
87C33×C4—41no—
88C33×C4—41no—
89C3×C3⋊Dic3—42no—
90C23×C3×C4C3×D314yescomplemented
91C22×#67—38no—
92C2×#235—94no—
93C2×#235—94no—
94C2×#235—98no—
95C2×#235—94no—
96C2×#235—98no—
97C2×#235—94no—
98C22×D12—34no—
99C22×C4×D3—38no—
100C22×Dic6—34no—
101Q8×A4—91no—
102D4×A4—92no—
103D4×A4—91no—
104C2×C4×A4—24no—
105C2×C4×A4—92no—
106C2×C4×A4C3⋊D311yescharacteristic, complemented
107C23×C32—21no—
108C23×C32—31no—
109C23×C32—31no—
110C23×C32—61no—
111C23×C32C23×C311yescharacteristic
112C22×C3⋊D3—22no—
113C22×C3⋊D3—32no—
114C22×C3⋊D3—32no—
115C22×C3⋊D3—62no—
116C2×C3×A4—28no—
117C2×C3×A4—24no—
118C2×C3×A4—24no—
119C2×C3×A4C22×D314yes—
120D3×A4—68no—
121C2×C32×C4—32no—
122C2×C32×C4—31no—
123C2×C32×C4—31no—
124C2×C32×C4—31no—
125C2×C32×C4—31no—
126C2×C32×C4—31no—
127C2×C32×C4—31no—
128C2×C32×C4—31no—
129C2×C32×C4—44no—
130C2×C32×C4—84no—
131C2×C32×C4C2×A411yescharacteristic, complemented
132#638—34no—
133#638—32no—
134#638—32no—
135#638—32no—
136#638C2×A412yescomplemented
137C2×C3⋊Dic3—32no—
138C2×C3⋊Dic3—32no—
139C4⋊(C3⋊D3)—32no—
140C4⋊(C3⋊D3)—32no—
141C4⋊(C3⋊D3)—31no—
142C4⋊(C3⋊D3)C2×A411yescharacteristic, complemented
143C4×C3⋊D3—34no—
144C4×C3⋊D3—32no—
145C4×C3⋊D3—32no—
146C4×C3⋊D3—32no—
147C4×C3⋊D3C2×A412yescomplemented
148#634—32no—
149#634—31no—
150#634—32no—
151#634C2×A411yescharacteristic, complemented
152C3×#67—128no—
153C3×D12—124no—
154C3×C4×D3—128no—
155C3×Dic6—124no—
156C22×#41—91noSylow
157C2×C33—41no—
158C2×C33—81no—
159C3×C3⋊D3—82no—
160C24×C3C2×C3×D314yes—
161C23×D3—38no—
162C22×A4—24no—
163C22×A4—92no—
164C22×A4C2×C3⋊D311yescharacteristic
165C3×#41—361no—
166C22×C3×C4—34no—
167C22×C3×C4—34no—
168C22×C3×C4—34no—
169C22×C3×C4—34no—
170C22×C3×C4C2×C3×D314yes—
171C22×C3×C4C2×C3×D314yes—
172C2×#67—98no—
173C2×#67—98no—
174C2×#67—98no—
175C2×#67—98no—
176C22×Dic3—38no—
177#235—34no—
178#235—94no—
179#235—98no—
180#235—94no—
181#235—98no—
182#235—916no—
183#235—916no—
184#235—98no—
185#235—98no—
186#235—98no—
187#235—94no—
188C2×D12—94no—
189C2×D12—98no—
190C2×D12—94no—
191C2×C4×D3—98no—
192C2×C4×D3—98no—
193C2×C4×D3—98no—
194C2×C4×D3—98no—
195C2×Dic6—94no—
196C2×Dic6—98no—
197C2×Dic6—94no—
198C4×A4—24no—
199C4×A4—24no—
200C4×A4—92no—
201C4×A4C2×C3⋊D311yescharacteristic
202C4×A4C2×C3⋊D311yescharacteristic
203C22×C32—31no—
204C22×C32—31no—
205C22×C32—31no—
206C22×C32—44no—
207C22×C32—61no—
208C22×C32—61no—
209C22×C32—61no—
210C22×C32—61no—
211C22×C32—84no—
212C22×C32C3×#4111yescharacteristic, complemented
213C22×C32C22×A411yescharacteristic
214C2×C3⋊D3—32no—
215C2×C3⋊D3—62no—
216C2×C3⋊D3—62no—
217C2×C3⋊D3—62no—
218C2×C3⋊D3—62no—
219C2×C3⋊D3C22×A412yes—
220C2×C3×D3—128no—
221C3×A4—24no—
222C3×A4#23514yescomplemented
223C32×C4—31no—
224C32×C4—31no—
225C32×C4—44no—
226C32×C4—44no—
227C32×C4—84no—
228C32×C4—84no—
229C32×C4C22×A411yescharacteristic
230C32×C4C22×A411yescharacteristic
231C3⋊Dic3—32no—
232C3⋊Dic3C22×A412yes—
233C3×Dic3—128no—
234C2×#41—271no—
235C2×#41—271no—
236C2×#41—272no—
237C2×#41—271no—
238C2×#41—272no—
239C2×#41—271no—
240C22×Q8—91no—
241C22×D4—92no—
242C22×D4—91no—
243C23×C4—92no—
244C23×C4C3×C3⋊D311yescharacteristic, complemented
245C33—41noSylow
246C23×C3—24no—
247C23×C3—34no—
248C23×C3—34no—
249C23×C3—64no—
250C23×C3C22×C3×D314yes—
251C22×D3—68no—
252C22×D3—98no—
253C22×D3—98no—
254C22×D3—188no—
255C2×A4—28no—
256C2×A4—21no—
257C2×A4—24no—
258C2×A4—182no—
259C2×A4C22×C3⋊D311yescharacteristic
260C3×Q8—361no—
261C3×D4—361no—
262C3×D4—362no—
263C2×C3×C4—38no—
264C2×C3×C4—34no—
265C2×C3×C4—34no—
266C2×C3×C4—34no—
267C2×C3×C4—34no—
268C2×C3×C4—34no—
269C2×C3×C4—34no—
270C2×C3×C4—34no—
271C2×C3×C4—41no—
272C2×C3×C4—84no—
273C2×C3×C4—362no—
274C2×C3×C4D3×A414yescomplemented
275#67—38no—
276#67—98no—
277#67—98no—
278#67—916no—
279#67—98no—
280C2×Dic3—98no—
281C2×Dic3—98no—
282D12—34no—
283D12—98no—
284D12—98no—
285D12—94no—
286C4×D3—38no—
287C4×D3—98no—
288C4×D3—98no—
289C4×D3—916no—
290C4×D3—98no—
291Dic6—34no—
292Dic6—98no—
293Dic6—98no—
294Dic6—94no—
295C2×C32—21no—
296C2×C32—31no—
297C2×C32—31no—
298C2×C32—44no—
299C2×C32—61no—
300C2×C32—84no—
301C2×C32—88no—
302C2×C32—84no—
303C2×C32C23×A411yescharacteristic
304C3⋊D3—22no—
305C3⋊D3—62no—
306C3×D3—248no—
307C24—92no—
308C24C2×C3×C3⋊D311yescharacteristic
309#41—91no—
310#41—272no—
311#41—272no—
312#41—271no—
313#41—271no—
314#41—274no—
315#41—274no—
316#41—272no—
317#41—272no—
318#41—272no—
319#41—271no—
320C2×Q8—271no—
321C2×Q8—271no—
322C2×Q8—272no—
323C2×D4—271no—
324C2×D4—271no—
325C2×D4—272no—
326C2×D4—272no—
327C2×D4—272no—
328C2×D4—272no—
329C2×D4—272no—
330C22×C4—31no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4—92no—
335C22×C4—272no—
336C22×C4—272no—
337C22×C4—272no—
338C22×C4—272no—
339C22×C4C2×C3×C3⋊D311yescharacteristic
340C22×C4C2×C3×C3⋊D311yescharacteristic
341C22×C3—34no—
342C22×C3—34no—
343C22×C3—34no—
344C22×C3—41no—
345C22×C3—64no—
346C22×C3—64no—
347C22×C3—64no—
348C22×C3—64no—
349C22×C3—84no—
350C22×C3—362no—
351C22×C3C3×#23514yescomplemented
352C22×C3C2×D3×A414yes—
353C2×D3—38no—
354C2×D3—98no—
355C2×D3—188no—
356C2×D3—188no—
357C2×D3—188no—
358C2×D3—188no—
359A4—24no—
360A4#382811yescharacteristic, direct product factor
361C3×C4—34no—
362C3×C4—34no—
363C3×C4—41no—
364C3×C4—41no—
365C3×C4—84no—
366C3×C4—84no—
367C3×C4—362no—
368C3×C4C2×D3×A414yes—
369C3×C4C2×D3×A414yes—
370Dic3—38no—
371Dic3—98no—
372C32—44no—
373C32—84no—
374C32A4×#4111yescharacteristic, complemented
375C23—21no—
376C23—31no—
377C23—31no—
378C23—61no—
379C23—182no—
380C23—272no—
381C23—272no—
382C23—542no—
383C23C22×C3×C3⋊D311yescharacteristic
384Q8—91no—
385Q8—272no—
386Q8—272no—
387Q8—271no—
388D4—92no—
389D4—91no—
390D4—272no—
391D4—272no—
392D4—272no—
393D4—272no—
394D4—274no—
395D4—272no—
396D4—271no—
397C2×C4—32no—
398C2×C4—31no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—31no—
402C2×C4—31no—
403C2×C4—31no—
404C2×C4—31no—
405C2×C4—92no—
406C2×C4—272no—
407C2×C4—272no—
408C2×C4—274no—
409C2×C4—272no—
410C2×C4—272no—
411C2×C4—272no—
412C2×C4A4×C3⋊D311yescharacteristic, complemented
413C2×C3—24no—
414C2×C3—34no—
415C2×C3—34no—
416C2×C3—41no—
417C2×C3—64no—
418C2×C3—81no—
419C2×C3—88no—
420C2×C3—84no—
421C2×C3—722no—
422C2×C3C22×D3×A414yes—
423D3—68no—
424D3—188no—
425C22—31no—
426C22—31no—
427C22—31no—
428C22—61no—
429C22—61no—
430C22—61no—
431C22—61no—
432C22—92no—
433C22—272no—
434C22—542no—
435C22—542no—
436C22—542no—
437C22—542no—
438C22C3×#382811yescharacteristic, complemented
439C22C2×A4×C3⋊D311yescharacteristic
440C4—31no—
441C4—31no—
442C4—92no—
443C4—272no—
444C4C2×A4×C3⋊D311yescharacteristic
445C4C2×A4×C3⋊D311yescharacteristic
446C3—41no—
447C3—84no—
448C3A4×#23514yescomplemented
449C2—21no—
450C2—31no—
451C2—31no—
452C2—61no—
453C2—182no—
454C2—542no—
455C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
centerC4C2×A4×C3⋊D3
commutator, socle, cosocleC23×C32C23×C3
FrattiniC2C22×A4×C3⋊D3
FittingC23×C32×C4C2×C3
hypercenterC2×C4A4×C3⋊D3
hypocenterC22×C32C3×#41
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×A4C2semidirect
2. A4×#638C2semidirect
3. A4×C4⋊(C3⋊D3)C2semidirect
4. C4×A4×C3⋊D3C2semidirect
5. A4×#634C2semidirect
6. C22×#3828C3semidirect
14. C23×C32×C4C2×C3semidirect
15. C22×#638C2×C3semidirect
22. C22×C4⋊(C3⋊D3)C2×C3semidirect
23. C22×C4×C3⋊D3C2×C3semidirect
24. C22×#634C2×C3semidirect
26. C2×C3×C4×A4D3semidirect
67. #3828A4direct (N × H)
86. C32×A4#41semidirect
90. C23×C3×C4C3×D3semidirect
106. C2×C4×A4C3⋊D3semidirect
131. C2×C32×C4C2×A4semidirect
136. #638C2×A4semidirect
142. C4⋊(C3⋊D3)C2×A4semidirect
147. C4×C3⋊D3C2×A4semidirect
151. #634C2×A4semidirect
212. C22×C32C3×#41semidirect
222. C3×A4#235semidirect
244. C23×C4C3×C3⋊D3semidirect
274. C2×C3×C4D3×A4semidirect
351. C22×C3C3×#235semidirect
360. A4#3828direct (N × H)
374. C32A4×#41semidirect
412. C2×C4A4×C3⋊D3semidirect
438. C22C3×#3828semidirect
448. C3A4×#235semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×A42
A4×#6382
A4×C4⋊(C3⋊D3)2
C4×A4×C3⋊D32
A4×#6342
C22×#38283
A4×#2353
C3×#38284