Finite Group Databasebeta

← A4×#3757 #13173191

Subgroups — A4×#3757 #13173191

Order 1728.

1270 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 1270 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C23×C32×A4C211yesmaximal, characteristic
2C2×A4×C3⋊Dic3C212yesmaximal, complemented
3C22×#3757C311yesmaximal, characteristic, complemented
4A4×#217—34nomaximal
5C22×C32×A4—22no—
6C22×C32×A4C412yescomplemented
7C22×C32×A4C2211yescharacteristic
8A4×C3⋊Dic3—22no—
9C3×#3757—41nomaximal
10C25×C32C2×C311yescharacteristic
11C23×C3×A4—24no—
12C23×C3×A4D314yes—
13C23×C3⋊Dic3C2×C312yescomplemented
14C2×Dic3×A4—38no—
15C2×#3757—31no—
16C2×#3757—31no—
17C2×#3757—32no—
18C23×C33—41no—
19C2×C32×A4—22no—
20C2×C32×A4C2×C411yescharacteristic
21C2×C32×A4D412yes—
22C2×C3×C3⋊Dic3—42no—
23C22×#217—34no—
24A4×C22⋊C4—91no—
25C24×C32—22no—
26C24×C32—31no—
27C24×C32—32no—
28C24×C32—31no—
29C24×C32—62no—
30C24×C32C3×C412yescomplemented
31C24×C32C22×C311yescharacteristic
32C22×C3×A4—216no—
33C22×C3×A4—28no—
34C22×C3×A4—24no—
35C22×C3×A4—28no—
36C22×C3×A4Dic318yescomplemented
37C22×C3×A4C2×D314yes—
38C22×C3⋊Dic3—22no—
39C22×C3⋊Dic3—32no—
40C22×C3⋊Dic3—32no—
41C22×C3⋊Dic3—62no—
42Dic3×A4—68no—
43#3757—32no—
44#3757—31no—
45#3757—32no—
46#3757A411yescharacteristic, direct product factor
47C3×#217—124no—
48C22×C33—41no—
49C22×C33—42no—
50C22×C33—82no—
51C32×A4C22⋊C411yescharacteristic, complemented
52C3×C3⋊Dic3—82no—
53C25×C3C3×D314yes—
54C23×A4—24no—
55C23×A4C3⋊D311yescharacteristic
56C23×Dic3—38no—
57C2×C4×A4—92no—
58C2×#217—94no—
59C2×#217—98no—
60C2×#217—94no—
61C23×C32—22no—
62C23×C32—31no—
63C23×C32—32no—
64C23×C32—31no—
65C23×C32—32no—
66C23×C32—32no—
67C23×C32—32no—
68C23×C32—32no—
69C23×C32—31no—
70C23×C32—44no—
71C23×C32—62no—
72C23×C32—62no—
73C23×C32—62no—
74C23×C32—64no—
75C23×C32—62no—
76C23×C32—62no—
77C23×C32—64no—
78C23×C32—84no—
79C23×C32C2×C3×C411yescharacteristic
80C23×C32C3×D412yes—
81C23×C32C2×A411yescharacteristic
82C2×C3×A4—216no—
83C2×C3×A4—28no—
84C2×C3×A4—24no—
85C2×C3×A4—28no—
86C2×C3×A4C2×Dic314yes—
87C2×C3×A4#6718yes—
88C2×C3⋊Dic3—32no—
89C2×C3⋊Dic3—62no—
90C2×C3⋊Dic3—62no—
91C2×C3⋊Dic3—62no—
92C2×C3⋊Dic3—62no—
93C2×C3⋊Dic3C2×A412yescomplemented
94C2×C3×Dic3—128no—
95C22×C22⋊C4—91noSylow
96C2×C33—41no—
97C2×C33—42no—
98C2×C33—82no—
99C24×C3—28no—
100C24×C3—38no—
101C24×C3—34no—
102C24×C3—34no—
103C24×C3—68no—
104C24×C3C3×Dic318yescomplemented
105C24×C3C2×C3×D314yes—
106C22×A4—216no—
107C22×A4—22no—
108C22×A4—28no—
109C22×A4—24no—
110C22×A4C3⋊Dic312yescomplemented
111C22×A4C2×C3⋊D311yescharacteristic
112C22×Dic3—68no—
113C22×Dic3—98no—
114C22×Dic3—98no—
115C22×Dic3—188no—
116C4×A4—182no—
117C3×C22⋊C4—361no—
118#217—34no—
119#217—98no—
120#217—98no—
121#217—94no—
122C22×C32—22no—
123C22×C32—31no—
124C22×C32—32no—
125C22×C32—32no—
126C22×C32—31no—
127C22×C32—31no—
128C22×C32—32no—
129C22×C32—32no—
130C22×C32—32no—
131C22×C32—44no—
132C22×C32—48no—
133C22×C32—62no—
134C22×C32—64no—
135C22×C32—62no—
136C22×C32—62no—
137C22×C32—62no—
138C22×C32—64no—
139C22×C32—62no—
140C22×C32—88no—
141C22×C32—816no—
142C22×C32—84no—
143C22×C32—88no—
144C22×C32C3×C22⋊C411yescharacteristic, complemented
145C22×C32C4×A412yescomplemented
146C22×C32C22×A411yescharacteristic
147C3×A4—24no—
148C3×A4#21714yescomplemented
149C3⋊Dic3—22no—
150C3⋊Dic3—62no—
151C3×Dic3—248no—
152C25C3×C3⋊D311yescharacteristic
153C23×C4—92no—
154C2×C22⋊C4—271no—
155C2×C22⋊C4—272no—
156C2×C22⋊C4—271no—
157C33—41noSylow
158C23×C3—28no—
159C23×C3—38no—
160C23×C3—34no—
161C23×C3—34no—
162C23×C3—38no—
163C23×C3—38no—
164C23×C3—34no—
165C23×C3—38no—
166C23×C3—38no—
167C23×C3—41no—
168C23×C3—68no—
169C23×C3—68no—
170C23×C3—68no—
171C23×C3—616no—
172C23×C3—68no—
173C23×C3—68no—
174C23×C3—616no—
175C23×C3—84no—
176C23×C3C2×C3×Dic314yes—
177C23×C3C3×#6718yes—
178C23×C3D3×A414yes—
179C2×A4—216no—
180C2×A4—22no—
181C2×A4—24no—
182C2×A4—28no—
183C2×A4C2×C3⋊Dic311yescharacteristic
184C2×A4#63812yes—
185C2×C3×C4—362no—
186C2×Dic3—38no—
187C2×Dic3—98no—
188C2×Dic3—188no—
189C2×Dic3—188no—
190C2×Dic3—188no—
191C2×Dic3—188no—
192C2×C32—22no—
193C2×C32—31no—
194C2×C32—32no—
195C2×C32—31no—
196C2×C32—44no—
197C2×C32—48no—
198C2×C32—62no—
199C2×C32—88no—
200C2×C32—816no—
201C2×C32—84no—
202C2×C32—88no—
203C2×C32C2×C4×A411yescharacteristic
204C2×C32D4×A412yes—
205C24—22no—
206C24—31no—
207C24—32no—
208C24—31no—
209C24—62no—
210C24C3×C3⋊Dic312yescomplemented
211C24C2×C3×C3⋊D311yescharacteristic
212C22×C4—182no—
213C22×C4—272no—
214C22×C4—272no—
215C22×C4—542no—
216C22⋊C4—91no—
217C22⋊C4—272no—
218C22⋊C4—272no—
219C22⋊C4—271no—
220C22×C3—28no—
221C22×C3—34no—
222C22×C3—38no—
223C22×C3—38no—
224C22×C3—34no—
225C22×C3—34no—
226C22×C3—38no—
227C22×C3—38no—
228C22×C3—38no—
229C22×C3—41no—
230C22×C3—42no—
231C22×C3—68no—
232C22×C3—616no—
233C22×C3—68no—
234C22×C3—68no—
235C22×C3—68no—
236C22×C3—616no—
237C22×C3—68no—
238C22×C3—82no—
239C22×C3—816no—
240C22×C3—84no—
241C22×C3—88no—
242C22×C3C3×#21714yescomplemented
243C22×C3Dic3×A418yescomplemented
244C22×C3C2×D3×A414yes—
245A4—24no—
246A4#375711yescharacteristic, direct product factor
247C3×C4—722no—
248Dic3—68no—
249Dic3—188no—
250C32—44no—
251C32—84no—
252C32A4×C22⋊C411yescharacteristic, complemented
253C23—22no—
254C23—32no—
255C23—31no—
256C23—31no—
257C23—32no—
258C23—32no—
259C23—31no—
260C23—32no—
261C23—32no—
262C23—62no—
263C23—62no—
264C23—62no—
265C23—64no—
266C23—62no—
267C23—62no—
268C23—64no—
269C23C2×C3×C3⋊Dic311yescharacteristic
270C23C3×#63812yes—
271C23A4×C3⋊D311yescharacteristic
272C2×C4—92no—
273C2×C4—272no—
274C2×C4—542no—
275C2×C4—542no—
276C2×C4—542no—
277C2×C4—542no—
278C2×C3—28no—
279C2×C3—34no—
280C2×C3—34no—
281C2×C3—38no—
282C2×C3—41no—
283C2×C3—42no—
284C2×C3—68no—
285C2×C3—82no—
286C2×C3—816no—
287C2×C3—84no—
288C2×C3—88no—
289C2×C3C2×Dic3×A414yes—
290C2×C3A4×#6718yes—
291C22—22no—
292C22—31no—
293C22—32no—
294C22—32no—
295C22—31no—
296C22—31no—
297C22—32no—
298C22—32no—
299C22—32no—
300C22—62no—
301C22—62no—
302C22—64no—
303C22—62no—
304C22—62no—
305C22—64no—
306C22—62no—
307C22C3×#375711yescharacteristic, complemented
308C22A4×C3⋊Dic312yescomplemented
309C22C2×A4×C3⋊D311yescharacteristic
310C4—182no—
311C4—542no—
312C3—41no—
313C3—84no—
314C3A4×#21714yescomplemented
315C2—22no—
316C2—31no—
317C2—31no—
318C2—32no—
319C2—62no—
320C2C2×A4×C3⋊Dic311yescharacteristic
321C2A4×#63812yes—
subgroupG/H
center, FrattiniC22C2×A4×C3⋊D3
commutatorC23×C32C2×C3×C4
FittingC25×C32C2×C3
socle, cosocleC24×C32C22×C3
prime radicalC23×C32×A4C2
hypercenterC23A4×C3⋊D3
hypocenterC22×C32C3×C22⋊C4
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×A4×C3⋊Dic3C2semidirect
3. C22×#3757C3semidirect
6. C22×C32×A4C4semidirect
13. C23×C3⋊Dic3C2×C3semidirect
30. C24×C32C3×C4semidirect
36. C22×C3×A4Dic3semidirect
46. #3757A4direct (N × H)
51. C32×A4C22⋊C4semidirect
93. C2×C3⋊Dic3C2×A4semidirect
104. C24×C3C3×Dic3semidirect
110. C22×A4C3⋊Dic3semidirect
144. C22×C32C3×C22⋊C4semidirect
145. C22×C32C4×A4semidirect
148. C3×A4#217semidirect
210. C24C3×C3⋊Dic3semidirect
242. C22×C3C3×#217semidirect
243. C22×C3Dic3×A4semidirect
246. A4#3757direct (N × H)
252. C32A4×C22⋊C4semidirect
307. C22C3×#3757semidirect
308. C22A4×C3⋊Dic3semidirect
314. C3A4×#217semidirect

Maximal subgroups

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

subgroupindex
C23×C32×A42
C2×A4×C3⋊Dic32
C22×#37573
A4×#2173
C3×#37574