Finite Group Databasebeta

← A4×#3832 #13181466

Subgroups — A4×#3832 #13181466

Order 1728.

1386 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 1386 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
1C32×Q8×A4C211yesmaximal, characteristic, complemented
2A4×C4⋊(C3⋊D3)C213yesmaximal, complemented
3C4×A4×C3⋊D3C213yesmaximal, complemented
4C22×#3832C311yesmaximal, characteristic, complemented
5A4×#239—34nomaximal
6C2×A4×C3⋊D3C2213yes—
7C3×#3832—41nomaximal
8C32×C4×A4C2213yes—
9A4×C3⋊Dic3C2211yescharacteristic
10C22×C32×Q8C2×C311yescharacteristic, complemented
11C2×#3832—31no—
12C2×#3832—33no—
13C2×#3832—33no—
14C2×#3832—31no—
15C22×C4⋊(C3⋊D3)C2×C313yescomplemented
16C22×C4×C3⋊D3C2×C313yescomplemented
17C3×Q8×A4—24no—
18C3×Q8×A4D314yescomplemented
19A4×D12—312no—
20C4×D3×A4—312no—
21C2×C32×A4C2311yescharacteristic
22A4×C3⋊D3—23no—
23C33×Q8—41no—
24C3×C4⋊(C3⋊D3)—43no—
25C3×C4×C3⋊D3—43no—
26C22×#239—34no—
27A4×#41—91no—
28C23×C3⋊D3C22×C313yes—
29C2×D3×A4—312no—
30C2×C32×Q8—31no—
31C2×C32×Q8—33no—
32C22×C32×C4C22×C313yes—
33C22×C3⋊Dic3C22×C311yescharacteristic
34#3832—36no—
35#3832—36no—
36#3832—32no—
37#3832—31no—
38#3832—33no—
39#3832—33no—
40#3832A411yescharacteristic, direct product factor
41C2×C4⋊(C3⋊D3)—33no—
42C2×C4⋊(C3⋊D3)—36no—
43C2×C4⋊(C3⋊D3)—33no—
44C2×C4×C3⋊D3—33no—
45C2×C4×C3⋊D3—33no—
46C2×C4×C3⋊D3—33no—
47C2×C4×C3⋊D3—33no—
48C3×#239—124no—
49C3×C4×A4—212no—
50C3×C4×A4C2×D3112yes—
51Dic3×A4—34no—
52C2×C3×C3⋊D3—43no—
53C32×A4#4111yescharacteristic, complemented
54C33×C4—43no—
55C3×C3⋊Dic3—41no—
56C22×C3×Q8C3×D314yescomplemented
57C2×#239—94no—
58C2×#239—912no—
59C2×#239—912no—
60C2×#239—94no—
61C22×D12—312no—
62C22×C4×D3—312no—
63Q8×A4—24no—
64Q8×A4C3⋊D311yescharacteristic, complemented
65D4×A4—93no—
66C2×C4×A4—93no—
67C23×C32C23×C311yescharacteristic
68C22×C3⋊D3—23no—
69C22×C3⋊D3—33no—
70C22×C3⋊D3—33no—
71C22×C3⋊D3—63no—
72C2×C3×A4—24no—
73C2×C3×A4C22×D314yes—
74D3×A4—612no—
75C32×Q8—32no—
76C32×Q8—33no—
77C32×Q8—44no—
78C32×Q8—84no—
79C32×Q8C2×A411yescharacteristic, complemented
80C2×C32×C4—33no—
81C2×C32×C4—33no—
82C2×C3⋊Dic3—31no—
83C2×C3⋊Dic3—31no—
84C4⋊(C3⋊D3)—36no—
85C4⋊(C3⋊D3)—36no—
86C4⋊(C3⋊D3)—33no—
87C4⋊(C3⋊D3)C2×A413yescomplemented
88C4×C3⋊D3—36no—
89C4×C3⋊D3—33no—
90C4×C3⋊D3—33no—
91C4×C3⋊D3—33no—
92C4×C3⋊D3C2×A413yescomplemented
93C3×D12—1212no—
94C3×C4×D3—1212no—
95C22×#41—91noSylow
96C2×C33—41no—
97C3×C3⋊D3—83no—
98C23×D3—312no—
99C22×A4—93no—
100C3×#41—361no—
101C2×C3×Q8—312no—
102C2×C3×Q8—34no—
103C22×C3×C4C2×C3×D3112yes—
104C22×Dic3—34no—
105#239—34no—
106#239—912no—
107#239—912no—
108#239—98no—
109#239—924no—
110#239—924no—
111#239—94no—
112C2×D12—912no—
113C2×D12—924no—
114C2×D12—912no—
115C2×C4×D3—912no—
116C2×C4×D3—912no—
117C2×C4×D3—912no—
118C2×C4×D3—912no—
119C4×A4—212no—
120C4×A4—91no—
121C4×A4C2×C3⋊D313yes—
122C22×C32—31no—
123C22×C32—31no—
124C22×C32C3×#4111yescharacteristic, complemented
125C2×C3⋊D3—33no—
126C2×C3⋊D3—63no—
127C2×C3⋊D3—63no—
128C2×C3⋊D3—63no—
129C2×C3⋊D3—63no—
130C2×C3⋊D3C22×A413yes—
131C2×C3×D3—1212no—
132C3×A4—24no—
133C3×A4#23914yescomplemented
134C32×C4—33no—
135C32×C4—412no—
136C32×C4—812no—
137C32×C4C22×A413yes—
138C3⋊Dic3—31no—
139C3⋊Dic3C22×A411yescharacteristic
140C3×Dic3—124no—
141C2×#41—271no—
142C2×#41—273no—
143C2×#41—273no—
144C2×#41—271no—
145C22×Q8C3×C3⋊D311yescharacteristic, complemented
146C22×D4—93no—
147C23×C4—93no—
148C33—41noSylow
149C23×C3C22×C3×D314yes—
150C22×D3—612no—
151C22×D3—912no—
152C22×D3—912no—
153C22×D3—1812no—
154C2×A4—24no—
155C2×A4—183no—
156C2×A4C22×C3⋊D311yescharacteristic
157C3×Q8—38no—
158C3×Q8—312no—
159C3×Q8—41no—
160C3×Q8—84no—
161C3×Q8D3×A414yescomplemented
162C3×D4—363no—
163C2×C3×C4—312no—
164C2×C3×C4—312no—
165C2×C3×C4—363no—
166C2×Dic3—94no—
167C2×Dic3—94no—
168D12—312no—
169D12—924no—
170D12—924no—
171D12—912no—
172C4×D3—312no—
173C4×D3—912no—
174C4×D3—912no—
175C4×D3—924no—
176C4×D3—912no—
177C2×C32—31no—
178C2×C32—31no—
179C2×C32—44no—
180C2×C32—84no—
181C2×C32C23×A411yescharacteristic
182C3⋊D3—23no—
183C3⋊D3—63no—
184C3×D3—2412no—
185C24—93no—
186#41—91no—
187#41—273no—
188#41—273no—
189#41—272no—
190#41—276no—
191#41—276no—
192#41—271no—
193C2×Q8—33no—
194C2×Q8—31no—
195C2×D4—273no—
196C2×D4—273no—
197C2×D4—276no—
198C22×C4—91no—
199C22×C4—273no—
200C22×C4—273no—
201C22×C4—273no—
202C22×C4—273no—
203C22×C4C2×C3×C3⋊D313yes—
204C22×C3—34no—
205C22×C3—34no—
206C22×C3—363no—
207C22×C3C3×#23914yescomplemented
208C2×D3—312no—
209C2×D3—912no—
210C2×D3—1812no—
211C2×D3—1812no—
212C2×D3—1812no—
213C2×D3—1812no—
214A4—24no—
215A4#383211yescharacteristic, direct product factor
216C3×C4—312no—
217C3×C4—43no—
218C3×C4—812no—
219C3×C4—361no—
220C3×C4C2×D3×A4112yes—
221Dic3—34no—
222Dic3—94no—
223C32—44no—
224C32—84no—
225C32A4×#4111yescharacteristic, complemented
226C23—183no—
227C23—273no—
228C23—273no—
229C23—543no—
230C23C22×C3×C3⋊D311yescharacteristic
231Q8—32no—
232Q8—33no—
233Q8A4×C3⋊D311yescharacteristic, complemented
234D4—93no—
235D4—276no—
236D4—276no—
237D4—273no—
238C2×C4—33no—
239C2×C4—33no—
240C2×C4—93no—
241C2×C4—273no—
242C2×C4—273no—
243C2×C4—276no—
244C2×C4—271no—
245C2×C4—273no—
246C2×C4—271no—
247C2×C3—34no—
248C2×C3—34no—
249C2×C3—41no—
250C2×C3—84no—
251C2×C3—723no—
252C2×C3C22×D3×A414yes—
253D3—612no—
254D3—1812no—
255C22—31no—
256C22—31no—
257C22—93no—
258C22—273no—
259C22—543no—
260C22—543no—
261C22—543no—
262C22—543no—
263C22C3×#383211yescharacteristic, complemented
264C4—33no—
265C4—91no—
266C4—271no—
267C4C2×A4×C3⋊D313yes—
268C3—41no—
269C3—84no—
270C3A4×#23914yescomplemented
271C2—31no—
272C2—31no—
273C2—183no—
274C2—543no—
275C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×A4×C3⋊D3
commutator, socle, cosocleC23×C32C23×C3
FittingC22×C32×Q8C2×C3
hypercenterQ8A4×C3⋊D3
hypocenterC22×C32C3×#41
LCSG ▷ C23×C32 ▷ C22×C32
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C32×Q8×A4C2semidirect
2. A4×C4⋊(C3⋊D3)C2semidirect
3. C4×A4×C3⋊D3C2semidirect
4. C22×#3832C3semidirect
10. C22×C32×Q8C2×C3semidirect
15. C22×C4⋊(C3⋊D3)C2×C3semidirect
16. C22×C4×C3⋊D3C2×C3semidirect
18. C3×Q8×A4D3semidirect
40. #3832A4direct (N × H)
53. C32×A4#41semidirect
56. C22×C3×Q8C3×D3semidirect
64. Q8×A4C3⋊D3semidirect
79. C32×Q8C2×A4semidirect
87. C4⋊(C3⋊D3)C2×A4semidirect
92. C4×C3⋊D3C2×A4semidirect
124. C22×C32C3×#41semidirect
133. C3×A4#239semidirect
145. C22×Q8C3×C3⋊D3semidirect
161. C3×Q8D3×A4semidirect
207. C22×C3C3×#239semidirect
215. A4#3832direct (N × H)
225. C32A4×#41semidirect
233. Q8A4×C3⋊D3semidirect
263. C22C3×#3832semidirect
270. C3A4×#239semidirect

Maximal subgroups

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

subgroupindex
C32×Q8×A42
A4×C4⋊(C3⋊D3)2
C4×A4×C3⋊D32
C22×#38323
A4×#2393
C3×#38324