Finite Group Databasebeta

← C3×A4×#126 #11919706

Subgroups — C3×A4×#126 #11919706

Order 1152.

760 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 160 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⋊C4C213yesmaximal, complemented
2C3×A4×C22⋊C4C213yesmaximal
3C3×C42×A4C211yesmaximal, characteristic, complemented
4C22×C3×#126C311yesmaximal, characteristic, complemented
5A4×#126C313yesmaximal, direct product factor
6C23×C3×A4C2211yescharacteristic
7C2×C3×C4×A4C2213yes—
8C2×C3×C4×A4C2213yes—
9C32×#126—41nomaximal
10C2×C3×#126—31no—
11C2×C3×#126—33no—
12C2×C3×#126—33no—
13C2×C3×#126—31no—
14C22×C3×C4⋊C4C2×C313yescomplemented
15C22×C3×C22⋊C4C2×C313yes—
16C22×C3×C42C2×C311yescharacteristic, complemented
17A4×C4⋊C4C2×C319yescomplemented
18A4×C22⋊C4C2×C319yes—
19C42×A4C2×C313yescomplemented
20C22×C3×A4—23no—
21C22×C3×A4C2311yescharacteristic
22C3×C4×A4—23no—
23C3×C4×A4—23no—
24C32×C4⋊C4—43no—
25C32×C22⋊C4—43no—
26C32×C42—41no—
27C22×#126C3211yescharacteristic, complemented, Sylow
28C25×C3C22×C311yescharacteristic
29C23×A4C22×C313yes—
30C23×C3×C4C22×C313yes—
31C23×C3×C4C22×C313yes—
32C2×C4×A4C22×C319yes—
33C2×C4×A4C22×C319yes—
34C3×#126—36no—
35C3×#126—32no—
36C3×#126—36no—
37C3×#126—33no—
38C3×#126—33no—
39C3×#126—31no—
40C3×#126—43no—
41C3×#126A411yescharacteristic, direct product factor
42C2×C3×C4⋊C4—33no—
43C2×C3×C4⋊C4—33no—
44C2×C3×C4⋊C4—36no—
45C2×C3×C22⋊C4—33no—
46C2×C3×C22⋊C4—33no—
47C2×C3×C22⋊C4—33no—
48C2×C3×C22⋊C4—33no—
49C2×C3×C42—31no—
50C2×C3×C42—33no—
51C23×C32—41no—
52C2×C3×A4—41no—
53C2×C3×A4#4113yes—
54C2×C32×C4—43no—
55C2×C32×C4—43no—
56C2×#126—31no—
57C2×#126—33no—
58C2×#126—33no—
59C2×#126—31no—
60C22×C4⋊C4C2×C3213yescomplemented
61C22×C22⋊C4C2×C3213yes—
62C22×C42C2×C3211yescharacteristic, complemented
63C24×C3—23no—
64C24×C3—31no—
65C24×C3—31no—
66C24×C3—63no—
67C24×C3C23×C311yescharacteristic
68C22×A4—29no—
69C22×A4C23×C313yes—
70C22×C3×C4—23no—
71C22×C3×C4—23no—
72C22×C3×C4—33no—
73C22×C3×C4—33no—
74C22×C3×C4—33no—
75C22×C3×C4—33no—
76C22×C3×C4—63no—
77C22×C3×C4—63no—
78C4×A4—29no—
79C4×A4—29no—
80C3×C4⋊C4—36no—
81C3×C4⋊C4—36no—
82C3×C4⋊C4—33no—
83C3×C4⋊C4—49no—
84C3×C4⋊C4C2×A413yescomplemented
85C3×C22⋊C4—36no—
86C3×C22⋊C4—33no—
87C3×C22⋊C4—33no—
88C3×C22⋊C4—33no—
89C3×C22⋊C4—49no—
90C3×C22⋊C4C2×A413yes—
91C3×C42—32no—
92C3×C42—33no—
93C3×C42—43no—
94C3×C42C2×A411yescharacteristic, complemented
95C22×C32—41no—
96C22×C32—83no—
97C3×A4#12611yescharacteristic, direct product factor
98C32×C4—83no—
99C32×C4—83no—
100C25C22×C3211yescharacteristic
101C23×C4C22×C3213yes—
102C23×C4C22×C3213yes—
103#126—36no—
104#126—32no—
105#126—36no—
106#126—33no—
107#126—33no—
108#126—31no—
109#126C3×A411yescharacteristic, direct product factor
110C2×C4⋊C4—33no—
111C2×C4⋊C4—33no—
112C2×C4⋊C4—36no—
113C2×C22⋊C4—33no—
114C2×C22⋊C4—33no—
115C2×C22⋊C4—33no—
116C2×C22⋊C4—33no—
117C2×C42—31no—
118C2×C42—33no—
119C23×C3—33no—
120C23×C3—31no—
121C23×C3—31no—
122C23×C3—41no—
123C23×C3—43no—
124C23×C3—63no—
125C23×C3—63no—
126C23×C3—63no—
127C23×C3—63no—
128C23×C3—123no—
129C23×C3—122no—
130C23×C3C3×#4113yes—
131C23×C3C22×A411yescharacteristic
132C2×A4—43no—
133C2×A4C3×#4119yes—
134C2×C3×C4—33no—
135C2×C3×C4—33no—
136C2×C3×C4—49no—
137C2×C3×C4—49no—
138C2×C3×C4—63no—
139C2×C3×C4—63no—
140C2×C3×C4—63no—
141C2×C3×C4—63no—
142C2×C3×C4—63no—
143C2×C3×C4—63no—
144C2×C3×C4—63no—
145C2×C3×C4—63no—
146C2×C3×C4C22×A413yes—
147C2×C3×C4C22×A413yes—
148C2×C32—43no—
149C2×C32—161no—
150C24—23no—
151C24—31no—
152C24—31no—
153C24—63no—
154C24C23×C3211yescharacteristic
155C22×C4—23no—
156C22×C4—23no—
157C22×C4—33no—
158C22×C4—33no—
159C22×C4—33no—
160C22×C4—33no—
161C22×C4—63no—
162C22×C4—63no—
163C4⋊C4—36no—
164C4⋊C4—36no—
165C4⋊C4—33no—
166C4⋊C4C2×C3×A413yescomplemented
167C22⋊C4—36no—
168C22⋊C4—33no—
169C22⋊C4—33no—
170C22⋊C4—33no—
171C22⋊C4C2×C3×A413yes—
172C42—32no—
173C42—33no—
174C42C2×C3×A411yescharacteristic, complemented
175C22×C3—23no—
176C22×C3—33no—
177C22×C3—32no—
178C22×C3—33no—
179C22×C3—33no—
180C22×C3—43no—
181C22×C3—63no—
182C22×C3—89no—
183C22×C3—121no—
184C22×C3—123no—
185C22×C3—121no—
186C22×C3—123no—
187C22×C3C3×#12611yescharacteristic, complemented
188C22×C3C23×A411yescharacteristic
189A4C3×#12613yesdirect product factor
190C3×C4—23no—
191C3×C4—23no—
192C3×C4—63no—
193C3×C4—63no—
194C3×C4—89no—
195C3×C4—89no—
196C32—41noSylow
197C23—33no—
198C23—31no—
199C23—31no—
200C23—41no—
201C23—63no—
202C23—63no—
203C23—63no—
204C23—63no—
205C23—123no—
206C23—122no—
207C23C32×#4113yes—
208C23C22×C3×A411yescharacteristic
209C2×C4—33no—
210C2×C4—33no—
211C2×C4—63no—
212C2×C4—63no—
213C2×C4—63no—
214C2×C4—63no—
215C2×C4—63no—
216C2×C4—63no—
217C2×C4—63no—
218C2×C4—63no—
219C2×C4C22×C3×A413yes—
220C2×C4C22×C3×A413yes—
221C2×C3—33no—
222C2×C3—31no—
223C2×C3—41no—
224C2×C3—49no—
225C2×C3—121no—
226C2×C3—163no—
227C2×C3A4×#4113yes—
228C22—23no—
229C22—33no—
230C22—32no—
231C22—33no—
232C22—33no—
233C22—63no—
234C22—121no—
235C22—123no—
236C22—121no—
237C22—123no—
238C22C32×#12611yescharacteristic, complemented
239C22C23×C3×A411yescharacteristic
240C4—23no—
241C4—23no—
242C4—63no—
243C4—63no—
244C3—43no—
245C3A4×#12611yescharacteristic, direct product factor
246C2—31no—
247C2—33no—
248C2—41no—
249C2—121no—
250C2C3×A4×#4113yes—
subgroupG/H
centerC22×C3C23×A4
commutator, cosocleC24C23×C32
FrattiniC22C23×C3×A4
FittingC22×C3×#126C3
socleC24×C3C23×C3
prime radicalC23×C3×A4C22
hypercenterC3×#126A4
hypocenterC22C32×#126
LCSG ▷ C24 ▷ C22
UCSC1 ◁ C22×C3 ◁ C3×#126

Complements

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

NH ≅ G/Ntype
1. C3×A4×C4⋊C4C2semidirect
3. C3×C42×A4C2semidirect
4. C22×C3×#126C3semidirect
5. A4×#126C3direct (N × H)
14. C22×C3×C4⋊C4C2×C3semidirect
16. C22×C3×C42C2×C3semidirect
17. A4×C4⋊C4C2×C3semidirect
19. C42×A4C2×C3semidirect
27. C22×#126C32semidirect
41. C3×#126A4direct (N × H)
60. C22×C4⋊C4C2×C32semidirect
62. C22×C42C2×C32semidirect
84. C3×C4⋊C4C2×A4semidirect
94. C3×C42C2×A4semidirect
97. C3×A4#126direct (N × H)
109. #126C3×A4direct (N × H)
166. C4⋊C4C2×C3×A4semidirect
174. C42C2×C3×A4semidirect
187. C22×C3C3×#126semidirect
189. A4C3×#126direct (N × H)
238. C22C32×#126semidirect
245. C3A4×#126direct (N × H)

Maximal subgroups

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

subgroupindex
C3×A4×C4⋊C42
C3×A4×C22⋊C42
C3×C42×A42
C22×C3×#1263
A4×#1263
C32×#1264