Finite Group Databasebeta

← C3×A4×#125 #11919702

Subgroups — C3×A4×#125 #11919702

Order 1152.

696 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 176 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⋊C4C214yesmaximal
2C3×A4×C4⋊C4C212yesmaximal
3C3×C42×A4C211yesmaximal, characteristic
4C22×C3×#125C311yesmaximal, characteristic, complemented
5A4×#125C313yesmaximal, direct product factor
6C2×C3×C4×A4C2212yes—
7C2×C3×C4×A4C2214yes—
8C2×C3×C4×A4C2211yescharacteristic
9C32×#125—41nomaximal
10C2×C3×#125—31no—
11C2×C3×#125—32no—
12C2×C3×#125—34no—
13C2×C3×#125—31no—
14C22×C3×C4⋊C4C2×C314yes—
15C22×C3×C4⋊C4C2×C312yes—
16C22×C3×C42C2×C311yescharacteristic
17A4×C4⋊C4C2×C3112yes—
18A4×C4⋊C4C2×C316yes—
19C42×A4C2×C313yes—
20C22×C3×A4C2311yescharacteristic
21C3×C4×A4—22no—
22C3×C4×A4—24no—
23C3×C4×A4Q812yes—
24C32×C4⋊C4—42no—
25C32×C4⋊C4—44no—
26C32×C42—41no—
27C22×#125C3211yescharacteristic, complemented, Sylow
28C23×C3×C4C22×C312yes—
29C23×C3×C4C22×C314yes—
30C23×C3×C4C22×C311yescharacteristic
31C2×C4×A4C22×C316yes—
32C2×C4×A4C22×C3112yes—
33C2×C4×A4C22×C313yes—
34C3×#125—38no—
35C3×#125—34no—
36C3×#125—32no—
37C3×#125—34no—
38C3×#125—32no—
39C3×#125—31no—
40C3×#125—43no—
41C3×#125A411yescharacteristic, direct product factor
42C2×C3×C4⋊C4—32no—
43C2×C3×C4⋊C4—34no—
44C2×C3×C4⋊C4—32no—
45C2×C3×C4⋊C4—34no—
46C2×C3×C4⋊C4—34no—
47C2×C3×C4⋊C4—38no—
48C2×C3×C42—31no—
49C2×C3×C42—31no—
50C2×C3×C42—32no—
51C2×C3×A4C2×Q811yescharacteristic
52C2×C3×A4#4112yes—
53C2×C32×C4—41no—
54C2×C32×C4—42no—
55C2×C32×C4—44no—
56C2×#125—31no—
57C2×#125—32no—
58C2×#125—34no—
59C2×#125—31no—
60C22×C4⋊C4C2×C3214yes—
61C22×C4⋊C4C2×C3212yes—
62C22×C42C2×C3211yescharacteristic
63C24×C3C23×C311yescharacteristic
64C22×A4C23×C313yes—
65C22×C3×C4—22no—
66C22×C3×C4—24no—
67C22×C3×C4—31no—
68C22×C3×C4—31no—
69C22×C3×C4—32no—
70C22×C3×C4—32no—
71C22×C3×C4—34no—
72C22×C3×C4—32no—
73C22×C3×C4—34no—
74C22×C3×C4—62no—
75C22×C3×C4—64no—
76C22×C3×C4C3×Q812yes—
77C4×A4—26no—
78C4×A4—212no—
79C4×A4C3×Q816yes—
80C3×C4⋊C4—34no—
81C3×C4⋊C4—38no—
82C3×C4⋊C4—34no—
83C3×C4⋊C4—38no—
84C3×C4⋊C4—34no—
85C3×C4⋊C4—32no—
86C3×C4⋊C4—46no—
87C3×C4⋊C4—412no—
88C3×C4⋊C4C2×A414yes—
89C3×C4⋊C4C2×A412yes—
90C3×C42—32no—
91C3×C42—32no—
92C3×C42—31no—
93C3×C42—43no—
94C3×C42C2×A411yescharacteristic
95C22×C32—41no—
96C3×A4#12511yescharacteristic, direct product factor
97C32×C4—42no—
98C32×C4—82no—
99C32×C4—84no—
100C23×C4C22×C3214yes—
101C23×C4C22×C3212yes—
102C23×C4C22×C3211yescharacteristic
103#125—38no—
104#125—34no—
105#125—32no—
106#125—34no—
107#125—32no—
108#125—31no—
109#125C3×A411yescharacteristic, direct product factor
110C2×C4⋊C4—32no—
111C2×C4⋊C4—34no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—34no—
114C2×C4⋊C4—34no—
115C2×C4⋊C4—38no—
116C2×C42—31no—
117C2×C42—31no—
118C2×C42—32no—
119C23×C3—31no—
120C23×C3—32no—
121C23×C3—31no—
122C23×C3C2×C3×Q811yescharacteristic
123C23×C3C3×#4112yes—
124C2×A4C2×C3×Q813yes—
125C2×A4C3×#4116yes—
126C2×C3×C4—32no—
127C2×C3×C4—31no—
128C2×C3×C4—32no—
129C2×C3×C4—32no—
130C2×C3×C4—32no—
131C2×C3×C4—32no—
132C2×C3×C4—34no—
133C2×C3×C4—43no—
134C2×C3×C4—46no—
135C2×C3×C4—412no—
136C2×C3×C4—62no—
137C2×C3×C4—64no—
138C2×C3×C4—62no—
139C2×C3×C4—64no—
140C2×C3×C4—62no—
141C2×C3×C4—64no—
142C2×C3×C4—62no—
143C2×C3×C4—64no—
144C2×C3×C4C22×A412yes—
145C2×C3×C4C22×A414yes—
146C2×C3×C4C22×A411yescharacteristic
147C2×C32—41no—
148C2×C32—42no—
149C24C23×C3211yescharacteristic
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—31no—
153C22×C4—31no—
154C22×C4—32no—
155C22×C4—32no—
156C22×C4—34no—
157C22×C4—32no—
158C22×C4—34no—
159C22×C4—64no—
160C22×C4—62no—
161C22×C4C32×Q812yes—
162C4⋊C4—34no—
163C4⋊C4—38no—
164C4⋊C4—34no—
165C4⋊C4—38no—
166C4⋊C4—34no—
167C4⋊C4—32no—
168C4⋊C4C2×C3×A414yes—
169C4⋊C4C2×C3×A412yes—
170C42—32no—
171C42—32no—
172C42—31no—
173C42C2×C3×A411yescharacteristic
174C22×C3—31no—
175C22×C3—32no—
176C22×C3—32no—
177C22×C3—31no—
178C22×C3—31no—
179C22×C3—32no—
180C22×C3—32no—
181C22×C3—43no—
182C22×C3C3×#12511yescharacteristic, complemented
183C22×C3C23×A411yescharacteristic
184A4C3×#12513yesdirect product factor
185C3×C4—22no—
186C3×C4—24no—
187C3×C4—32no—
188C3×C4—46no—
189C3×C4—62no—
190C3×C4—64no—
191C3×C4—86no—
192C3×C4—812no—
193C3×C4Q8×A412yes—
194C32—41noSylow
195C23—31no—
196C23—32no—
197C23—31no—
198C23C2×C32×Q811yescharacteristic
199C23C32×#4112yes—
200C2×C4—32no—
201C2×C4—31no—
202C2×C4—32no—
203C2×C4—32no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—34no—
207C2×C4—62no—
208C2×C4—64no—
209C2×C4—62no—
210C2×C4—64no—
211C2×C4—62no—
212C2×C4—64no—
213C2×C4—62no—
214C2×C4—64no—
215C2×C4C22×C3×A412yes—
216C2×C4C22×C3×A414yes—
217C2×C4C22×C3×A411yescharacteristic
218C2×C3—31no—
219C2×C3—32no—
220C2×C3—31no—
221C2×C3—43no—
222C2×C3—46no—
223C2×C3C2×Q8×A411yescharacteristic
224C2×C3A4×#4112yes—
225C22—31no—
226C22—32no—
227C22—32no—
228C22—31no—
229C22—31no—
230C22—32no—
231C22—32no—
232C22C32×#12511yescharacteristic, complemented
233C22C23×C3×A411yescharacteristic
234C4—22no—
235C4—24no—
236C4—32no—
237C4—62no—
238C4—64no—
239C4C3×Q8×A412yes—
240C3—43no—
241C3A4×#12511yescharacteristic, direct product factor
242C2—31no—
243C2—31no—
244C2—32no—
245C2C2×C3×Q8×A411yescharacteristic
246C2C3×A4×#4112yes—
subgroupG/H
centerC22×C3C23×A4
commutator, cosocleC24C23×C32
FrattiniC22C23×C3×A4
FittingC22×C3×#125C3
socleC24×C3C23×C3
prime radicalC22×C3×A4C23
hypercenterC3×#125A4
hypocenterC22C32×#125
LCSG ▷ C24 ▷ C22
UCSC1 ◁ C22×C3 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
4. C22×C3×#125C3semidirect
5. A4×#125C3direct (N × H)
27. C22×#125C32semidirect
41. C3×#125A4direct (N × H)
96. C3×A4#125direct (N × H)
109. #125C3×A4direct (N × H)
182. C22×C3C3×#125semidirect
184. A4C3×#125direct (N × H)
232. C22C32×#125semidirect
241. C3A4×#125direct (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×C42×A42
C22×C3×#1253
A4×#1253
C32×#1254