Finite Group Databasebeta

← C2×C3×C4×C22⋊A4 #11919886

Subgroups — C2×C3×C4×C22⋊A4 #11919886

Order 1152.

3816 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 3816 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C22×C3×C22⋊A4C211yesmaximal, characteristic
2C3×C4×C22⋊A4C212yesmaximal, direct product factor
3C25×C3×C4C311yesmaximal, characteristic, complemented
4C2×C4×C22⋊A4C313yesmaximal, direct product factor
5C2×C3×C22⋊A4C412yesdirect product factor
6C2×C3×C22⋊A4C2211yescharacteristic
7C2×C3×C4×A4—45nomaximal
8C26×C3C2×C311yescharacteristic
9C22×C22⋊A4C2×C313yes—
10C24×C3×C4—310no—
11C24×C3×C4—35no—
12C24×C3×C4—35no—
13C24×C3×C4C2×C312yescomplemented
14C4×C22⋊A4C2×C316yesdirect product factor
15C3×C22⋊A4C2×C411yescharacteristic, direct product factor
16C22×C3×A4—45no—
17C3×C4×A4—410no—
18C25×C4C3211yescharacteristic, complemented, Sylow
19C25×C3—35no—
20C25×C3—310no—
21C25×C3—35no—
22C25×C3C3×C412yescomplemented
23C25×C3C22×C311yescharacteristic
24C2×C22⋊A4C3×C416yesdirect product factor
25C2×C22⋊A4C22×C313yes—
26C23×C3×C4—360no—
27C23×C3×C4—360no—
28C23×C3×C4—330no—
29C23×C3×C4—310no—
30C23×C3×C4—310no—
31C23×C3×C4—35no—
32C23×C3×C4—310no—
33C23×C3×C4—310no—
34C23×C3×C4—310no—
35C23×C3×C4A415yescomplemented
36C2×C4×A4—415no—
37C2×C3×A4—410no—
38C2×C3×A4—45no—
39C2×C32×C4—161no—
40C26C2×C3211yescharacteristic
41C24×C4—35no—
42C24×C4—35no—
43C24×C4—310no—
44C24×C4C2×C3212yescomplemented
45C24×C3—310no—
46C24×C3—35no—
47C24×C3—360no—
48C24×C3—360no—
49C24×C3—310no—
50C24×C3—330no—
51C24×C3—310no—
52C24×C3—310no—
53C24×C3—310no—
54C24×C3—35no—
55C24×C3—35no—
56C24×C3C2×C3×C411yescharacteristic, complemented
57C24×C3C2×A415yes—
58C22⋊A4C2×C3×C413yesdirect product factor
59C22×A4—415no—
60C22×C3×C4—330no—
61C22×C3×C4—3120no—
62C22×C3×C4—360no—
63C22×C3×C4—360no—
64C22×C3×C4—330no—
65C22×C3×C4—360no—
66C22×C3×C4—320no—
67C22×C3×C4—310no—
68C22×C3×C4—35no—
69C22×C3×C4—35no—
70C22×C3×C4—310no—
71C22×C3×C4C2×A4110yescomplemented
72C4×A4—430no—
73C22×C32—161no—
74C3×A4—45no—
75C32×C4—162no—
76C25—35no—
77C25—310no—
78C25—35no—
79C25C32×C412yescomplemented
80C25C22×C3211yescharacteristic
81C23×C4—360no—
82C23×C4—360no—
83C23×C4—330no—
84C23×C4—310no—
85C23×C4—310no—
86C23×C4—35no—
87C23×C4—310no—
88C23×C4—310no—
89C23×C4—310no—
90C23×C4C3×A415yescomplemented
91C23×C3—35no—
92C23×C3—310no—
93C23×C3—330no—
94C23×C3—310no—
95C23×C3—360no—
96C23×C3—360no—
97C23×C3—330no—
98C23×C3—330no—
99C23×C3—3120no—
100C23×C3—360no—
101C23×C3—320no—
102C23×C3—310no—
103C23×C3—35no—
104C23×C3—35no—
105C23×C3—35no—
106C23×C3C4×A4110yescomplemented
107C23×C3C22×A415yes—
108C2×A4—415no—
109C2×A4—430no—
110C2×C3×C4—360no—
111C2×C3×C4—310no—
112C2×C3×C4—360no—
113C2×C3×C4—35no—
114C2×C3×C4—310no—
115C2×C3×C4—310no—
116C2×C3×C4—310no—
117C2×C3×C4—163no—
118C2×C3×C4C22⋊A411yescharacteristic, direct product factor
119C2×C32—162no—
120C2×C32—161no—
121C24—310no—
122C24—35no—
123C24—330no—
124C24—310no—
125C24—360no—
126C24—360no—
127C24—310no—
128C24—35no—
129C24—35no—
130C24—310no—
131C24—310no—
132C24C2×C32×C411yescharacteristic, complemented
133C24C2×C3×A415yes—
134C22×C4—330no—
135C22×C4—3120no—
136C22×C4—360no—
137C22×C4—360no—
138C22×C4—330no—
139C22×C4—320no—
140C22×C4—360no—
141C22×C4—310no—
142C22×C4—35no—
143C22×C4—35no—
144C22×C4—310no—
145C22×C4C2×C3×A4110yescomplemented
146C22×C3—310no—
147C22×C3—35no—
148C22×C3—310no—
149C22×C3—360no—
150C22×C3—360no—
151C22×C3—330no—
152C22×C3—310no—
153C22×C3—310no—
154C22×C3—310no—
155C22×C3—35no—
156C22×C3—35no—
157C22×C3—163no—
158C22×C3C2×C4×A415yescomplemented
159C22×C3C2×C22⋊A411yescharacteristic
160A4—415no—
161C3×C4—310no—
162C3×C4—166no—
163C3×C4C2×C22⋊A412yesdirect product factor
164C32—161noSylow
165C23—35no—
166C23—310no—
167C23—310no—
168C23—330no—
169C23—330no—
170C23—330no—
171C23—360no—
172C23—360no—
173C23—3120no—
174C23—320no—
175C23—360no—
176C23—35no—
177C23—310no—
178C23—35no—
179C23—35no—
180C23C3×C4×A4110yescomplemented
181C23C22×C3×A415yes—
182C2×C4—360no—
183C2×C4—360no—
184C2×C4—310no—
185C2×C4—35no—
186C2×C4—310no—
187C2×C4—310no—
188C2×C4—310no—
189C2×C4C3×C22⋊A411yescharacteristic, direct product factor
190C2×C3—310no—
191C2×C3—35no—
192C2×C3—35no—
193C2×C3—166no—
194C2×C3—163no—
195C2×C3C4×C22⋊A412yesdirect product factor
196C2×C3C22×C22⋊A411yescharacteristic
197C22—35no—
198C22—310no—
199C22—310no—
200C22—330no—
201C22—360no—
202C22—310no—
203C22—360no—
204C22—310no—
205C22—310no—
206C22—35no—
207C22—35no—
208C22C2×C3×C4×A415yescomplemented
209C22C2×C3×C22⋊A411yescharacteristic
210C4—310no—
211C4C2×C3×C22⋊A412yesdirect product factor
212C3—163no—
213C3C2×C4×C22⋊A411yescharacteristic, direct product factor
214C2—35no—
215C2—35no—
216C2—310no—
217C2C3×C4×C22⋊A412yesdirect product factor
218C2C22×C3×C22⋊A411yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4C22⋊A4
commutator, hypocenterC24C2×C32×C4
FrattiniC2C22×C3×C22⋊A4
FittingC25×C3×C4C3
socleC26×C3C2×C3
prime radicalC22×C3×C22⋊A4C2
cosocleC25C22×C32
LCSG ▷ C24
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×C22⋊A4C2direct (N × H)
3. C25×C3×C4C3semidirect
4. C2×C4×C22⋊A4C3direct (N × H)
5. C2×C3×C22⋊A4C4direct (N × H)
13. C24×C3×C4C2×C3semidirect
14. C4×C22⋊A4C2×C3direct (N × H)
15. C3×C22⋊A4C2×C4direct (N × H)
18. C25×C4C32semidirect
22. C25×C3C3×C4semidirect
24. C2×C22⋊A4C3×C4direct (N × H)
35. C23×C3×C4A4semidirect
44. C24×C4C2×C32semidirect
56. C24×C3C2×C3×C4semidirect
58. C22⋊A4C2×C3×C4direct (N × H)
71. C22×C3×C4C2×A4semidirect
79. C25C32×C4semidirect
90. C23×C4C3×A4semidirect
106. C23×C3C4×A4semidirect
118. C2×C3×C4C22⋊A4direct (N × H)
132. C24C2×C32×C4semidirect
145. C22×C4C2×C3×A4semidirect
158. C22×C3C2×C4×A4semidirect
163. C3×C4C2×C22⋊A4direct (N × H)
180. C23C3×C4×A4semidirect
189. C2×C4C3×C22⋊A4direct (N × H)
195. C2×C3C4×C22⋊A4direct (N × H)
208. C22C2×C3×C4×A4semidirect
211. C4C2×C3×C22⋊A4direct (N × H)
213. C3C2×C4×C22⋊A4direct (N × H)
217. C2C3×C4×C22⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊A42
C3×C4×C22⋊A42
C25×C3×C43
C2×C4×C22⋊A43
C2×C3×C4×A44