Finite Group Databasebeta

← C4⋊C4×S4 #85306

Subgroups — C4⋊C4×S4 #85306

Order 384.

419 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×C4×S4C212yesmaximal
2C2×C4×S4C211yesmaximal, characteristic
3A4×C4⋊C4C211yesmaximal, characteristic, complemented
4C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5SL(2,ℤ/4)⋊C4C212yesmaximal, complemented
6D4×C4⋊C4—31nomaximal, Sylow
7C22×S4C2211yescharacteristic
8C2×C4×A4C2212yes—
9C2×C4×A4C2211yescharacteristic
10C2×SL(2,ℤ/4)C2212yes—
11C2×SL(2,ℤ/4)C2211yescharacteristic
12C4×S4—24no—
13C4×S4C414yescomplemented
14D3×C4⋊C4—41nomaximal
15C2×C4×D4—31no—
16C2×C4×D4—32no—
17C22×C4⋊C4—31no—
18C22×C4⋊C4D311yescharacteristic, complemented
19#389—32no—
20#385—32no—
21#385—32no—
22#384—31no—
23#380—31no—
24#380—31no—
25C4×C4⋊C4—31no—
26C22×A4C2311yescharacteristic
27C2×S4C2×C412yes—
28C2×S4D412yes—
29C2×S4Q812yes—
30C2×C4×D3—41no—
31C2×C4×D3—42no—
32C4×A4—22no—
33C4×A4C2×C412yescomplemented
34SL(2,ℤ/4)—22no—
35SL(2,ℤ/4)C2×C412yescomplemented
36C3×C4⋊C4—41no—
37C4⋊Dic3—41no—
38Dic3⋊C4—42no—
39C22×D4—31no—
40C23×C4—31no—
41C23×C4—32no—
42C23×C4C2×D312yes—
43C23×C4C2×D311yescharacteristic
44C4×D4—34no—
45C4×D4—34no—
46C4×D4—64no—
47C4×D4—64no—
48C2×C4⋊C4—31no—
49C2×C4⋊C4—31no—
50C2×C4⋊C4—31no—
51C2×C4⋊C4—32no—
52C2×C4⋊C4—31no—
53C2×C4⋊C4—32no—
54C2×C4⋊C4—32no—
55C2×C4⋊C4—31no—
56C2×C4⋊C4—32no—
57C2×C4⋊C4—62no—
58C2×C4⋊C4—61no—
59C2×C4⋊C4—61no—
60C2×C22⋊C4—31no—
61C2×C22⋊C4—31no—
62C2×C22⋊C4—32no—
63C2×C22⋊C4—32no—
64C2×C42—31no—
65C2×C42—32no—
66#95—32no—
67#95—32no—
68#95—32no—
69C22×D3—41no—
70C2×A4C22×C411yescharacteristic
71C2×A4C2×D411yescharacteristic
72C2×A4C2×Q811yescharacteristic
73S4C4⋊C414yesdirect product factor
74C2×C3×C4—41no—
75C2×C3×C4—42no—
76C2×Dic3—41no—
77C2×Dic3—42no—
78C4×D3—44no—
79C4×D3—84no—
80C24—31no—
81C24C22×D311yescharacteristic
82C2×D4—32no—
83C2×D4—32no—
84C2×D4—32no—
85C2×D4—32no—
86C2×D4—32no—
87C2×D4—32no—
88C22×C4—22no—
89C22×C4—31no—
90C22×C4—31no—
91C22×C4—31no—
92C22×C4—32no—
93C22×C4—31no—
94C22×C4—31no—
95C22×C4—34no—
96C22×C4—32no—
97C22×C4—32no—
98C22×C4—32no—
99C22×C4—32no—
100C22×C4—61no—
101C22×C4—62no—
102C22×C4—62no—
103C22×C4—61no—
104C22×C4—62no—
105C22×C4—64no—
106C22×C4—62no—
107C22×C4—122no—
108C22×C4C4×D312yescomplemented
109C4⋊C4—34no—
110C4⋊C4—32no—
111C4⋊C4—31no—
112C4⋊C4—61no—
113C4⋊C4—62no—
114C4⋊C4—61no—
115C4⋊C4—62no—
116C4⋊C4—64no—
117C4⋊C4—62no—
118C4⋊C4—61no—
119C4⋊C4—122no—
120C4⋊C4S411yescharacteristic, direct product factor
121C22⋊C4—34no—
122C22⋊C4—32no—
123C22⋊C4—32no—
124C22⋊C4—64no—
125C22⋊C4—62no—
126C22⋊C4—62no—
127C42—34no—
128C42—64no—
129C22×C3—41no—
130C2×D3—42no—
131C2×D3—42no—
132C2×D3—42no—
133A4C2×C4⋊C411yescharacteristic, complemented
134C3×C4—42no—
135C3×C4—82no—
136Dic3—42no—
137Dic3—82no—
138C23—31no—
139C23—31no—
140C23—31no—
141C23—31no—
142C23—32no—
143C23—32no—
144C23—32no—
145C23—61no—
146C23—61no—
147C23—61no—
148C23—61no—
149C23C2×C4×D311yescharacteristic
150C23D3×D411yescharacteristic
151C23D3×Q811yescharacteristic
152D4—34no—
153D4—34no—
154D4—34no—
155D4—34no—
156C2×C4—32no—
157C2×C4—32no—
158C2×C4—32no—
159C2×C4—31no—
160C2×C4—32no—
161C2×C4—32no—
162C2×C4—32no—
163C2×C4—32no—
164C2×C4—32no—
165C2×C4—32no—
166C2×C4—62no—
167C2×C4—62no—
168C2×C4—61no—
169C2×C4—64no—
170C2×C4—61no—
171C2×C4—61no—
172C2×C4—61no—
173C2×C4—61no—
174C2×C4—64no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—124no—
184C2×C4—124no—
185C2×C4C2×S412yes—
186C2×C4C2×S411yescharacteristic
187C2×C3—41no—
188C2×C3—41no—
189C2×C3—41no—
190D3—44no—
191C22—31no—
192C22—31no—
193C22—31no—
194C22—31no—
195C22—31no—
196C22—31no—
197C22—31no—
198C22—31no—
199C22—31no—
200C22—34no—
201C22—61no—
202C22—62no—
203C22—62no—
204C22—62no—
205C22—62no—
206C22—62no—
207C22—62no—
208C22D3×C4⋊C411yescharacteristic, complemented
209C22C22×S411yescharacteristic
210C4—22no—
211C4—34no—
212C4—32no—
213C4—62no—
214C4—62no—
215C4—62no—
216C4—122no—
217C4—122no—
218C4C4×S412yescomplemented
219C3—41noSylow
220C2—31no—
221C2—31no—
222C2—31no—
223C2—31no—
224C2—64no—
225C2C2×C4×S411yescharacteristic
226C2D4×S411yescharacteristic
227C2Q8×S411yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutatorC2×A4C22×C4
FittingC22×C4⋊C4D3
socleC24C22×D3
prime radicalC22×S4C22
cosocleC22×A4C23
hypercenterC4⋊C4S4
hypocenterA4C2×C4⋊C4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. A4×C4⋊C4C2semidirect
4. C4⋊SL(2,ℤ/4)C2semidirect
5. SL(2,ℤ/4)⋊C4C2semidirect
13. C4×S4C4semidirect
18. C22×C4⋊C4D3semidirect
33. C4×A4C2×C4semidirect
35. SL(2,ℤ/4)C2×C4semidirect
73. S4C4⋊C4direct (N × H)
108. C22×C4C4×D3semidirect
120. C4⋊C4S4direct (N × H)
133. A4C2×C4⋊C4semidirect
208. C22D3×C4⋊C4semidirect
218. C4C4×S4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×S42
A4×C4⋊C42
C4⋊SL(2,ℤ/4)2
SL(2,ℤ/4)⋊C42
D4×C4⋊C43
D3×C4⋊C44