Finite Group Databasebeta

← C4×C4⋊S4 #85288

Subgroups — C4×C4⋊S4 #85288

Order 384.

433 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 58 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⋊S4C211yesmaximal, characteristic
2C2×C4×S4C212yesmaximal, complemented
3C42×A4C211yesmaximal, characteristic, complemented
4(C2×S4)⋊C4C212yesmaximal, complemented
5C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6C4×#121—31nomaximal, Sylow
7C22×S4C2212yes—
8C2×C4×A4C2211yescharacteristic
9C2×C4×A4C2211yescharacteristic
10C2×C4×A4C2211yescharacteristic
11C2×SL(2,ℤ/4)C2212yes—
12C4⋊S4C414yesdirect product factor
13C4×S4—24no—
14C4×D12—41nomaximal
15C2×#121—31no—
16C2×C4×D4—31no—
17C2×C4×D4—32no—
18C22×C42D311yescharacteristic, complemented
19#390—31no—
20#389—31no—
21#388—32no—
22#386—32no—
23#380—31no—
24C4×C4⋊C4—31no—
25C4×C22⋊C4—32no—
26C22×A4C2311yescharacteristic
27C2×S4—22no—
28C2×S4—22no—
29C2×S4C2×C414yescomplemented
30C2×D12—41no—
31C2×C4×D3—42no—
32C4×A4—21no—
33C4×A4C2×C412yescomplemented
34C4×A4D412yescomplemented
35SL(2,ℤ/4)—22no—
36C3×C42—41no—
37(C2×D3)⋊C4—42no—
38C4⋊Dic3—41no—
39C22×D4—31no—
40C22×D4—32no—
41C23×C4—32no—
42C23×C4C2×D311yescharacteristic
43C23×C4C2×D311yescharacteristic
44C23×C4C2×D311yescharacteristic
45#121—34no—
46#121—34no—
47C4×D4—61no—
48C4×D4—61no—
49C4×D4—61no—
50C4×D4—64no—
51C4×D4—64no—
52C4×D4—61no—
53C2×C4⋊C4—31no—
54C2×C4⋊C4—31no—
55C2×C4⋊C4—32no—
56C2×C4⋊C4—31no—
57C2×C22⋊C4—32no—
58C2×C22⋊C4—32no—
59C2×C22⋊C4—32no—
60C2×C22⋊C4—32no—
61C2×C22⋊C4—32no—
62C2×C42—31no—
63C2×C42—31no—
64C2×C42—31no—
65C2×C42—31no—
66C2×C42—32no—
67#95—32no—
68#95—32no—
69#95—32no—
70C22×D3—42no—
71C2×A4C22×C411yescharacteristic
72C2×A4C2×D411yescharacteristic
73C2×A4#4111yescharacteristic
74S4—24no—
75C2×C3×C4—41no—
76C2×C3×C4—41no—
77C2×C3×C4—41no—
78C2×Dic3—42no—
79D12—44no—
80C4×D3—84no—
81C24—32no—
82C24C22×D311yescharacteristic
83C2×D4—34no—
84C2×D4—34no—
85C2×D4—34no—
86C2×D4—61no—
87C2×D4—62no—
88C2×D4—62no—
89C2×D4—62no—
90C2×D4—62no—
91C2×D4—62no—
92C2×D4—61no—
93C22×C4—21no—
94C22×C4—31no—
95C22×C4—31no—
96C22×C4—31no—
97C22×C4—31no—
98C22×C4—31no—
99C22×C4—32no—
100C22×C4—32no—
101C22×C4—32no—
102C22×C4—31no—
103C22×C4—32no—
104C22×C4—32no—
105C22×C4—61no—
106C22×C4—62no—
107C22×C4—64no—
108C22×C4—64no—
109C22×C4—62no—
110C22×C4C4×D312yescomplemented
111C22×C4D1212yescomplemented
112C4⋊C4—34no—
113C4⋊C4—62no—
114C4⋊C4—61no—
115C4⋊C4—64no—
116C4⋊C4—61no—
117C22⋊C4—38no—
118C22⋊C4—64no—
119C22⋊C4—64no—
120C22⋊C4—62no—
121C22⋊C4—62no—
122C22⋊C4—62no—
123C22⋊C4—62no—
124C42—31no—
125C42—31no—
126C42—31no—
127C42—61no—
128C42—64no—
129C42S411yescharacteristic, complemented
130C22×C3—41no—
131C2×D3—44no—
132C2×D3—82no—
133C2×D3—82no—
134A4C4×D411yescharacteristic, complemented
135C3×C4—42no—
136C3×C4—42no—
137C3×C4—81no—
138Dic3—82no—
139C23—31no—
140C23—31no—
141C23—31no—
142C23—31no—
143C23—34no—
144C23—62no—
145C23—62no—
146C23—62no—
147C23—62no—
148C23—62no—
149C23—62no—
150C23C2×C4×D311yescharacteristic
151C23C2×D1211yescharacteristic
152C23#23511yescharacteristic
153D4—64no—
154D4—64no—
155D4—64no—
156D4—64no—
157D4—64no—
158D4—64no—
159C2×C4—31no—
160C2×C4—31no—
161C2×C4—32no—
162C2×C4—32no—
163C2×C4—32no—
164C2×C4—32no—
165C2×C4—32no—
166C2×C4—32no—
167C2×C4—34no—
168C2×C4—32no—
169C2×C4—31no—
170C2×C4—61no—
171C2×C4—61no—
172C2×C4—61no—
173C2×C4—61no—
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—61no—
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×S411yescharacteristic
186C2×C4C2×S411yescharacteristic
187C2×C4C2×S411yescharacteristic
188C2×C3—41no—
189C2×C3—41no—
190C2×C3—41no—
191D3—84no—
192C22—31no—
193C22—31no—
194C22—31no—
195C22—31no—
196C22—31no—
197C22—31no—
198C22—31no—
199C22—31no—
200C22—31no—
201C22—61no—
202C22—64no—
203C22—64no—
204C22—64no—
205C22—122no—
206C22—122no—
207C22—122no—
208C22—122no—
209C22C4×D1211yescharacteristic, complemented
210C22C22×S411yescharacteristic
211C4—21no—
212C4—32no—
213C4—32no—
214C4—64no—
215C4—61no—
216C4—122no—
217C4—122no—
218C4C4×S412yescomplemented
219C4C4⋊S412yesdirect product factor
220C3—41noSylow
221C2—31no—
222C2—31no—
223C2—31no—
224C2—31no—
225C2—124no—
226C2C2×C4×S411yescharacteristic
227C2C2×C4⋊S411yescharacteristic
228C2#591211yescharacteristic
subgroupG/H
centerC2×C4C2×S4
commutatorC2×A4C22×C4
FrattiniC22C22×S4
FittingC22×C42D3
socleC24C22×D3
prime radicalC2×C4⋊S4C2
cosocleC22×A4C23
hypercenterC42S4
hypocenterA4C4×D4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×C4×S4C2semidirect
3. C42×A4C2semidirect
4. (C2×S4)⋊C4C2semidirect
5. C4⋊SL(2,ℤ/4)C2semidirect
12. C4⋊S4C4direct (N × H)
18. C22×C42D3semidirect
29. C2×S4C2×C4semidirect
33. C4×A4C2×C4semidirect
34. C4×A4D4semidirect
110. C22×C4C4×D3semidirect
111. C22×C4D12semidirect
129. C42S4semidirect
134. A4C4×D4semidirect
209. C22C4×D12semidirect
218. C4C4×S4semidirect
219. C4C4⋊S4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊S42
C2×C4×S42
C42×A42
(C2×S4)⋊C42
C4⋊SL(2,ℤ/4)2
C4×#1213
C4×D124