Finite Group Databasebeta

← C22⋊C4×S4 #85295

Subgroups — C22⋊C4×S4 #85295

Order 384.

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

normal subgroups only

This group has 657 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
1C23×S4C211yesmaximal, characteristic
2C2×C4×S4C212yesmaximal, complemented
3A4×C22⋊C4C211yesmaximal, characteristic, complemented
4#5432C211yesmaximal, characteristic, complemented
5(C2×S4)⋊C4C212yesmaximal, complemented
6D4×C22⋊C4—31nomaximal, Sylow
7C23×A4C2211yescharacteristic
8C22×S4—24no—
9C22×S4C414yescomplemented
10C22×S4C2211yescharacteristic
11C22×S4C2211yescharacteristic
12C2×C4×A4C2212yescomplemented
13C2×SL(2,ℤ/4)C2212yescomplemented
14C4×S4—24no—
15D3×C22⋊C4—41nomaximal
16C23×D4—31no—
17C2×C4×D4—32no—
18C22×C22⋊C4—31no—
19C22×C22⋊C4D311yescharacteristic, complemented
20#390—32no—
21#386—32no—
22#386—32no—
23#380—31no—
24#379—31no—
25#379—31no—
26C4×C22⋊C4—31no—
27C23×D3—41no—
28C22×A4—22no—
29C22×A4C2×C412yescomplemented
30C22×A4C2311yescharacteristic
31C2×S4—22no—
32C2×S4—28no—
33C2×S4C2×C412yes—
34C2×S4C2×C412yescomplemented
35C2×S4D414yes—
36C2×C4×D3—42no—
37C4×A4—22no—
38SL(2,ℤ/4)—22no—
39C3×C22⋊C4—41no—
40#217—41no—
41(C2×D3)⋊C4—42no—
42C25—31no—
43C25C2×D311yescharacteristic
44C22×D4—31no—
45C22×D4—34no—
46C22×D4—31no—
47C22×D4—31no—
48C22×D4—34no—
49C22×D4—31no—
50C22×D4—64no—
51C22×D4—64no—
52C23×C4—31no—
53C23×C4—32no—
54C23×C4C2×D312yescomplemented
55C4×D4—64no—
56C4×D4—64no—
57C2×C4⋊C4—32no—
58C2×C22⋊C4—31no—
59C2×C22⋊C4—31no—
60C2×C22⋊C4—32no—
61C2×C22⋊C4—31no—
62C2×C22⋊C4—32no—
63C2×C22⋊C4—32no—
64C2×C22⋊C4—31no—
65C2×C22⋊C4—32no—
66C2×C22⋊C4—32no—
67C2×C22⋊C4—32no—
68C2×C22⋊C4—61no—
69C2×C22⋊C4—62no—
70C2×C22⋊C4—61no—
71C2×C42—32no—
72#95—32no—
73#95—32no—
74C23×C3—41no—
75C22×D3—41no—
76C22×D3—41no—
77C22×D3—44no—
78C22×D3—84no—
79C2×A4—22no—
80C2×A4C22×C411yescharacteristic
81C2×A4C2×D412yes—
82S4—22no—
83S4C22⋊C414yesdirect product factor
84C2×C3×C4—42no—
85C2×Dic3—42no—
86C4×D3—84no—
87C24—22no—
88C24—31no—
89C24—31no—
90C24—31no—
91C24—32no—
92C24—31no—
93C24—34no—
94C24—62no—
95C24—61no—
96C24—64no—
97C24—62no—
98C24—61no—
99C24—122no—
100C24C4×D312yescomplemented
101C24C22×D311yescharacteristic
102C2×D4—34no—
103C2×D4—32no—
104C2×D4—32no—
105C2×D4—34no—
106C2×D4—34no—
107C2×D4—32no—
108C2×D4—34no—
109C2×D4—32no—
110C2×D4—64no—
111C2×D4—68no—
112C2×D4—68no—
113C2×D4—616no—
114C2×D4—62no—
115C2×D4—64no—
116C2×D4—62no—
117C22×C4—22no—
118C22×C4—31no—
119C22×C4—31no—
120C22×C4—34no—
121C22×C4—32no—
122C22×C4—32no—
123C22×C4—32no—
124C22×C4—32no—
125C22×C4—64no—
126C22×C4—62no—
127C22×C4—62no—
128C22×C4—64no—
129C22×C4—62no—
130C22×C4—122no—
131C4⋊C4—64no—
132C22⋊C4—31no—
133C22⋊C4—32no—
134C22⋊C4—61no—
135C22⋊C4—62no—
136C22⋊C4—62no—
137C22⋊C4—61no—
138C22⋊C4—64no—
139C22⋊C4—62no—
140C22⋊C4—62no—
141C22⋊C4—61no—
142C22⋊C4—62no—
143C22⋊C4—122no—
144C22⋊C4—122no—
145C22⋊C4S411yescharacteristic, direct product factor
146C42—64no—
147C22×C3—42no—
148C22×C3—41no—
149C22×C3—82no—
150C2×D3—44no—
151C2×D3—42no—
152C2×D3—42no—
153C2×D3—88no—
154C2×D3—82no—
155A4C2×C22⋊C411yescharacteristic, complemented
156C3×C4—82no—
157Dic3—82no—
158C23—22no—
159C23—31no—
160C23—32no—
161C23—31no—
162C23—34no—
163C23—32no—
164C23—32no—
165C23—32no—
166C23—32no—
167C23—31no—
168C23—32no—
169C23—62no—
170C23—62no—
171C23—61no—
172C23—62no—
173C23—62no—
174C23—61no—
175C23—62no—
176C23—64no—
177C23—62no—
178C23—62no—
179C23—62no—
180C23—68no—
181C23—61no—
182C23—64no—
183C23—61no—
184C23—64no—
185C23—61no—
186C23—124no—
187C23—122no—
188C23—124no—
189C23—128no—
190C23—124no—
191C23C2×C4×D311yescharacteristic
192C23D3×D412yes—
193C23C2×S411yescharacteristic
194D4—34no—
195D4—38no—
196D4—34no—
197D4—68no—
198D4—62no—
199D4—64no—
200D4—68no—
201D4—62no—
202C2×C4—34no—
203C2×C4—32no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—68no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—124no—
219C2×C4—124no—
220C2×C4C2×S412yescomplemented
221C2×C3—41no—
222C2×C3—42no—
223C2×C3—82no—
224D3—44no—
225D3—82no—
226C22—22no—
227C22—31no—
228C22—32no—
229C22—31no—
230C22—31no—
231C22—32no—
232C22—32no—
233C22—34no—
234C22—32no—
235C22—62no—
236C22—64no—
237C22—62no—
238C22—61no—
239C22—62no—
240C22—62no—
241C22—64no—
242C22—62no—
243C22—62no—
244C22—62no—
245C22—62no—
246C22—64no—
247C22—62no—
248C22—62no—
249C22—128no—
250C22—122no—
251C22—122no—
252C22—128no—
253C22—122no—
254C22D3×C22⋊C411yescharacteristic, complemented
255C22C4×S412yescomplemented
256C22C22×S411yescharacteristic
257C4—22no—
258C4—34no—
259C4—62no—
260C4—62no—
261C4—122no—
262C4—122no—
263C3—41noSylow
264C2—22no—
265C2—31no—
266C2—32no—
267C2—31no—
268C2—62no—
269C2—64no—
270C2—122no—
271C2C2×C4×S411yescharacteristic
272C2D4×S412yes—
subgroupG/H
center, FrattiniC22C22×S4
commutatorC2×A4C22×C4
FittingC22×C22⋊C4D3
socleC24C22×D3
prime radicalC23×S4C2
cosocleC22×A4C23
hypercenterC22⋊C4S4
hypocenterA4C2×C22⋊C4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×S4C2semidirect
3. A4×C22⋊C4C2semidirect
4. #5432C2semidirect
5. (C2×S4)⋊C4C2semidirect
9. C22×S4C4semidirect
12. C2×C4×A4C22semidirect
13. C2×SL(2,ℤ/4)C22semidirect
19. C22×C22⋊C4D3semidirect
29. C22×A4C2×C4semidirect
34. C2×S4C2×C4semidirect
54. C23×C4C2×D3semidirect
83. S4C22⋊C4direct (N × H)
100. C24C4×D3semidirect
145. C22⋊C4S4direct (N × H)
155. A4C2×C22⋊C4semidirect
220. C2×C4C2×S4semidirect
254. C22D3×C22⋊C4semidirect
255. C22C4×S4semidirect

Maximal subgroups

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

subgroupindex
C23×S42
C2×C4×S42
A4×C22⋊C42
#54322
(C2×S4)⋊C42
D4×C22⋊C43
D3×C22⋊C44