Finite Group Databasebeta

← C2×C4×A4⋊C8 #11712541

Subgroups — C2×C4×A4⋊C8 #11712541

Order 768.

905 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 905 conjugacy classes of subgroups — too many to draw. Only its 189 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×C42×A4C211yesmaximal, characteristic
2C22×A4⋊C8C212yesmaximal
3C4×A4⋊C8C214yesmaximal, direct product factor
4C2×C4×C22⋊C8—31nomaximal, Sylow
5C22×C4×A4C412yes—
6C22×C4×A4C2211yescharacteristic
7C42×A4C412yes—
8C42×A4C2212yes—
9C2×A4⋊C8C418yesdirect product factor
10C2×A4⋊C8C2214yes—
11C2×C4×C3⋊C8—41nomaximal
12C23×C42D311yescharacteristic
13C22×C22⋊C8—32no—
14C22×C4×C8—31no—
15C22×C4×C8—31no—
16C4×C22⋊C8—34no—
17C4×C22⋊C8—34no—
18C2×#336—32no—
19C23×A4C2×C411yescharacteristic
20C2×C4×A4C818yescomplemented
21C2×C4×A4C2×C414yes—
22C2×C4×A4C2×C411yescharacteristic
23C2×C4×A4C2×C414yes—
24C2×C4×A4C2311yescharacteristic
25C2×C3×C42—41no—
26C22×C3⋊C8—42no—
27A4⋊C8C2×C418yesdirect product factor
28C4×C3⋊C8—44no—
29C24×C4Dic312yes—
30C24×C4C2×D311yescharacteristic
31C23×C8—32no—
32C23×C8—32no—
33C22×C42—31no—
34C22×C42—32no—
35C22×C42—32no—
36C22×C42—31no—
37C22×C42—32no—
38C22×C42Dic312yes—
39C22×C42C2×D312yes—
40C2×C22⋊C8—38no—
41C2×C22⋊C8—34no—
42C2×C22⋊C8—38no—
43C2×C22⋊C8—34no—
44C2×C4×C8—34no—
45C2×C4×C8—34no—
46C2×C4×C8—62no—
47C2×C4×C8—61no—
48C2×C4×C8—62no—
49C2×C4×C8—62no—
50C2×C4×C8—61no—
51#336—38no—
52C22×A4C4212yes—
53C22×A4C2×C814yes—
54C22×A4C22×C411yescharacteristic
55C22×C3×C4—42no—
56C22×C3×C4—41no—
57C4×A4C4212yes—
58C4×A4C2×C818yescomplemented
59C4×A4C22×C412yes—
60C3×C42—42no—
61C3×C42—42no—
62C2×C3⋊C8—48no—
63C2×C3⋊C8—44no—
64C25C2×Dic311yescharacteristic
65C23×C4—32no—
66C23×C4—31no—
67C23×C4—38no—
68C23×C4—32no—
69C23×C4—34no—
70C23×C4—34no—
71C23×C4—31no—
72C23×C4—31no—
73C23×C4—31no—
74C23×C4C3⋊C818yescomplemented
75C23×C4C4×D314yes—
76C23×C4C2×Dic314yes—
77C23×C4C2×Dic311yescharacteristic
78C23×C4C22×D311yescharacteristic
79C22×C8—34no—
80C22×C8—34no—
81C22×C8—38no—
82C22×C8—38no—
83C22×C8—64no—
84C22×C8—64no—
85C22×C8—62no—
86C22×C8—62no—
87C22×C8—62no—
88C22×C8—62no—
89C2×C42—32no—
90C2×C42—31no—
91C2×C42—32no—
92C2×C42—32no—
93C2×C42—32no—
94C2×C42—32no—
95C2×C42—32no—
96C2×C42—32no—
97C2×C42—34no—
98C2×C42—34no—
99C2×C42—64no—
100C2×C42—61no—
101C2×C42—61no—
102C2×C42—61no—
103C2×C42S411yescharacteristic
104C22⋊C8—316no—
105C22⋊C8—38no—
106C22⋊C8—38no—
107C4×C8—68no—
108C4×C8—64no—
109C4×C8—64no—
110C23×C3—41no—
111C2×A4C4×C814yescomplemented
112C2×A4C2×C4211yescharacteristic
113C2×A4C22×C812yes—
114C2×C3×C4—48no—
115C2×C3×C4—44no—
116C2×C3×C4—44no—
117C2×C3×C4—41no—
118C2×C3×C4—41no—
119C3⋊C8—48no—
120C24—31no—
121C24—31no—
122C24—34no—
123C24—32no—
124C24C2×C3⋊C814yes—
125C24C4×Dic312yes—
126C24C22×Dic311yescharacteristic
127C22×C4—31no—
128C22×C4—32no—
129C22×C4—38no—
130C22×C4—38no—
131C22×C4—34no—
132C22×C4—34no—
133C22×C4—34no—
134C22×C4—31no—
135C22×C4—31no—
136C22×C4—34no—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—34no—
140C22×C4—32no—
141C22×C4—34no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—38no—
145C22×C4—316no—
146C22×C4—34no—
147C22×C4—34no—
148C22×C4—38no—
149C22×C4—32no—
150C22×C4—62no—
151C22×C4—68no—
152C22×C4—62no—
153C22×C4—61no—
154C22×C4—62no—
155C22×C4—64no—
156C22×C4—62no—
157C22×C4C2×C3⋊C818yescomplemented
158C22×C4C4×Dic312yes—
159C22×C4SL(2,ℤ/4)12yes—
160C22×C4C2×C4×D312yes—
161C22×C4C2×S411yescharacteristic
162C2×C8—38no—
163C2×C8—38no—
164C2×C8—64no—
165C2×C8—68no—
166C2×C8—64no—
167C2×C8—68no—
168C2×C8—64no—
169C2×C8—68no—
170C2×C8—68no—
171C2×C8—64no—
172C42—34no—
173C42—34no—
174C42—32no—
175C42—32no—
176C42—62no—
177C42—62no—
178C42SL(2,ℤ/4)12yes—
179C42C2×S412yes—
180C22×C3—44no—
181C22×C3—42no—
182C22×C3—41no—
183A4C2×C4×C811yescharacteristic, complemented
184C3×C4—42no—
185C3×C4—48no—
186C3×C4—42no—
187C23—34no—
188C23—34no—
189C23—32no—
190C23—32no—
191C23—31no—
192C23—31no—
193C23—34no—
194C23—32no—
195C23—31no—
196C23—32no—
197C23—38no—
198C23—34no—
199C23—62no—
200C23—61no—
201C23—64no—
202C23C4×C3⋊C814yescomplemented
203C23C22×C3⋊C812yes—
204C23C2×C4×Dic311yescharacteristic
205C23C2×SL(2,ℤ/4)11yescharacteristic
206C2×C4—38no—
207C2×C4—34no—
208C2×C4—31no—
209C2×C4—34no—
210C2×C4—31no—
211C2×C4—32no—
212C2×C4—32no—
213C2×C4—34no—
214C2×C4—38no—
215C2×C4—38no—
216C2×C4—316no—
217C2×C4—34no—
218C2×C4—32no—
219C2×C4—32no—
220C2×C4—38no—
221C2×C4—32no—
222C2×C4—32no—
223C2×C4—64no—
224C2×C4—68no—
225C2×C4—61no—
226C2×C4—64no—
227C2×C4—61no—
228C2×C4A4⋊C818yesdirect product factor
229C2×C4C4×S414yes—
230C2×C4C2×SL(2,ℤ/4)14yes—
231C2×C4C2×SL(2,ℤ/4)11yescharacteristic
232C2×C4C22×S411yescharacteristic
233C8—68no—
234C8—68no—
235C2×C3—44no—
236C2×C3—42no—
237C2×C3—41no—
238C22—34no—
239C22—32no—
240C22—31no—
241C22—34no—
242C22—38no—
243C22—34no—
244C22—32no—
245C22—34no—
246C22—32no—
247C22—31no—
248C22—31no—
249C22—32no—
250C22—64no—
251C22—62no—
252C22—61no—
253C22C2×C4×C3⋊C811yescharacteristic, complemented
254C22C2×A4⋊C814yes—
255C22C4×SL(2,ℤ/4)12yes—
256C22C22×SL(2,ℤ/4)11yescharacteristic
257C4—38no—
258C4—32no—
259C4—32no—
260C4C2×A4⋊C818yesdirect product factor
261C4C4×SL(2,ℤ/4)12yes—
262C4C2×C4×S412yes—
263C3—41noSylow
264C2—31no—
265C2—34no—
266C2—32no—
267C2—31no—
268C2C4×A4⋊C814yesdirect product factor
269C2C22×A4⋊C812yes—
270C2C2×C4×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC2×C42S4
commutator, hypocenterA4C2×C4×C8
FrattiniC2×C4C22×S4
FittingC23×C42D3
socleC25C2×Dic3
prime radicalC23×A4C2×C4
cosocleC2×C4×A4C23
LCSG ▷ A4
UCSC1 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
3. C4×A4⋊C8C2direct (N × H)
9. C2×A4⋊C8C4direct (N × H)
20. C2×C4×A4C8semidirect
27. A4⋊C8C2×C4direct (N × H)
58. C4×A4C2×C8semidirect
74. C23×C4C3⋊C8semidirect
111. C2×A4C4×C8semidirect
157. C22×C4C2×C3⋊C8semidirect
183. A4C2×C4×C8semidirect
202. C23C4×C3⋊C8semidirect
228. C2×C4A4⋊C8direct (N × H)
253. C22C2×C4×C3⋊C8semidirect
260. C4C2×A4⋊C8direct (N × H)
268. C2C4×A4⋊C8direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C42×A42
C22×A4⋊C82
C4×A4⋊C82
C2×C4×C22⋊C83
C2×C4×C3⋊C84