Finite Group Databasebeta

← #85308

Subgroups — #85308

Order 384.

427 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 54 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
3A4×C4⋊C4C211yesmaximal, characteristic, complemented
4(C2×S4)⋊C4C212yesmaximal, complemented
5C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6#2361—31nomaximal, Sylow
7C22×S4C2212yes—
8C2×C4×A4C2212yes—
9C2×C4×A4C2211yescharacteristic
10C2×SL(2,ℤ/4)C2211yescharacteristic
11C2×SL(2,ℤ/4)C2211yescharacteristic
12C4⋊S4C414yescomplemented
13C4×S4—24no—
14#893—41nomaximal
15C2×#121—31no—
16C2×C4×D4—31no—
17C2×C4×D4—32no—
18C22×C4⋊C4D311yescharacteristic, complemented
19#390—31no—
20#389—31no—
21#388—32no—
22#386—32no—
23#385—32no—
24C4×C4⋊C4—31no—
25C4×C22⋊C4—31no—
26C22×A4C2311yescharacteristic
27C2×S4—22no—
28C2×S4—22no—
29C2×S4C2×C414yescomplemented
30C2×D12—41no—
31C2×C4×D3—42no—
32C4×A4—22no—
33C4×A4C2×C412yescomplemented
34SL(2,ℤ/4)—21no—
35SL(2,ℤ/4)D412yescomplemented
36C3×C4⋊C4—41no—
37(C2×D3)⋊C4—42no—
38C4×Dic3—41no—
39C22×D4—31no—
40C22×D4—32no—
41C23×C4—32no—
42C23×C4C2×D312yes—
43C23×C4C2×D311yescharacteristic
44#121—34no—
45#121—34no—
46C4×D4—64no—
47C4×D4—64no—
48C4×D4—62no—
49C4×D4—61no—
50C4×D4—61no—
51C2×C4⋊C4—31no—
52C2×C4⋊C4—31no—
53C2×C4⋊C4—32no—
54C2×C4⋊C4—31no—
55C2×C4⋊C4—32no—
56C2×C22⋊C4—32no—
57C2×C22⋊C4—31no—
58C2×C22⋊C4—32no—
59C2×C22⋊C4—32no—
60C2×C22⋊C4—31no—
61C2×C22⋊C4—32no—
62C2×C42—31no—
63C2×C42—32no—
64C2×C42—31no—
65#95—31no—
66#95—31no—
67#95—32no—
68#95—32no—
69C22×D3—42no—
70C2×A4C22×C411yescharacteristic
71C2×A4C2×D411yescharacteristic
72C2×A4#4111yescharacteristic
73S4—24no—
74C2×C3×C4—41no—
75C2×C3×C4—42no—
76C2×Dic3—41no—
77C2×Dic3—41no—
78D12—44no—
79C4×D3—84no—
80C24—32no—
81C24C22×D311yescharacteristic
82C2×D4—34no—
83C2×D4—34no—
84C2×D4—34no—
85C2×D4—61no—
86C2×D4—62no—
87C2×D4—62no—
88C2×D4—62no—
89C2×D4—62no—
90C2×D4—62no—
91C2×D4—61no—
92C22×C4—22no—
93C22×C4—31no—
94C22×C4—31no—
95C22×C4—32no—
96C22×C4—32no—
97C22×C4—31no—
98C22×C4—31no—
99C22×C4—32no—
100C22×C4—32no—
101C22×C4—31no—
102C22×C4—31no—
103C22×C4—64no—
104C22×C4—62no—
105C22×C4—62no—
106C22×C4—62no—
107C22×C4—122no—
108C22×C4C4×D312yescomplemented
109C4⋊C4—34no—
110C4⋊C4—32no—
111C4⋊C4—31no—
112C4⋊C4—61no—
113C4⋊C4—64no—
114C4⋊C4S411yescharacteristic, complemented
115C22⋊C4—32no—
116C22⋊C4—38no—
117C22⋊C4—32no—
118C22⋊C4—64no—
119C22⋊C4—62no—
120C22⋊C4—61no—
121C22⋊C4—61no—
122C22⋊C4—62no—
123C22⋊C4—122no—
124C42—62no—
125C42—61no—
126C42—64no—
127C42—61no—
128C22×C3—41no—
129C2×D3—44no—
130C2×D3—82no—
131C2×D3—82no—
132A4C4×D411yescharacteristic, complemented
133C3×C4—42no—
134C3×C4—82no—
135Dic3—42no—
136Dic3—81no—
137C23—31no—
138C23—31no—
139C23—31no—
140C23—31no—
141C23—34no—
142C23—62no—
143C23—62no—
144C23—62no—
145C23—62no—
146C23—62no—
147C23—62no—
148C23C2×C4×D311yescharacteristic
149C23D3×D411yescharacteristic
150C23#23911yescharacteristic
151D4—64no—
152D4—64no—
153D4—64no—
154D4—64no—
155D4—64no—
156D4—64no—
157C2×C4—31no—
158C2×C4—32no—
159C2×C4—32no—
160C2×C4—32no—
161C2×C4—32no—
162C2×C4—34no—
163C2×C4—32no—
164C2×C4—32no—
165C2×C4—62no—
166C2×C4—62no—
167C2×C4—61no—
168C2×C4—62no—
169C2×C4—61no—
170C2×C4—61no—
171C2×C4—61no—
172C2×C4—62no—
173C2×C4—61no—
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—61no—
177C2×C4—61no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—61no—
181C2×C4—61no—
182C2×C4—124no—
183C2×C4—124no—
184C2×C4C2×S412yes—
185C2×C4C2×S411yescharacteristic
186C2×C3—41no—
187C2×C3—41no—
188C2×C3—41no—
189D3—84no—
190C22—31no—
191C22—31no—
192C22—31no—
193C22—31no—
194C22—31no—
195C22—31no—
196C22—31no—
197C22—31no—
198C22—31no—
199C22—61no—
200C22—64no—
201C22—64no—
202C22—64no—
203C22—122no—
204C22—122no—
205C22—122no—
206C22—122no—
207C22#89311yescharacteristic, complemented
208C22C22×S411yescharacteristic
209C4—22no—
210C4—32no—
211C4—62no—
212C4—64no—
213C4—62no—
214C4—62no—
215C4—121no—
216C4—121no—
217C4C4×S412yescomplemented
218C3—41noSylow
219C2—31no—
220C2—31no—
221C2—31no—
222C2—31no—
223C2—124no—
224C2C2×C4×S411yescharacteristic
225C2D4×S411yescharacteristic
226C2#591911yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutatorC2×A4C22×C4
FittingC22×C4⋊C4D3
socleC24C22×D3
prime radicalC2×C4⋊S4C2
cosocleC22×A4C23
hypercenterC4⋊C4S4
hypocenterA4C4×D4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×S4C2semidirect
3. A4×C4⋊C4C2semidirect
4. (C2×S4)⋊C4C2semidirect
5. C4×SL(2,ℤ/4)C2semidirect
12. C4⋊S4C4semidirect
18. C22×C4⋊C4D3semidirect
29. C2×S4C2×C4semidirect
33. C4×A4C2×C4semidirect
35. SL(2,ℤ/4)D4semidirect
108. C22×C4C4×D3semidirect
114. C4⋊C4S4semidirect
132. A4C4×D4semidirect
207. C22#893semidirect
217. 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
C2×C4×S42
A4×C4⋊C42
(C2×S4)⋊C42
C4×SL(2,ℤ/4)2
#23613
#8934