Finite Group Databasebeta

← #85491

Subgroups — #85491

Order 384.

427 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 44 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×D4×A4C211yesmaximal, characteristic
2C22×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
3#5432C212yesmaximal, complemented
4#5432C211yesmaximal, characteristic, complemented
5SL(2,ℤ/4)⋊C4C212yesmaximal, complemented
6#2405—31nomaximal, Sylow
7C23×A4C2211yescharacteristic
8C23×A4C2211yescharacteristic
9D4×A4—22no—
10C2×C4×A4C2211yescharacteristic
11C2×SL(2,ℤ/4)—22no—
12C2×SL(2,ℤ/4)C2212yes—
13C2×SL(2,ℤ/4)C2212yes—
14#935—41nomaximal
15C23×D4D311yescharacteristic
16C2×#123—31no—
17C2×#123—31no—
18C22×C22⋊C4—31no—
19#394—32no—
20#394—32no—
21#386—31no—
22#386—31no—
23#385—32no—
24#379—31no—
25#379—32no—
26C22×A4—22no—
27C22×A4—21no—
28C22×A4—22no—
29C22×A4D412yes—
30C22×A4C2311yescharacteristic
31C2×C3×D4—41no—
32C22×Dic3—41no—
33C4×A4—21no—
34SL(2,ℤ/4)—22no—
35SL(2,ℤ/4)—22no—
36#217—41no—
37#217—42no—
38Dic3⋊C4—42no—
39C25C2×D311yescharacteristic
40C25C2×D311yescharacteristic
41C22×D4—22no—
42C22×D4—31no—
43C22×D4—31no—
44C22×D4—31no—
45C22×D4—31no—
46C22×D4—62no—
47C23×C4—31no—
48C23×C4—31no—
49C23×C4C2×D311yescharacteristic
50#123—64no—
51#123—61no—
52#123—61no—
53#123—61no—
54#123—61no—
55C2×C4⋊C4—32no—
56C2×C4⋊C4—32no—
57C2×C22⋊C4—32no—
58C2×C22⋊C4—31no—
59C2×C22⋊C4—31no—
60C2×C22⋊C4—32no—
61C2×C22⋊C4—32no—
62C2×C22⋊C4—32no—
63C2×C22⋊C4—32no—
64C2×C22⋊C4—32no—
65C2×C22⋊C4—32no—
66C2×C22⋊C4—32no—
67C2×C22⋊C4—62no—
68C2×C22⋊C4—62no—
69#95—32no—
70#95—32no—
71C23×C3—41no—
72C23×C3—41no—
73C2×A4—22no—
74C2×A4—41no—
75C2×A4C2×D411yescharacteristic
76C2×A4#4112yes—
77C3×D4—82no—
78C2×C3×C4—41no—
79C2×Dic3—42no—
80C2×Dic3—42no—
81C2×Dic3—82no—
82C24—22no—
83C24—22no—
84C24—21no—
85C24—31no—
86C24—31no—
87C24—31no—
88C24—32no—
89C24—31no—
90C24—61no—
91C24—62no—
92C24—62no—
93C24#6712yes—
94C24C22×D311yescharacteristic
95C2×D4—34no—
96C2×D4—31no—
97C2×D4—31no—
98C2×D4—31no—
99C2×D4—61no—
100C2×D4—62no—
101C2×D4—62no—
102C2×D4—62no—
103C2×D4—62no—
104C2×D4—62no—
105C2×D4—62no—
106C2×D4—62no—
107C2×D4S411yescharacteristic
108C22×C4—21no—
109C22×C4—31no—
110C22×C4—31no—
111C22×C4—32no—
112C22×C4—32no—
113C22×C4—32no—
114C22×C4—32no—
115C22×C4—61no—
116C22×C4—62no—
117C22×C4—62no—
118C22×C4—62no—
119C22×C4—61no—
120C22×C4—61no—
121C22×C4—122no—
122C4⋊C4—64no—
123C4⋊C4—62no—
124C4⋊C4—62no—
125C22⋊C4—62no—
126C22⋊C4—62no—
127C22⋊C4—64no—
128C22⋊C4—62no—
129C22⋊C4—64no—
130C22⋊C4—61no—
131C22⋊C4—62no—
132C22⋊C4—62no—
133C22⋊C4—62no—
134C22⋊C4—61no—
135C22⋊C4—62no—
136C22⋊C4—122no—
137C22⋊C4—122no—
138C22⋊C4—122no—
139C22×C3—42no—
140C22×C3—41no—
141C22×C3—82no—
142C22×C3—81no—
143C22×C3—82no—
144A4#12311yescharacteristic, complemented
145C3×C4—81no—
146Dic3—82no—
147Dic3—82no—
148C23—22no—
149C23—31no—
150C23—31no—
151C23—32no—
152C23—32no—
153C23—32no—
154C23—32no—
155C23—31no—
156C23—32no—
157C23—31no—
158C23—41no—
159C23—62no—
160C23—62no—
161C23—62no—
162C23—61no—
163C23—61no—
164C23—62no—
165C23—61no—
166C23—62no—
167C23—62no—
168C23—62no—
169C23—64no—
170C23—62no—
171C23—62no—
172C23—61no—
173C23—122no—
174C23—121no—
175C23—122no—
176C23—122no—
177C23#23712yes—
178C23C2×#6711yescharacteristic
179C23C2×S411yescharacteristic
180C23C2×S411yescharacteristic
181D4—22no—
182D4—62no—
183D4—62no—
184D4—64no—
185D4—62no—
186C2×C4—32no—
187C2×C4—31no—
188C2×C4—61no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—122no—
204C2×C4—124no—
205C2×C4—122no—
206C2×C4C2×S411yescharacteristic
207C2×C3—42no—
208C2×C3—41no—
209C2×C3—82no—
210C2×C3—161no—
211C22—22no—
212C22—22no—
213C22—21no—
214C22—32no—
215C22—31no—
216C22—32no—
217C22—32no—
218C22—31no—
219C22—31no—
220C22—32no—
221C22—62no—
222C22—62no—
223C22—64no—
224C22—61no—
225C22—62no—
226C22—62no—
227C22—64no—
228C22—62no—
229C22—62no—
230C22—61no—
231C22—121no—
232C22—121no—
233C22—122no—
234C22—122no—
235C22—121no—
236C22—121no—
237C22#93511yescharacteristic, complemented
238C22GL(2,ℤ/4)12yes—
239C22C22×S411yescharacteristic
240C4—21no—
241C4—61no—
242C4—122no—
243C4—122no—
244C4—122no—
245C4—122no—
246C3—41noSylow
247C2—22no—
248C2—31no—
249C2—32no—
250C2—31no—
251C2—41no—
252C2—62no—
253C2—121no—
254C2#591412yes—
255C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, cosocleC22×A4C23
FittingC23×D4D3
socleC24C22×D3
prime radicalC2×D4×A4C2
hypercenterC2×D4S4
hypocenterA4#123
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×SL(2,ℤ/4)C2semidirect
3. #5432C2semidirect
4. #5432C2semidirect
5. SL(2,ℤ/4)⋊C4C2semidirect
144. A4#123semidirect
237. C22#935semidirect

Maximal subgroups

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

subgroupindex
C2×D4×A42
C22×SL(2,ℤ/4)2
#54322
SL(2,ℤ/4)⋊C42
#24053
#9354